{"id":159,"date":"2020-08-13T11:56:33","date_gmt":"2020-08-13T10:56:33","guid":{"rendered":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/?post_type=chapter&#038;p=159"},"modified":"2020-08-13T13:32:10","modified_gmt":"2020-08-13T12:32:10","slug":"propagation-of-disturbances","status":"publish","type":"chapter","link":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/chapter\/propagation-of-disturbances\/","title":{"raw":"Propagation of disturbances","rendered":"Propagation of disturbances"},"content":{"raw":"There are four ways for disturbance to pass from the source to the victim, via:\r\n<ul>\r\n \t<li>galvanic coupling,<\/li>\r\n \t<li>capacitive (electric) coupling,<\/li>\r\n \t<li>inductive (magnetic) coupling,<\/li>\r\n \t<li>wave coupling.<\/li>\r\n<\/ul>\r\nThe first three accompany transmission of power or signals via conductors. It can proceed unsymmetrically or symmetrically.\r\n\r\nUnsymmetrical transmission comes about when return conductor is common for more circuits and nonzero current as an overlay of currents of all circuits drives in the common return path. Three examples are shown in Fig.<a href=\"#Unsym_line\" data-reference-type=\"ref\" data-reference=\"Unsym_line\">4.1<\/a>. The return path can be identical with feeding lines or can differ with them.\r\n<figure><img src=\"Propagation\/Figures\/Unsym_line.png\" id=\"Unsym_line\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_162\" align=\"aligncenter\" width=\"739\"]<img class=\"wp-image-162\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances.png\" width=\"739\" height=\"391\" alt=\"\" \/> Figure 4.1: Three examples of unsymmetrical lines.[\/caption]<\/figure>\r\nSymmetrical transmission is carried with a pair of identical conductors isolated from the surrounding. Examples are shown in Fig.<a href=\"#Sym_line\" data-reference-type=\"ref\" data-reference=\"Sym_line\">4.2<\/a>. There is always better or worse conducting layer beneath (striped plane) which builds the ground reference.\r\n<figure><img src=\"Propagation\/Figures\/Sym_line.png\" id=\"Sym_line\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_163\" align=\"aligncenter\" width=\"824\"]<img class=\"wp-image-163\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-1.png\" width=\"824\" height=\"336\" alt=\"\" \/> Figure 4.2: Two examples of symmetrical lines.[\/caption]<\/figure>\r\nUnsymmetrical transmission is potentially accompanied with galvanic, capacitive and inductive coupling. Symmetrical transmission only with capacitive and inductive. They must be considered simultaneously. Here the phenomena will be presented separately in order to catch its essence. Such approach is violation of circuit theory, provide only one type of coupling is dominant, remaining negligible. It is case dependent.\r\n<h2 id=\"examples-of-unsymmetrical-and-symmetrical-transmission\">Examples of unsymmetrical and symmetrical transmission<\/h2>\r\nAn example of unsymmetrical transmission is cabling of a sensor as shown in Fig.<a href=\"#Galvanic_sensor\" data-reference-type=\"ref\" data-reference=\"Galvanic_sensor\">4.3<\/a> in which supply circuit $latex V_{cc+}$ and signal circuit have the same return path $latex GND$.\r\n<figure><img src=\"Propagation\/Figures\/Galvanic_sensor.png\" id=\"Galvanic_sensor\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_164\" align=\"aligncenter\" width=\"415\"]<img class=\"wp-image-164\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-2.png\" width=\"415\" height=\"125\" alt=\"\" \/> Figure 4.3: An example of galvanic coupling.[\/caption]<\/figure>\r\nAnother example is electrical installation in vehicles which very often is single ended. Return path is vehicle\u2019s chassis.\r\n<figure>\r\n\r\n[caption id=\"attachment_165\" align=\"aligncenter\" width=\"565\"]<img class=\"wp-image-165\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-3.png\" width=\"565\" height=\"366\" alt=\"\" \/> Fihure 4.4: Surge due to galvanic coupling in TN-C mains supply system, by direct strike in the external lightning protection system of the building.[\/caption]<\/figure>\r\nMind that three phase supply system shown in Fig.<a href=\"#Unsym_line\" data-reference-type=\"ref\" data-reference=\"Unsym_line\">4.1<\/a>a) with Neutral (blue line) as return path is an example of unsymmetrical transmission as soon as current is driven in the neutral line. One case common for house installation is presented below.\r\n\r\nLow voltage AC power is delivered to a house as shown in Fig.<a href=\"#Lemp_1\" data-reference-type=\"ref\" data-reference=\"Lemp_1\">[Lemp_1]<\/a> in chapter <a href=\"#Thunderstorm\" data-reference-type=\"ref\" data-reference=\"Thunderstorm\">[Thunderstorm]<\/a>. Regulations forces to carry out the supply line between the transformer and the building input as the TN-C system <span class=\"citation\" data-cites=\"IEC-60364-1\">[@IEC-60364-1]<\/span>. It is four line supply system in which the Neutral line $latex N$ acts just as well as the Protective Earth line $latex PE$, therefore it is called $latex PEN$ line.\r\n\r\nIn Fig.<a href=\"#Galvanic_TNC\" data-reference-type=\"ref\" data-reference=\"Galvanic_TNC\">4.4<\/a> electric circuit of the transformer with Line to Neutral EMF: $latex E_A$, $latex E_B$ and $latex E_C$ feeding the building with the four line system $latex L_1$, $latex L_2$, $latex L_3$ and $latex PEN$ is shown. $latex Z_A$, $latex Z_B$ and $latex Z_C$ represents impedances of the whole building installation seen from the feeding point of the building. $latex Z_{PE}^B$ and $latex Z_{PE}^T$ are earthing impedances of the star point of the building and the transformer respectively. By direct strike of the lightning electromagnetic pulse $latex LEMP$ in the external lightning protection system of the building the current is driven to the star point of the building where it is split as shown with the red arrows in Fig.<a href=\"#Galvanic_TNC\" data-reference-type=\"ref\" data-reference=\"Galvanic_TNC\">4.4<\/a>. The biggest part of the current drives to the Earth through earthing impedance of the building $latex Z_{PE}^B$ but due to the fact that between the building and the transformer there is common $latex PEN$ line, part of the current is driven as the surge in the whole installation inside the building and between the building and the transformer.\r\n\r\nUsually there are more customers supplied from one transformer, as shown in Fig.<a href=\"#Lemp_1\" data-reference-type=\"ref\" data-reference=\"Lemp_1\">[Lemp_1]<\/a>. Due to galvanic connection of all of them with the star point of the transformer, all customers experiences the surge with different degree.\r\n\r\nMore resistive against surges would be the TN-S system <span class=\"citation\" data-cites=\"IEC-60364-1\">[@IEC-60364-1]<\/span>. It is five line supply system which has separate the Neutral $latex N$ and the Protective Earth $latex PE$ line.\r\n\r\nExample of symmetrical transmission can be:\r\n<ul>\r\n \t<li>power or signal cables of house installation layouted in the walls, ceilings or floors,<\/li>\r\n \t<li>cables layouted on the mounting plate in the control cabinet of a system,<\/li>\r\n \t<li>signal paths on the printed circuit board.<\/li>\r\n<\/ul>\r\n<h2 id=\"galvanic_coupling\">Galvanic coupling<\/h2>\r\nGalvanic coupling is illustrated in Fig.<a href=\"#Galvanic_PCB\" data-reference-type=\"ref\" data-reference=\"Galvanic_PCB\">4.5<\/a> with very simple situation. There is a circuit composed of two meshes. In mesh 1 there is a source of power or a signal with electromotive force $latex EMF_1$ and internal resistance $latex R_{S1}$ and load $latex R_{L1}$. The second with $latex EMF_2$ and $latex R_{L2}$. Intension is to deliver electric power or signal from $latex EMF_1$ to $latex R_{L1}$ and from $latex EMF_2$ to $latex R_{L2}$. However for some reason the two meshes have common return path represented with $latex R_{GND}$ and $latex L_{GND}$ which build the impedance $latex Z_{GND}=R_{GND} + j \\omega L_{GND}$.\r\n<figure><img src=\"Propagation\/Figures\/Galvanic_PCB.png\" id=\"Galvanic_PCB\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_166\" align=\"aligncenter\" width=\"443\"]<img class=\"wp-image-166\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-4.png\" width=\"443\" height=\"305\" alt=\"\" \/> Figure 4.5: Explanation of galvanic coupling.[\/caption]<\/figure>\r\nEach loop has self-inductance. It is composed of external inductance depending on wires layout and consequently on the area of the loop they build. Internal inductance of wires from which the loops are built also contribute to the total inductance. Obviously inductance must be assigned to the mesh but it is justified to extract the internal inductance of the common return path $latex L_{GND}$ as shown in Fig.<a href=\"#Galvanic_PCB\" data-reference-type=\"ref\" data-reference=\"Galvanic_PCB\">4.5<\/a> because it is the attribute of the return path. External inductances of both meshes and internal inductances of feeding wires are disregarded as declared earlier.\r\n\r\nVoltage $latex U_{GND}$ across the common return path represented with impedance $latex Z_{GND}$ is as follows\r\n\r\n$latex U_{GND} =\r\n-\\frac{ \\frac{EMF_1}{RS1+RL1} + \\frac{EMF_2}{RS2+RL2} }\r\n{ \\frac{1}{R_{S1}+R_{L1}} + \\frac{1}{R_{S2}+R_{L2}} + \\frac{1}{Z_{GND}}}\r\n\\label{U_GND}\\tag{4.1} $\r\n\r\nVoltage $latex U_{L2}$ across the load resistance $latex R_{L2}$ in mesh 2 is as follows\r\n\r\n$latex U_{L2} = \\frac{ R_{L2}} {R_{S2}+R_{L2}}\r\n\\left( EMF_2 - U_{GND} \\right)\r\n\\label{U_L2}\\tag{4.2} $\r\n\r\nElectromotive force $latex EMF_1$ contributes to the voltage $latex U_{L2}$ across the load of mesh 2 as follows\r\n\r\n$latex U_{L2}^{EMF_1} = - \\frac{ R_{L2} \\cdot EMF_1}\r\n{\r\n(R_{S1}+R_{L1}) (R_{S2}+R_{L2})\r\n\\left(\r\n\\frac{1}{R_{S1}+R_{L1}} + \\frac{1}{R_{S2}+R_{L2}} + \\frac{1}{Z_{GND}}\r\n\\right)\r\n}\r\n\\label{U_2}\\tag{4.3} $\r\n\r\nThe common return path causes unintentional contribution of $latex EMF_1$ to voltage across $latex R_{L2}$ and vice versa, via galvanic coupling.\r\n\r\nObviously, coupled voltage $latex U_{L2}^{EMF_1}$ would disappear by zero voltage $latex U_{GND}$ across the common return path. It can happen in two cases:\r\n<ul>\r\n \t<li>by zero impedance of the return path $latex Z_{GND}=0\\Omega$. That is unrealistic to achieve.<\/li>\r\n \t<li>by total symmetry of the meshes i.e.\u00a0if electromotive forces have the same amplitude and opposite phases. Moreover impedances of both paths are the same. In most cases it does not have sense by energy as well signal transmission.<\/li>\r\n<\/ul>\r\nMoreover perfect galvanic decoupling will happen with replacement of electromotive forces $latex EMF_1$ and $latex EMF_2$ with current sources. By power transportation it is hardly to imagine but transmission of current signals is used very often for that reason.\r\n\r\nIntensity of galvanic coupling rises with increased impedance of the return path. By $latex Z_{GND}$ tending to infinity, $latex I_{GND}$ tends to zero and coupled voltage $latex U_{L2}$ approaches maximum expressed with the formula below\r\n\r\n$latex U_{L2_{MAX}}^{EMF_1} = \\lim_{Z_{GND} \\rightarrow \\infty} U_{L2}^{EMF_1} = \\frac{ R_{L2} \\cdot EMF_1} {R_{S1}+R_{S2}+R_{L1}+R_{L2} }\r\n\\label{U_2max}\\tag{4.4} $\r\n\r\nAs galvanic coupling rises with increased internal impedance $latex Z_{GND}$ of the common return path it is valuable to investigate frequency dependence of impedance $latex Z_{GND}$. This impedance depends on the shape of the cross section of the conductor. Considered are two cases for: round wire and rectangle cross section.\r\n<h3 id=\"internal-impedance-of-round-wire\">Internal impedance of round wire<\/h3>\r\nConsidered is idealized case of infinitely long strait wire with circular cross-section with radius $latex r$ placed in free space in order to neglect proximity effects. Segment of such wire with length $latex l$ is shown in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>a).\r\n<figure><img src=\"Propagation\/Figures\/Round_wire.png\" id=\"Round_wire\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_167\" align=\"aligncenter\" width=\"698\"]<img class=\"wp-image-167\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-5.png\" width=\"698\" height=\"335\" alt=\"\" \/> Figure 4.6: Frequency dependence of round wire impedance per unit length.[\/caption]<\/figure>\r\nImpedance of such wire per unit length yields\r\n\r\n$latex Z(f) = \\frac{k(f)}{2 \\pi r \\sigma} \\frac{J_0[k(f)r]}{J_1[k(f)r]}\r\n\\label{Z_f}\\tag{4.5} $\r\n\r\nwhere $latex J_0[k(f)r]$ and $latex J_1[k(f)r]$ are Bessel\u2019s functions of zero and first order, $latex \\sigma$ and $latex \\mu$ is conductivity and magnetic permeability of wire material respectively and $latex k(f)$ is the wave number in the wire material\r\n\r\n$latex k(f) = \\frac{1-j}{\\delta (f)}\r\n\\label{k_f}\\tag{4.6} $\r\n\r\n$latex \\delta(f)$ is called the skin depth\r\n\r\n$latex \\delta(f) = \\sqrt{\\frac{2}{\\omega \\mu \\sigma}}\r\n\\label{delta}\\tag{4.7} $\r\n\r\nwhich expresses depth in which current density is $latex e$ times smaller than on the surface of the wire, where $latex e$ is Euler\u2019s number. To be precise it is valid for plane wave facing infinite conducting medium oriented perpendicularly to the direction of the wave propagation.\r\n\r\nBy low frequencies the real part of impedance $latex Z(f)$, Eq.(4.5) is nothing else but DC resistance of the round wire\r\n\r\n$latex R_{LF} = \\frac{1}{\\pi \\sigma r^2}\r\n\\label{R_LF}\\tag{4.8} $\r\n\r\nBy high frequencies the real part of impedance $latex Z(f)$, Eq.(4.5) can be solved as if the current flowed uniformly through a layer of thickness equal to skin depth $latex \\delta$. The effective cross-sectional area for driving the current would be then approximately equal to skin depth $latex \\delta$ times the conductor\u2019s circumference $latex 2r-\\delta \\approx 2r$. Thus HF resistance of round wire is approximately equal to DC resistance of a hollow tube with wall thickness $latex 2r-\\delta \\approx 2r$ carrying direct current\r\n\r\n$latex R_{HF}(f) = \\frac{1}{\\pi \\sigma [2r-\\delta(f)]\\delta(f)} \\approx \\frac{1}{2 \\pi \\sigma r \\delta(f)}\r\n\\label{R_HF}\\tag{4.9} $\r\n\r\nStatic and low frequency inductance of the round wire, accompanying uniformly flowed current is independent on wire radius $latex r$ <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>\r\n\r\n$latex L_{DC} = \\frac{\\mu}{8 \\pi }\r\n\\label{L_DC}\\tag{4.18} $\r\n\r\nThus the imaginary part of impedance $latex Z(f)$, Eq.(4.5) i.e. reactance $latex X_{LF}(f)$ in low frequencies depends linearly on frequency $latex X_{LF}(f) = 2 \\pi f L_{DC}\r\n\\label{X_LF}\\tag{4.11} $\r\n\r\nBy high frequencies reactance and resistance are equal $latex X_{HF}(f) = R_{HF}(f)$. It can be explained with the plane electromagnetic wave penetrating good conducting medium. The wave impedance has then equal real and imaginary part.\r\n\r\nFrequency dependence of the internal impedance of the round wire in double logarithmic scale is illustrated in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>b). Breaking frequency $latex f_0$ marked there, which is border between low and high frequency regions can be established equating LF resistance and reactance\r\n\r\n$latex f_0 = \\frac{4}{\\pi \\mu \\sigma r^2}\r\n\\label{f_0}\\tag{4.12} $\r\n\r\nMind that low frequency reactance is proportional to frequency, therefore it is strait line with the slope $latex 20dB\/dec$, compare Eq.(4.11). The HF resistance and reactance are reciprocally proportional to the skin depth $latex \\delta(f)$, see Eq.(4.9) therefore they increase proportionally to the square root of frequency. In other words they have slopes $latex 10dB\/dec$. Module of high frequency impedance $latex |Z_{GND}(f)|$ is by factor $latex \\sqrt{2}$ bigger than HF resistance or reactance, because they are equal one to another. In double logarithmic scale it is bigger about $latex 3dB$.\r\n\r\nEq.(4.12) gives the hint how to keep impedance of the round wire small up to relatively high frequencies. The trick consists in keeping the radius $latex r$ of wire relatively small. It is done in so called HF litz-wire in which wire is composed of bundle of individually isolated strands, as shown in Fig.<a href=\"#HF_Litz_wire\" data-reference-type=\"ref\" data-reference=\"HF_Litz_wire\">4.7<\/a>.\r\n<figure><img src=\"Propagation\/Figures\/HF_Litz-wire.png\" id=\"HF_Litz_wire\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_168\" align=\"aligncenter\" width=\"268\"]<img class=\"wp-image-168\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-6.png\" width=\"268\" height=\"206\" alt=\"\" \/> Figure 4.6: High frequency litz-wire built of seven isolated strains.[\/caption]<\/figure>\r\nIn such wire, in spite of the cross-section equal to sum of cross-sections of individual strains, the breaking frequency Eq.(4.12) is correlated with individual strain<a href=\"#fn1\" class=\"footnote-ref\" id=\"fnref1\" role=\"doc-noteref\"><sup>1<\/sup><\/a>.\r\n<h3 id=\"internal-impedance-of-conductor-with-rectangular-cross-section\">Internal impedance of conductor with rectangular cross section<\/h3>\r\nConsidered is idealized case of infinitely long strait conductor with rectangular cross section with width $latex w$ and thickness $latex t$ placed in free space in order to neglect proximity effects. Segment of such wire with length $latex l$ is shown in Fig.<a href=\"#Rectangle\" data-reference-type=\"ref\" data-reference=\"Rectangle\">4.8<\/a>a).\r\n\r\nUnlike by round wire no analytical formula exist for such conductor.\r\n\r\nThe low frequency resistance yields\r\n\r\n$latex R_{LF} = \\frac{1}{\\sigma w t}\r\n\\label{R_LF_wt}\\tag{4.13} $\r\n\r\nBy high frequencies resistance can be solved as by round wire i.e.\u00a0as if the current flowed uniformly through a layer of thickness equal to skin depth $latex \\delta$. The effective cross-sectional area for driving the current would be then approximately equal to skin depth $latex \\delta$ times the conductor\u2019s circumference $latex 2 (w + t)$. Thus HF resistance of round wire is approximately equal to DC resistance of a hollow rectangular tube with wall thickness $latex \\delta$ carrying direct current <span class=\"citation\" data-cites=\"Clayton_MTL\">[@Clayton_MTL]<\/span>\r\n\r\n$latex R_{HF}(f) = \\frac{1}{2 \\sigma \\delta(f) (w+t)}\r\n\\label{R_HF_wt}\\tag{4.14} $\r\n<figure><img src=\"Propagation\/Figures\/Rectangle_wire.png\" id=\"Rectangle\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_169\" align=\"aligncenter\" width=\"680\"]<img class=\"wp-image-169\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-7.png\" width=\"680\" height=\"316\" alt=\"\" \/> Figure 4.8: Frequency dependence of impedance of conductor with rectangular cross section.[\/caption]<\/figure>\r\nPrime to derivation of static and low frequency inductance of the conductor, distribution of magnetic field inside the conductor must be considered. If in the whole cross section uniformly distributed current $latex I$ is driven, then circulation of magnetic field along the rectangular circumference shown with dashed line in Fig.<a href=\"#Circulation\" data-reference-type=\"ref\" data-reference=\"Circulation\">4.9<\/a> yields\r\n\r\n$latex 2\\cdot H_x(y) \\cdot 2x + 2\\cdot H_y(x) \\cdot 2y = \\frac{I}{wt } \\cdot 4xy\r\n\\label{HxHy}\\tag{4.16} $\r\n\r\n$latex H_x(y) = \\frac{I}{wt } \\cdot y\r\n\\label{HxHy}\\tag{4.16} $\r\n\r\nThis is valid only if $latex H_x(y)$ is constant along the whole length of horizontal sides and contribution of circulation along vertical sides can be neglected. These rough simplification are justified provide $latex w \\gg t$.\r\n<figure><img src=\"Propagation\/Figures\/Cyrkulacja.png\" id=\"Circulation\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_170\" align=\"aligncenter\" width=\"492\"]<img class=\"wp-image-170\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-8.png\" width=\"492\" height=\"202\" alt=\"\" \/> Figure 4.9: $latex \\stackrel{\\rightarrow}{H}(x,y)$ circulation inside the conductor with rectangular cross section.[\/caption]<\/figure>\r\nInternal inductance can be directly computed from energy relation by equating magnetic energy stored inside the conductor expressed with magnetic field strength $latex \\stackrel{\\rightarrow}{H}(x,y)$ to the same energy expressed with the internal inductance $latex L_{DC}$\r\n\r\n$latex \\frac{1}{2} L_{DC} I^2 = \\frac{\\mu}{2} \\int_{S} |H(x,y)|^2 dS = \\frac{\\mu}{2} \\cdot \\frac{I^2}{w^2 t^2} \\int_{-\\frac{w}{2}}^{-\\frac{w}{2}}dy\r\n\\int_{-\\frac{t}{2}}^{-\\frac{t}{2}}x^2dx\r\n\\label{Hx}\\tag{4.17} $\r\n\r\nFinally\r\n\r\n$latex L_{DC} = \\frac{\\mu}{12 } \\cdot {t}{w}\r\n\\label{L_DC}\\tag{4.18} $\r\n\r\nBreaking frequency $latex f_0$ can be established as by the round wire with equating LF resistance and reactance\r\n\r\n$latex f_0 = \\frac{6}{\\pi \\mu \\sigma t^2}\r\n\\label{f_0_xy}\\tag{4.19} $\r\n\r\nFrequency dependence of the internal impedance of conductor with rectangle cross section in double logarithmic scale is illustrated in Fig.<a href=\"#Rectangle\" data-reference-type=\"ref\" data-reference=\"Rectangle\">4.8<\/a>b). Qualitatively it is identical with behavior of the round wire shown in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>b).\r\n<figure><img src=\"Propagation\/Figures\/Current_wt_mod.png\" id=\"J_wt\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_171\" align=\"aligncenter\" width=\"753\"]<img class=\"wp-image-171\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-9.png\" width=\"753\" height=\"587\" alt=\"\" \/> Figure 4.10: Distribution of current density in the rectangular conductor with width $latex w = 381\\mu m$ and thickness $latex t = 35.56 \\mu m$.[\/caption]<\/figure>\r\nAn example of distribution of current density in the rectangular conductor in frequency dependence, sourced from <span class=\"citation\" data-cites=\"Clayton_MTL\">[@Clayton_MTL]<\/span> is shown in Fig.<a href=\"#J_wt\" data-reference-type=\"ref\" data-reference=\"J_wt\">4.10<\/a>. Features notable in it are summarized below:\r\n<ul>\r\n \t<li>current crowding toward the outer edge is remarkable only in direction for which the skin depth $latex \\delta(f)$ is much smaller than the side of the rectangle,<\/li>\r\n \t<li>current which is the integral of current density $latex J(x,y)$ over the cross section surface is the smaller the bigger frequency is, because impedance of conductor rises,<\/li>\r\n \t<li>proportionality of the skin depth $latex \\delta(f)$ to the square root of frequency $latex \\sqrt f$ can be confirmed.<\/li>\r\n<\/ul>\r\n<h3 id=\"precedence-of-rectangular-conductor-over-round-wire\">Precedence of rectangular conductor over round wire<\/h3>\r\nIt is desired to compare impedance $latex Z_{GND}$ of the common path for different shape of the cross section because as presented in previous sections it is shape dependent.\r\n\r\nBreaking frequency is the upper bound of the frequency by which internal impedance is kept relatively deep i.e.\u00a0impedance is only slightly bigger than DC resistance. Comparison of Eq.(4.12) and Eq.(4.19) shows unequivocally that $latex f_0$ is much bigger for conductor with rectangle cross section. Indeed in case of rectangle cross section the number in numerator is bigger than by round wire, 6 instead of 4 but crucial is thickness in square in denominator which by $latex w \\gg t$ is much smaller than square of radius $latex r$ in case of round wire. By comparison of HF litz-wire and rectangle conductor the break frequencies can be similar but costs of HF litz-wire exceeds pretty costs of ordinary solid conductor.\r\n\r\nThis rationalises precedence of rectangular conductor over round wire. The wider conductor with rectangle cross section by unchanged thickness the smaller low frequency resistance $latex R_{LF}$. Therefore as shown in Fig.<span id=\"Unsym_line\" label=\"Unsym_line\">[Unsym_line]<\/span>c) by double layers\u2019 PCBs the return path are as wide as practical and by multilayers\u2019 PCBs separate layer or even more layers are dedicated to the return paths.\r\n<h2 id=\"LC_coupling\">Electric (capacitive) and magnetic (inductive) coupling<\/h2>\r\n<h3 id=\"capacitive-coupling\">Capacitive coupling<\/h3>\r\n<figure><img src=\"Propagation\/Figures\/Capacitive_coupling.png\" id=\"Coupling_C\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_172\" align=\"aligncenter\" width=\"696\"]<img class=\"wp-image-172\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-10.png\" width=\"696\" height=\"274\" alt=\"\" \/> Figure 4.11: Illustration of capacitive coupling by unsymmetrical transmission: cross sectional view a), top view b).[\/caption]<\/figure>\r\nTotal voltage across load conductance $latex G_{L2}$ yields\r\n\r\n$latex U_{L2_{total}} = \\frac{ G_{S2}(G_{L1} + G_{S1} + Y_1 + Y_{12}) \\cdot EMF_2 + G_{S1} Y_{12} \\cdot EMF_1}\r\n{\r\n(G_{L1}+G_{S1} + Y_1) (G_{L2}+G_{S2} + Y_2) +\r\nY_{12} (G_{L1}+G_{L2} + G_{S1} + G_{S2} +Y_1 + Y_2)\r\n}\r\n\\label{U_2C_tot}\\tag{4.20} $\r\n\r\nwhere $latex Y = j \\omega C$ is admittance of a capacitor\r\n\r\nElectromotive force $latex EMF_1$ contributes to the voltage $latex U_{L2}$ across the load conductance $latex G_{L2}$ of mesh 2 as follows\r\n\r\n$latex U_{L2} = \\frac{ G_{S1} Y_{12} \\cdot EMF_1}\r\n{\r\n(G_{L1}+G_{S1} + Y_1) (G_{L2}+G_{S2} + Y_2) +\r\nY_{12} (G_{L1}+G_{L2} + G_{S1} + G_{S2} +Y_1 + Y_2)\r\n}\r\n\\label{U_2C}\\tag{4.21} $\r\n\r\nObviously, coupled voltage $latex U_{L2}$ would be zero by zero mutual capacitance $latex C_{12}$, $latex Y_{12}=0 \\frac{1}{\\Omega}$. That is unrealistic. There exist always capacitance between two metallic objects as signal or power lines. Even replacement of electromotive forces $latex EMF_1$ and $latex EMF_2$ with current sources does not liberate from capacitive coupling.\r\n<h3 id=\"inductive-coupling\">Inductive coupling<\/h3>\r\n<figure>\r\n\r\n[caption id=\"attachment_173\" align=\"aligncenter\" width=\"642\"]<img class=\"wp-image-173\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-11.png\" width=\"642\" height=\"285\" alt=\"\" \/> Figure 4.12: Illustration of inductive coupling by unsymmetrical transmission: cross sectional view a), top view b).[\/caption]<\/figure>\r\nTotal voltage across load resistance $latex R_{L2}$ yields\r\n\r\n$latex U_{L2_{total}} = \\frac{R_{L2}}{R_{L2}+ R_{S2} + Z_2+Z_{12}}\r\n\\left[EMF_2 -\r\n\\frac{\r\n\\frac{EMF_1}{R_{L1} + R_{S1} + Z_1+Z_{12}} + \\frac{EMF_2}{R_{L2} + R_{S2} + Z_2+Z_{12}}}\r\n{\\frac{1}{R_{L1} + R_{S1} + Z_1+Z_{12}} + {\\frac{1}{R_{L2} + R_{S2} + Z_2+Z_{12}} - \\frac{1}{Z_{12}} }}\r\n\\right]\r\n\\label{U_2L_tot}\\tag{4.22} $\r\n\r\nwhere $latex Z_1 = j \\omega L_1$ and $latex Z_2 = j \\omega L_2$ are impedances of self inductances $latex L_1$ and $latex L_2$ respectively. $latex Z_{12} = j \\omega M_{12}$ is impedance of a mutual inductance $latex M_{12}$. It depends on self inductances as follows $latex M_{12} = k \\sqrt{L_1 L_2}$ where $latex 0&lt;k&lt;1$ is coupling factor.\r\n\r\nElectromotive force $latex EMF_1$ contributes to the voltage $latex U_{L2}$ across the load resistance $latex R_{L2}$ of mesh 2 as follows\r\n\r\n$latex U_{L2} =\r\n\\frac{\r\n-\\frac{R_{L2}}{R_{L2}+ R_{S2} + Z_2+Z_{12}} \\cdot EMF_1 }\r\n{\\left[ {\\frac{1}{R_{L1} + R_{S1} + Z_1+Z_{12}} + {\\frac{1}{R_{L2} + R_{S2} + Z_2+Z_{12}} - \\frac{1}{Z_{12}} }} \\right]\r\n\\left( R_{L1}+R_{S1} + Z_1+Z_{12}\\right)}\r\n\\label{U_2L}\\tag{4.23} $\r\n\r\nObviously, coupled voltage $latex U_{L2}$ would be zero by zeroised coupling factor $latex k$. Though it can be kept on very low level, it is never equal to zero.\r\n\r\nReplacement of electromotive forces $latex EMF_1$ and $latex EMF_2$ with current sources liberate from inductive coupling.\r\n<h3 id=\"dependence-of-capacitance-and-external-inductance-on-cross-sections-shape\">Dependence of capacitance and external inductance on cross section\u2019s shape<\/h3>\r\n<figure><img src=\"Propagation\/Figures\/L_ext.png\" id=\"LC_ext\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_174\" align=\"aligncenter\" width=\"633\"]<img class=\"wp-image-174\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-12.png\" width=\"633\" height=\"172\" alt=\"\" \/> Figure 4.13: Illustration for consideration of capacitance and external inductance.[\/caption]<\/figure>\r\nCapacitance per unit length of a infinitely long strait wire with circular cross section, layouted in air parallel to infinite conducting plane, as shown in Fig.<a href=\"#LC_ext\" data-reference-type=\"ref\" data-reference=\"LC_ext\">4.13<\/a>a) yields <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>\r\n\r\n$latex C_{\\bigcirc} = \\frac{2\\pi\\varepsilon_0}{\\operatorname{ar\\,cosh} \\left(\\frac{ h}{r} \\right)}\r\n\\label{C_r}\\tag{4.24} $\r\n\r\nwhere $latex \\operatorname{ar\\,cosh} (x) = \\ln \\left[ \\sqrt{x^2-1} + x \\right] \\approx \\ln(2x)$ for $latex x \\gg 1$. Therefore for a wire that is sufficiently far from the conducting plane, $latex h \\gg r$ this simplifies to\r\n\r\n$latex \\begin{array} {ll}\r\nC_{\\bigcirc} \\approx \\frac{2\\pi\\varepsilon_0}{\\ln \\left(\\frac{ 2h}{r} \\right)} &amp; \\text{for}~~h \\gg r \r\n\\end{array}\r\n\\label{C_r_approx}\\tag{4.25} $\r\n\r\nAs explained in chapter <a href=\"#El%20size\" data-reference-type=\"ref\" data-reference=\"El size\">[El size]<\/a>, wave propagates along a transmission line if it is electrically long. Velocity of propagation is the quantity that binds parameters $latex \\mu$ and $latex \\varepsilon$ of the surrounding medium with per length parameters $latex L$ and $latex C$ of the transmission line. On one side it is given as in Eq.(<a href=\"#v_propagation\" data-reference-type=\"ref\" data-reference=\"v_propagation\">[v_propagation]<\/a>), on the other side as $latex v = \\frac{1}{\\sqrt{LC}}$. Therefore by homogeneous surrounding of transmission line holds\r\n\r\n$latex L \\: C = \\mu \\: \\varepsilon\r\n\\label{LC}\\tag{4.26} $\r\n\r\nThis straightforwardly leads to formulas for external inductance of the round wire\r\n\r\n$latex L_{\\bigcirc}^{EXT} = \\frac{\\mu_0}{2\\pi} \\operatorname{ar\\,cosh} \\left(\\frac{ h}{r} \\right)\r\n\\label{L_r}\\tag{4.27} $\r\n\r\n$latex \\begin{array} {ll}\r\nL_{\\bigcirc}^{EXT} \\approx \\frac{\\mu_0}{2\\pi} \\ln{\\left (\\frac{2h}{r} \\right)} &amp; \\text{for}~~h \\gg r \r\n\\end{array}\r\n\\label{L_r_approx}\\tag{4.28} $\r\n\r\nThere exist no general analytical formula for parameters of conductor with rectangular cross section. Cited here is approximate relation for the case shown in Fig.<a href=\"#LC_ext\" data-reference-type=\"ref\" data-reference=\"LC_ext\">4.13<\/a>b), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>, assuming zero thickness of conductor\r\n\r\n$latex \\begin{array} {ll}\r\nC_{\\framebox[0.15in]{}} \\approx \\varepsilon_0 \\left[ \\frac{w}{h} + 1.393 + 0.667 \\ln{\\left (\\frac{w}{h} + 1.444 \\right)} \\right] &amp;  \\text{for}~~w \\geq h\r\n\\end{array}\r\n\\label{C_rect}\\tag{4.29} $\r\n\r\n$latex \\begin{array} {ll}\r\nL_{\\framebox[0.15in]{}}^{EXT} \\approx \\frac{ \\mu_0}{ \\frac{w}{h} + 1.393 + 0.667 \\ln{\\left (\\frac{w}{h} + 1.444 \\right)}} &amp;  \\text{for}~~w \\geq h \r\n\\end{array}\r\n\\label{L_rect}\\tag{4.30} $\r\n\r\nCapacitance of the round wire with $latex 2.5mm^2$ cross section area i.e. with radius $latex =0.892mm$ layouted $latex h=1cm$ above ground plane, according to Eq.(4.25) amounts to $latex C_{\\bigcirc}=17.89pF\/m$. Rectangular conductor with the same cross section area and longer size $latex w=2cm$ has shorter size $latex t=0.125mm$<a href=\"#fn2\" class=\"footnote-ref\" id=\"fnref2\" role=\"doc-noteref\"><sup>2<\/sup><\/a>. According to Eq.(4.29) its capacitance is $latex C_{\\framebox[0.15in]{}}=37.35 pF\/m$.\r\n\r\nAnalogue is with external inductance. By the same geometry relations, according to Eq.(4.28) external inductance of the round wire amounts to $latex L_{\\bigcirc}^{EXT}=0.622 \\mu H\/m$ and by rectangular conductor is $latex L_{\\framebox[0.15in]{}}^{EXT}=0.298 \\mu H\/m$, according to Eq.(4.30).\r\n\r\nIn numerical example above capacitance\/external inductance is smaller\/ bigger for a round wire\/rectangular conductor. This example is realistic and the relation can be generalized. Rationale of it is rooted in the Euclidean geometry. It is taught there, that from all plane figures, the circle has the smallest ratio of circumference to surface area.\r\n\r\nLet us compare capacitances. The Gauss\u2019s flux theorem is a law relating the distribution of electric field to charges originating it. The electric flux through any hypothetical closed surface $latex S$ is equal to the net electric charge $latex Q$ within that closed surface $latex \\varepsilon_0 \\oiint_S \\overrightarrow{E}\\cdot \\overrightarrow{dS} = \\sum Q $. For infinite strait conductor the electric flux can be calculated through the surface per unit length and therefore it is reduced to the integral along arbitrary closed loop enclosing the conductor.\r\n\r\nLet us compare round wire and rectangular conductor with the same surface area, layouted on the same height $latex h$ above the ground plane. For round wire the integration path is shorter than for rectangular conductor, due to ratio of circumference to area. Therefore if in both cases net charge is the same, electric field strength distributed around rectangular conductor is smaller.\r\n\r\nVoltage across conductor and the ground plane is given by the line integral of electric field along arbitrary path between them $latex U = \\int_l \\overrightarrow{E}\\cdot \\overrightarrow{dl} $. Consequently it is smaller in case of rectangular conductor. Capacitance is ratio of free charge on the conductor or the ground plane to voltage necessary for gathering this amount of charge $latex C=Q\/U$. Conclusion is that, for gathering particular net charge less voltage is needed in case of rectangular conductor than of round wire. Finally, capacitance of rectangular conductor is bigger than of round wire.\r\n\r\nSimilarly can be proceeded by comparing external inductances. The Ampere\u2019s circular low relates the distribution of magnetic field strength to current causing it. The line integral of magnetic field strength $latex \\overrightarrow{H}$ along any hypothetical closed loop $latex l$ is equal to the net current $latex I$ enclosed in this loop $latex \\oint_l \\overrightarrow{H}\\cdot \\overrightarrow{dl} = \\sum I $.\r\n\r\nLet us compare round wire and rectangular conductor with the same surface area, layouted on the same height $latex h$ above the ground plane. For round wire the integration path is shorter than for rectangular conductor, due to ratio of circumference to area. Therefore if in both cases enclosed net current is the same, magnetic field strength distributed around rectangular conductor is smaller.\r\n\r\nMagnetic flux through any hypothetical surface built by the loop driving current $latex I$ is given by $latex \\Phi = \\mu_0 \\oiint_S \\overrightarrow{H}\\cdot \\overrightarrow{dS} $. For infinite loop composed of round wire or rectangular conductor, load in infinity and the ground plane as return path, only flux per unit length makes sense. It is smaller in case of rectangular conductor due to smaller field strength. Inductance is ratio of magnetic flux to current necessary for generating it $latex L=\\Phi\/I$. It means that, by driving the same particular net current less flux is generated in case of rectangular conductor than of round wire. Finally, inductance of rectangular conductor is smaller than of round wire.\r\n\r\nThis statement can be rationalised alternatively starting with comparison of capacitances along with conclusion from Eq.(4.26).\r\n\r\nNotice, that capacitance\/external inductance of the round wire rises\/decays slower with increased radius $latex r$ than capacitance\/external inductance of the rectangular conductor with its width $latex w$. In the first case the change is compressed because radius is argument of natural logarithm. In the second case, dependence on width $latex w$ is direct.\r\n<h4 id=\"capacitive-and-inductive-coupling-by-symmetrical-transmission\">Capacitive and inductive coupling by symmetrical transmission<\/h4>\r\nUp to now only unsymmetrical transmission was covered. The only message concerning symmetrical transmission discussed here is the size of zone surrounding transmission line in which risk of capacitive or inductive coupling exist.\r\n<figure><img src=\"Propagation\/Figures\/C_unsym_sym_round.png\" id=\"C_unsym_sym_round\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_175\" align=\"aligncenter\" width=\"724\"]<img class=\"wp-image-175\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-13.png\" width=\"724\" height=\"234\" alt=\"\" \/> Figure 4.14: Module of electric field strength around unsymmeric a) and symmetric b) transmission line with round wire.[\/caption]<\/figure>\r\nIn Fig.<a href=\"#C_unsym_sym_round\" data-reference-type=\"ref\" data-reference=\"C_unsym_sym_round\">4.14<\/a> distribution of module of electric field strength around unsymmeric and symmetric transmission line with round wire is illustrated. The same voltage is applied and spectrum of colours has the same scale in both cases.\r\n\r\nEvidently the zone polluted with the electric field in case of symmetrical transmission is smaller. Module of electric field strength by symmetrical line decays stronger versus distance from the line in horizontal as well vertical direction. Rationale for it is the fact that space between feeding and return line in case of symmetrical transmission is much smaller than in case of unsymmetrical transmission. Fields outside the line cancel one another. As a consequence of it, zone with practically total cancellation of field is closer to the line in the case of symmetrical transmission.\r\n\r\nThe same can be concluded for unsymmetrical and symmetrical lines build of conductors with rectangular cross sections as shown in Fig.<a href=\"#C_unsym_sym_rectangle\" data-reference-type=\"ref\" data-reference=\"C_unsym_sym_rectangle\">4.15<\/a>.\r\n\r\nIn Fig.<a href=\"#L_unsym_sym_round\" data-reference-type=\"ref\" data-reference=\"L_unsym_sym_round\">4.16<\/a> distribution of module of magnetic field strength around unsymmeric and symmetric transmission line with round wire is illustrated. The same current is driven and spectrum of colours has the same scale in both cases.\r\n\r\nEvidently the zone polluted with the magnetic field in case of symmetrical transmission is smaller.\r\n<figure><img src=\"Propagation\/Figures\/C_unsym_sym_rectangle.png\" id=\"C_unsym_sym_rectangle\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_176\" align=\"aligncenter\" width=\"800\"]<img class=\"wp-image-176\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-14.png\" width=\"800\" height=\"284\" alt=\"\" \/> Figure 4.15: Module of electric field strength around unsymmeric a) and symmetric b) transmission line with conductor having rectangle cross section.[\/caption]<\/figure>\r\n<figure><img src=\"Propagation\/Figures\/L_unsym_sym_round.png\" id=\"L_unsym_sym_round\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_177\" align=\"aligncenter\" width=\"823\"]<img class=\"wp-image-177\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-15.png\" width=\"823\" height=\"267\" alt=\"\" \/> Figure 4.16: Module of magnetic field strength around unsymmeric a) and symmetric b) transmission line with round wire.[\/caption]<\/figure>\r\n<figure><img src=\"Propagation\/Figures\/L_unsym_sym_rectangle.png\" id=\"L_unsym_sym_rectangle\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_178\" align=\"aligncenter\" width=\"818\"]<img class=\"wp-image-178\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-16.png\" width=\"818\" height=\"257\" alt=\"\" \/> Figure 4.17: Module of magnetic field strength around unsymmeric a) and symmetric b) transmission line with conductor having rectangle cross section.[\/caption]\r\n\r\n<figcaption><\/figcaption><\/figure>\r\nThe same can be concluded for unsymmetrical and symmetrical lines build of conductors with rectangular cross sections as shown in Fig.<a href=\"#L_unsym_sym_rectangle\" data-reference-type=\"ref\" data-reference=\"L_unsym_sym_rectangle\">4.17<\/a>.\r\n<h3 id=\"TCL_chapter\">Transverse to longitudinal conversion<\/h3>\r\nEquivalent scheme of symmetric transmission is shown in Fig.<a href=\"#TCL_Fig\" data-reference-type=\"ref\" data-reference=\"TCL_Fig\">4.18<\/a>. Energy or signal should be delivered from the source $latex EMF$ with internal resistance $latex R_S$ to the load $latex R_L$ via symmetrical line. This line is layouted above the ground. Each line i.e.\u00a0feeding: 1-3 and return: 2-4 has parasitic capacitance related to the ground. They are represented with capacitances $latex C_{10}$ and $latex C_{20}$ by the source and $latex C_{35}$ and $latex C_{45}$ by the load. There are also parasitic capacitances of one to another line. They are represented with capacitances $latex C_{12}$ by the source and $latex C_{34}$ by the load. Moreover each line builds inductance represented with $latex L_{13}$ and $latex L_{24}$.\r\n\r\nSymmetric transmission means that parasitic parameters are in equilibrium consisted in following identities $latex \\begin{array} {rcl}\r\nC_{10} = C_{20} \\nonumber \\\\\r\nC_{35} = C_{45} \\nonumber \\\\\r\nL_{13} = L_{24} \\nonumber\r\n\\end{array}$\r\n\r\nIn such situation $latex U_{50} = 0$. Transmission line is perfectly balanced.\r\n<figure><img src=\"Propagation\/Figures\/TCL.png\" id=\"TCL_Fig\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_179\" align=\"aligncenter\" width=\"568\"]<img class=\"wp-image-179\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-17.png\" width=\"568\" height=\"331\" alt=\"\" \/> Figure 4.18: Symmetrical transmission over the ground plane, top view.[\/caption]<\/figure>\r\nElectromotive force $latex EMF$, which is oriented crosswise to the direction of transmission generates voltage $latex U_{50}$ along direction of transmission only if the circuit is out balance.\r\n\r\nRatio of transverse electromotive force $latex EMF$ to longitudinal voltage $latex U_{50}$\r\n\r\n$latex \\begin{array} {rcl}\r\nL_{TC} &amp; = &amp; \\frac{EMF}{U_{50}} \\\\\r\nL_{TC(dB)} &amp; = &amp; 20 \\log \\left( \\frac{EMF}{U_{50}} \\right)\r\n\\end{array}\r\n\\label{TCL}\\tag{4.35} $\r\n\r\nis called loss of transverse (to longitudinal) conversion, alternatively Transverse Conversion Loss TCL. It is used for rating deviation from the ideal balance by symmetrical transmission. By approaching perfect equilibrium, $latex L_{TC}$ tends to infinity.\r\n\r\nThere are two reasons for transverse conversion to be undesired: deterioration of integrity of transmitted signal and enlargement of the zone around the line polluted with the electromagnetic field, similarly as by unsymmetrical transmission.\r\n<h4 id=\"decomposition-of-currents-into-common-and-differential-mode\">Decomposition of currents into common and differential mode<\/h4>\r\nIn this subsection currents in the transmission line shown in Fig.<a href=\"#TCL_Fig\" data-reference-type=\"ref\" data-reference=\"TCL_Fig\">4.18<\/a> will be derived. For suppressing complexity of the formulas, source resistance $latex R_S$ will be omitted.\r\n\r\nVoltage $latex U_{50}$ unequal to zero means current $latex I_{50}$ driven through the ground impedance $latex Z_{GND}$. It can be calculated with the Thevenin\u2019s Theorem. Open circuit voltage $latex U_{50}^{Th}$ by removing branch $latex Z_{GND}$ yields\r\n\r\n$latex U_{50}^{Th} = \\frac{Y_{13} \\left[Y_{34}(Y_{35} + Y_{45}) + Y_{35}(Y_{24} + Y_{45})\\right] \\cdot EMF}\r\n{(Y_{13} + Y_{24})\\left[Y_{34}(Y_{35} + Y_{45}) + Y_{35} Y_{45} \\right] + Y_{13} Y_{24}(Y_{35} + Y_{45})} - \\frac{Y_{10}\\cdot EMF}{Y_{10} + Y_{20}}\r\n\\label{U50}\\tag{4.36} $\r\n\r\nwhere $latex Y_{34} = \\frac{R_L + \\frac{1}{j\\omega C_{34}}}{R_L \\frac{1}{j\\omega C_{34}}} $\r\n\r\nFor derivation of the Thevenin\u2019s impedance seen between poles 5-0, transposition of triangle 3-4-5 to the star with the star node 0\u2019 as shown in Fig.<a href=\"#TCL_Z_Th\" data-reference-type=\"ref\" data-reference=\"TCL_Z_Th\">4.19<\/a> must be performed\r\n\r\n$latex \\begin{array} {rcl}\r\nZ_{30'} = \\frac{Z_{34} Z_{35}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber \\\\\r\n\\nonumber \\\\\r\nZ_{40'} = \\frac{Z_{34} Z_{45}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber \\\\\r\n\\nonumber \\\\\r\nZ_{50'} = \\frac{Z_{35} Z_{45}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber\r\n\\end{array}$\r\n<figure><img src=\"Propagation\/Figures\/TCL_Z_Th.png\" id=\"TCL_Z_Th\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_180\" align=\"aligncenter\" width=\"594\"]<img class=\"wp-image-180\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-18.png\" width=\"594\" height=\"314\" alt=\"\" \/> Figure 4.19: Circuit for calculation of the Thevenin\u2019s impedance seen between poles 0-5.[\/caption]<\/figure>\r\nCurrent $latex I_{50}$ driven through the ground impedance $latex Z_{GND}$ is called common mode current $latex I^{CM}$. It yields\r\n\r\n$latex I^{CM} = I_{50} = \\frac{U_{50}^{Th}} {Z_{50}^{Th} + Z_{GND}}\r\n\\label{I50}\\tag{4.38} $\r\n\r\nIn node 0 it is split to current $latex I_{13}^{CM}$ and $latex I_{24}^{CM}$ driven in line 1-3 and 2-4 respectively. Ratio of currents $latex I_{13}^{CM}$ and $latex I_{24}^{CM}$ depends on relation between impedances of paths 0-1-3-5 and 0-2-4-5.\r\n\r\nAlong with the common mode component the differential mode current $latex I^{DM}$ is driven in the feeding and return line.\r\n\r\nIt can be calculated after removing branch with the ground impedance $latex Z_{GND}$\r\n\r\n$latex I^{DM} = I_{50} = \\frac{EMF} {Z_{13} + Z_{24} + \\frac{(Z_{35} + Z_{45}) Z_{34}}{Z_{35} + Z_{45} + Z_{34}}}\r\n\\label{IDM}\\tag{4.39} $\r\n\r\nFinally actual currents in the transmission lines expressed with the common and differential components yields\r\n\r\n$latex \\begin{array} {rcl}\r\nI_{13} &amp; = &amp; I^{DM} + I_{13}^{CM} \\\\\r\n\\\\\r\nI_{42} &amp; = &amp; I^{DM} - I_{24}^{CM}\r\n\\end{array} $\r\n\r\nCalculation of capacitances $latex C_{10}$, $latex C_{20}$ and $latex C_{12}$ as well $latex C_{35}$, $latex C_{45}$ and $latex C_{34}$ can be done only numerically because both triple of capacitors are linked. Any change of one of them causes changes of the rest.\r\n\r\nThe same concerns inductances. Notice that $latex L_{13}$ is serial connection of two inductances because line 1-3 is part of the loop 1-3-5-0 and 1-3-4-2.\r\n<h2 id=\"wave_coupling\">Wave coupling<\/h2>\r\n<h3 id=\"radiation\">Radiated waves<\/h3>\r\n<h4 id=\"idealized-entities-as-elementary-radiators\">Idealized entities as elementary radiators<\/h4>\r\n<ul>\r\n \t<li><em>Isotropic antenna.<\/em> It is omni directional radiator. In other words it radiates the same power in all directions.<\/li>\r\n \t<li><em>Electric (Hertzian) dipole.<\/em> Imagine electrically short, infinitesimally thin strait conducting segment carrying a current represented with the phasor $latex \\bf{I}$ that is assumed to be constant (as to magnitude and phase) at all points along the segment. If the segment length $latex l$ tends to zero whereas the current $latex \\bf{I}$ infinitely growths so that the quantity $latex {\\bf{p}} = l{\\bf{I}}$ remains finite and constant, then this product constitutes magnitude of the vector called the dipole moment. Its direction is along the segment and the sense according to the current direction $latex \\overrightarrow{1}_l$\r\n\r\n$latex \\overrightarrow{\\bf{p}} = \\lim \\limits_{l \\to 0, I \\to \\infty} (l{\\bf{I}}) \\overrightarrow{1}_l\r\n\\label{p_dipole}\\tag{4.41} $<\/li>\r\n \t<li><em>Magnetic (Fitzgeraldian) dipole.<\/em> Imagine electrically small, infinitesimally thin ring with azimuthal current $latex \\bf{I}$ flowing in it which does not depend on the angle. If the ring radius $latex a$ tends to zero whereas the current $latex \\bf{I}$ infinitely growths so that the product of the ring area and current $latex {\\bf{m}} = \\pi a^2 {\\bf{I}}$ remains finite and constant, then this product constitutes magnitude of the vector called the dipole moment. Its direction is orthogonal to the plane of the ring and the sense results from the vector product of unit radius vector $latex \\overrightarrow{1}_r$ and unit current density vector $latex \\overrightarrow{1}_j$\r\n\r\n$latex \\overrightarrow{\\bf{m}} = \\lim \\limits_{\r\n\\pi a^2 \\to 0, I \\to \\infty} (\\pi a^2 {\\bf{I}}) \\overrightarrow{1}_r \\times \\overrightarrow{1}_j\r\n\\label{m_dipole}\\tag{4.42} $<\/li>\r\n<\/ul>\r\nAll these entities are lossless. None of them exist in reality but they are useful in understanding the antenna theory including unintentional antennas such as cables connected to the EUT.\r\n<figure><img src=\"Propagation\/Figures\/xyz.png\" id=\"xyz\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_181\" align=\"aligncenter\" width=\"709\"]<img class=\"wp-image-181\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-19.png\" width=\"709\" height=\"283\" alt=\"\" \/> Figure 4.20: Cartesian orthogonal 3D system of coordinates a), electric dipole with only $latex {\\bf{p}}_z$ component of dipole moment b), magnetic dipole with only $latex {\\bf{m}}_z$ component of dipole moment.[\/caption]<\/figure>\r\nThe best way of describing the 3D fields generated by dipoles is the spherical system of co-ordinates in which the spacial position of the point under interest is described with three following numbers: $latex r, \\theta, \\varphi$, as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>a). The radius $latex r$ is distance from the origin of the co-ordinates system, $latex \\theta$ elevation angle between the $latex z$ axis and radius $latex r$, $latex \\varphi$ is the azimuthal angle between $latex x$ axis and projection of the radius $latex r$ on the $latex x0y$ plane.\r\n\r\nComponents of the electric field strength of the electric dipole oriented as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>b), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span> are as follows\r\n\r\n$latex {\\bf{E}}_r (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0^2} {2 \\pi} \\cos{(\\theta)}\r\n\\left(\r\n\\frac{1}{\\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{eE_r} \\tag{4.43} $\r\n\r\n$latex {\\bf{E}}_\\theta (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0^2} {4 \\pi} \\sin{(\\theta)}\r\n\\left(\r\nj \\frac{1}{\\beta_0 r} + \\frac{1} { \\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{eE_theta} \\tag{4.44} $\r\n\r\n$latex {\\bf{E_\\varphi}}= 0\r\n\\label{eE_phi} \\tag{4.45} $\r\n\r\nand of magnetic field strength\r\n\r\n$latex {\\bf{H}}_r= 0\r\n\\label{eH_r} \\tag{4.46} $\r\n\r\n$latex {\\bf{H}}_\\theta = 0\r\n\\label{eH_theta} \\tag{4.47} $\r\n\r\n$latex {\\bf{H}}_\\varphi(r,\\theta)= \\frac{{\\bf{p}}_z \\beta_0^2} {4 \\pi} \\sin{(\\theta)}\r\n\\left(\r\nj\\frac{1}{\\beta_0 r} + \\frac{1}{\\beta_0^2 r^2}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{eH_phi} \\tag{4.48} $\r\n\r\nwhere $latex Z_0$ is the intrinsic impedance of the medium in which the dipole is placed. It depends on the magnetic and dielectric constants $latex \\mu_0$ and $latex \\epsilon_0$ of the medium as follows\r\n\r\n$latex Z_0 = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}}\r\n\\label{Z_0}\\tag{4.49} $\r\n\r\nand $latex \\beta_0$ is the phase constant of the wave expressed with the formula\r\n\r\n$latex \\beta_0 = \\omega \\sqrt{\\mu_0 \\epsilon_0} = \\frac{2 \\pi}{\\lambda_0}\r\n\\label{beta}\\tag{4.50} $\r\n\r\nwhere $latex \\omega$ is angular frequency of the propagated wave and $latex \\lambda_0$ is the wave length.\r\n\r\nFar away from the electric dipole, regardless of direction disappears dependence of electric field on the second and the third power of radius $latex r$, refer to Eqs. (4.43) and (4.44) and dependence of $latex {\\bf{H}}_ {\\varphi}$ component of magnetic field on the second power of radius $latex r$, refer to Eq.\u00a0(4.48). Solely summands reciprocally proportional to the distance $latex r$ remains.\r\n\r\nIn the immediate vicinity of the electric dipole dominant is dependence of electric field on the third power of radius $latex r$, refer to Eqs. (4.43), (4.44). Zone within such solid is the near field zone. Magnetic field $latex {\\bf{H}}_{\\varphi}$ does not exist in it because it does not possess summand with radius $latex r$ in the third power. Emerges only electric field distributed as in static case. Therefore another name of the zone is the static zone.\r\n\r\nBetween the near field and the far field zone there is the buffer called the reactive zone.\r\n\r\nComponents of the electric field strength of the magnetic dipole oriented as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>c), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span> are as follows\r\n\r\n$latex {\\bf{E}}_r = 0\r\n\\label{mE_r} \\tag{4.51} $\r\n\r\n$latex {\\bf{E}}_\\theta = 0\r\n\\label{mE_theta} \\tag{4.52} $\r\n\r\n$latex {\\bf{E}}_\\varphi (r,\\theta)=-j \\frac{ {\\bf{m}}_z \\omega \\mu_0 \\beta_0^2 } {4 \\pi} \\sin{(\\theta)}\r\n\\left(\r\nj\\frac{1}{\\beta_0 r} + \\frac{1}{\\beta_0^2 r^2}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{mE_phi} \\tag{4.53} $\r\n\r\nand of magnetic field strength\r\n\r\n$latex {\\bf{H}}_r (r,\\theta)= j \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0^2} {2 \\pi Z_0 } \\cos{(\\theta)}\r\n\\left(\r\n\\frac{1}{\\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{mH_r} \\tag{4.54} $\r\n\r\n$latex {\\bf{H}}_\\theta (r,\\theta)= j \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0^2 } {4 \\pi Z_0} \\sin{(\\theta)}\r\n\\left(\r\nj \\frac{1}{\\beta_0 r} + \\frac{1} { \\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}\r\n\\right) e^{-j\\beta_0 r}\r\n\\label{mH_theta} \\tag{4.55} $\r\n\r\n$latex {\\bf{H}}_\\varphi = 0\r\n\\label{mH_phi} \\tag{4.56} $\r\n\r\nDefinition of the far, reactive and near field zones are valid also for the magnetic dipole.\r\n\r\nPhasor of total strength of electric and magnetic field in the far field zone of electric dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b) dependent on locus of point yields\r\n\r\n$latex \\label{Phasor_p}\r\n{\\bf{E}}_\\theta (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}\r\n\\cdot \\frac{ e^{j \\left( \\frac{\\pi}{2} - \\beta_0 r \\right)}} {r} \\tag{4.57} $\r\n\r\n$latex {\\bf{H}}_\\varphi( r,\\theta)= \\frac{{\\bf{p}}_z \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}\r\n\\cdot \\frac{ e^{j \\left( \\frac{\\pi}{2} - \\beta_0 r \\right)}} {r}\r\n\\label{Phasor_e2} \\tag{4.58} $\r\n\r\nPhasor of total strength of electric and magnetic field in the far field zone of magnetic dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c) dependent on locus of point yields\r\n\r\n$latex \\label{Phasor_m0}\r\n{\\bf{E}}_\\varphi (\\omega t, r,\\theta)= \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}\r\n\\cdot \\frac{ e^{-j \\beta_0 r}} {r} \\tag{4.59} $\r\n\r\n$latex {\\bf{H}}_\\theta(\\omega t, r,\\theta)= - \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0 } {4 \\pi Z_0 } \\cdot \\sin{(\\theta)}\r\n\\cdot \\frac{ e^{-j \\beta_0 r }} {r}\r\n\\label{Phasor_m} \\tag{4.60} $\r\n\r\nTotal strength of electric and magnetic field in the far field zone of electric dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b), dependent on time and locus of point yields\r\n\r\n$latex \\label{Time_e}\r\nE_\\theta (\\omega t, r,\\theta)= \\frac{p_z Z_0 \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}\r\n\\cdot \\sin{\\left( \\omega t + \\frac{\\pi}{2} - \\beta_0 r \\right)} \\tag{4.61} $\r\n\r\n$latex H_\\varphi(\\omega t, r,\\theta)= \\frac{p_z \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}\r\n\\cdot \\sin{\\left( \\omega t + \\frac{\\pi}{2} - \\beta_0 r \\right)}\r\n\\label{Time_e1} \\tag{4.62} $\r\n\r\nTotal strength of electric and magnetic field in the far field zone of magnetic dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c), dependent on time and locus of point yields\r\n\r\n$latex \\label{Time_m0}\r\nE_\\varphi (\\omega t, r,\\theta)= \\frac{m_z \\omega \\mu_0 \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}\r\n\\cdot \\sin{\\left( \\omega t - \\beta_0 r \\right)} \\tag{4.63} $\r\n\r\n$latex H_\\theta(\\omega t, r,\\theta)= - \\frac{m_z \\omega \\mu_0 \\beta_0 } {4 \\pi Z_0 } \\cdot \\frac{ \\sin{\\theta}}{r}\r\n\\cdot \\sin{\\left( \\omega t - \\beta_0 r \\right)}\r\n\\label{Time_m} \\tag{4.64} $\r\n\r\nOne of features of waves in the far field zone is the wavefront called also equiphase surface. It is the set of points at which field strength has the same phase at a given instant<a href=\"#fn3\" class=\"footnote-ref\" id=\"fnref3\" role=\"doc-noteref\"><sup>3<\/sup><\/a>. Evidently the equiphase surface of the isotropic antenna is the sphere.\r\n\r\nThe wavefront constraint of the electric dipole means constant phase in Eq.\u00a0(4.61) and in (4.62) $latex \\omega t + \\frac{\\pi}{2} - \\beta_0 r = const$ and for magnetic dipole in Eq\u00a0(4.63) and in (4.64) $latex \\omega t - \\beta_0 r = const$ . Hence the wavefront of both dipoles must be also the sphere $latex r=const$. The wavefront constraint includes also information about movement direction of the wavefront. In order to fulfill the constraint, radius $latex r$ must increase with passage of the time i.e.\u00a0the wavefront moves outwards.\r\n\r\nIn arbitrary place of the space surrounding any antenna ratio of magnitudes of vectors of phasors of electric and magnetic field $latex {\\bf{Z}}_w (r,\\theta,\\varphi)= \\frac{|\\overrightarrow{{\\bf{E}}} (r,\\theta, \\varphi)|}\r\n{|\\overrightarrow{{\\bf{H}}} (r,\\theta,\\varphi)|}\r\n\\label{Zw}\\tag{4.65} $\r\n\r\nis called wave impedance.\r\n\r\nEven for the electric and magnetic dipole general formula for the wave impedance are complicated but its constrains to the symmetry plane $latex x0y$ is agreeable. In the formulas below the distance $latex r$ is scaled with the wavelength $latex \\lambda$ transmitted or received by the dipole. $latex r\/\\lambda$ can be seen as the \"electrical distance\" which is in line with definition of electrical size presented in chapter <a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a>\r\n\r\n$latex \\label{Zw_e}\r\n{\\bf{Z}}_w^{(e)} \\left(\\frac{r}{\\lambda},90^{\\circ},\\varphi \\right)= \r\nZ_0 \\frac{1+j\\frac{4\\pi^2 \\left( \\frac{r}{\\lambda} \\right)^2-1}{2\\pi \\left( \\frac{r}{\\lambda} \\right) }}{1+j 2\\pi \\left( \\frac{r}{\\lambda} \\right)} \\tag{4.66} $\r\n\r\n$latex \\label{Zw_m}\r\n{\\bf{Z}}_w^{(m)} \\left( \\frac{r}{\\lambda},90^{\\circ},\\varphi \\right)= \r\nZ_0 \\frac{1+j 2\\pi \\left( \\frac{r}{\\lambda} \\right)}{1+j\\frac{4\\pi^2 \\left( \\frac{r}{\\lambda} \\right)^2-1}{2\\pi \\left( \\frac{r}{\\lambda} \\right)}} \\tag{4.67} $\r\n\r\nMagnitude and phase angle of the wave impedances formulated in Eqs.\u00a0(4.66) and (4.67) are shown in Fig.\u00a0<a href=\"#Zw_em_Fig\" data-reference-type=\"ref\" data-reference=\"Zw_em_Fig\">4.21<\/a>.\r\n<figure><img src=\"Propagation\/Figures\/Z_w.png\" id=\"Zw_em_Fig\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_182\" align=\"aligncenter\" width=\"736\"]<img class=\"wp-image-182\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-20.png\" width=\"736\" height=\"277\" alt=\"\" \/> Figure 4.21: Magnitude and phase angle of the wave impedance $latex {\\bf{Z}}_w$ of the electric and magnetic dipole in the symmetry plane $latex x0y$ versus electrical distance $latex r\/\\lambda$ from the dipole.[\/caption]<\/figure>\r\nAccording to the rule of thumb it can be stated that the near field zone is within electrical distance $latex r\/\\lambda&lt;0.1$. By that distance the phase angle rises to $latex -80^{\\circ}$ for electric dipole and decays to $latex 80^{\\circ}$ for magnetic dipole. Within that distance field is practically pure electric or magnetic. The same rule of thumb tells us to presume the far field zone in an electrical distance $latex r\/\\lambda&gt;1$. Memorize that the wave impedance in the far field zone regardless the direction approaches the intrinsic impedance Eq.\u00a0(4.49)<a href=\"#fn4\" class=\"footnote-ref\" id=\"fnref4\" role=\"doc-noteref\"><sup>4<\/sup><\/a>, which for the vacuum and approximately for the air is $latex Z_0 = 120 \\pi$\u00a0$latex \\Omega \\approx 377$\u00a0$latex \\Omega$. The wave impedance in the far field zone has resistive character and it represents real radiated power.\r\n\r\nIt is evident why the zone between $latex 0.1&lt;r\/\\lambda&lt;1$ is called reactive. The phase shifting between electric and magnetic field means reactive power commuting between the antenna and point of space under consideration. The real part of impedance in the reactive zone is responsible for real power to be radiated.\r\n\r\nThe far field zone by real antennas depend on their construction and is usually few times bigger than $latex r\/\\lambda=1$.\r\n<h4 id=\"sssec:Field_probes\">Field probes<\/h4>\r\nProbes are used for measurements of field strength in near and reactive zone. They should not distort incident field thats why they must be electrically small. This rationalizes their drawback namely moderate signal of response.\r\n\r\nConstruction of the sensor in the field probe imitate electric or magnetic dipole with finite dimensions. Equivalent circuits are shown in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>. $latex C_S$ and $latex L_S$ are capacitance and inductance of the electric and magnetic sensor respectively. $latex R_L$ is resistance of the circuitry terminating the sensor.\r\n<figure><img src=\"Propagation\/Figures\/D-dot_B-dot.png\" id=\"DB_circuits\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_183\" align=\"aligncenter\" width=\"685\"]<img class=\"wp-image-183\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-21.png\" width=\"685\" height=\"174\" alt=\"\" \/> Figure 4.22: Equivalent circuits of electric sensor a), magnetic sensor b).[\/caption]<\/figure>\r\nOutput voltage $latex u_L(t)$ of the electric sensor is linked with the component of the density of the displacement current $latex \\overset{\\bullet}{D}_p(t)$ in the space point in which the sensor is placed and matched with the orientation of the probe, see Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>a)\r\n\r\n$latex C_S \\frac{du_L(t)}{dt} + \\frac{u_L(t)}{R_L} = A_e \\frac{dD_p(t)}{dt} = A_e \\overset{\\bullet}{D}_p(t)\r\n\\label{u_e}\\tag{4.68} $\r\n\r\n$latex A_e$ is parameter of the probe transferring the component of the density of the displacement current $latex \\overset{\\bullet}{D}_p(t)$ to the current source in the equivalent circuit in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>a), expressed in area units. It is called equivalent surface.\r\n\r\nIn the frequency band of applications current driven through the capacitance $latex C_S$ is negligibly small and can be omitted. Then response $latex u_L(t)$ of the sensor is proportional to the component of the density of the displacement current $latex \\overset{\\bullet}{D}_p(t)$. The sensor is a current source with magnetomotive force $latex A_e \\overset{\\bullet}{D}_p(t)$.\r\n\r\nCurrent $latex i(t)$ in the mesh in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>b) is linked with the time derivative of the component of magnetic inductance $latex \\overset{\\bullet}{B}_p(t)$ in the space point in which the sensor is placed and matched with the orientation of the sensor\r\n\r\n$latex L_S \\frac{di(t)}{dt} + R_L i(t) = A_m \\frac{dB_p(t)}{dt} = A_m \\overset{\\bullet}{B}_p(t)\r\n\\label{i_m}\\tag{4.69} $\r\n\r\n$latex A_m$ is parameter of the probe transferring the time derivative of the component of magnetic inductance $latex \\overset{\\bullet}{B}_p(t)$ to the voltage source in the equivalent circuit in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>b), expressed in area units. It is called equivalent surface.\r\n\r\nIn the frequency band of applications voltage across the inductance $latex L_S$ is negligibly small and can be omitted. Then response $latex u_L(t)$ of the sensor is proportional to the time derivative of the component of magnetic inductance $latex \\overset{\\bullet}{B}_p(t)$. The sensor is a voltage source with electromotive force $latex A_m \\overset{\\bullet}{B}_p(t)$.\r\n\r\nOn the market are available directive and isotropic probes. The first sense only field component mached with the probe orientation, the second sense three orthogonal components of field in space.\r\n\r\nThere is variety of probes capable of capturing amplitude in overall frequency band of application. Directly by the sensor they have RF detector, most frequently diode one. Thereafter there is A\/C converter and the signal is delivered either to the display integrated with the sensor in case of autonomous probe or is converted to light and transmitted via the fiber glass to the computer interface. Increasingly the fiber glass is used simultaneously for powering electronic by the sensor.\r\n\r\nProbes are able to measure faithfully only CW i.e.\u00a0single frequency fields. They are inapplicable for any modulated or transient fields.\r\n\r\nExample of autonomous isotropic probe for electric and magnetic field with the bandwidth from DC to 400\u00a0kHz is shown in Fig.\u00a0<a href=\"#Probes\" data-reference-type=\"ref\" data-reference=\"Probes\">4.23<\/a>a) and isotropic electric probe with optical transmission of signal and power applicable in the bandwidth from 10\u00a0kHz to 6\u00a0GHz in Fig.\u00a0<a href=\"#Probes\" data-reference-type=\"ref\" data-reference=\"Probes\">4.23<\/a>b).\r\n<figure><img src=\"Propagation\/Figures\/Probes.png\" id=\"Probes\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_184\" align=\"aligncenter\" width=\"668\"]<img class=\"wp-image-184\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-22.png\" width=\"668\" height=\"202\" alt=\"\" \/> Figure 4.23: Isotropic autonomous electric and magnetic field probe a), electric field probe with optical transmission of signal and power b).[\/caption]<\/figure>\r\nThere are other types of probes called in jargon D-dot and B-dot probes. They are applicable for measurements of pulsed fields. In such probes digital processing of the sensor response, which is proportional to the field derivative, see Eq.\u00a0(4.68) and Eq.\u00a0(4.69), is preceded with the time integration. They are exclusively directional probes.\r\n\r\nExample of free space directional D-dot probe with the bandwidth from from 100\u00a0kHz to 3.5\u00a0GHz is shown in Fig.\u00a0<a href=\"#DB-dot\" data-reference-type=\"ref\" data-reference=\"DB-dot\">4.24<\/a>a) and free space directional B-dot probe with the bandwidth from 100\u00a0kHz to 2\u00a0GHz in Fig.\u00a0<a href=\"#DB-dot\" data-reference-type=\"ref\" data-reference=\"DB-dot\">4.24<\/a>b).\r\n<figure><img src=\"Propagation\/Figures\/DB-dot.png\" id=\"DB-dot\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_185\" align=\"aligncenter\" width=\"700\"]<img class=\"wp-image-185\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-23.png\" width=\"700\" height=\"213\" alt=\"\" \/> Figure 4.24: Free space directional D-dot probe a), Free space directional B-dot probe b).[\/caption]<\/figure>\r\n<h4 id=\"antennas\">Antennas<\/h4>\r\nIn technical terminology terms electric and magnetic antenna are used. The first concerns antennas with the wave impedance in the near field zone bigger than intrinsic impedance.\r\n\r\nAntennas are used for selective measurements of stationary fields in the far field zone. It can be assumed that they do not distort incident field. Analogue voltage signal induced in the antenna is transmitted from the antenna terminal to the spectrum analyzer or other radio frequency selective measurement receiver which records frequency spectrum of the measured field.\r\n\r\nAntennas for the measurement of electric fields are variations of electric dipoles. Dipole can be interpreted as unloaded symmetrical transmission line with straightened out conductors. The tips of dipole\u2019s arms are nodes for current distribution and antinodes for voltage distribution. Current vanishes there and voltage varies from plus to minus amplitude due to open circuit condition.\r\n\r\nBy $latex \\lambda\/2$ dipole as shown in Fig.\u00a0<a href=\"#ePol_dipol\" data-reference-type=\"ref\" data-reference=\"ePol_dipol\">4.25<\/a>a) the current distribution, the blue area is half of the approximately cosine function<a href=\"#fn5\" class=\"footnote-ref\" id=\"fnref5\" role=\"doc-noteref\"><sup>5<\/sup><\/a> with amplitude in the midpoint between the arms i.e. by antenna terminal. Voltage distribution, the orange needles is half of the sine function with fixed zero value by antenna terminal and varying between plus and minus amplitudes at the tips.\r\n\r\nIn Fig.\u00a0<a href=\"#ePol_dipol\" data-reference-type=\"ref\" data-reference=\"ePol_dipol\">4.25<\/a>\u00a0b) a half dipole antenna is shown. Such miniature antennas are used as grounded probes of electric fields or much bigger as monopole (rod) receiving antennas.\r\n\r\nDipole radiate efficiently if its length is matched with approximately multiple of the half of the wavelength $latex \\lambda\/2$. This efficiency is very sensitive on mismatching. In order to expand the frequency bandwidth rods in the ordinary dipole are replaced with conus (bi-conical antenna), triangle (bow-tie antenna) or more complex shapes by brad band antennas.\r\n\r\nIn Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0b) the bow-tie-log-periodic antenna is shown. Its frequency of operation extends from 30\u00a0MHz do 1500\u00a0MHz. The bow-tie section of the antenna, big triangles next to the feeding point covers frequency band from 30\u00a0MHz to about 300\u00a0MHz. It is connected parallelly with the Log-Periodic Dipole Antenna LPDA ahead. The LPDA is group of dipole antennas of varying sizes strung together. The dipole antennas diminish in size from the back to the front. The element at the back of the array which is the largest is tuned to frequency about 300\u00a0MHz and that at the front is a half wavelength at the highest frequency of operation i.e at 1500\u00a0MHz.\r\n\r\nIn the antenna boom between coaxial junction and bow-tie section a black box with the balun is mounted. Input of the dipole antenna is symmetric but the feeding point coaxial. The balun is a two port adapting symmetric to coaxial terminal. The name is a cluster of two words balanced-unbalanced.\r\n<figure><img src=\"Propagation\/Figures\/Pol_dipol.png\" id=\"ePol_dipol\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_186\" align=\"aligncenter\" width=\"461\"]<img class=\"wp-image-186\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-24.png\" width=\"461\" height=\"267\" alt=\"\" \/> Figure 4.25: Electric dipole with the length matched to the $latex \\lambda\/2$ wavelength a) and monopole matched to the $latex \\lambda\/4$ wavelength.[\/caption]\r\n\r\n<figcaption><\/figcaption><\/figure>\r\n<figure><img src=\"Propagation\/Figures\/Balun_2.png\" id=\"Balun_2\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_187\" align=\"aligncenter\" width=\"714\"]<img class=\"wp-image-187\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-25.png\" width=\"714\" height=\"396\" alt=\"\" \/> Figure 4.26: Illustration how the current balun operates.[\/caption]<\/figure>\r\nDesired signal of electric dipole stems from capacitance between the arms. It is $latex C_{12}$ in Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0a). Additionally each arm has its own capacitance to the ground. They are $latex C_{10}$ and $latex C_{20}$. There is still one more capacitance that must be taken into account. Namely between the arm connected to the core of the coaxial cable and the shield. It is arm 1 and capacitance $latex C_{1S}$ in Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0a). Capacitance between the arm 2 and the cable shield is short circuited. $latex Z$ is the input impedance of the measurement receiver.\r\n\r\nEquivalent circuit of the dipole is composed of parallel circuits of the current sources with their capacitances as it was explained for the free space electric sensor shown in Fig.<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>\u00a0a)<a href=\"#fn6\" class=\"footnote-ref\" id=\"fnref6\" role=\"doc-noteref\"><sup>6<\/sup><\/a>. In Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0b) branch with $latex Z_C$ represents impedances of cable core, $latex Z_S$ cable shield and $latex Z_{GND}$ the ground reference. If capacitances $latex C_{10}$ and $latex C_{20}$ are different then they contribute to the differential current driven through the core, the measurement receiver and the shield. Moreover probable is also common mode current returning through the ground. Differential contribution of capacitances $latex C_{10}$ and $latex C_{20}$ cannot be eliminated. It overlays with desired signal $latex I_{12}$ causing distortion.\r\n\r\nThe current balun is simply ferrite mounted on the coaxial cable direct by junction with symmetrical antenna output. The common mode choke<a href=\"#fn7\" class=\"footnote-ref\" id=\"fnref7\" role=\"doc-noteref\"><sup>7<\/sup><\/a> $latex L^{CM}$ shown in Fig.\u00a0<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a> suppresses common mode contribution of capacitances $latex C_{10}$ and $latex C_{20}$ but it does not eliminate it totally. Total elimination of differential and common mode distortion caused by capacitances $latex C_{10}$ and $latex C_{20}$ would happen if they were equal. Practically this condition is fulfilled by horizontal polarization of the antenna and very high above the ground reference, regardless polarization<a href=\"#fn8\" class=\"footnote-ref\" id=\"fnref8\" role=\"doc-noteref\"><sup>8<\/sup><\/a>.\r\n\r\nThere is still another parasitic capacitance $latex C_{1S}$ which contributes differentially to the desired signal, see Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0b) and cannot be eliminated. The only way is to keep the capacitance small and place the common mode choke $latex L^{CM}$ as close as possible by antenna feeding.\r\n<figure><img src=\"Propagation\/Figures\/Balun_3.png\" id=\"Balun_3\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_188\" align=\"aligncenter\" width=\"564\"]<img class=\"wp-image-188\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-26.png\" width=\"564\" height=\"415\" alt=\"\" \/> Figure 4.27: Illustration how the voltage balun operates.[\/caption]<\/figure>\r\nMore sophisticated are voltage baluns, based on transformers. Example of one of them is shown in Fig.\u00a0<a href=\"#Balun_3\" data-reference-type=\"ref\" data-reference=\"Balun_3\">4.27<\/a>. Transformer separates galvanically symmetrical terminal of the antenna from coaxial cable connector. Additionally cable shield tapes the centre of the transformer winding on the symmetrical side. It has triple advantage:\r\n<ul>\r\n \t<li>cancellation of the common mode distortion due to capacitances $latex C_{10}$ and $latex C_{20}$, thanks galvanic separation,<\/li>\r\n \t<li>both arms of the dipole are liberated from the potential of the cable shield, thanks galvanic separation,<\/li>\r\n \t<li>capacitances of both dipole arms to the cable shield $latex C_{1S}$ and $latex C_{2S}$ are equal one to another due to introducing the potential of the cable shield to the midpoint between the arms.<\/li>\r\n<\/ul>\r\nContribution of the differential component in the distortion caused by unequal capacitances $latex C_{10}$ and $latex C_{20}$ is ruled in the same way same as by the current balun.\r\n\r\nThe balun in the antenna shown in Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0b) is necessary because of the bow-tie section. The LPDAs\u2019 antennas are not finished with them. The adjacent dipoles in a LPDA antenna are connected to the symmetrical line routed inside the antenna (red and blue lines in Fig.\u00a0<a href=\"#LPDA_twist\" data-reference-type=\"ref\" data-reference=\"LPDA_twist\">4.28<\/a>) alternately. Consequently averaged capacitances of upper and lower arms to the cable shield and to the ground reference are equalized, minimizing distortion of the measured signal.\r\n<figure><img src=\"Propagation\/Figures\/LPDA_twist.png\" id=\"LPDA_twist\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_189\" align=\"aligncenter\" width=\"301\"]<img class=\"wp-image-189\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-27.png\" width=\"301\" height=\"276\" alt=\"\" \/> Figure 4.28: Alternately fed dipoles in the LPDA.[\/caption]<\/figure>\r\nAll precautions concerning symmetrization in the antenna can be spoiled with incorrect routing of the coaxial cable behind the antenna. In the document <span class=\"citation\" data-cites=\"CISPR-16-1-4\">[@CISPR-16-1-4]<\/span> two recommendations are formulated:\r\n<ul>\r\n \t<li>by EMC testing of radiated emission the coaxial cable behind the antenna should be maintained horizontal, i.e.\u00a0parallel to the ground plane, for a distance of approximately 1 m or more to the rear of the antenna before dropping to the ground plane,<\/li>\r\n \t<li>by verification of the anechoic chambers the coaxial cable behind the antenna should be oriented horizontally behind the antenna for a distance as close to 2 m as physically possible.<\/li>\r\n<\/ul>\r\nThe LPDA antennas available on the market have booms with about 1 m length.\r\n<figure><img src=\"Propagation\/Figures\/Pol_dipol_m.png\" id=\"mPol_dipol\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_190\" align=\"aligncenter\" width=\"540\"]<img class=\"wp-image-190\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-28.png\" width=\"540\" height=\"207\" alt=\"\" \/> Figure 4.29: Magnetic dipole with the circumference matched to the $latex \\lambda\/2$ wavelength a) and half dipole with the circumference matched to the $latex \\lambda\/4$ wavelength.[\/caption]<\/figure>\r\n<figure><img src=\"Propagation\/Figures\/Antennas.png\" id=\"Loop_bilog\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_191\" align=\"aligncenter\" width=\"698\"]<img class=\"wp-image-191\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-29.png\" width=\"698\" height=\"513\" alt=\"\" \/> Figure 4.30: Broad band loop antenna for measurements of magnetic field a) and broad band bow-tie-log-periodic antenna for measurements of electric field.[\/caption]<\/figure>\r\nAntennas for the measurement of magnetic fields are variations of magnetic dipoles. Magnetic dipole can be interpreted as short circuited symmetrical transmission line with the circle shape. The junction point of upper and lower conductor is the node for voltage distribution and antinode for current distribution. Voltage vanishes there and current varies from plus to minus amplitude due to short circuit condition.\r\n\r\nBy $latex \\lambda\/2$ loop dipole<a href=\"#fn9\" class=\"footnote-ref\" id=\"fnref9\" role=\"doc-noteref\"><sup>9<\/sup><\/a> as shown in Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>\u00a0a) the current distribution, the blue area is half of the approximately cosine function with amplitude in the junction point of upper and lower conductor. Voltage distribution, the orange needles is half of the sine function with fixed zero value by the junction point of upper and lower conductor and varying between plus and minus amplitudes at the antenna feeding.\r\n\r\nIn Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>\u00a0b) a half loop antenna is shown. Such loops with miniature size are used as grounded probes of magnetic fields.\r\n\r\nLoop antennas are used in the frequency range up to 30\u00a0MHz. Their response is very week. The bigger radius the stronger response due to its dependence on magnetic flux streaming through the loop area. However lower location sensitivity by the measurements of inhomogeneous fields due to averaging of the flux with the loop area. By the loop antennas signal sensitivity and averaging effect must be always compromised. By active loop antenna with built in amplifier size of the loop can be reduced.\r\n\r\nIn Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0a) the loop antenna with frequency of operation from 9\u00a0kHz to 30\u00a0MHz is shown. In the box by the tripod the amplifier is placed. At the top of the antenna a black ring is visible. Loop antennas have metal tube around, which plays a role of electric shield. Thanks the shield picking up unwanted responses steaming from electric fields can be avoided. This shield however must be broken to prevent the metal tube from acting like a shorted turn. The plastic ring positions the free ends of the broken metal tube.\r\n<h5 id=\"antenna-factor.\">Antenna factor.<\/h5>\r\nBy measurements of radiated disturbances with antennas it is necessary to recalculate the signal at the input of the measurement receiver to the field strength to which the antenna is exposed. The antenna factor facilitates this conversion.\r\n\r\nIf receiving antenna is exposed to the field, electromotive force $latex EMF$ is induced in it. Antennas are not sensing the field module but the component matched with their orientation, $latex E_z$ or $latex H_x$ in Fig.\u00a0<a href=\"#F_A\" data-reference-type=\"ref\" data-reference=\"F_A\">4.31<\/a>. In other words induced $latex EMF$ depends on the projection of the field vector on the E-plane for electric and H-plane for magnetic antenna. $latex Z_s$ is the antenna impedance seen from its terminals.\r\n<figure><img src=\"Propagation\/Figures\/Antenna_factor.png\" id=\"F_A\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_192\" align=\"aligncenter\" width=\"565\"]<img class=\"wp-image-192\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-30.png\" width=\"565\" height=\"263\" alt=\"\" \/> Figure 4.31: Definition of the antenna factor.[\/caption]<\/figure>\r\nBy determining the antenna factor the antenna is oriented to the plane matching the field with known strength $latex E_z$ or $latex H_x$. The instrument with $latex 50$\u00a0$latex \\Omega$ input resistance capable of HF voltage measurements, such as: power meter with insertion unit or power sensor, spectrum analyzer or measurement receiver, is connected to the terminals of the antenna. Ratio of field strength and voltage at the measuring instrument is the antenna factor.\r\n\r\n$latex \\label{F_A}\r\nAF^{(E)}\\left[\\frac{1}{m} \\right] = \\frac{E_z}{U} \\hspace{3cm}\r\nAF^{(H)}\\left[\\frac{1}{\\Omega m} \\right] = \\frac{H_x}{U}\\tag{4.70} $\r\n\r\nIn the $latex dB$ scale the units are $latex dB[1\/m]$ and $latex dB[1\/(\\Omega m)]$ respectively. The antennas\u2019 factors are determined for the far field zone in the air where wave and intrinsic impedances are identical $latex E\/H = Z_0 \\approx 377$\u00a0$latex \\Omega$. Alternatively manufacturers and calibration laboratories presents the factor for magnetic antennas as follows\r\n\r\n$latex AF^{(H)}\\left[\\frac{1}{ m} \\right] = AF^{(H)} \\left[dB\\left(\\frac{1}{\\Omega m}\\right) \\right] + 51.5 [dB(\\Omega)] $\r\n<h5 id=\"surface-power-density.\">Surface power density.<\/h5>\r\nVector product of phasors of electric field strength and conjugate of magnetic field strength at arbitrary point in space builds the phasor of the Poynting\u2019s vector\r\n\r\n$latex \\overrightarrow{\\bf{S}} = \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^*\r\n\\label{Poynting_1}\\tag{4.72} $\r\n\r\nSurface integral of the phasor of the Poynting\u2019s vector over any closed surface $latex s$, oriented outwards versus solid enclosing some antennas gives the phasor of total apparent power $latex {\\bf{P}}_{app}$ on this surface <span class=\"citation\" data-cites=\"Hammond_2\">[@Hammond_2]<\/span>\r\n\r\n$latex {\\bf{P}}_{app}=\\oiint_s \\left( \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^* \\right) \\cdot \\overrightarrow{ds}\r\n\\label{Power_app} \\tag{4.73} $\r\n\r\ncrucial is that $latex \\overrightarrow{ds}$ is the surface versor oriented outward versus the solid with antennas.\r\n\r\nIf phasors in Eq.\u00a0(4.72) are scaled with RMS values then Poynting\u2019s vector $latex \\overrightarrow{\\bf{S}}$ gives density per surface unit of apparent power at the point of interest<a href=\"#fn10\" class=\"footnote-ref\" id=\"fnref10\" role=\"doc-noteref\"><sup>10<\/sup><\/a>. Sense of the Pointing\u2019s vector shows direction of power transportation in case of electric antennas and oposite direction in case of magnetic antennas. In the near and reactive field zone the imaginary part of the Poynting\u2019s vector represents density of reactive power traveling back and forth between the antenna and the point of interest and real part represents density of real power radiated out of the antenna.\r\n<figure><img src=\"Propagation\/Figures\/Poynting.png\" id=\"Poynting\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_193\" align=\"aligncenter\" width=\"742\"]<img class=\"wp-image-193\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-31.png\" width=\"742\" height=\"319\" alt=\"\" \/> Figure 4.32: The triple field vectors on the equiphase surface by electric antenna a), by magnetic antenna b) .[\/caption]<\/figure>\r\nIn the far field zone the imaginary part of the Poynting\u2019s vector decays. It is actually attribute of the far field zone. There is no phase shift between fields\u2019 strengths. Magnitude of the Poynting\u2019s vector represents solely density of radiated power.\r\n\r\nEach antenna, even geometrically complex is seen, from the observation point located at the wavefront, as the point radiator. Therefore three features of the dipoles\u2019 fields in the far field zone can be extended to antennas with any shape. Two of them are listed below:\r\n<ul>\r\n \t<li>the equiphase surface is the sphere,<\/li>\r\n \t<li>the field strength decreases in the far field zone reciprocally proportional to the distance.<\/li>\r\n<\/ul>\r\nFor formulation of the third one, additional definition is necessary. It will be introduced later in this paragraph.\r\n\r\nLet us imagine wavefront of an antenna with electric field strength polarized in the $latex \\varphi = const$ plane and magnetic field strength polarized in the $latex z = const$ plane, green coloured in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> a). The triple of the vectors can be oriented as shown there. The magnitude of the Poynting\u2019s vector yields\r\n\r\n$latex S_r(r,\\theta) = E_\\theta(r,\\theta) H_\\varphi(r,\\theta) = \\frac{{E_\\theta^2(r,\\theta)}}{Z_0} = Z_0 {H_\\varphi^2(r,\\theta)}\r\n\\label{Poynting_2}\\tag{4.74} $\r\n\r\nLet us imagine wavefront of an antenna with magnetic field strength polarized in the $latex \\varphi = const$ plane and electric field strength polarized in the $latex z = const$ plane, green coloured in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> b). The triple of the vectors can be oriented as shown there. The magnitude of the Poynting\u2019s vector yields\r\n\r\n$latex S_r(r,\\theta) = E_\\varphi(r,\\theta) H_\\theta(r,\\theta) = \\frac{{E_\\varphi^2(r,\\theta)}}{Z_0} = Z_0 {H_\\theta^2(r,\\theta)}\r\n\\label{Poynting_3}\\tag{4.75} $\r\n\r\nIf the antenna in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> a) is elementary electric dipole with strengths expressed with Eqs.\u00a0(4.57) and (4.58) then\r\n\r\n$latex S^{(E)}_r(r,\\theta) = \\frac{p_z^2 \\beta_0^2 Z_0 } {16 \\pi^2 } \\cdot \\frac{\\sin^2{\\theta}}{r^2}\r\n\\label{Poynting_4}\\tag{4.76} $\r\n\r\nTotal power radiated by the electric dipole is equal to the surface integral of the Poynting vector Eq.\u00a0(4.73) and can be calculated over equiphase sphere with arbitrary radius $latex r$.\r\n\r\n$latex P_{rad}^{(E)}=\\unicode{x222F}_s S_r^{(E)} \\overrightarrow{1}_r \\cdot \\overrightarrow{ds} =\r\n\\unicode{x222F}_s S_r^{(E)} r^2 \\sin{\\theta} d\\theta d\\varphi =  $\r\n\r\n$latex =\\frac{p_z^2 \\beta_0^2 Z_0 } {16 \\pi^2 }\r\n{\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sin^3{\\theta} d\\theta} {\\int_{0}^{2 \\pi} d\\varphi} = \\nonumber $\r\n\r\n$latex = \\frac{p_z^2 \\beta_0^2 Z_0 } {6 \\pi }\r\n\\label{Power_rad} \\tag{4.77} $\r\n\r\nIf the antenna in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> b) is elementary magnetic dipole with strengths expressed with Eqs.\u00a0(4.59) and (4.60) then\r\n\r\n$latex S^{(H)}_r(r,\\theta) = -\\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {16 \\pi^2 Z_0 } \\cdot \\frac{\\sin^2{\\theta}}{r^2}\r\n\\label{Poynting_5}\\tag{4.78} $\r\n\r\nTotal power radiated by the electric dipole is equal to the surface integral of the Poynting vector Eq.\u00a0(4.73) and can be calculated over equiphase sphere with arbitrary radius $latex r$.\r\n\r\n$latex P_{rad}^{(H)}=\\unicode{x222F}_s S_r^{(H)} \\left( -\\overrightarrow{1}_r \\right) \\cdot \\overrightarrow{ds} =\r\n\\unicode{x222F}_s S_r r^2 \\sin{\\theta} d\\theta d\\varphi = \\nonumber $\r\n\r\n$latex =-\\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {16 \\pi^2 Z_0 }\r\n{\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sin^3{\\theta} d\\theta} {\\int_{0}^{2 \\pi} d\\varphi} = \\nonumber $\r\n\r\n$latex  = \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 }{6 \\pi Z_0 }\r\n\\label{Power_rad_H} \\tag{4.79} $\r\n\r\nIn Eqs.\u00a0(4.76) and (4.78) is apparent that the surface power density in the far field zone of elementary radiators decreases reciprocally proportional to the distance $latex r$ in square. This is the third feature valid also for realised antennas with any shape and size.\r\n<h5 id=\"radiation-pattern.\">Radiation pattern.<\/h5>\r\nAbility of antennas to radiate electromagnetic energy in different directions is portrayed with radiation pattern. It is defined for electric, magnetic field strength as well as for radiated power in far field zone. The radiation pattern is a function built over equiphase surface. Its value represents vector magnitude or power density. For electric\/magnetic antenna oriented along z-axis, as shown in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b)\/Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c) electric\/magnetic field has only $latex \\theta$ component dependent on all co-ordinates $latex r$, $latex \\theta$ and $latex \\varphi$. Patterns normalized with maximal value are objective measures for comparison of antennas\u2019 performance. They are independent on distance $latex r$\r\n\r\n$latex F^{(E)}(\\theta,\\varphi) = \\frac{E_\\theta(r,\\theta,\\varphi)}{ E_{\\theta_{max}}} \\hspace{1.5cm}\r\nF^{(H)}(\\theta,\\varphi) = \\frac{H_\\theta(r,\\theta,\\varphi)}{ H_{\\theta_{max}}}\r\n\\label{F_EH}\\tag{4.80} $\r\n\r\nObviously normalized pattern of power radiation yields, see Eq.\u00a0( 4.74)\r\n\r\n$latex F_P^{(E)}(\\theta,\\varphi)=\\frac{S_r^{(E)}(r,\\theta,\\varphi)}{ S_{r_{max}}^{(E)}} = \\left[ F^{(E)} (\\theta,\\varphi) \\right]^2 $\r\n\r\n$latex \\nonumber $\r\n\r\n$latex F_P^{(H)}(\\theta,\\varphi)= \\frac{S_r^{(H)}(r,\\theta,\\varphi)}{ S_{r_{max}}^{(H)}} = \\left[ F^{(H)} (\\theta,\\varphi) \\right]^2\r\n\\label{F_S} \\tag{4.82} $\r\n\r\nRadiation pattern of field strength of the electric and magnetic dipole according to Eq.\u00a0(4.57) and Eq.\u00a0( 4.60) are identical\r\n\r\n$latex F^{(E)}(\\theta) = F^{(H)}(\\theta) = F(\\theta) = \\sin{(\\theta)}\r\n\\label{F_pm}\\tag{4.84} $\r\n<figure><img src=\"Propagation\/Figures\/Pattern_E.png\" id=\"Pattern_E\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_194\" align=\"aligncenter\" width=\"733\"]<img class=\"wp-image-194\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-32.png\" width=\"733\" height=\"288\" alt=\"\" \/> Figure 4.33: Radiation pattern of electric\/magnetic field strength round the electric\/magnetic dipole.[\/caption]<\/figure>\r\nand of surface power density\r\n\r\n$latex F^{(E)}_P(\\theta) = F^{(H)}_P(\\theta) = F_P(\\theta) = \\sin^2{(\\theta)}\r\n\\label{F_pma}\\tag{4.84} $\r\n<figure><img src=\"Propagation\/Figures\/Pattern_S.png\" id=\"Pattern_S\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_195\" align=\"aligncenter\" width=\"771\"]<img class=\"wp-image-195\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-33.png\" width=\"771\" height=\"260\" alt=\"\" \/> Figure 4.34: Radiation pattern of surface power density round the electric\/magnetic dipole.[\/caption]<\/figure>\r\nRadiation pattern of field strength of the electric and magnetic dipole in the spherical system of coordinates $latex r, \\theta, \\varphi$ is function of one variable $latex \\theta$. In the system of Cartesian coordinates $latex x, y, z$ it is function of all variables and can be depicted as a map of colours on the surface of the sphere with radius $latex r=1$, as shown in Fig.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0a). From the engineer point of view such picture is useless.\r\n\r\nAntennas\u2019 engineers are accustomed with mapping the one variable pattern $latex F(\\theta)$ into the two variables pattern $latex F(r,\\theta)$ in which $latex r = \\sin{(\\theta)}$. Such surface is the torus (doughnut) without hole in centre with circular cross-section in the plane $latex \\theta = 90^{\\circ}$ with radius $latex r = 1$ and two excentrically placed touching circles with radii equal $latex 0.5$. Distance between the origin of the Cartesian system of coordinates and the point at the pattern\u2019s surface is equal to the value of the field strength by that elevation $latex \\theta$, see Fig.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0b). From such portrait radiation ability can be directly determined quantitatively.\r\n\r\nThe same concerns the pattern of the surface power density which is identical for both dipoles. In the spherical system of coordinates $latex r, \\theta, \\varphi$ it is function of one variable $latex \\theta$. In the system of Cartesian coordinates $latex x, y, z$ it is function of all variables and can be depicted as a map of colours on the surface of the sphere with radius $latex r=1$, as shown in Fig.\u00a0<a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0a). Usually it is mapped into the two variables pattern $latex F(r,\\theta)$ in which $latex r = \\sin^2{(\\theta)}$. Such surface is the vertically squeezed torus (doughnut) without hole in centre with circular cross-section in the plane $latex \\theta = 90^{\\circ}$ with radius $latex r = 1$ and two excentrically placed touching, deformed circles. Distance between the origin of the Cartesian system of coordinates and the point at the pattern\u2019s surface is equal to the value of the surface power density by that elevation $latex \\theta$, see Fig.\u00a0<a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0b).\r\n\r\nSolid representing the two variables radiation pattern $latex F(r,\\theta)$ is called the lobe. The electric as well as magnetic dipole has only one lobe which is the torus. The realized antennas have so called main lobe roundabout the desired radiation direction and usually more than one side lobe which is undesired, side effect causing waste of radiated power.\r\n\r\nDipoles and antennas presented up to now are oriented along z-axis of the global system of coordinates i.e.\u00a0vertically. For presentation of radiation pattern at the plane, as it is usually the case in the antennas\u2019data sheets, it is more convenient to use system of coordinates tied to the antenna. Then the cross-section including only electric field component or magnetic field component is called E-plane or H-plane respectively. For the dipole in Figs.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0b) and <a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0b), the presented cross-section is the E-plane if it is electric dipole and H-plane if it is magnetic dipole. The cross-section $latex \\theta = 90^{\\circ}$ would be H-plane for electric dipole and E-plane for magnetic dipole.\r\n<h5 id=\"directive-gain-and-directivity.\">Directive gain and directivity.<\/h5>\r\nCompared is antenna under interest with isotropic antenna radiating the same power. Directive gain $latex D(r,\\theta)$ is ratio of the surface radiated power of the first and the second. Directivity is maximal value of the directive gain. Both parameters gives the measure of squeezing the power beam referred to the omnidirectional radiation.\r\n\r\nLet us derive these parameters for the electric and magnetic dipole as shown in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b) and c). Division of total radiated power of the electric dipole Eq.\u00a0(4.77) and magnetic dipole Eq.\u00a0(4.79) by the solid angle of the sphere $latex 4\\pi r^2$ gives surface power density of the isotropic antenna\r\n\r\n$latex S_i(r) = \\frac{p_z^2 \\beta_0^2 Z_0 } {24 \\pi^2 r^2 } \\hspace{2cm} S_i(r) = \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {24 \\pi^2 Z_0 r^2 }\r\n\\label{Poynting_i}\\tag{4.85} $\r\n\r\nAccording to the definition, the directive gain is fraction of surface power density of the dipole and density of isotropic radiator Eq.\u00a0(4.85). The numerator in this fraction is Eq.\u00a0(4.76) by the electric dipole and Eq.\u00a0(4.78) by the magnetic dipole\r\n\r\n$latex D(\\theta) = \\frac{3 } {2 } \\sin^2{\\theta}\r\n\\label{Directivity_1}\\tag{4.86} $\r\n\r\nand directivity\r\n\r\n$latex D_{max} = D(90^{\\circ}) = \\frac{3 } {2 }\r\n\\label{Directivity_2}\\tag{4.87} $\r\n\r\nIn the $latex dB$ scale $latex D_{max} \\approx 1.8$\u00a0$latex dB_i$<a href=\"#fn11\" class=\"footnote-ref\" id=\"fnref11\" role=\"doc-noteref\"><sup>11<\/sup><\/a>. In the direction of the maximal radiation, the dipole radiates 50% or $latex 1.8$\u00a0$latex dB_i$ more power then isotropic radiator with the same totally radiated power.\r\n<h5 id=\"energetic-gain\">(Energetic) gain,<\/h5>\r\nusually named simply gain is the directivity diminished by energetic efficiency of the antenna $latex \\eta$ defined as ratio of total power radiated by the antenna $latex P_{rad}$ and power at the antenna feeding port $latex P_{in}$\r\n\r\n$latex G_i = \\eta D_{max} = \\frac{P_{rad}}{P_{in}} D_{max}\r\n\\label{Gain}\\tag{4.88} $\r\n\r\nBy interpretation of power at the antenna feeding port $latex P_{in}$ ambiguity can creep in. If net power<a href=\"#fn12\" class=\"footnote-ref\" id=\"fnref12\" role=\"doc-noteref\"><sup>12<\/sup><\/a> is introduced as the input power $latex P_{in} = P_{fwr} - P_{rev}$ then the gain is preceded with the adjective absolute. If forward power sent from the signal source is introduced as the input power $latex P_{in} = P_{fwr}$ then the gain is preceded with the adjective realised. By absolute gain, unlike by realised gain mismatching at the feeding point is taken into account. Be prudent by evaluation of catalogue data of an antenna in respect of its gain.\r\n\r\nMore practical meaning has the gain $latex G_d$ referred to the half wave dipole. Since the gain of the half wave dipole referred to the isotropic antenna is $latex 1.64$ in the linear scale and $latex 2.15$\u00a0$latex dB_i$, see <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>\r\n\r\n$latex G_d=\\frac{1}{1.64} G_i $\r\n\r\n$latex &amp; \\nonumber $\r\n\r\n$latex G_{{dB}_d}=G_{{dB}_i} - 2.15dB_i\r\n\\label{G_d} \\tag{4.90} $\r\n<h5 id=\"equivalent-isotropically-radiated-power-eirp\">Equivalent isotropically radiated power EIRP<\/h5>\r\nis the input power at the terminal of the isotropic radiator necessary for radiation with the surface power density equal to the maximum of the magnitude of the surface power density $latex S(r)$ of the considered antenna in the far field zone. It is product of this value and solid angle of the sphere\r\n\r\n$latex EIRP = 4 \\pi r^2 |S(r)|_{max}\r\n\\label{EIRP}\\tag{4.91} $\r\n\r\nThe EIRP does not depend on the distance $latex r$. Indeed the EIRP is explicitly proportional to the distance $latex r$ in square but reciprocal proportionality to distance $latex r$ in square is implied in the $latex |S(r)|_{max}$ due to attribute of the far field zone.\r\n\r\nFor the electric dipole calculation is based on Eq.\u00a0(4.76) and for magnetic dipole on Eq.\u00a0(4.78) with setting $latex \\theta = 90^{\\circ}$\r\n\r\n$latex EIRP^{(E)} = \\frac{p_z^2 \\beta_0^2 Z_0 }{4 \\pi } \\hspace {1.5cm} EIRP^{(H)} = \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 }{4 \\pi Z_0 }\r\n\\label{EIRP_EH}\\tag{4.92} $\r\n\r\nLet us derive the measurement procedure of the EIRP for an electric antenna. The set up is established in the full anechoic chamber in which walls, ceiling and floor are lined with the absorbers marked with pyramids in Fig.\u00a0<a href=\"#EIRP_Fig\" data-reference-type=\"ref\" data-reference=\"EIRP_Fig\">4.35<\/a>, simulating reflection free space. The investigated antenna $latex T_x$ and receiving antenna $latex R_x$ are placed at the same height above the floor in order to match their elevations with $latex \\theta = 90^{\\circ}$. The investigated antenna must be rotated about the vertical axis to the azimuth maximizing signal $latex U_R$ in the measurement receiver and fixed in this position.\r\n<figure><img src=\"Propagation\/Figures\/EIRP.png\" id=\"EIRP_Fig\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_196\" align=\"aligncenter\" width=\"522\"]<img class=\"wp-image-196\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-34.png\" width=\"522\" height=\"291\" alt=\"\" \/> Figure 4.35: Ascertainment of the EIRP in the full anechoic chamber.[\/caption]<\/figure>\r\nApplied must be Eq.\u00a0(4.74) for surface power density in the direction of maximal radiation i.e.\u00a0by elevation $latex \\theta = 90^{\\circ}$ and Eq.\u00a0(<a href=\"#F_A\" data-reference-type=\"ref\" data-reference=\"F_A\">4.31<\/a>) for antenna factor of an electric antenna\r\n\r\n$latex S_r(r,90^{\\circ}) = \\frac{E_{\\theta}^2(r,90^{\\circ})}{Z_0} = \\frac{\\left( AF^{(E)} \\right)^2}{Z_0} U_A^2(r)\r\n\\label{EIRP_2}\\tag{4.93} $\r\n\r\nFor EIRP this formula must be multiplied by the solid angle of the sphere\r\n\r\n$latex EIRP = \\frac{4 \\pi r^2}{Z_0} \\left( AF^{(E)} \\right)^2 U_A^2(r)\r\n\\label{EIRP_3}\\tag{4.94} $\r\n\r\nRequested unit for the EIRP is $latex mW$ and for voltage at the antenna terminal $latex \\mu V$. Suitable formula yields\r\n\r\n$latex EIRP \\left[ mW\\right] = \\frac{4 \\pi r^2 \\left[m^2 \\right]}{377 [\\Omega]} \\left( AF^{(E)} \\right)^2 \\left[ \\frac{1}{m^2}\\right] U_A^2(r) \\left[(\\mu V)^2 \\right] 10^{-9}\r\n\\label{EIRP_4}\\tag{4.95} $\r\n\r\nBoth sides of equation are logarithmised and multiplied by 10\r\n\r\n$latex EIRP_{_{dB(mW)}}(r)= 10\\log{ \\left( \\frac{4 \\pi r^2 \\left[ m^2 \\right] }{377 [\\Omega]} \\right)} - 90_{_{dB}}+ 20\\log{ \\left\\{ AF^{(E)}\\left[ \\frac{1}{m} \\right] \\right\\}} + \\nonumber $\r\n\r\n$latex \\nonumber $\r\n\r\n$latex &amp; +A_{C_{dB}} + 20\\log{ \\left\\{ U_R(r) \\left[ \\mu V \\right] \\right\\} }\r\n\\label{EIRP_5} \\tag{4.96} $\r\n\r\nVoltage at the feeding terminal of the receiving antenna $latex U_{A_{dB(\\mu V)}}$ is greater than voltage measured by the receiver $latex U_{R_{dB(\\mu V)}}$ about attenuation $latex A_{C_{dB}}$ of the measurement path (cables, feed through) $latex U_{A_{dB(\\mu V)}} (r) = A_{C_{dB}} + U_{R_{dB(\\mu V)}} (r) $\r\n\r\n$latex EIRP_{_{dB(mW)}}(r)= \\left[10\\log{ \\left( \\frac{4 \\pi r^2 }{377} \\right)} - 90_{_{dB}} \\right]+ AF_{_{dB(1\/m)}}^{(E)} + \\nonumber $\r\n\r\n$latex \\nonumber $\r\n\r\n$latex &amp; +A_{C_{dB}} + U_{R_{dB(\\mu V)}} (r)\r\n\\label{EIRP_6} \\tag{4.98} $\r\n\r\nThe EIRP is independent on distance $latex r$. Indeed there is radius in square in the numerator in Eqs. from (4.94) to (4.98) but measurement is done in the far field zone in which surface power density decreases reciprocally proportional to the distance in square. Consequently the same concerns the voltage at the antenna terminal. By increasing distance e.g.\u00a0three times the term in square bracket in Eq.\u00a0(4.98) increases about 9.5\u00a0dB but in the same time the voltage at the antenna terminal $latex U_{A_{dB(\\mu V)}}$ and consequently at the receiver input $latex U_{R_{dB(\\mu V)}}$ decreases about the same amount.\r\n\r\nTypical distance for performing the measurements of the EIRP is ($latex r= 3$\u00a0$latex m$). By doing so correction factor in the square brackets is $latex -95.2_{{dB}}$.\r\n<h5 id=\"radiation-resistance\">Radiation resistance<\/h5>\r\nTotal radiated power is represented in the equivalent circuit of the antenna with radiation resistance $latex R_{rad}$ through which current $latex I$ from the source feeding antenna is driven $latex R_{rad}I^2$.\r\n\r\nLet us calculate this resistance for the electric dipole. Total radiated power is ruled with Eq.\u00a0(4.77). Replacing the magnitude of dipole moment with the product of current magnitude $latex I$ and infinitesimal segment $latex l$ Eq.\u00a0(4.41), moreover with implementing the wavelength $latex \\lambda_0$ instead of the phase constant $latex \\beta_0$, according to Eq.\u00a0(4.50) yields\r\n\r\n$latex R_{rad}^{(E)} = \\frac{2 \\pi Z_0}{ 3} \\left(\\frac{l}{\\lambda_0} \\right)^2\r\n\\label{R_rad}\\tag{4.99} $\r\n\r\nIn vacuum and approximately in air this resistance is $latex R_{rad}^{(E)} \\approx 789.57 \\left( \\frac{l}{\\lambda_0} \\right)^2 [\\Omega] $ if the dipole length $latex l$ and the wavelength $latex \\lambda_0$ have the same units.\r\n\r\nIn feasible short dipole relation $latex l\/\\lambda_0$ and consequently radiation resistance is very small. Therefore radiation ability is very weak<a href=\"#fn13\" class=\"footnote-ref\" id=\"fnref13\" role=\"doc-noteref\"><sup>13<\/sup><\/a>.\r\n\r\nFor example, for a length ($latex l = 1$\u00a0cm) and a frequency of 300 MHz ($latex \\lambda_0 = 1$\u00a0m), the radiation resistance is $latex R_{rad}^{(E)} \\approx 79 [m\\Omega] $ In order to radiate 1\u00a0W of power, required is current of 3.6 A!\r\n\r\nIf the frequency is changed to 3 MHz ($latex \\lambda_0 = 100$\u00a0m), the radiation resistance is $latex R_{rad}^{(E)} \\approx 7.9 \\left[ \\mu \\Omega \\right] $ and the current required to radiate 1\u00a0W is 356 A!\r\n\r\nMoreover resistance $latex R_{rad}$ is serially circuited with the element representing antenna outlet to the lossless space. Depending on electrical length of the dipole $latex l\/\\lambda_0$ it can be capacitance, shortcircuit or inductance but for a short dipole it is huge capacitance limiting current driven through radiation resistance. It is the second factor contributing to very small radiation efficiency of the Hertzian as well as short dipole.\r\n\r\nLet us proceed now to the radiation resistance of magnetic dipole. Total radiated power is ruled with Eq.\u00a0(4.79). Replacing the magnitude of dipole moment with the product of current magnitude $latex I$ and infinitesimal loop area $latex \\pi a^2$ Eq.\u00a0(4.42), moreover with implementing the wavelength $latex \\lambda_0$ instead of the phase constant $latex \\beta_0$, according to Eq.\u00a0(4.50) and relation $latex \\omega = 2 \\pi v\/\\lambda_0$ between angular frequency, wave velocity $latex v$ and wavelength yields $latex \\omega \\beta_0 = \\frac{4 \\pi^2}{\\lambda_0^2} v $\r\n\r\nFinally\r\n\r\n$latex R_{rad}^{(H)} = \\frac{8 \\pi^3 \\mu_0^2 v^2}{ 3 Z_0} \\left( \\frac{\\pi a^2}{\\lambda_0^2} \\right)^2\r\n\\label{R_rad_H}\\tag{4.104} $\r\n\r\nValue of the wave speed in vacuum is given in Eq.(<a href=\"#v_0\" data-reference-type=\"ref\" data-reference=\"v_0\">[v_0]<\/a>) therefore this resistance in air is approximately $latex R_{rad}^{(H)} \\approx 31'170.91 \\left( \\frac{\\pi a^2}{\\lambda_0^2} \\right)^2 [\\Omega] $\r\n\r\nif radius $latex a$ and wavelength $latex \\lambda_0$ have the same units.\r\n\r\nConsider a loop of radius ($latex a= 1$\u00a0cm). At 300 MHz the wavelength ($latex \\lambda_0 = 1$\u00a0m) and the radiation resistance is $latex R_{rad}^{(H)} \\approx 3.08 \\mu \\Omega $ In order to radiate 1\u00a0W, the loop requires a current of 18\u00a0A!\r\n\r\nAt 3 MHz the wavelength ($latex \\lambda_0 = 100$\u00a0m) and the radiation resistance is $latex R_{rad}^{(H)} \\approx 30.8 p \\Omega $ and current required to radiate 1 W is 180\u00a0kA!\r\n\r\nBoth elementary dipoles radiate very weakly but their comparison shows that magnetic dipole is overwhelmingly worse radiator than electric. In practice they are scarcely met as unintentional antennas.\r\n<h5 id=\"half-power-beamwidth-hpbw\">Half power beamWidth HPBW<\/h5>\r\ncalled also 3 dB beamwidth is an angle span, either in azimuth or in elevation between two directions encompassing set of directions with the surface power density decaying not more then to the half of the maximum. In other words not reduced about more then 3 dB.\r\n<figure><img src=\"Propagation\/Figures\/HPBW.png\" id=\"HPBW\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_197\" align=\"aligncenter\" width=\"551\"]<img class=\"wp-image-197\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-35.png\" width=\"551\" height=\"264\" alt=\"\" \/> Figure 4.36: Illustration of the half power beamwidth of electric and magnetic dipole.[\/caption]<\/figure>\r\nThe electrical dipole is omnidirectional in respect to azimuth $latex \\varphi$, therefore determining the HPBW on the plane $latex \\theta = 0^{\\circ}$ is groundless. For the elevation<a href=\"#fn14\" class=\"footnote-ref\" id=\"fnref14\" role=\"doc-noteref\"><sup>14<\/sup><\/a> the upper bound is $latex \\theta = 45^{\\circ}$ and lower bound $latex \\theta = 135^{\\circ}$ therefore $latex HPBW = 90^{\\circ}$, see Fig.\u00a0(<a href=\"#HPBW\" data-reference-type=\"ref\" data-reference=\"HPBW\">4.36<\/a>).\r\n\r\nHPBW expresses spreading of the power beam. The narrower HPBW, the better concentration of radiated power roundabout desired direction.\r\n<h4 id=\"unintentional-antennas\">Unintentional antennas<\/h4>\r\nIt is mentioned frequently, that any cable can be either transmitting or receiving antenna.\r\n\r\nBy the measurement of radiated emission of the table top equipments in the Semi Anechoic Chamber SAC, the EUT must be placed on the table with 80\u00a0cm height. The mains cord should be stretched towards the table edge, dropped down towards the floor and plugged in the supply socket under the metal floor. Typical arrangement of the mains cord is shown in Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0a). Such cable is a monopole antenna while the EUT case is its ground reference. If the common mode current is driven in the cord and the cord length is matched with the wavelength, then the cable radiate and it is very often the case.\r\n<figure><img src=\"Propagation\/Figures\/Cable_as_antenna.png\" id=\"Cable_antenna\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_198\" align=\"aligncenter\" width=\"756\"]<img class=\"wp-image-198\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-36.png\" width=\"756\" height=\"420\" alt=\"\" \/> Figure 4.37: Examples of the mains cable as an unintentional antenna: monopole antenna a), loop antenna b) excess cable folded to and forth c).[\/caption]<\/figure>\r\nIn Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0b) excess cable is arranged as it should not be. It is wound in few turns building the loop antenna. If the common mode current is driven in the cord and the loop diameter is matched with the wavelength, then it can radiate. Excess cable should be folded to and forth in order to form a low inductive bundle as shown in Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0c).\r\n\r\nLet\u2019s consider the metal case with the primary plane wave inside, propagating towards the wall with vector\u2019s orientation of: electric field $latex \\overrightarrow{E}_p$, magnetic field $latex \\overrightarrow{H}_p$ and Poynting vector $latex \\overrightarrow{S}_p$ as shown in the side projection in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b). By continuous metal wall of the case, surface current would be induced on the internal surface of the wall. The slot enables the current to leak outside. It flows on the external surface of the wall, in surrounding of the slot as shown with reddish region in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a). Black arrows on the edges of the slot shows direction of the driven current. It is accompanied with electric field in the vicinity of the slot oriented as vector $latex \\overrightarrow{E}$ in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b). Imagine that the edges of the slot build two dipoles shortcircuited at the top of the arms. Such entity can hardly radiate due to cancellation in the far field zone. But as shown with the reddish region in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a) current is driven also on the external surface around the edges.\r\n<figure><img src=\"Propagation\/Figures\/Slot_antenna.png\" id=\"Slot_antenna\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_199\" align=\"aligncenter\" width=\"668\"]<img class=\"wp-image-199\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-37.png\" width=\"668\" height=\"181\" alt=\"\" \/> Figure 4.38: Half wavelength slot antenna.[\/caption]<\/figure>\r\nTherefore the secondary wave oriented as shown with the vectors of: electric field $latex \\overrightarrow{E}_s$, magnetic field $latex \\overrightarrow{H}_s$ and Poynting vector $latex \\overrightarrow{S}_s$ in the side projection in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b) propagates from the case, particularly by matching the slot length. Example is $latex \\lambda\/2$ matching shown in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a). Memorize that orientation of the propagated field is dual to the electric dipole. Horizontally oriented slot propagates vertically oriented electric field.\r\n<figure><img src=\"Propagation\/Figures\/Slot_antenna_2.png\" id=\"Slot_antenna_2\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_200\" align=\"aligncenter\" width=\"740\"]<img class=\"wp-image-200\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-38.png\" width=\"740\" height=\"244\" alt=\"\" \/> Figure 4.39: Gap by point wise fixing of two pieces of the case a), window in a case as the aperture antenna b).[\/caption]<\/figure>\r\nBy point wise techniques of joining parts of the metal cases with screws, nuts or spot welding, gaps between places tied together, marked with yellowish area in Fig.\u00a0<a href=\"#Slot_antenna_2\" data-reference-type=\"ref\" data-reference=\"Slot_antenna_2\">4.39<\/a>\u00a0a) are formed. Things like incertitude of joined edges and deformations caused by excessed contact pressing force are contribution factors of it. In order to reduce effect of secondary radiation from such gaps distances between fixed spots must ensure length mismatching or\/and EMC gaskets obstructing electromagnetically the gaps must be applied.\r\n\r\nWindows and openings in cases are indispensable. They serves for mounting displays, cameras, switches, buttons, as sight hole or for ventilation and manipulation of the arrangement. Due to the same rules as by the gaps in the metal case the surface currents flow on the external surface of the wall as shown with the reddish area in Fig.\u00a0<a href=\"#Slot_antenna_2\" data-reference-type=\"ref\" data-reference=\"Slot_antenna_2\">4.39<\/a>\u00a0b). The window can build aperture antenna enabling the secondary wave to radiate out of the case.\r\n\r\nThere is alway maximal frequency called cut off, up to which the window is not capable to radiate. It is associated with the biggest wavelength which can be radiated. Above the cut off frequency the wavelength for the rectangular window with sides $latex ab$ and the circular window with the radius $latex r$ are approximately ruled with following inequalities\r\n\r\n$latex \\begin{array} {cc}\r\n\\lambda_{\\framebox[0.15in]{}} &lt; 2 \\sqrt{\\pi a b} &amp; \\hspace{2cm} \\lambda_{\\bigcirc} &lt; 2 \\pi^{\\frac{3}{2}} r \\approx 11.14 r\r\n\\end{array}\r\n\\label{cut_off_1}\\tag{4.108} $\r\n\r\nInequality for the circular window is convergent with matching of the circumference of the magnetic dipole to the $latex \\lambda\/2$ wavelength $latex \\lambda \\approx 12.57 r$ as illustrated in Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>. It is also in line with condition of not being electrically small i.e. bigger than $latex 0.1 \\lambda$, see subsection <a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a>.\r\n<h3 id=\"wave-guides\">Guided waves<\/h3>\r\nSo far in the chapter <a href=\"#wave_coupling\" data-reference-type=\"ref\" data-reference=\"wave_coupling\">4.4<\/a> we have been discussing conveying energy, signals and disturbances with radiation. By this art of transportation both field vectors $latex \\overrightarrow{E}$ as well as $latex \\overrightarrow{H}$ are perpendicular to the direction of propagation. This type of waves is called TEM (transverse electric magnetic) mode. The same TEM mode is present in transmission lines elaborated by galvanic coupling in chapter <a href=\"#galvanic_coupling\" data-reference-type=\"ref\" data-reference=\"galvanic_coupling\">4.2<\/a> as well as by electric and magnetic coupling in chapter <a href=\"#LC_coupling\" data-reference-type=\"ref\" data-reference=\"LC_coupling\">4.3<\/a>. Transportation passed along the line and fields are perpendicular to it<a href=\"#fn15\" class=\"footnote-ref\" id=\"fnref15\" role=\"doc-noteref\"><sup>15<\/sup><\/a>. Therefore TEM mode is often referred to as the transmission line mode.\r\n<figure><img src=\"Propagation\/Figures\/WaveGuide.png\" id=\"waveguide\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_201\" align=\"aligncenter\" width=\"451\"]<img class=\"wp-image-201\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-39.png\" width=\"451\" height=\"296\" alt=\"\" \/> Figure 4.40: A rectangle waveguide.[\/caption]<\/figure>\r\nLet us consider tubular structure with rectangular cross-section, built of perfectly conducting material $latex \\sigma = \\infty$ filled with lossless dielectric $latex \\sigma = 0$ with dielectric permittivity $latex \\epsilon_0$ and magnetic permeability $latex \\mu_0$ as shown in Fig.\u00a0<a href=\"#waveguide\" data-reference-type=\"ref\" data-reference=\"waveguide\">4.40<\/a>. Since $latex \\overrightarrow{E}$ must be always perpendicular and $latex \\overrightarrow{H}$ always tangential to the boundary, it is not possible to excite TEM in such structure.\r\n\r\nLet us assume that in such structure propagates wave with three component of phasor\u2019s electric field: two cross-sectional $latex {\\bf{E}}_x(x,y,z)$, $latex {\\bf{E}}_y(x,y,z)$ and longitudinal $latex {\\bf{E}}_z(x,y,z)$. The last is governed with the partial differential equation called the wave equation\r\n\r\n$latex \\nabla^2 {\\bf{E}}_z + \\omega^2 \\mu_0 \\epsilon_0 {\\bf{E}}_z = 0\r\n\\label{Ez_1}\\tag{4.109} $\r\n\r\nThis equation can be solved by the method of separation of variables $latex {\\bf{E}}_z(x,y,z) = {\\bf{X}}(x) {\\bf{Y}}(y) {\\bf{Z}}(z)$\r\n\r\n$latex \\frac{{\\bf{X}}^{''}}{{\\bf{X}}} + \\frac{{\\bf{Y}}^{''}}{{\\bf{Y}}} + \\frac{{\\bf{Z}}^{''}}{{\\bf{Z}}} + \\beta_0^2 = 0\r\n\\label{Ez_2}\\tag{4.110} $\r\n\r\nwhere $latex \\beta_0$ is phase constant as defined in Eq.\u00a0(4.50).\r\n\r\nEq.\u00a0(4.110) can be split into three independent ordinary differential equations\r\n\r\n$latex \\label{X}\r\n{\\bf{X}}^{''} - {\\bf\\gamma}_x^2 {\\bf{X}} = 0 \\tag{4.111} $\r\n\r\n$latex \\label{Y}\r\n{\\bf{Y}}^{''} - {\\bf\\gamma}_y^2 {\\bf{Y}} = 0 \\tag{4.112} $\r\n\r\n$latex \\label{Z}\r\n{\\bf{Z}}^{''} - {\\bf\\gamma}_z^2 {\\bf{Z}} = 0 \\tag{4.113} $\r\n\r\nalong with dispersion equation\r\n\r\n$latex {\\bf\\gamma}_x^2 + {\\bf\\gamma}_y^2 + {\\bf\\gamma}_z^2 + \\beta_0^2 = 0\r\n\\label{Ez_4}\\tag{4.114} $\r\n\r\nwhere $latex {\\bf\\gamma}_x$, $latex {\\bf\\gamma}_y$ and $latex {\\bf\\gamma}_z$ are separations\u2019 constants.\r\n\r\nSolution of Eq.\u00a0(4.111) is following linear combination of waves $latex {\\bf{X}}(x) = A e^{{\\bf\\gamma}_x x} + B e^{-{\\bf\\gamma}_x x} $ traveling forth and back along x axis. $latex {\\bf\\gamma}_x x = \\alpha_x + j\\beta_x$ can be a complex quantity. As $latex E_z$ is tangential to the walls of the waveguide but tangential component of electric field must be zero on perfectly conducting metal wall, $latex {\\bf{X}}(x)$ must disappear by $latex x =0$ and $latex x= a$. This is possible only if $latex \\alpha_x = 0$. Consequently magnitude $latex X(x)$ is a standing wave with the nodes on walls\r\n\r\n$latex X_m(x) = (A_m - B_m) \\sin{(\\beta_{x,m}}x) = X_{0,m} \\sin{(\\beta_{x,m} x)} $\r\n\r\n$latex \\beta_{x,m} = m \\frac{\\pi}{a}\r\n\\label{beta_x}\\tag{4.117} $\r\n\r\nwhere $latex m$ is integer number.\r\n\r\nIdentical procedure applied to Eq.\u00a0(4.112) yields $latex Y_n(y) = Y_{0,n} \\sin{(\\beta_{y,n} y)} $ along with\r\n\r\n$latex \\beta_{y,n} = n \\frac{\\pi}{b}\r\n\\label{beta_y}\\tag{4.119} $\r\n\r\nwhere $latex n$ is integer number.\r\n\r\nSolution of Eq.\u00a0(4.113) is following linear combination of waves $latex {\\bf{Z}}(z) = C e^{{j\\gamma}_z z} + D e^{{-j\\gamma}_z z} $ traveling back $latex C e^{{j\\gamma}_z z }$ and forth $latex D e^{{-j\\gamma}_z z}$ along the z-axis. The waveguide is infinite in respect to z-axis. Therefore physical meaning has only one wave.\r\n\r\nThe source of field is placed somewhere in the waveguide. From that location on, two waves propagate: one forwards another backwards in respect to z-axis. Assuming that source of field is placed by $latex z&lt;0$ so that by $latex z=0$ regular propagation takes place, then by $latex z \\ge 0$ propagates only\r\n\r\n$latex {\\bf{Z}}_{mn}(z) = D_{mn} e^{-{\\bf\\gamma}_{z,mn} z}\r\n\\label{Z_2}\\tag{4.121} $\r\n\r\nwhere\r\n\r\n$latex {\\bf\\gamma}_{z,mn}^2 = - \\beta_0^2 - \\left( {\\bf\\gamma}_{x,mn}^2 + {\\bf\\gamma}_{y,mn}^2 \\right) = \\beta_{x,mn}^2 + \\beta_{y,mn}^2 - \\beta_0^2\r\n\\label{gamma_z}\\tag{4.125} $\r\n\r\n$latex {\\bf\\gamma}_{z,mn}$ can be real, imaginary or can have zero value by the break point which corresponds to the cut-off frequency $latex f_{c,mn}$\r\n\r\n$latex f_{c,mn} = \\frac{v_0}{2} \\sqrt{\\left( \\frac{m}{a} \\right)^2 + \\left( \\frac{n}{b}\\right)^2}\r\n\\label{gamma_zz}\\tag{4.123} $\r\n\r\nwhere $latex v_0$ is wave velocity in vacuum and indexes $latex mn$ corresponds to Eqs.\u00a0(4.117) and (4.119).\r\n\r\nFinally separation constant $latex {\\bf{\\gamma}}_{z,mn}$ which is identical with propagation constant in z direction is\r\n\r\n$latex {\\bf{\\gamma}}_{z,mn} = \\left\\{ \\begin{array} {ll}\r\n\\frac{\\omega}{v_0}\\sqrt{ \\left( \\frac{f_{c,mn}}{f} \\right)^2 - 1} &amp;  \\text{for}~~f &lt; f_{c,mn}\\\\\r\n\\\\\r\nj\\frac{\\omega}{v_0} \\sqrt{ 1 - \\left( \\frac{f_{c,mn}}{f} \\right)^2 } &amp;  \\text{for}~~f &gt; f_{c,mn}\r\n\\end{array}\r\n\\right.\r\n\\label{gamma_z2}\\tag{4.124} $\r\n\r\nand z component of the phasor of electric field\r\n\r\n$latex {\\bf{E}}_{z,mn}(x,y,z) = E_{z,mn}(0) \\sin{\\left(\\frac{ m\\pi }{a}x \\right)} \\sin{\\left(\\frac{n \\pi }{b}y \\right)} e^{-\\gamma_{z,mn} z}\r\n\\label{gamma_z}\\tag{4.125} $\r\n\r\n$latex E_{z,mn}(0)$ is product of $latex X_{m}0$, $latex Y_{n}0$ and $latex D_{mn}$. For unique solution the value $latex E_{z,mn}(0)$ must be known.\r\n\r\nDerivation of remaining field components can be found for instance in <span class=\"citation\" data-cites=\"Hammond_2\">[@Hammond_2]<\/span>.\r\n\r\nEq.\u00a0(4.125) can be rearranged with introduction of phase constant by propagation in free space with cut-off frequency $latex \\beta_0(f_{c,mn})$\r\n\r\n$latex {\\bf{\\gamma}}_{z,mn} = \\left\\{ \\begin{array} {ll}\r\n\\beta_0({f_{c,mn}}) \\sqrt{ 1 - \\left( \\frac{f}{f_{c,mn}} \\right)^2 } &amp;  \\text{for}~~\\frac{f}{f_{c,mn}} &lt; 1\\\\\r\n\\\\\r\nj\\beta_0({f_{c,mn}}) \\sqrt{ \\left( \\frac{f}{f_{c,mn}} \\right)^2 - 1} &amp;  \\text{for}~~\\frac{f}{f_{c,mn}} &gt; 1\r\n\\end{array}\r\n\\right.\r\n\\label{gamma_z3}\\tag{4.126} $\r\n\r\nDependence of the module of the propagation constant $latex |\\gamma_z|$ versus frequency related to the cut-off frequency $latex f\/f_{c,mn}$ is shown in Fig.<a href=\"#gamma_plot\" data-reference-type=\"ref\" data-reference=\"gamma_plot\">4.41<\/a>.\r\n<figure><img src=\"Propagation\/Figures\/Gamma_z.png\" id=\"gamma_plot\" alt=\"\" \/>\r\n\r\n[caption id=\"attachment_202\" align=\"aligncenter\" width=\"485\"]<img class=\"wp-image-202\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-40.png\" width=\"485\" height=\"303\" alt=\"\" \/> Figure 4.41: Frequency dependence of the module of the propagation constant $latex \\gamma_z$.[\/caption]<\/figure>\r\nBelow the cut-off frequency (yellowish region) the propagation constant is real $latex \\gamma_z = \\alpha_z$. The wave is attenuated. Above the cut-off frequency (greenish region) the propagation constant is imaginary $latex \\gamma_z = j\\beta_z$. The wave is able to propagate. The propagation constant in this region approaches asymptote $latex \\beta_{z,mn} = \\beta_0 \\left(f_{c,mn} \\right) \\frac{f}{f_{c,mn}} $\r\n\r\nIt is blue strait line in Fig.<a href=\"#gamma_plot\" data-reference-type=\"ref\" data-reference=\"gamma_plot\">4.41<\/a>.\r\n\r\nThe type of fields discussed up to now are so called E-mode named equivalently TM-mode. Notice that the lowest mode of this type of fields can be $latex E_{11}$ ($latex TM_{11}$) due to constraint on tangential component of E field on the walls.\r\n\r\nSimilarly $latex H_z$ component for the H mode (TE) mode can be derived\r\n\r\n$latex {\\bf{H}}_{z,mn}(x,y,z) = H_{z,mn}(0) \\cos{\\left(\\frac{ m\\pi }{a}x \\right)} \\cos{\\left(\\frac{n \\pi }{b}y \\right)} e^{-\\gamma_{z,mn} z}\r\n\\label{gamma_z4}\\tag{4.128} $\r\n\r\nDependence in x and y direction is cosinusoidal since H field must be tangential on the metal walls i.e. $latex \\frac{\\partial H_z}{\\partial n} = 0$. Consequently the lowest mode that can be propagated can have one number equal zero i.e.\u00a0$latex H_{10}$, $latex H_{01}$ ($latex TE_{10}$, $latex TE_{01}$).\r\n\r\n<section class=\"footnotes\" role=\"doc-endnotes\">\r\n\r\n<hr \/>\r\n\r\n<ol>\r\n \t<li id=\"fn1\" role=\"doc-endnote\">Proximity effect between strains makes the exact analysis of the problem more complex but the tendency is unambiguous.<a href=\"#fnref1\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn2\" role=\"doc-endnote\">$latex w \\gg t$<a href=\"#fnref2\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn3\" role=\"doc-endnote\">If in a given instant the field strength at one point of the equiphase surface has e.g.\u00a0positive magnitude, then it also has at all remaining points.<a href=\"#fnref3\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn4\" role=\"doc-endnote\">$latex {\\bf{E}}_r$ component by electric dipole Eq.\u00a0(4.43) and $latex {\\bf{H}}_r$ component by magnetic dipole Eq.\u00a0(4.54) disappears due to lack of summand with the first power of radius $latex r$.<a href=\"#fnref4\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn5\" role=\"doc-endnote\">Exactly cosinusoidal distribution is valid for lossless symmetrical transmission line with parallelly layouted forward and return conductors. Per unit length parameter $latex LC$ of such line are constant. By straightened out conductors $latex LC$ parameters depend on position along the arms.<a href=\"#fnref5\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn6\" role=\"doc-endnote\">By field probes the parasitic capacitances can be neglected due to small electrical size. It is not the case by antennas.<a href=\"#fnref6\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn7\" role=\"doc-endnote\">Common mode chokes are covered in chapter <a href=\"#Filters\" data-reference-type=\"ref\" data-reference=\"Filters\">[Filters]<\/a><a href=\"#fnref7\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn8\" role=\"doc-endnote\">By EMC measurements of radiated emission, antenna height must be scanned from 1\u00a0m to 4\u00a0m. Up to the height 2\u00a0m the capacitances are hardly equal.<a href=\"#fnref8\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn9\" role=\"doc-endnote\">It is a circle with the radius $latex r = \\lambda\/(4\\pi)$.<a href=\"#fnref9\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn10\" role=\"doc-endnote\">By scaling with the magnitudes, power density is $latex \\frac{1}{2} \\left( \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^* \\right)$.<a href=\"#fnref10\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn11\" role=\"doc-endnote\">Index $latex i$ points out the isotropic radiator as the reference.<a href=\"#fnref11\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn12\" role=\"doc-endnote\">Difference of power sent from the source and power reflected at the antenna feeding terminal by mismatching.<a href=\"#fnref12\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn13\" role=\"doc-endnote\">Notice that it is in line with definition of electrical dimension and with the rule about radiation inability of electrically small objects, see chapter\u00a0<a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a>.<a href=\"#fnref13\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn14\" role=\"doc-endnote\">Introduce $latex F_P(\\theta) = 1\/2$ into equation on power radiation pattern of the dipole Eq.\u00a0(4.84).<a href=\"#fnref14\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n \t<li id=\"fn15\" role=\"doc-endnote\">It is apart from small region next to the conductors\u2019 surfaces where the Poynting\u2019s vector is leant toward the inside of conductors where the wave energy is converted into heat.<a href=\"#fnref15\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<p>There are four ways for disturbance to pass from the source to the victim, via:<\/p>\n<ul>\n<li>galvanic coupling,<\/li>\n<li>capacitive (electric) coupling,<\/li>\n<li>inductive (magnetic) coupling,<\/li>\n<li>wave coupling.<\/li>\n<\/ul>\n<p>The first three accompany transmission of power or signals via conductors. It can proceed unsymmetrically or symmetrically.<\/p>\n<p>Unsymmetrical transmission comes about when return conductor is common for more circuits and nonzero current as an overlay of currents of all circuits drives in the common return path. Three examples are shown in Fig.<a href=\"#Unsym_line\" data-reference-type=\"ref\" data-reference=\"Unsym_line\">4.1<\/a>. The return path can be identical with feeding lines or can differ with them.<\/p>\n<figure><img src=\"Propagation\/Figures\/Unsym_line.png\" id=\"Unsym_line\" alt=\"\" \/><\/p>\n<figure id=\"attachment_162\" aria-describedby=\"caption-attachment-162\" style=\"width: 739px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-162\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances.png\" width=\"739\" height=\"391\" alt=\"\" \/><figcaption id=\"caption-attachment-162\" class=\"wp-caption-text\">Figure 4.1: Three examples of unsymmetrical lines.<\/figcaption><\/figure>\n<\/figure>\n<p>Symmetrical transmission is carried with a pair of identical conductors isolated from the surrounding. Examples are shown in Fig.<a href=\"#Sym_line\" data-reference-type=\"ref\" data-reference=\"Sym_line\">4.2<\/a>. There is always better or worse conducting layer beneath (striped plane) which builds the ground reference.<\/p>\n<figure><img src=\"Propagation\/Figures\/Sym_line.png\" id=\"Sym_line\" alt=\"\" \/><\/p>\n<figure id=\"attachment_163\" aria-describedby=\"caption-attachment-163\" style=\"width: 824px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-163\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-1.png\" width=\"824\" height=\"336\" alt=\"\" \/><figcaption id=\"caption-attachment-163\" class=\"wp-caption-text\">Figure 4.2: Two examples of symmetrical lines.<\/figcaption><\/figure>\n<\/figure>\n<p>Unsymmetrical transmission is potentially accompanied with galvanic, capacitive and inductive coupling. Symmetrical transmission only with capacitive and inductive. They must be considered simultaneously. Here the phenomena will be presented separately in order to catch its essence. Such approach is violation of circuit theory, provide only one type of coupling is dominant, remaining negligible. It is case dependent.<\/p>\n<h2 id=\"examples-of-unsymmetrical-and-symmetrical-transmission\">Examples of unsymmetrical and symmetrical transmission<\/h2>\n<p>An example of unsymmetrical transmission is cabling of a sensor as shown in Fig.<a href=\"#Galvanic_sensor\" data-reference-type=\"ref\" data-reference=\"Galvanic_sensor\">4.3<\/a> in which supply circuit [latex]V_{cc+}[\/latex] and signal circuit have the same return path [latex]GND[\/latex].<\/p>\n<figure><img src=\"Propagation\/Figures\/Galvanic_sensor.png\" id=\"Galvanic_sensor\" alt=\"\" \/><\/p>\n<figure id=\"attachment_164\" aria-describedby=\"caption-attachment-164\" style=\"width: 415px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-164\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-2.png\" width=\"415\" height=\"125\" alt=\"\" \/><figcaption id=\"caption-attachment-164\" class=\"wp-caption-text\">Figure 4.3: An example of galvanic coupling.<\/figcaption><\/figure>\n<\/figure>\n<p>Another example is electrical installation in vehicles which very often is single ended. Return path is vehicle\u2019s chassis.<\/p>\n<figure>\n<figure id=\"attachment_165\" aria-describedby=\"caption-attachment-165\" style=\"width: 565px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-165\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-3.png\" width=\"565\" height=\"366\" alt=\"\" \/><figcaption id=\"caption-attachment-165\" class=\"wp-caption-text\">Fihure 4.4: Surge due to galvanic coupling in TN-C mains supply system, by direct strike in the external lightning protection system of the building.<\/figcaption><\/figure>\n<\/figure>\n<p>Mind that three phase supply system shown in Fig.<a href=\"#Unsym_line\" data-reference-type=\"ref\" data-reference=\"Unsym_line\">4.1<\/a>a) with Neutral (blue line) as return path is an example of unsymmetrical transmission as soon as current is driven in the neutral line. One case common for house installation is presented below.<\/p>\n<p>Low voltage AC power is delivered to a house as shown in Fig.<a href=\"#Lemp_1\" data-reference-type=\"ref\" data-reference=\"Lemp_1\">[Lemp_1]<\/a> in chapter <a href=\"#Thunderstorm\" data-reference-type=\"ref\" data-reference=\"Thunderstorm\">[Thunderstorm]<\/a>. Regulations forces to carry out the supply line between the transformer and the building input as the TN-C system <span class=\"citation\" data-cites=\"IEC-60364-1\">[@IEC-60364-1]<\/span>. It is four line supply system in which the Neutral line [latex]N[\/latex] acts just as well as the Protective Earth line [latex]PE[\/latex], therefore it is called [latex]PEN[\/latex] line.<\/p>\n<p>In Fig.<a href=\"#Galvanic_TNC\" data-reference-type=\"ref\" data-reference=\"Galvanic_TNC\">4.4<\/a> electric circuit of the transformer with Line to Neutral EMF: [latex]E_A[\/latex], [latex]E_B[\/latex] and [latex]E_C[\/latex] feeding the building with the four line system [latex]L_1[\/latex], [latex]L_2[\/latex], [latex]L_3[\/latex] and [latex]PEN[\/latex] is shown. [latex]Z_A[\/latex], [latex]Z_B[\/latex] and [latex]Z_C[\/latex] represents impedances of the whole building installation seen from the feeding point of the building. [latex]Z_{PE}^B[\/latex] and [latex]Z_{PE}^T[\/latex] are earthing impedances of the star point of the building and the transformer respectively. By direct strike of the lightning electromagnetic pulse [latex]LEMP[\/latex] in the external lightning protection system of the building the current is driven to the star point of the building where it is split as shown with the red arrows in Fig.<a href=\"#Galvanic_TNC\" data-reference-type=\"ref\" data-reference=\"Galvanic_TNC\">4.4<\/a>. The biggest part of the current drives to the Earth through earthing impedance of the building [latex]Z_{PE}^B[\/latex] but due to the fact that between the building and the transformer there is common [latex]PEN[\/latex] line, part of the current is driven as the surge in the whole installation inside the building and between the building and the transformer.<\/p>\n<p>Usually there are more customers supplied from one transformer, as shown in Fig.<a href=\"#Lemp_1\" data-reference-type=\"ref\" data-reference=\"Lemp_1\">[Lemp_1]<\/a>. Due to galvanic connection of all of them with the star point of the transformer, all customers experiences the surge with different degree.<\/p>\n<p>More resistive against surges would be the TN-S system <span class=\"citation\" data-cites=\"IEC-60364-1\">[@IEC-60364-1]<\/span>. It is five line supply system which has separate the Neutral [latex]N[\/latex] and the Protective Earth [latex]PE[\/latex] line.<\/p>\n<p>Example of symmetrical transmission can be:<\/p>\n<ul>\n<li>power or signal cables of house installation layouted in the walls, ceilings or floors,<\/li>\n<li>cables layouted on the mounting plate in the control cabinet of a system,<\/li>\n<li>signal paths on the printed circuit board.<\/li>\n<\/ul>\n<h2 id=\"galvanic_coupling\">Galvanic coupling<\/h2>\n<p>Galvanic coupling is illustrated in Fig.<a href=\"#Galvanic_PCB\" data-reference-type=\"ref\" data-reference=\"Galvanic_PCB\">4.5<\/a> with very simple situation. There is a circuit composed of two meshes. In mesh 1 there is a source of power or a signal with electromotive force [latex]EMF_1[\/latex] and internal resistance [latex]R_{S1}[\/latex] and load [latex]R_{L1}[\/latex]. The second with [latex]EMF_2[\/latex] and [latex]R_{L2}[\/latex]. Intension is to deliver electric power or signal from [latex]EMF_1[\/latex] to [latex]R_{L1}[\/latex] and from [latex]EMF_2[\/latex] to [latex]R_{L2}[\/latex]. However for some reason the two meshes have common return path represented with [latex]R_{GND}[\/latex] and [latex]L_{GND}[\/latex] which build the impedance [latex]Z_{GND}=R_{GND} + j \\omega L_{GND}[\/latex].<\/p>\n<figure><img src=\"Propagation\/Figures\/Galvanic_PCB.png\" id=\"Galvanic_PCB\" alt=\"\" \/><\/p>\n<figure id=\"attachment_166\" aria-describedby=\"caption-attachment-166\" style=\"width: 443px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-166\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-4.png\" width=\"443\" height=\"305\" alt=\"\" \/><figcaption id=\"caption-attachment-166\" class=\"wp-caption-text\">Figure 4.5: Explanation of galvanic coupling.<\/figcaption><\/figure>\n<\/figure>\n<p>Each loop has self-inductance. It is composed of external inductance depending on wires layout and consequently on the area of the loop they build. Internal inductance of wires from which the loops are built also contribute to the total inductance. Obviously inductance must be assigned to the mesh but it is justified to extract the internal inductance of the common return path [latex]L_{GND}[\/latex] as shown in Fig.<a href=\"#Galvanic_PCB\" data-reference-type=\"ref\" data-reference=\"Galvanic_PCB\">4.5<\/a> because it is the attribute of the return path. External inductances of both meshes and internal inductances of feeding wires are disregarded as declared earlier.<\/p>\n<p>Voltage [latex]U_{GND}[\/latex] across the common return path represented with impedance [latex]Z_{GND}[\/latex] is as follows<\/p>\n<p>[latex]U_{GND} =  -\\frac{ \\frac{EMF_1}{RS1+RL1} + \\frac{EMF_2}{RS2+RL2} }  { \\frac{1}{R_{S1}+R_{L1}} + \\frac{1}{R_{S2}+R_{L2}} + \\frac{1}{Z_{GND}}}  \\label{U_GND}\\tag{4.1}[\/latex]<\/p>\n<p>Voltage [latex]U_{L2}[\/latex] across the load resistance [latex]R_{L2}[\/latex] in mesh 2 is as follows<\/p>\n<p>[latex]U_{L2} = \\frac{ R_{L2}} {R_{S2}+R_{L2}}  \\left( EMF_2 - U_{GND} \\right)  \\label{U_L2}\\tag{4.2}[\/latex]<\/p>\n<p>Electromotive force [latex]EMF_1[\/latex] contributes to the voltage [latex]U_{L2}[\/latex] across the load of mesh 2 as follows<\/p>\n<p>[latex]U_{L2}^{EMF_1} = - \\frac{ R_{L2} \\cdot EMF_1}  {  (R_{S1}+R_{L1}) (R_{S2}+R_{L2})  \\left(  \\frac{1}{R_{S1}+R_{L1}} + \\frac{1}{R_{S2}+R_{L2}} + \\frac{1}{Z_{GND}}  \\right)  }  \\label{U_2}\\tag{4.3}[\/latex]<\/p>\n<p>The common return path causes unintentional contribution of [latex]EMF_1[\/latex] to voltage across [latex]R_{L2}[\/latex] and vice versa, via galvanic coupling.<\/p>\n<p>Obviously, coupled voltage [latex]U_{L2}^{EMF_1}[\/latex] would disappear by zero voltage [latex]U_{GND}[\/latex] across the common return path. It can happen in two cases:<\/p>\n<ul>\n<li>by zero impedance of the return path [latex]Z_{GND}=0\\Omega[\/latex]. That is unrealistic to achieve.<\/li>\n<li>by total symmetry of the meshes i.e.\u00a0if electromotive forces have the same amplitude and opposite phases. Moreover impedances of both paths are the same. In most cases it does not have sense by energy as well signal transmission.<\/li>\n<\/ul>\n<p>Moreover perfect galvanic decoupling will happen with replacement of electromotive forces [latex]EMF_1[\/latex] and [latex]EMF_2[\/latex] with current sources. By power transportation it is hardly to imagine but transmission of current signals is used very often for that reason.<\/p>\n<p>Intensity of galvanic coupling rises with increased impedance of the return path. By [latex]Z_{GND}[\/latex] tending to infinity, [latex]I_{GND}[\/latex] tends to zero and coupled voltage [latex]U_{L2}[\/latex] approaches maximum expressed with the formula below<\/p>\n<p>[latex]U_{L2_{MAX}}^{EMF_1} = \\lim_{Z_{GND} \\rightarrow \\infty} U_{L2}^{EMF_1} = \\frac{ R_{L2} \\cdot EMF_1} {R_{S1}+R_{S2}+R_{L1}+R_{L2} }  \\label{U_2max}\\tag{4.4}[\/latex]<\/p>\n<p>As galvanic coupling rises with increased internal impedance [latex]Z_{GND}[\/latex] of the common return path it is valuable to investigate frequency dependence of impedance [latex]Z_{GND}[\/latex]. This impedance depends on the shape of the cross section of the conductor. Considered are two cases for: round wire and rectangle cross section.<\/p>\n<h3 id=\"internal-impedance-of-round-wire\">Internal impedance of round wire<\/h3>\n<p>Considered is idealized case of infinitely long strait wire with circular cross-section with radius [latex]r[\/latex] placed in free space in order to neglect proximity effects. Segment of such wire with length [latex]l[\/latex] is shown in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>a).<\/p>\n<figure><img src=\"Propagation\/Figures\/Round_wire.png\" id=\"Round_wire\" alt=\"\" \/><\/p>\n<figure id=\"attachment_167\" aria-describedby=\"caption-attachment-167\" style=\"width: 698px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-167\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-5.png\" width=\"698\" height=\"335\" alt=\"\" \/><figcaption id=\"caption-attachment-167\" class=\"wp-caption-text\">Figure 4.6: Frequency dependence of round wire impedance per unit length.<\/figcaption><\/figure>\n<\/figure>\n<p>Impedance of such wire per unit length yields<\/p>\n<p>[latex]Z(f) = \\frac{k(f)}{2 \\pi r \\sigma} \\frac{J_0[k(f)r]}{J_1[k(f)r]}  \\label{Z_f}\\tag{4.5}[\/latex]<\/p>\n<p>where [latex]J_0[k(f)r][\/latex] and [latex]J_1[k(f)r][\/latex] are Bessel\u2019s functions of zero and first order, [latex]\\sigma[\/latex] and [latex]\\mu[\/latex] is conductivity and magnetic permeability of wire material respectively and [latex]k(f)[\/latex] is the wave number in the wire material<\/p>\n<p>[latex]k(f) = \\frac{1-j}{\\delta (f)}  \\label{k_f}\\tag{4.6}[\/latex]<\/p>\n<p>[latex]\\delta(f)[\/latex] is called the skin depth<\/p>\n<p>[latex]\\delta(f) = \\sqrt{\\frac{2}{\\omega \\mu \\sigma}}  \\label{delta}\\tag{4.7}[\/latex]<\/p>\n<p>which expresses depth in which current density is [latex]e[\/latex] times smaller than on the surface of the wire, where [latex]e[\/latex] is Euler\u2019s number. To be precise it is valid for plane wave facing infinite conducting medium oriented perpendicularly to the direction of the wave propagation.<\/p>\n<p>By low frequencies the real part of impedance [latex]Z(f)[\/latex], Eq.(4.5) is nothing else but DC resistance of the round wire<\/p>\n<p>[latex]R_{LF} = \\frac{1}{\\pi \\sigma r^2}  \\label{R_LF}\\tag{4.8}[\/latex]<\/p>\n<p>By high frequencies the real part of impedance [latex]Z(f)[\/latex], Eq.(4.5) can be solved as if the current flowed uniformly through a layer of thickness equal to skin depth [latex]\\delta[\/latex]. The effective cross-sectional area for driving the current would be then approximately equal to skin depth [latex]\\delta[\/latex] times the conductor\u2019s circumference [latex]2r-\\delta \\approx 2r[\/latex]. Thus HF resistance of round wire is approximately equal to DC resistance of a hollow tube with wall thickness [latex]2r-\\delta \\approx 2r[\/latex] carrying direct current<\/p>\n<p>[latex]R_{HF}(f) = \\frac{1}{\\pi \\sigma [2r-\\delta(f)]\\delta(f)} \\approx \\frac{1}{2 \\pi \\sigma r \\delta(f)}  \\label{R_HF}\\tag{4.9}[\/latex]<\/p>\n<p>Static and low frequency inductance of the round wire, accompanying uniformly flowed current is independent on wire radius [latex]r[\/latex] <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span><\/p>\n<p>[latex]L_{DC} = \\frac{\\mu}{8 \\pi }  \\label{L_DC}\\tag{4.18}[\/latex]<\/p>\n<p>Thus the imaginary part of impedance [latex]Z(f)[\/latex], Eq.(4.5) i.e. reactance [latex]X_{LF}(f)[\/latex] in low frequencies depends linearly on frequency [latex]X_{LF}(f) = 2 \\pi f L_{DC}  \\label{X_LF}\\tag{4.11}[\/latex]<\/p>\n<p>By high frequencies reactance and resistance are equal [latex]X_{HF}(f) = R_{HF}(f)[\/latex]. It can be explained with the plane electromagnetic wave penetrating good conducting medium. The wave impedance has then equal real and imaginary part.<\/p>\n<p>Frequency dependence of the internal impedance of the round wire in double logarithmic scale is illustrated in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>b). Breaking frequency [latex]f_0[\/latex] marked there, which is border between low and high frequency regions can be established equating LF resistance and reactance<\/p>\n<p>[latex]f_0 = \\frac{4}{\\pi \\mu \\sigma r^2}  \\label{f_0}\\tag{4.12}[\/latex]<\/p>\n<p>Mind that low frequency reactance is proportional to frequency, therefore it is strait line with the slope [latex]20dB\/dec[\/latex], compare Eq.(4.11). The HF resistance and reactance are reciprocally proportional to the skin depth [latex]\\delta(f)[\/latex], see Eq.(4.9) therefore they increase proportionally to the square root of frequency. In other words they have slopes [latex]10dB\/dec[\/latex]. Module of high frequency impedance [latex]|Z_{GND}(f)|[\/latex] is by factor [latex]\\sqrt{2}[\/latex] bigger than HF resistance or reactance, because they are equal one to another. In double logarithmic scale it is bigger about [latex]3dB[\/latex].<\/p>\n<p>Eq.(4.12) gives the hint how to keep impedance of the round wire small up to relatively high frequencies. The trick consists in keeping the radius [latex]r[\/latex] of wire relatively small. It is done in so called HF litz-wire in which wire is composed of bundle of individually isolated strands, as shown in Fig.<a href=\"#HF_Litz_wire\" data-reference-type=\"ref\" data-reference=\"HF_Litz_wire\">4.7<\/a>.<\/p>\n<figure><img src=\"Propagation\/Figures\/HF_Litz-wire.png\" id=\"HF_Litz_wire\" alt=\"\" \/><\/p>\n<figure id=\"attachment_168\" aria-describedby=\"caption-attachment-168\" style=\"width: 268px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-168\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-6.png\" width=\"268\" height=\"206\" alt=\"\" \/><figcaption id=\"caption-attachment-168\" class=\"wp-caption-text\">Figure 4.6: High frequency litz-wire built of seven isolated strains.<\/figcaption><\/figure>\n<\/figure>\n<p>In such wire, in spite of the cross-section equal to sum of cross-sections of individual strains, the breaking frequency Eq.(4.12) is correlated with individual strain<a href=\"#fn1\" class=\"footnote-ref\" id=\"fnref1\" role=\"doc-noteref\"><sup>1<\/sup><\/a>.<\/p>\n<h3 id=\"internal-impedance-of-conductor-with-rectangular-cross-section\">Internal impedance of conductor with rectangular cross section<\/h3>\n<p>Considered is idealized case of infinitely long strait conductor with rectangular cross section with width [latex]w[\/latex] and thickness [latex]t[\/latex] placed in free space in order to neglect proximity effects. Segment of such wire with length [latex]l[\/latex] is shown in Fig.<a href=\"#Rectangle\" data-reference-type=\"ref\" data-reference=\"Rectangle\">4.8<\/a>a).<\/p>\n<p>Unlike by round wire no analytical formula exist for such conductor.<\/p>\n<p>The low frequency resistance yields<\/p>\n<p>[latex]R_{LF} = \\frac{1}{\\sigma w t}  \\label{R_LF_wt}\\tag{4.13}[\/latex]<\/p>\n<p>By high frequencies resistance can be solved as by round wire i.e.\u00a0as if the current flowed uniformly through a layer of thickness equal to skin depth [latex]\\delta[\/latex]. The effective cross-sectional area for driving the current would be then approximately equal to skin depth [latex]\\delta[\/latex] times the conductor\u2019s circumference [latex]2 (w + t)[\/latex]. Thus HF resistance of round wire is approximately equal to DC resistance of a hollow rectangular tube with wall thickness [latex]\\delta[\/latex] carrying direct current <span class=\"citation\" data-cites=\"Clayton_MTL\">[@Clayton_MTL]<\/span><\/p>\n<p>[latex]R_{HF}(f) = \\frac{1}{2 \\sigma \\delta(f) (w+t)}  \\label{R_HF_wt}\\tag{4.14}[\/latex]<\/p>\n<figure><img src=\"Propagation\/Figures\/Rectangle_wire.png\" id=\"Rectangle\" alt=\"\" \/><\/p>\n<figure id=\"attachment_169\" aria-describedby=\"caption-attachment-169\" style=\"width: 680px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-169\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-7.png\" width=\"680\" height=\"316\" alt=\"\" \/><figcaption id=\"caption-attachment-169\" class=\"wp-caption-text\">Figure 4.8: Frequency dependence of impedance of conductor with rectangular cross section.<\/figcaption><\/figure>\n<\/figure>\n<p>Prime to derivation of static and low frequency inductance of the conductor, distribution of magnetic field inside the conductor must be considered. If in the whole cross section uniformly distributed current [latex]I[\/latex] is driven, then circulation of magnetic field along the rectangular circumference shown with dashed line in Fig.<a href=\"#Circulation\" data-reference-type=\"ref\" data-reference=\"Circulation\">4.9<\/a> yields<\/p>\n<p>[latex]2\\cdot H_x(y) \\cdot 2x + 2\\cdot H_y(x) \\cdot 2y = \\frac{I}{wt } \\cdot 4xy  \\label{HxHy}\\tag{4.16}[\/latex]<\/p>\n<p>[latex]H_x(y) = \\frac{I}{wt } \\cdot y  \\label{HxHy}\\tag{4.16}[\/latex]<\/p>\n<p>This is valid only if [latex]H_x(y)[\/latex] is constant along the whole length of horizontal sides and contribution of circulation along vertical sides can be neglected. These rough simplification are justified provide [latex]w \\gg t[\/latex].<\/p>\n<figure><img src=\"Propagation\/Figures\/Cyrkulacja.png\" id=\"Circulation\" alt=\"\" \/><\/p>\n<figure id=\"attachment_170\" aria-describedby=\"caption-attachment-170\" style=\"width: 492px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-170\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-8.png\" width=\"492\" height=\"202\" alt=\"\" \/><figcaption id=\"caption-attachment-170\" class=\"wp-caption-text\">Figure 4.9: [latex]\\stackrel{\\rightarrow}{H}(x,y)[\/latex] circulation inside the conductor with rectangular cross section.<\/figcaption><\/figure>\n<\/figure>\n<p>Internal inductance can be directly computed from energy relation by equating magnetic energy stored inside the conductor expressed with magnetic field strength [latex]\\stackrel{\\rightarrow}{H}(x,y)[\/latex] to the same energy expressed with the internal inductance [latex]L_{DC}[\/latex]<\/p>\n<p>[latex]\\frac{1}{2} L_{DC} I^2 = \\frac{\\mu}{2} \\int_{S} |H(x,y)|^2 dS = \\frac{\\mu}{2} \\cdot \\frac{I^2}{w^2 t^2} \\int_{-\\frac{w}{2}}^{-\\frac{w}{2}}dy  \\int_{-\\frac{t}{2}}^{-\\frac{t}{2}}x^2dx  \\label{Hx}\\tag{4.17}[\/latex]<\/p>\n<p>Finally<\/p>\n<p>[latex]L_{DC} = \\frac{\\mu}{12 } \\cdot {t}{w}  \\label{L_DC}\\tag{4.18}[\/latex]<\/p>\n<p>Breaking frequency [latex]f_0[\/latex] can be established as by the round wire with equating LF resistance and reactance<\/p>\n<p>[latex]f_0 = \\frac{6}{\\pi \\mu \\sigma t^2}  \\label{f_0_xy}\\tag{4.19}[\/latex]<\/p>\n<p>Frequency dependence of the internal impedance of conductor with rectangle cross section in double logarithmic scale is illustrated in Fig.<a href=\"#Rectangle\" data-reference-type=\"ref\" data-reference=\"Rectangle\">4.8<\/a>b). Qualitatively it is identical with behavior of the round wire shown in Fig.<a href=\"#Round_wire\" data-reference-type=\"ref\" data-reference=\"Round_wire\">4.6<\/a>b).<\/p>\n<figure><img src=\"Propagation\/Figures\/Current_wt_mod.png\" id=\"J_wt\" alt=\"\" \/><\/p>\n<figure id=\"attachment_171\" aria-describedby=\"caption-attachment-171\" style=\"width: 753px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-171\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-9.png\" width=\"753\" height=\"587\" alt=\"\" \/><figcaption id=\"caption-attachment-171\" class=\"wp-caption-text\">Figure 4.10: Distribution of current density in the rectangular conductor with width [latex]w = 381\\mu m[\/latex] and thickness [latex]t = 35.56 \\mu m[\/latex].<\/figcaption><\/figure>\n<\/figure>\n<p>An example of distribution of current density in the rectangular conductor in frequency dependence, sourced from <span class=\"citation\" data-cites=\"Clayton_MTL\">[@Clayton_MTL]<\/span> is shown in Fig.<a href=\"#J_wt\" data-reference-type=\"ref\" data-reference=\"J_wt\">4.10<\/a>. Features notable in it are summarized below:<\/p>\n<ul>\n<li>current crowding toward the outer edge is remarkable only in direction for which the skin depth [latex]\\delta(f)[\/latex] is much smaller than the side of the rectangle,<\/li>\n<li>current which is the integral of current density [latex]J(x,y)[\/latex] over the cross section surface is the smaller the bigger frequency is, because impedance of conductor rises,<\/li>\n<li>proportionality of the skin depth [latex]\\delta(f)[\/latex] to the square root of frequency [latex]\\sqrt f[\/latex] can be confirmed.<\/li>\n<\/ul>\n<h3 id=\"precedence-of-rectangular-conductor-over-round-wire\">Precedence of rectangular conductor over round wire<\/h3>\n<p>It is desired to compare impedance [latex]Z_{GND}[\/latex] of the common path for different shape of the cross section because as presented in previous sections it is shape dependent.<\/p>\n<p>Breaking frequency is the upper bound of the frequency by which internal impedance is kept relatively deep i.e.\u00a0impedance is only slightly bigger than DC resistance. Comparison of Eq.(4.12) and Eq.(4.19) shows unequivocally that [latex]f_0[\/latex] is much bigger for conductor with rectangle cross section. Indeed in case of rectangle cross section the number in numerator is bigger than by round wire, 6 instead of 4 but crucial is thickness in square in denominator which by [latex]w \\gg t[\/latex] is much smaller than square of radius [latex]r[\/latex] in case of round wire. By comparison of HF litz-wire and rectangle conductor the break frequencies can be similar but costs of HF litz-wire exceeds pretty costs of ordinary solid conductor.<\/p>\n<p>This rationalises precedence of rectangular conductor over round wire. The wider conductor with rectangle cross section by unchanged thickness the smaller low frequency resistance [latex]R_{LF}[\/latex]. Therefore as shown in Fig.<span>[Unsym_line]<\/span>c) by double layers\u2019 PCBs the return path are as wide as practical and by multilayers\u2019 PCBs separate layer or even more layers are dedicated to the return paths.<\/p>\n<h2 id=\"LC_coupling\">Electric (capacitive) and magnetic (inductive) coupling<\/h2>\n<h3 id=\"capacitive-coupling\">Capacitive coupling<\/h3>\n<figure><img src=\"Propagation\/Figures\/Capacitive_coupling.png\" id=\"Coupling_C\" alt=\"\" \/><\/p>\n<figure id=\"attachment_172\" aria-describedby=\"caption-attachment-172\" style=\"width: 696px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-172\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-10.png\" width=\"696\" height=\"274\" alt=\"\" \/><figcaption id=\"caption-attachment-172\" class=\"wp-caption-text\">Figure 4.11: Illustration of capacitive coupling by unsymmetrical transmission: cross sectional view a), top view b).<\/figcaption><\/figure>\n<\/figure>\n<p>Total voltage across load conductance [latex]G_{L2}[\/latex] yields<\/p>\n<p>[latex]U_{L2_{total}} = \\frac{ G_{S2}(G_{L1} + G_{S1} + Y_1 + Y_{12}) \\cdot EMF_2 + G_{S1} Y_{12} \\cdot EMF_1}  {  (G_{L1}+G_{S1} + Y_1) (G_{L2}+G_{S2} + Y_2) +  Y_{12} (G_{L1}+G_{L2} + G_{S1} + G_{S2} +Y_1 + Y_2)  }  \\label{U_2C_tot}\\tag{4.20}[\/latex]<\/p>\n<p>where [latex]Y = j \\omega C[\/latex] is admittance of a capacitor<\/p>\n<p>Electromotive force [latex]EMF_1[\/latex] contributes to the voltage [latex]U_{L2}[\/latex] across the load conductance [latex]G_{L2}[\/latex] of mesh 2 as follows<\/p>\n<p>[latex]U_{L2} = \\frac{ G_{S1} Y_{12} \\cdot EMF_1}  {  (G_{L1}+G_{S1} + Y_1) (G_{L2}+G_{S2} + Y_2) +  Y_{12} (G_{L1}+G_{L2} + G_{S1} + G_{S2} +Y_1 + Y_2)  }  \\label{U_2C}\\tag{4.21}[\/latex]<\/p>\n<p>Obviously, coupled voltage [latex]U_{L2}[\/latex] would be zero by zero mutual capacitance [latex]C_{12}[\/latex], [latex]Y_{12}=0 \\frac{1}{\\Omega}[\/latex]. That is unrealistic. There exist always capacitance between two metallic objects as signal or power lines. Even replacement of electromotive forces [latex]EMF_1[\/latex] and [latex]EMF_2[\/latex] with current sources does not liberate from capacitive coupling.<\/p>\n<h3 id=\"inductive-coupling\">Inductive coupling<\/h3>\n<figure>\n<figure id=\"attachment_173\" aria-describedby=\"caption-attachment-173\" style=\"width: 642px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-173\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-11.png\" width=\"642\" height=\"285\" alt=\"\" \/><figcaption id=\"caption-attachment-173\" class=\"wp-caption-text\">Figure 4.12: Illustration of inductive coupling by unsymmetrical transmission: cross sectional view a), top view b).<\/figcaption><\/figure>\n<\/figure>\n<p>Total voltage across load resistance [latex]R_{L2}[\/latex] yields<\/p>\n<p>[latex]U_{L2_{total}} = \\frac{R_{L2}}{R_{L2}+ R_{S2} + Z_2+Z_{12}}  \\left[EMF_2 -  \\frac{  \\frac{EMF_1}{R_{L1} + R_{S1} + Z_1+Z_{12}} + \\frac{EMF_2}{R_{L2} + R_{S2} + Z_2+Z_{12}}}  {\\frac{1}{R_{L1} + R_{S1} + Z_1+Z_{12}} + {\\frac{1}{R_{L2} + R_{S2} + Z_2+Z_{12}} - \\frac{1}{Z_{12}} }}  \\right]  \\label{U_2L_tot}\\tag{4.22}[\/latex]<\/p>\n<p>where [latex]Z_1 = j \\omega L_1[\/latex] and [latex]Z_2 = j \\omega L_2[\/latex] are impedances of self inductances [latex]L_1[\/latex] and [latex]L_2[\/latex] respectively. [latex]Z_{12} = j \\omega M_{12}[\/latex] is impedance of a mutual inductance [latex]M_{12}[\/latex]. It depends on self inductances as follows [latex]M_{12} = k \\sqrt{L_1 L_2}[\/latex] where [latex]0Dependence of capacitance and external inductance on cross section\u2019s shape  <\/p>\n<figure><img src=\"Propagation\/Figures\/L_ext.png\" id=\"LC_ext\" alt=\"\" \/>    [caption id=\"attachment_174\" align=\"aligncenter\" width=\"633\"]<img class=\"wp-image-174\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-12.png\" width=\"633\" height=\"172\" alt=\"\" \/> Figure 4.13: Illustration for consideration of capacitance and external inductance.[\/caption]<\/figure>\n<p>  Capacitance per unit length of a infinitely long strait wire with circular cross section, layouted in air parallel to infinite conducting plane, as shown in Fig.<a href=\"#LC_ext\" data-reference-type=\"ref\" data-reference=\"LC_ext\">4.13<\/a>a) yields <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>    [latex]C_{\\bigcirc} = \\frac{2\\pi\\varepsilon_0}{\\operatorname{ar\\,cosh} \\left(\\frac{ h}{r} \\right)}  \\label{C_r}\\tag{4.24}[\/latex]<\/p>\n<p>where [latex]\\operatorname{ar\\,cosh} (x) = \\ln \\left[ \\sqrt{x^2-1} + x \\right] \\approx \\ln(2x)[\/latex] for [latex]x \\gg 1[\/latex]. Therefore for a wire that is sufficiently far from the conducting plane, [latex]h \\gg r[\/latex] this simplifies to<\/p>\n<p>[latex]\\begin{array} {ll}  C_{\\bigcirc} \\approx \\frac{2\\pi\\varepsilon_0}{\\ln \\left(\\frac{ 2h}{r} \\right)} & \\text{for}~~h \\gg r   \\end{array}  \\label{C_r_approx}\\tag{4.25}[\/latex]<\/p>\n<p>As explained in chapter <a href=\"#El%20size\" data-reference-type=\"ref\" data-reference=\"El size\">[El size]<\/a>, wave propagates along a transmission line if it is electrically long. Velocity of propagation is the quantity that binds parameters [latex]\\mu[\/latex] and [latex]\\varepsilon[\/latex] of the surrounding medium with per length parameters [latex]L[\/latex] and [latex]C[\/latex] of the transmission line. On one side it is given as in Eq.(<a href=\"#v_propagation\" data-reference-type=\"ref\" data-reference=\"v_propagation\">[v_propagation]<\/a>), on the other side as [latex]v = \\frac{1}{\\sqrt{LC}}[\/latex]. Therefore by homogeneous surrounding of transmission line holds<\/p>\n<p>[latex]L \\: C = \\mu \\: \\varepsilon  \\label{LC}\\tag{4.26}[\/latex]<\/p>\n<p>This straightforwardly leads to formulas for external inductance of the round wire<\/p>\n<p>[latex]L_{\\bigcirc}^{EXT} = \\frac{\\mu_0}{2\\pi} \\operatorname{ar\\,cosh} \\left(\\frac{ h}{r} \\right)  \\label{L_r}\\tag{4.27}[\/latex]<\/p>\n<p>[latex]\\begin{array} {ll}  L_{\\bigcirc}^{EXT} \\approx \\frac{\\mu_0}{2\\pi} \\ln{\\left (\\frac{2h}{r} \\right)} & \\text{for}~~h \\gg r   \\end{array}  \\label{L_r_approx}\\tag{4.28}[\/latex]<\/p>\n<p>There exist no general analytical formula for parameters of conductor with rectangular cross section. Cited here is approximate relation for the case shown in Fig.<a href=\"#LC_ext\" data-reference-type=\"ref\" data-reference=\"LC_ext\">4.13<\/a>b), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span>, assuming zero thickness of conductor<\/p>\n<p>[latex]\\begin{array} {ll}  C_{\\framebox[0.15in]{}} \\approx \\varepsilon_0 \\left[ \\frac{w}{h} + 1.393 + 0.667 \\ln{\\left (\\frac{w}{h} + 1.444 \\right)} \\right] &  \\text{for}~~w \\geq h  \\end{array}  \\label{C_rect}\\tag{4.29}[\/latex]<\/p>\n<p>[latex]\\begin{array} {ll}  L_{\\framebox[0.15in]{}}^{EXT} \\approx \\frac{ \\mu_0}{ \\frac{w}{h} + 1.393 + 0.667 \\ln{\\left (\\frac{w}{h} + 1.444 \\right)}} &  \\text{for}~~w \\geq h   \\end{array}  \\label{L_rect}\\tag{4.30}[\/latex]<\/p>\n<p>Capacitance of the round wire with [latex]2.5mm^2[\/latex] cross section area i.e. with radius [latex]=0.892mm[\/latex] layouted [latex]h=1cm[\/latex] above ground plane, according to Eq.(4.25) amounts to [latex]C_{\\bigcirc}=17.89pF\/m[\/latex]. Rectangular conductor with the same cross section area and longer size [latex]w=2cm[\/latex] has shorter size [latex]t=0.125mm[\/latex]<a href=\"#fn2\" class=\"footnote-ref\" id=\"fnref2\" role=\"doc-noteref\"><sup>2<\/sup><\/a>. According to Eq.(4.29) its capacitance is [latex]C_{\\framebox[0.15in]{}}=37.35 pF\/m[\/latex].<\/p>\n<p>Analogue is with external inductance. By the same geometry relations, according to Eq.(4.28) external inductance of the round wire amounts to [latex]L_{\\bigcirc}^{EXT}=0.622 \\mu H\/m[\/latex] and by rectangular conductor is [latex]L_{\\framebox[0.15in]{}}^{EXT}=0.298 \\mu H\/m[\/latex], according to Eq.(4.30).<\/p>\n<p>In numerical example above capacitance\/external inductance is smaller\/ bigger for a round wire\/rectangular conductor. This example is realistic and the relation can be generalized. Rationale of it is rooted in the Euclidean geometry. It is taught there, that from all plane figures, the circle has the smallest ratio of circumference to surface area.<\/p>\n<p>Let us compare capacitances. The Gauss\u2019s flux theorem is a law relating the distribution of electric field to charges originating it. The electric flux through any hypothetical closed surface [latex]S[\/latex] is equal to the net electric charge [latex]Q[\/latex] within that closed surface [latex]\\varepsilon_0 \\oiint_S \\overrightarrow{E}\\cdot \\overrightarrow{dS} = \\sum Q[\/latex]. For infinite strait conductor the electric flux can be calculated through the surface per unit length and therefore it is reduced to the integral along arbitrary closed loop enclosing the conductor.<\/p>\n<p>Let us compare round wire and rectangular conductor with the same surface area, layouted on the same height [latex]h[\/latex] above the ground plane. For round wire the integration path is shorter than for rectangular conductor, due to ratio of circumference to area. Therefore if in both cases net charge is the same, electric field strength distributed around rectangular conductor is smaller.<\/p>\n<p>Voltage across conductor and the ground plane is given by the line integral of electric field along arbitrary path between them [latex]U = \\int_l \\overrightarrow{E}\\cdot \\overrightarrow{dl}[\/latex]. Consequently it is smaller in case of rectangular conductor. Capacitance is ratio of free charge on the conductor or the ground plane to voltage necessary for gathering this amount of charge [latex]C=Q\/U[\/latex]. Conclusion is that, for gathering particular net charge less voltage is needed in case of rectangular conductor than of round wire. Finally, capacitance of rectangular conductor is bigger than of round wire.<\/p>\n<p>Similarly can be proceeded by comparing external inductances. The Ampere\u2019s circular low relates the distribution of magnetic field strength to current causing it. The line integral of magnetic field strength [latex]\\overrightarrow{H}[\/latex] along any hypothetical closed loop [latex]l[\/latex] is equal to the net current [latex]I[\/latex] enclosed in this loop [latex]\\oint_l \\overrightarrow{H}\\cdot \\overrightarrow{dl} = \\sum I[\/latex].<\/p>\n<p>Let us compare round wire and rectangular conductor with the same surface area, layouted on the same height [latex]h[\/latex] above the ground plane. For round wire the integration path is shorter than for rectangular conductor, due to ratio of circumference to area. Therefore if in both cases enclosed net current is the same, magnetic field strength distributed around rectangular conductor is smaller.<\/p>\n<p>Magnetic flux through any hypothetical surface built by the loop driving current [latex]I[\/latex] is given by [latex]\\Phi = \\mu_0 \\oiint_S \\overrightarrow{H}\\cdot \\overrightarrow{dS}[\/latex]. For infinite loop composed of round wire or rectangular conductor, load in infinity and the ground plane as return path, only flux per unit length makes sense. It is smaller in case of rectangular conductor due to smaller field strength. Inductance is ratio of magnetic flux to current necessary for generating it [latex]L=\\Phi\/I[\/latex]. It means that, by driving the same particular net current less flux is generated in case of rectangular conductor than of round wire. Finally, inductance of rectangular conductor is smaller than of round wire.<\/p>\n<p>This statement can be rationalised alternatively starting with comparison of capacitances along with conclusion from Eq.(4.26).<\/p>\n<p>Notice, that capacitance\/external inductance of the round wire rises\/decays slower with increased radius [latex]r[\/latex] than capacitance\/external inductance of the rectangular conductor with its width [latex]w[\/latex]. In the first case the change is compressed because radius is argument of natural logarithm. In the second case, dependence on width [latex]w[\/latex] is direct.<\/p>\n<h4 id=\"capacitive-and-inductive-coupling-by-symmetrical-transmission\">Capacitive and inductive coupling by symmetrical transmission<\/h4>\n<p>Up to now only unsymmetrical transmission was covered. The only message concerning symmetrical transmission discussed here is the size of zone surrounding transmission line in which risk of capacitive or inductive coupling exist.<\/p>\n<figure><img src=\"Propagation\/Figures\/C_unsym_sym_round.png\" id=\"C_unsym_sym_round\" alt=\"\" \/><\/p>\n<figure id=\"attachment_175\" aria-describedby=\"caption-attachment-175\" style=\"width: 724px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-175\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-13.png\" width=\"724\" height=\"234\" alt=\"\" \/><figcaption id=\"caption-attachment-175\" class=\"wp-caption-text\">Figure 4.14: Module of electric field strength around unsymmeric a) and symmetric b) transmission line with round wire.<\/figcaption><\/figure>\n<\/figure>\n<p>In Fig.<a href=\"#C_unsym_sym_round\" data-reference-type=\"ref\" data-reference=\"C_unsym_sym_round\">4.14<\/a> distribution of module of electric field strength around unsymmeric and symmetric transmission line with round wire is illustrated. The same voltage is applied and spectrum of colours has the same scale in both cases.<\/p>\n<p>Evidently the zone polluted with the electric field in case of symmetrical transmission is smaller. Module of electric field strength by symmetrical line decays stronger versus distance from the line in horizontal as well vertical direction. Rationale for it is the fact that space between feeding and return line in case of symmetrical transmission is much smaller than in case of unsymmetrical transmission. Fields outside the line cancel one another. As a consequence of it, zone with practically total cancellation of field is closer to the line in the case of symmetrical transmission.<\/p>\n<p>The same can be concluded for unsymmetrical and symmetrical lines build of conductors with rectangular cross sections as shown in Fig.<a href=\"#C_unsym_sym_rectangle\" data-reference-type=\"ref\" data-reference=\"C_unsym_sym_rectangle\">4.15<\/a>.<\/p>\n<p>In Fig.<a href=\"#L_unsym_sym_round\" data-reference-type=\"ref\" data-reference=\"L_unsym_sym_round\">4.16<\/a> distribution of module of magnetic field strength around unsymmeric and symmetric transmission line with round wire is illustrated. The same current is driven and spectrum of colours has the same scale in both cases.<\/p>\n<p>Evidently the zone polluted with the magnetic field in case of symmetrical transmission is smaller.<\/p>\n<figure><img src=\"Propagation\/Figures\/C_unsym_sym_rectangle.png\" id=\"C_unsym_sym_rectangle\" alt=\"\" \/><\/p>\n<figure id=\"attachment_176\" aria-describedby=\"caption-attachment-176\" style=\"width: 800px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-176\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-14.png\" width=\"800\" height=\"284\" alt=\"\" \/><figcaption id=\"caption-attachment-176\" class=\"wp-caption-text\">Figure 4.15: Module of electric field strength around unsymmeric a) and symmetric b) transmission line with conductor having rectangle cross section.<\/figcaption><\/figure>\n<\/figure>\n<figure><img src=\"Propagation\/Figures\/L_unsym_sym_round.png\" id=\"L_unsym_sym_round\" alt=\"\" \/><\/p>\n<figure id=\"attachment_177\" aria-describedby=\"caption-attachment-177\" style=\"width: 823px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-177\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-15.png\" width=\"823\" height=\"267\" alt=\"\" \/><figcaption id=\"caption-attachment-177\" class=\"wp-caption-text\">Figure 4.16: Module of magnetic field strength around unsymmeric a) and symmetric b) transmission line with round wire.<\/figcaption><\/figure>\n<\/figure>\n<figure><img src=\"Propagation\/Figures\/L_unsym_sym_rectangle.png\" id=\"L_unsym_sym_rectangle\" alt=\"\" \/><\/p>\n<figure id=\"attachment_178\" aria-describedby=\"caption-attachment-178\" style=\"width: 818px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-178\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-16.png\" width=\"818\" height=\"257\" alt=\"\" \/><figcaption id=\"caption-attachment-178\" class=\"wp-caption-text\">Figure 4.17: Module of magnetic field strength around unsymmeric a) and symmetric b) transmission line with conductor having rectangle cross section.<\/figcaption><\/figure><figcaption><\/figcaption><\/figure>\n<p>The same can be concluded for unsymmetrical and symmetrical lines build of conductors with rectangular cross sections as shown in Fig.<a href=\"#L_unsym_sym_rectangle\" data-reference-type=\"ref\" data-reference=\"L_unsym_sym_rectangle\">4.17<\/a>.<\/p>\n<h3 id=\"TCL_chapter\">Transverse to longitudinal conversion<\/h3>\n<p>Equivalent scheme of symmetric transmission is shown in Fig.<a href=\"#TCL_Fig\" data-reference-type=\"ref\" data-reference=\"TCL_Fig\">4.18<\/a>. Energy or signal should be delivered from the source [latex]EMF[\/latex] with internal resistance [latex]R_S[\/latex] to the load [latex]R_L[\/latex] via symmetrical line. This line is layouted above the ground. Each line i.e.\u00a0feeding: 1-3 and return: 2-4 has parasitic capacitance related to the ground. They are represented with capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] by the source and [latex]C_{35}[\/latex] and [latex]C_{45}[\/latex] by the load. There are also parasitic capacitances of one to another line. They are represented with capacitances [latex]C_{12}[\/latex] by the source and [latex]C_{34}[\/latex] by the load. Moreover each line builds inductance represented with [latex]L_{13}[\/latex] and [latex]L_{24}[\/latex].<\/p>\n<p>Symmetric transmission means that parasitic parameters are in equilibrium consisted in following identities [latex]\\begin{array} {rcl}  C_{10} = C_{20} \\nonumber \\\\  C_{35} = C_{45} \\nonumber \\\\  L_{13} = L_{24} \\nonumber  \\end{array}[\/latex]<\/p>\n<p>In such situation [latex]U_{50} = 0[\/latex]. Transmission line is perfectly balanced.<\/p>\n<figure><img src=\"Propagation\/Figures\/TCL.png\" id=\"TCL_Fig\" alt=\"\" \/><\/p>\n<figure id=\"attachment_179\" aria-describedby=\"caption-attachment-179\" style=\"width: 568px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-179\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-17.png\" width=\"568\" height=\"331\" alt=\"\" \/><figcaption id=\"caption-attachment-179\" class=\"wp-caption-text\">Figure 4.18: Symmetrical transmission over the ground plane, top view.<\/figcaption><\/figure>\n<\/figure>\n<p>Electromotive force [latex]EMF[\/latex], which is oriented crosswise to the direction of transmission generates voltage [latex]U_{50}[\/latex] along direction of transmission only if the circuit is out balance.<\/p>\n<p>Ratio of transverse electromotive force [latex]EMF[\/latex] to longitudinal voltage [latex]U_{50}[\/latex]<\/p>\n<p>[latex]\\begin{array} {rcl}  L_{TC} & = & \\frac{EMF}{U_{50}} \\\\  L_{TC(dB)} & = & 20 \\log \\left( \\frac{EMF}{U_{50}} \\right)  \\end{array}  \\label{TCL}\\tag{4.35}[\/latex]<\/p>\n<p>is called loss of transverse (to longitudinal) conversion, alternatively Transverse Conversion Loss TCL. It is used for rating deviation from the ideal balance by symmetrical transmission. By approaching perfect equilibrium, [latex]L_{TC}[\/latex] tends to infinity.<\/p>\n<p>There are two reasons for transverse conversion to be undesired: deterioration of integrity of transmitted signal and enlargement of the zone around the line polluted with the electromagnetic field, similarly as by unsymmetrical transmission.<\/p>\n<h4 id=\"decomposition-of-currents-into-common-and-differential-mode\">Decomposition of currents into common and differential mode<\/h4>\n<p>In this subsection currents in the transmission line shown in Fig.<a href=\"#TCL_Fig\" data-reference-type=\"ref\" data-reference=\"TCL_Fig\">4.18<\/a> will be derived. For suppressing complexity of the formulas, source resistance [latex]R_S[\/latex] will be omitted.<\/p>\n<p>Voltage [latex]U_{50}[\/latex] unequal to zero means current [latex]I_{50}[\/latex] driven through the ground impedance [latex]Z_{GND}[\/latex]. It can be calculated with the Thevenin\u2019s Theorem. Open circuit voltage [latex]U_{50}^{Th}[\/latex] by removing branch [latex]Z_{GND}[\/latex] yields<\/p>\n<p>[latex]U_{50}^{Th} = \\frac{Y_{13} \\left[Y_{34}(Y_{35} + Y_{45}) + Y_{35}(Y_{24} + Y_{45})\\right] \\cdot EMF}  {(Y_{13} + Y_{24})\\left[Y_{34}(Y_{35} + Y_{45}) + Y_{35} Y_{45} \\right] + Y_{13} Y_{24}(Y_{35} + Y_{45})} - \\frac{Y_{10}\\cdot EMF}{Y_{10} + Y_{20}}  \\label{U50}\\tag{4.36}[\/latex]<\/p>\n<p>where [latex]Y_{34} = \\frac{R_L + \\frac{1}{j\\omega C_{34}}}{R_L \\frac{1}{j\\omega C_{34}}}[\/latex]<\/p>\n<p>For derivation of the Thevenin\u2019s impedance seen between poles 5-0, transposition of triangle 3-4-5 to the star with the star node 0\u2019 as shown in Fig.<a href=\"#TCL_Z_Th\" data-reference-type=\"ref\" data-reference=\"TCL_Z_Th\">4.19<\/a> must be performed<\/p>\n<p>[latex]\\begin{array} {rcl}  Z_{30'} = \\frac{Z_{34} Z_{35}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber \\\\  \\nonumber \\\\  Z_{40'} = \\frac{Z_{34} Z_{45}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber \\\\  \\nonumber \\\\  Z_{50'} = \\frac{Z_{35} Z_{45}}{Z_{34} + Z_{35} + Z_{45}} \\nonumber  \\end{array}[\/latex]<\/p>\n<figure><img src=\"Propagation\/Figures\/TCL_Z_Th.png\" id=\"TCL_Z_Th\" alt=\"\" \/><\/p>\n<figure id=\"attachment_180\" aria-describedby=\"caption-attachment-180\" style=\"width: 594px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-180\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-18.png\" width=\"594\" height=\"314\" alt=\"\" \/><figcaption id=\"caption-attachment-180\" class=\"wp-caption-text\">Figure 4.19: Circuit for calculation of the Thevenin\u2019s impedance seen between poles 0-5.<\/figcaption><\/figure>\n<\/figure>\n<p>Current [latex]I_{50}[\/latex] driven through the ground impedance [latex]Z_{GND}[\/latex] is called common mode current [latex]I^{CM}[\/latex]. It yields<\/p>\n<p>[latex]I^{CM} = I_{50} = \\frac{U_{50}^{Th}} {Z_{50}^{Th} + Z_{GND}}  \\label{I50}\\tag{4.38}[\/latex]<\/p>\n<p>In node 0 it is split to current [latex]I_{13}^{CM}[\/latex] and [latex]I_{24}^{CM}[\/latex] driven in line 1-3 and 2-4 respectively. Ratio of currents [latex]I_{13}^{CM}[\/latex] and [latex]I_{24}^{CM}[\/latex] depends on relation between impedances of paths 0-1-3-5 and 0-2-4-5.<\/p>\n<p>Along with the common mode component the differential mode current [latex]I^{DM}[\/latex] is driven in the feeding and return line.<\/p>\n<p>It can be calculated after removing branch with the ground impedance [latex]Z_{GND}[\/latex]<\/p>\n<p>[latex]I^{DM} = I_{50} = \\frac{EMF} {Z_{13} + Z_{24} + \\frac{(Z_{35} + Z_{45}) Z_{34}}{Z_{35} + Z_{45} + Z_{34}}}  \\label{IDM}\\tag{4.39}[\/latex]<\/p>\n<p>Finally actual currents in the transmission lines expressed with the common and differential components yields<\/p>\n<p>[latex]\\begin{array} {rcl}  I_{13} & = & I^{DM} + I_{13}^{CM} \\\\  \\\\  I_{42} & = & I^{DM} - I_{24}^{CM}  \\end{array}[\/latex]<\/p>\n<p>Calculation of capacitances [latex]C_{10}[\/latex], [latex]C_{20}[\/latex] and [latex]C_{12}[\/latex] as well [latex]C_{35}[\/latex], [latex]C_{45}[\/latex] and [latex]C_{34}[\/latex] can be done only numerically because both triple of capacitors are linked. Any change of one of them causes changes of the rest.<\/p>\n<p>The same concerns inductances. Notice that [latex]L_{13}[\/latex] is serial connection of two inductances because line 1-3 is part of the loop 1-3-5-0 and 1-3-4-2.<\/p>\n<h2 id=\"wave_coupling\">Wave coupling<\/h2>\n<h3 id=\"radiation\">Radiated waves<\/h3>\n<h4 id=\"idealized-entities-as-elementary-radiators\">Idealized entities as elementary radiators<\/h4>\n<ul>\n<li><em>Isotropic antenna.<\/em> It is omni directional radiator. In other words it radiates the same power in all directions.<\/li>\n<li><em>Electric (Hertzian) dipole.<\/em> Imagine electrically short, infinitesimally thin strait conducting segment carrying a current represented with the phasor [latex]\\bf{I}[\/latex] that is assumed to be constant (as to magnitude and phase) at all points along the segment. If the segment length [latex]l[\/latex] tends to zero whereas the current [latex]\\bf{I}[\/latex] infinitely growths so that the quantity [latex]{\\bf{p}} = l{\\bf{I}}[\/latex] remains finite and constant, then this product constitutes magnitude of the vector called the dipole moment. Its direction is along the segment and the sense according to the current direction [latex]\\overrightarrow{1}_l[\/latex]\n<p>[latex]\\overrightarrow{\\bf{p}} = \\lim \\limits_{l \\to 0, I \\to \\infty} (l{\\bf{I}}) \\overrightarrow{1}_l  \\label{p_dipole}\\tag{4.41}[\/latex]<\/li>\n<li><em>Magnetic (Fitzgeraldian) dipole.<\/em> Imagine electrically small, infinitesimally thin ring with azimuthal current [latex]\\bf{I}[\/latex] flowing in it which does not depend on the angle. If the ring radius [latex]a[\/latex] tends to zero whereas the current [latex]\\bf{I}[\/latex] infinitely growths so that the product of the ring area and current [latex]{\\bf{m}} = \\pi a^2 {\\bf{I}}[\/latex] remains finite and constant, then this product constitutes magnitude of the vector called the dipole moment. Its direction is orthogonal to the plane of the ring and the sense results from the vector product of unit radius vector [latex]\\overrightarrow{1}_r[\/latex] and unit current density vector [latex]\\overrightarrow{1}_j[\/latex]\n<p>[latex]\\overrightarrow{\\bf{m}} = \\lim \\limits_{  \\pi a^2 \\to 0, I \\to \\infty} (\\pi a^2 {\\bf{I}}) \\overrightarrow{1}_r \\times \\overrightarrow{1}_j  \\label{m_dipole}\\tag{4.42}[\/latex]<\/li>\n<\/ul>\n<p>All these entities are lossless. None of them exist in reality but they are useful in understanding the antenna theory including unintentional antennas such as cables connected to the EUT.<\/p>\n<figure><img src=\"Propagation\/Figures\/xyz.png\" id=\"xyz\" alt=\"\" \/><\/p>\n<figure id=\"attachment_181\" aria-describedby=\"caption-attachment-181\" style=\"width: 709px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-181\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-19.png\" width=\"709\" height=\"283\" alt=\"\" \/><figcaption id=\"caption-attachment-181\" class=\"wp-caption-text\">Figure 4.20: Cartesian orthogonal 3D system of coordinates a), electric dipole with only [latex]{\\bf{p}}_z[\/latex] component of dipole moment b), magnetic dipole with only [latex]{\\bf{m}}_z[\/latex] component of dipole moment.<\/figcaption><\/figure>\n<\/figure>\n<p>The best way of describing the 3D fields generated by dipoles is the spherical system of co-ordinates in which the spacial position of the point under interest is described with three following numbers: [latex]r, \\theta, \\varphi[\/latex], as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>a). The radius [latex]r[\/latex] is distance from the origin of the co-ordinates system, [latex]\\theta[\/latex] elevation angle between the [latex]z[\/latex] axis and radius [latex]r[\/latex], [latex]\\varphi[\/latex] is the azimuthal angle between [latex]x[\/latex] axis and projection of the radius [latex]r[\/latex] on the [latex]x0y[\/latex] plane.<\/p>\n<p>Components of the electric field strength of the electric dipole oriented as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>b), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span> are as follows<\/p>\n<p>[latex]{\\bf{E}}_r (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0^2} {2 \\pi} \\cos{(\\theta)}  \\left(  \\frac{1}{\\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}  \\right) e^{-j\\beta_0 r}  \\label{eE_r} \\tag{4.43}[\/latex]<\/p>\n<p>[latex]{\\bf{E}}_\\theta (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0^2} {4 \\pi} \\sin{(\\theta)}  \\left(  j \\frac{1}{\\beta_0 r} + \\frac{1} { \\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}  \\right) e^{-j\\beta_0 r}  \\label{eE_theta} \\tag{4.44}[\/latex]<\/p>\n<p>[latex]{\\bf{E_\\varphi}}= 0  \\label{eE_phi} \\tag{4.45}[\/latex]<\/p>\n<p>and of magnetic field strength<\/p>\n<p>[latex]{\\bf{H}}_r= 0  \\label{eH_r} \\tag{4.46}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\theta = 0  \\label{eH_theta} \\tag{4.47}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\varphi(r,\\theta)= \\frac{{\\bf{p}}_z \\beta_0^2} {4 \\pi} \\sin{(\\theta)}  \\left(  j\\frac{1}{\\beta_0 r} + \\frac{1}{\\beta_0^2 r^2}  \\right) e^{-j\\beta_0 r}  \\label{eH_phi} \\tag{4.48}[\/latex]<\/p>\n<p>where [latex]Z_0[\/latex] is the intrinsic impedance of the medium in which the dipole is placed. It depends on the magnetic and dielectric constants [latex]\\mu_0[\/latex] and [latex]\\epsilon_0[\/latex] of the medium as follows<\/p>\n<p>[latex]Z_0 = \\sqrt{\\frac{\\mu_0}{\\epsilon_0}}  \\label{Z_0}\\tag{4.49}[\/latex]<\/p>\n<p>and [latex]\\beta_0[\/latex] is the phase constant of the wave expressed with the formula<\/p>\n<p>[latex]\\beta_0 = \\omega \\sqrt{\\mu_0 \\epsilon_0} = \\frac{2 \\pi}{\\lambda_0}  \\label{beta}\\tag{4.50}[\/latex]<\/p>\n<p>where [latex]\\omega[\/latex] is angular frequency of the propagated wave and [latex]\\lambda_0[\/latex] is the wave length.<\/p>\n<p>Far away from the electric dipole, regardless of direction disappears dependence of electric field on the second and the third power of radius [latex]r[\/latex], refer to Eqs. (4.43) and (4.44) and dependence of [latex]{\\bf{H}}_ {\\varphi}[\/latex] component of magnetic field on the second power of radius [latex]r[\/latex], refer to Eq.\u00a0(4.48). Solely summands reciprocally proportional to the distance [latex]r[\/latex] remains.<\/p>\n<p>In the immediate vicinity of the electric dipole dominant is dependence of electric field on the third power of radius [latex]r[\/latex], refer to Eqs. (4.43), (4.44). Zone within such solid is the near field zone. Magnetic field [latex]{\\bf{H}}_{\\varphi}[\/latex] does not exist in it because it does not possess summand with radius [latex]r[\/latex] in the third power. Emerges only electric field distributed as in static case. Therefore another name of the zone is the static zone.<\/p>\n<p>Between the near field and the far field zone there is the buffer called the reactive zone.<\/p>\n<p>Components of the electric field strength of the magnetic dipole oriented as shown in Fig.<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>c), according to <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span> are as follows<\/p>\n<p>[latex]{\\bf{E}}_r = 0  \\label{mE_r} \\tag{4.51}[\/latex]<\/p>\n<p>[latex]{\\bf{E}}_\\theta = 0  \\label{mE_theta} \\tag{4.52}[\/latex]<\/p>\n<p>[latex]{\\bf{E}}_\\varphi (r,\\theta)=-j \\frac{ {\\bf{m}}_z \\omega \\mu_0 \\beta_0^2 } {4 \\pi} \\sin{(\\theta)}  \\left(  j\\frac{1}{\\beta_0 r} + \\frac{1}{\\beta_0^2 r^2}  \\right) e^{-j\\beta_0 r}  \\label{mE_phi} \\tag{4.53}[\/latex]<\/p>\n<p>and of magnetic field strength<\/p>\n<p>[latex]{\\bf{H}}_r (r,\\theta)= j \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0^2} {2 \\pi Z_0 } \\cos{(\\theta)}  \\left(  \\frac{1}{\\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}  \\right) e^{-j\\beta_0 r}  \\label{mH_r} \\tag{4.54}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\theta (r,\\theta)= j \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0^2 } {4 \\pi Z_0} \\sin{(\\theta)}  \\left(  j \\frac{1}{\\beta_0 r} + \\frac{1} { \\beta_0^2 r^2} -j \\frac{1}{\\beta_0^3 r^3}  \\right) e^{-j\\beta_0 r}  \\label{mH_theta} \\tag{4.55}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\varphi = 0  \\label{mH_phi} \\tag{4.56}[\/latex]<\/p>\n<p>Definition of the far, reactive and near field zones are valid also for the magnetic dipole.<\/p>\n<p>Phasor of total strength of electric and magnetic field in the far field zone of electric dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b) dependent on locus of point yields<\/p>\n<p>[latex]\\label{Phasor_p}  {\\bf{E}}_\\theta (r,\\theta)= \\frac{{\\bf{p}}_z Z_0 \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}  \\cdot \\frac{ e^{j \\left( \\frac{\\pi}{2} - \\beta_0 r \\right)}} {r} \\tag{4.57}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\varphi( r,\\theta)= \\frac{{\\bf{p}}_z \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}  \\cdot \\frac{ e^{j \\left( \\frac{\\pi}{2} - \\beta_0 r \\right)}} {r}  \\label{Phasor_e2} \\tag{4.58}[\/latex]<\/p>\n<p>Phasor of total strength of electric and magnetic field in the far field zone of magnetic dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c) dependent on locus of point yields<\/p>\n<p>[latex]\\label{Phasor_m0}  {\\bf{E}}_\\varphi (\\omega t, r,\\theta)= \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0 } {4 \\pi } \\cdot \\sin{(\\theta)}  \\cdot \\frac{ e^{-j \\beta_0 r}} {r} \\tag{4.59}[\/latex]<\/p>\n<p>[latex]{\\bf{H}}_\\theta(\\omega t, r,\\theta)= - \\frac{{\\bf{m}}_z \\omega \\mu_0 \\beta_0 } {4 \\pi Z_0 } \\cdot \\sin{(\\theta)}  \\cdot \\frac{ e^{-j \\beta_0 r }} {r}  \\label{Phasor_m} \\tag{4.60}[\/latex]<\/p>\n<p>Total strength of electric and magnetic field in the far field zone of electric dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b), dependent on time and locus of point yields<\/p>\n<p>[latex]\\label{Time_e}  E_\\theta (\\omega t, r,\\theta)= \\frac{p_z Z_0 \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}  \\cdot \\sin{\\left( \\omega t + \\frac{\\pi}{2} - \\beta_0 r \\right)} \\tag{4.61}[\/latex]<\/p>\n<p>[latex]H_\\varphi(\\omega t, r,\\theta)= \\frac{p_z \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}  \\cdot \\sin{\\left( \\omega t + \\frac{\\pi}{2} - \\beta_0 r \\right)}  \\label{Time_e1} \\tag{4.62}[\/latex]<\/p>\n<p>Total strength of electric and magnetic field in the far field zone of magnetic dipole, oriented as in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c), dependent on time and locus of point yields<\/p>\n<p>[latex]\\label{Time_m0}  E_\\varphi (\\omega t, r,\\theta)= \\frac{m_z \\omega \\mu_0 \\beta_0 } {4 \\pi } \\cdot \\frac{\\sin{\\theta}}{r}  \\cdot \\sin{\\left( \\omega t - \\beta_0 r \\right)} \\tag{4.63}[\/latex]<\/p>\n<p>[latex]H_\\theta(\\omega t, r,\\theta)= - \\frac{m_z \\omega \\mu_0 \\beta_0 } {4 \\pi Z_0 } \\cdot \\frac{ \\sin{\\theta}}{r}  \\cdot \\sin{\\left( \\omega t - \\beta_0 r \\right)}  \\label{Time_m} \\tag{4.64}[\/latex]<\/p>\n<p>One of features of waves in the far field zone is the wavefront called also equiphase surface. It is the set of points at which field strength has the same phase at a given instant<a href=\"#fn3\" class=\"footnote-ref\" id=\"fnref3\" role=\"doc-noteref\"><sup>3<\/sup><\/a>. Evidently the equiphase surface of the isotropic antenna is the sphere.<\/p>\n<p>The wavefront constraint of the electric dipole means constant phase in Eq.\u00a0(4.61) and in (4.62) [latex]\\omega t + \\frac{\\pi}{2} - \\beta_0 r = const[\/latex] and for magnetic dipole in Eq\u00a0(4.63) and in (4.64) [latex]\\omega t - \\beta_0 r = const[\/latex] . Hence the wavefront of both dipoles must be also the sphere [latex]r=const[\/latex]. The wavefront constraint includes also information about movement direction of the wavefront. In order to fulfill the constraint, radius [latex]r[\/latex] must increase with passage of the time i.e.\u00a0the wavefront moves outwards.<\/p>\n<p>In arbitrary place of the space surrounding any antenna ratio of magnitudes of vectors of phasors of electric and magnetic field [latex]{\\bf{Z}}_w (r,\\theta,\\varphi)= \\frac{|\\overrightarrow{{\\bf{E}}} (r,\\theta, \\varphi)|}  {|\\overrightarrow{{\\bf{H}}} (r,\\theta,\\varphi)|}  \\label{Zw}\\tag{4.65}[\/latex]<\/p>\n<p>is called wave impedance.<\/p>\n<p>Even for the electric and magnetic dipole general formula for the wave impedance are complicated but its constrains to the symmetry plane [latex]x0y[\/latex] is agreeable. In the formulas below the distance [latex]r[\/latex] is scaled with the wavelength [latex]\\lambda[\/latex] transmitted or received by the dipole. [latex]r\/\\lambda[\/latex] can be seen as the \"electrical distance\" which is in line with definition of electrical size presented in chapter <a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a><\/p>\n<p>[latex]\\label{Zw_e}  {\\bf{Z}}_w^{(e)} \\left(\\frac{r}{\\lambda},90^{\\circ},\\varphi \\right)=   Z_0 \\frac{1+j\\frac{4\\pi^2 \\left( \\frac{r}{\\lambda} \\right)^2-1}{2\\pi \\left( \\frac{r}{\\lambda} \\right) }}{1+j 2\\pi \\left( \\frac{r}{\\lambda} \\right)} \\tag{4.66}[\/latex]<\/p>\n<p>[latex]\\label{Zw_m}  {\\bf{Z}}_w^{(m)} \\left( \\frac{r}{\\lambda},90^{\\circ},\\varphi \\right)=   Z_0 \\frac{1+j 2\\pi \\left( \\frac{r}{\\lambda} \\right)}{1+j\\frac{4\\pi^2 \\left( \\frac{r}{\\lambda} \\right)^2-1}{2\\pi \\left( \\frac{r}{\\lambda} \\right)}} \\tag{4.67}[\/latex]<\/p>\n<p>Magnitude and phase angle of the wave impedances formulated in Eqs.\u00a0(4.66) and (4.67) are shown in Fig.\u00a0<a href=\"#Zw_em_Fig\" data-reference-type=\"ref\" data-reference=\"Zw_em_Fig\">4.21<\/a>.<\/p>\n<figure><img src=\"Propagation\/Figures\/Z_w.png\" id=\"Zw_em_Fig\" alt=\"\" \/><\/p>\n<figure id=\"attachment_182\" aria-describedby=\"caption-attachment-182\" style=\"width: 736px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-182\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-20.png\" width=\"736\" height=\"277\" alt=\"\" \/><figcaption id=\"caption-attachment-182\" class=\"wp-caption-text\">Figure 4.21: Magnitude and phase angle of the wave impedance [latex]{\\bf{Z}}_w[\/latex] of the electric and magnetic dipole in the symmetry plane [latex]x0y[\/latex] versus electrical distance [latex]r\/\\lambda[\/latex] from the dipole.<\/figcaption><\/figure>\n<\/figure>\n<p>According to the rule of thumb it can be stated that the near field zone is within electrical distance [latex]r\/\\lambda<0.1[\/latex]. By that distance the phase angle rises to [latex]-80^{\\circ}[\/latex] for electric dipole and decays to [latex]80^{\\circ}[\/latex] for magnetic dipole. Within that distance field is practically pure electric or magnetic. The same rule of thumb tells us to presume the far field zone in an electrical distance [latex]r\/\\lambda>1[\/latex]. Memorize that the wave impedance in the far field zone regardless the direction approaches the intrinsic impedance Eq.\u00a0(4.49)<a href=\"#fn4\" class=\"footnote-ref\" id=\"fnref4\" role=\"doc-noteref\"><sup>4<\/sup><\/a>, which for the vacuum and approximately for the air is [latex]Z_0 = 120 \\pi[\/latex]\u00a0[latex]\\Omega \\approx 377[\/latex]\u00a0[latex]\\Omega[\/latex]. The wave impedance in the far field zone has resistive character and it represents real radiated power.<\/p>\n<p>It is evident why the zone between [latex]0.1Field probes  Probes are used for measurements of field strength in near and reactive zone. They should not distort incident field thats why they must be electrically small. This rationalizes their drawback namely moderate signal of response.    Construction of the sensor in the field probe imitate electric or magnetic dipole with finite dimensions. Equivalent circuits are shown in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>. [latex]C_S[\/latex] and [latex]L_S[\/latex] are capacitance and inductance of the electric and magnetic sensor respectively. [latex]R_L[\/latex] is resistance of the circuitry terminating the sensor.<\/p>\n<figure><img src=\"Propagation\/Figures\/D-dot_B-dot.png\" id=\"DB_circuits\" alt=\"\" \/><\/p>\n<figure id=\"attachment_183\" aria-describedby=\"caption-attachment-183\" style=\"width: 685px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-183\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-21.png\" width=\"685\" height=\"174\" alt=\"\" \/><figcaption id=\"caption-attachment-183\" class=\"wp-caption-text\">Figure 4.22: Equivalent circuits of electric sensor a), magnetic sensor b).<\/figcaption><\/figure>\n<\/figure>\n<p>Output voltage [latex]u_L(t)[\/latex] of the electric sensor is linked with the component of the density of the displacement current [latex]\\overset{\\bullet}{D}_p(t)[\/latex] in the space point in which the sensor is placed and matched with the orientation of the probe, see Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>a)<\/p>\n<p>[latex]C_S \\frac{du_L(t)}{dt} + \\frac{u_L(t)}{R_L} = A_e \\frac{dD_p(t)}{dt} = A_e \\overset{\\bullet}{D}_p(t)  \\label{u_e}\\tag{4.68}[\/latex]<\/p>\n<p>[latex]A_e[\/latex] is parameter of the probe transferring the component of the density of the displacement current [latex]\\overset{\\bullet}{D}_p(t)[\/latex] to the current source in the equivalent circuit in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>a), expressed in area units. It is called equivalent surface.<\/p>\n<p>In the frequency band of applications current driven through the capacitance [latex]C_S[\/latex] is negligibly small and can be omitted. Then response [latex]u_L(t)[\/latex] of the sensor is proportional to the component of the density of the displacement current [latex]\\overset{\\bullet}{D}_p(t)[\/latex]. The sensor is a current source with magnetomotive force [latex]A_e \\overset{\\bullet}{D}_p(t)[\/latex].<\/p>\n<p>Current [latex]i(t)[\/latex] in the mesh in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>b) is linked with the time derivative of the component of magnetic inductance [latex]\\overset{\\bullet}{B}_p(t)[\/latex] in the space point in which the sensor is placed and matched with the orientation of the sensor<\/p>\n<p>[latex]L_S \\frac{di(t)}{dt} + R_L i(t) = A_m \\frac{dB_p(t)}{dt} = A_m \\overset{\\bullet}{B}_p(t)  \\label{i_m}\\tag{4.69}[\/latex]<\/p>\n<p>[latex]A_m[\/latex] is parameter of the probe transferring the time derivative of the component of magnetic inductance [latex]\\overset{\\bullet}{B}_p(t)[\/latex] to the voltage source in the equivalent circuit in Fig.\u00a0<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>b), expressed in area units. It is called equivalent surface.<\/p>\n<p>In the frequency band of applications voltage across the inductance [latex]L_S[\/latex] is negligibly small and can be omitted. Then response [latex]u_L(t)[\/latex] of the sensor is proportional to the time derivative of the component of magnetic inductance [latex]\\overset{\\bullet}{B}_p(t)[\/latex]. The sensor is a voltage source with electromotive force [latex]A_m \\overset{\\bullet}{B}_p(t)[\/latex].<\/p>\n<p>On the market are available directive and isotropic probes. The first sense only field component mached with the probe orientation, the second sense three orthogonal components of field in space.<\/p>\n<p>There is variety of probes capable of capturing amplitude in overall frequency band of application. Directly by the sensor they have RF detector, most frequently diode one. Thereafter there is A\/C converter and the signal is delivered either to the display integrated with the sensor in case of autonomous probe or is converted to light and transmitted via the fiber glass to the computer interface. Increasingly the fiber glass is used simultaneously for powering electronic by the sensor.<\/p>\n<p>Probes are able to measure faithfully only CW i.e.\u00a0single frequency fields. They are inapplicable for any modulated or transient fields.<\/p>\n<p>Example of autonomous isotropic probe for electric and magnetic field with the bandwidth from DC to 400\u00a0kHz is shown in Fig.\u00a0<a href=\"#Probes\" data-reference-type=\"ref\" data-reference=\"Probes\">4.23<\/a>a) and isotropic electric probe with optical transmission of signal and power applicable in the bandwidth from 10\u00a0kHz to 6\u00a0GHz in Fig.\u00a0<a href=\"#Probes\" data-reference-type=\"ref\" data-reference=\"Probes\">4.23<\/a>b).<\/p>\n<figure><img src=\"Propagation\/Figures\/Probes.png\" id=\"Probes\" alt=\"\" \/><\/p>\n<figure id=\"attachment_184\" aria-describedby=\"caption-attachment-184\" style=\"width: 668px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-184\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-22.png\" width=\"668\" height=\"202\" alt=\"\" \/><figcaption id=\"caption-attachment-184\" class=\"wp-caption-text\">Figure 4.23: Isotropic autonomous electric and magnetic field probe a), electric field probe with optical transmission of signal and power b).<\/figcaption><\/figure>\n<\/figure>\n<p>There are other types of probes called in jargon D-dot and B-dot probes. They are applicable for measurements of pulsed fields. In such probes digital processing of the sensor response, which is proportional to the field derivative, see Eq.\u00a0(4.68) and Eq.\u00a0(4.69), is preceded with the time integration. They are exclusively directional probes.<\/p>\n<p>Example of free space directional D-dot probe with the bandwidth from from 100\u00a0kHz to 3.5\u00a0GHz is shown in Fig.\u00a0<a href=\"#DB-dot\" data-reference-type=\"ref\" data-reference=\"DB-dot\">4.24<\/a>a) and free space directional B-dot probe with the bandwidth from 100\u00a0kHz to 2\u00a0GHz in Fig.\u00a0<a href=\"#DB-dot\" data-reference-type=\"ref\" data-reference=\"DB-dot\">4.24<\/a>b).<\/p>\n<figure><img src=\"Propagation\/Figures\/DB-dot.png\" id=\"DB-dot\" alt=\"\" \/><\/p>\n<figure id=\"attachment_185\" aria-describedby=\"caption-attachment-185\" style=\"width: 700px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-185\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-23.png\" width=\"700\" height=\"213\" alt=\"\" \/><figcaption id=\"caption-attachment-185\" class=\"wp-caption-text\">Figure 4.24: Free space directional D-dot probe a), Free space directional B-dot probe b).<\/figcaption><\/figure>\n<\/figure>\n<h4 id=\"antennas\">Antennas<\/h4>\n<p>In technical terminology terms electric and magnetic antenna are used. The first concerns antennas with the wave impedance in the near field zone bigger than intrinsic impedance.<\/p>\n<p>Antennas are used for selective measurements of stationary fields in the far field zone. It can be assumed that they do not distort incident field. Analogue voltage signal induced in the antenna is transmitted from the antenna terminal to the spectrum analyzer or other radio frequency selective measurement receiver which records frequency spectrum of the measured field.<\/p>\n<p>Antennas for the measurement of electric fields are variations of electric dipoles. Dipole can be interpreted as unloaded symmetrical transmission line with straightened out conductors. The tips of dipole\u2019s arms are nodes for current distribution and antinodes for voltage distribution. Current vanishes there and voltage varies from plus to minus amplitude due to open circuit condition.<\/p>\n<p>By [latex]\\lambda\/2[\/latex] dipole as shown in Fig.\u00a0<a href=\"#ePol_dipol\" data-reference-type=\"ref\" data-reference=\"ePol_dipol\">4.25<\/a>a) the current distribution, the blue area is half of the approximately cosine function<a href=\"#fn5\" class=\"footnote-ref\" id=\"fnref5\" role=\"doc-noteref\"><sup>5<\/sup><\/a> with amplitude in the midpoint between the arms i.e. by antenna terminal. Voltage distribution, the orange needles is half of the sine function with fixed zero value by antenna terminal and varying between plus and minus amplitudes at the tips.<\/p>\n<p>In Fig.\u00a0<a href=\"#ePol_dipol\" data-reference-type=\"ref\" data-reference=\"ePol_dipol\">4.25<\/a>\u00a0b) a half dipole antenna is shown. Such miniature antennas are used as grounded probes of electric fields or much bigger as monopole (rod) receiving antennas.<\/p>\n<p>Dipole radiate efficiently if its length is matched with approximately multiple of the half of the wavelength [latex]\\lambda\/2[\/latex]. This efficiency is very sensitive on mismatching. In order to expand the frequency bandwidth rods in the ordinary dipole are replaced with conus (bi-conical antenna), triangle (bow-tie antenna) or more complex shapes by brad band antennas.<\/p>\n<p>In Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0b) the bow-tie-log-periodic antenna is shown. Its frequency of operation extends from 30\u00a0MHz do 1500\u00a0MHz. The bow-tie section of the antenna, big triangles next to the feeding point covers frequency band from 30\u00a0MHz to about 300\u00a0MHz. It is connected parallelly with the Log-Periodic Dipole Antenna LPDA ahead. The LPDA is group of dipole antennas of varying sizes strung together. The dipole antennas diminish in size from the back to the front. The element at the back of the array which is the largest is tuned to frequency about 300\u00a0MHz and that at the front is a half wavelength at the highest frequency of operation i.e at 1500\u00a0MHz.<\/p>\n<p>In the antenna boom between coaxial junction and bow-tie section a black box with the balun is mounted. Input of the dipole antenna is symmetric but the feeding point coaxial. The balun is a two port adapting symmetric to coaxial terminal. The name is a cluster of two words balanced-unbalanced.<\/p>\n<figure><img src=\"Propagation\/Figures\/Pol_dipol.png\" id=\"ePol_dipol\" alt=\"\" \/><\/p>\n<figure id=\"attachment_186\" aria-describedby=\"caption-attachment-186\" style=\"width: 461px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-186\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-24.png\" width=\"461\" height=\"267\" alt=\"\" \/><figcaption id=\"caption-attachment-186\" class=\"wp-caption-text\">Figure 4.25: Electric dipole with the length matched to the [latex]\\lambda\/2[\/latex] wavelength a) and monopole matched to the [latex]\\lambda\/4[\/latex] wavelength.<\/figcaption><\/figure><figcaption><\/figcaption><\/figure>\n<figure><img src=\"Propagation\/Figures\/Balun_2.png\" id=\"Balun_2\" alt=\"\" \/><\/p>\n<figure id=\"attachment_187\" aria-describedby=\"caption-attachment-187\" style=\"width: 714px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-187\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-25.png\" width=\"714\" height=\"396\" alt=\"\" \/><figcaption id=\"caption-attachment-187\" class=\"wp-caption-text\">Figure 4.26: Illustration how the current balun operates.<\/figcaption><\/figure>\n<\/figure>\n<p>Desired signal of electric dipole stems from capacitance between the arms. It is [latex]C_{12}[\/latex] in Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0a). Additionally each arm has its own capacitance to the ground. They are [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex]. There is still one more capacitance that must be taken into account. Namely between the arm connected to the core of the coaxial cable and the shield. It is arm 1 and capacitance [latex]C_{1S}[\/latex] in Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0a). Capacitance between the arm 2 and the cable shield is short circuited. [latex]Z[\/latex] is the input impedance of the measurement receiver.<\/p>\n<p>Equivalent circuit of the dipole is composed of parallel circuits of the current sources with their capacitances as it was explained for the free space electric sensor shown in Fig.<a href=\"#DB_circuits\" data-reference-type=\"ref\" data-reference=\"DB_circuits\">4.22<\/a>\u00a0a)<a href=\"#fn6\" class=\"footnote-ref\" id=\"fnref6\" role=\"doc-noteref\"><sup>6<\/sup><\/a>. In Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0b) branch with [latex]Z_C[\/latex] represents impedances of cable core, [latex]Z_S[\/latex] cable shield and [latex]Z_{GND}[\/latex] the ground reference. If capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] are different then they contribute to the differential current driven through the core, the measurement receiver and the shield. Moreover probable is also common mode current returning through the ground. Differential contribution of capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] cannot be eliminated. It overlays with desired signal [latex]I_{12}[\/latex] causing distortion.<\/p>\n<p>The current balun is simply ferrite mounted on the coaxial cable direct by junction with symmetrical antenna output. The common mode choke<a href=\"#fn7\" class=\"footnote-ref\" id=\"fnref7\" role=\"doc-noteref\"><sup>7<\/sup><\/a> [latex]L^{CM}[\/latex] shown in Fig.\u00a0<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a> suppresses common mode contribution of capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] but it does not eliminate it totally. Total elimination of differential and common mode distortion caused by capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] would happen if they were equal. Practically this condition is fulfilled by horizontal polarization of the antenna and very high above the ground reference, regardless polarization<a href=\"#fn8\" class=\"footnote-ref\" id=\"fnref8\" role=\"doc-noteref\"><sup>8<\/sup><\/a>.<\/p>\n<p>There is still another parasitic capacitance [latex]C_{1S}[\/latex] which contributes differentially to the desired signal, see Fig.<a href=\"#Balun_2\" data-reference-type=\"ref\" data-reference=\"Balun_2\">4.26<\/a>\u00a0b) and cannot be eliminated. The only way is to keep the capacitance small and place the common mode choke [latex]L^{CM}[\/latex] as close as possible by antenna feeding.<\/p>\n<figure><img src=\"Propagation\/Figures\/Balun_3.png\" id=\"Balun_3\" alt=\"\" \/><\/p>\n<figure id=\"attachment_188\" aria-describedby=\"caption-attachment-188\" style=\"width: 564px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-188\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-26.png\" width=\"564\" height=\"415\" alt=\"\" \/><figcaption id=\"caption-attachment-188\" class=\"wp-caption-text\">Figure 4.27: Illustration how the voltage balun operates.<\/figcaption><\/figure>\n<\/figure>\n<p>More sophisticated are voltage baluns, based on transformers. Example of one of them is shown in Fig.\u00a0<a href=\"#Balun_3\" data-reference-type=\"ref\" data-reference=\"Balun_3\">4.27<\/a>. Transformer separates galvanically symmetrical terminal of the antenna from coaxial cable connector. Additionally cable shield tapes the centre of the transformer winding on the symmetrical side. It has triple advantage:<\/p>\n<ul>\n<li>cancellation of the common mode distortion due to capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex], thanks galvanic separation,<\/li>\n<li>both arms of the dipole are liberated from the potential of the cable shield, thanks galvanic separation,<\/li>\n<li>capacitances of both dipole arms to the cable shield [latex]C_{1S}[\/latex] and [latex]C_{2S}[\/latex] are equal one to another due to introducing the potential of the cable shield to the midpoint between the arms.<\/li>\n<\/ul>\n<p>Contribution of the differential component in the distortion caused by unequal capacitances [latex]C_{10}[\/latex] and [latex]C_{20}[\/latex] is ruled in the same way same as by the current balun.<\/p>\n<p>The balun in the antenna shown in Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0b) is necessary because of the bow-tie section. The LPDAs\u2019 antennas are not finished with them. The adjacent dipoles in a LPDA antenna are connected to the symmetrical line routed inside the antenna (red and blue lines in Fig.\u00a0<a href=\"#LPDA_twist\" data-reference-type=\"ref\" data-reference=\"LPDA_twist\">4.28<\/a>) alternately. Consequently averaged capacitances of upper and lower arms to the cable shield and to the ground reference are equalized, minimizing distortion of the measured signal.<\/p>\n<figure><img src=\"Propagation\/Figures\/LPDA_twist.png\" id=\"LPDA_twist\" alt=\"\" \/><\/p>\n<figure id=\"attachment_189\" aria-describedby=\"caption-attachment-189\" style=\"width: 301px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-189\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-27.png\" width=\"301\" height=\"276\" alt=\"\" \/><figcaption id=\"caption-attachment-189\" class=\"wp-caption-text\">Figure 4.28: Alternately fed dipoles in the LPDA.<\/figcaption><\/figure>\n<\/figure>\n<p>All precautions concerning symmetrization in the antenna can be spoiled with incorrect routing of the coaxial cable behind the antenna. In the document <span class=\"citation\" data-cites=\"CISPR-16-1-4\">[@CISPR-16-1-4]<\/span> two recommendations are formulated:<\/p>\n<ul>\n<li>by EMC testing of radiated emission the coaxial cable behind the antenna should be maintained horizontal, i.e.\u00a0parallel to the ground plane, for a distance of approximately 1 m or more to the rear of the antenna before dropping to the ground plane,<\/li>\n<li>by verification of the anechoic chambers the coaxial cable behind the antenna should be oriented horizontally behind the antenna for a distance as close to 2 m as physically possible.<\/li>\n<\/ul>\n<p>The LPDA antennas available on the market have booms with about 1 m length.<\/p>\n<figure><img src=\"Propagation\/Figures\/Pol_dipol_m.png\" id=\"mPol_dipol\" alt=\"\" \/><\/p>\n<figure id=\"attachment_190\" aria-describedby=\"caption-attachment-190\" style=\"width: 540px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-190\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-28.png\" width=\"540\" height=\"207\" alt=\"\" \/><figcaption id=\"caption-attachment-190\" class=\"wp-caption-text\">Figure 4.29: Magnetic dipole with the circumference matched to the [latex]\\lambda\/2[\/latex] wavelength a) and half dipole with the circumference matched to the [latex]\\lambda\/4[\/latex] wavelength.<\/figcaption><\/figure>\n<\/figure>\n<figure><img src=\"Propagation\/Figures\/Antennas.png\" id=\"Loop_bilog\" alt=\"\" \/><\/p>\n<figure id=\"attachment_191\" aria-describedby=\"caption-attachment-191\" style=\"width: 698px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-191\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-29.png\" width=\"698\" height=\"513\" alt=\"\" \/><figcaption id=\"caption-attachment-191\" class=\"wp-caption-text\">Figure 4.30: Broad band loop antenna for measurements of magnetic field a) and broad band bow-tie-log-periodic antenna for measurements of electric field.<\/figcaption><\/figure>\n<\/figure>\n<p>Antennas for the measurement of magnetic fields are variations of magnetic dipoles. Magnetic dipole can be interpreted as short circuited symmetrical transmission line with the circle shape. The junction point of upper and lower conductor is the node for voltage distribution and antinode for current distribution. Voltage vanishes there and current varies from plus to minus amplitude due to short circuit condition.<\/p>\n<p>By [latex]\\lambda\/2[\/latex] loop dipole<a href=\"#fn9\" class=\"footnote-ref\" id=\"fnref9\" role=\"doc-noteref\"><sup>9<\/sup><\/a> as shown in Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>\u00a0a) the current distribution, the blue area is half of the approximately cosine function with amplitude in the junction point of upper and lower conductor. Voltage distribution, the orange needles is half of the sine function with fixed zero value by the junction point of upper and lower conductor and varying between plus and minus amplitudes at the antenna feeding.<\/p>\n<p>In Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>\u00a0b) a half loop antenna is shown. Such loops with miniature size are used as grounded probes of magnetic fields.<\/p>\n<p>Loop antennas are used in the frequency range up to 30\u00a0MHz. Their response is very week. The bigger radius the stronger response due to its dependence on magnetic flux streaming through the loop area. However lower location sensitivity by the measurements of inhomogeneous fields due to averaging of the flux with the loop area. By the loop antennas signal sensitivity and averaging effect must be always compromised. By active loop antenna with built in amplifier size of the loop can be reduced.<\/p>\n<p>In Fig.\u00a0<a href=\"#Loop_bilog\" data-reference-type=\"ref\" data-reference=\"Loop_bilog\">4.30<\/a>\u00a0a) the loop antenna with frequency of operation from 9\u00a0kHz to 30\u00a0MHz is shown. In the box by the tripod the amplifier is placed. At the top of the antenna a black ring is visible. Loop antennas have metal tube around, which plays a role of electric shield. Thanks the shield picking up unwanted responses steaming from electric fields can be avoided. This shield however must be broken to prevent the metal tube from acting like a shorted turn. The plastic ring positions the free ends of the broken metal tube.<\/p>\n<h5 id=\"antenna-factor.\">Antenna factor.<\/h5>\n<p>By measurements of radiated disturbances with antennas it is necessary to recalculate the signal at the input of the measurement receiver to the field strength to which the antenna is exposed. The antenna factor facilitates this conversion.<\/p>\n<p>If receiving antenna is exposed to the field, electromotive force [latex]EMF[\/latex] is induced in it. Antennas are not sensing the field module but the component matched with their orientation, [latex]E_z[\/latex] or [latex]H_x[\/latex] in Fig.\u00a0<a href=\"#F_A\" data-reference-type=\"ref\" data-reference=\"F_A\">4.31<\/a>. In other words induced [latex]EMF[\/latex] depends on the projection of the field vector on the E-plane for electric and H-plane for magnetic antenna. [latex]Z_s[\/latex] is the antenna impedance seen from its terminals.<\/p>\n<figure><img src=\"Propagation\/Figures\/Antenna_factor.png\" id=\"F_A\" alt=\"\" \/><\/p>\n<figure id=\"attachment_192\" aria-describedby=\"caption-attachment-192\" style=\"width: 565px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-192\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-30.png\" width=\"565\" height=\"263\" alt=\"\" \/><figcaption id=\"caption-attachment-192\" class=\"wp-caption-text\">Figure 4.31: Definition of the antenna factor.<\/figcaption><\/figure>\n<\/figure>\n<p>By determining the antenna factor the antenna is oriented to the plane matching the field with known strength [latex]E_z[\/latex] or [latex]H_x[\/latex]. The instrument with [latex]50[\/latex]\u00a0[latex]\\Omega[\/latex] input resistance capable of HF voltage measurements, such as: power meter with insertion unit or power sensor, spectrum analyzer or measurement receiver, is connected to the terminals of the antenna. Ratio of field strength and voltage at the measuring instrument is the antenna factor.<\/p>\n<p>[latex]\\label{F_A}  AF^{(E)}\\left[\\frac{1}{m} \\right] = \\frac{E_z}{U} \\hspace{3cm}  AF^{(H)}\\left[\\frac{1}{\\Omega m} \\right] = \\frac{H_x}{U}\\tag{4.70}[\/latex]<\/p>\n<p>In the [latex]dB[\/latex] scale the units are [latex]dB[1\/m][\/latex] and [latex]dB[1\/(\\Omega m)][\/latex] respectively. The antennas\u2019 factors are determined for the far field zone in the air where wave and intrinsic impedances are identical [latex]E\/H = Z_0 \\approx 377[\/latex]\u00a0[latex]\\Omega[\/latex]. Alternatively manufacturers and calibration laboratories presents the factor for magnetic antennas as follows<\/p>\n<p>[latex]AF^{(H)}\\left[\\frac{1}{ m} \\right] = AF^{(H)} \\left[dB\\left(\\frac{1}{\\Omega m}\\right) \\right] + 51.5 [dB(\\Omega)][\/latex]<\/p>\n<h5 id=\"surface-power-density.\">Surface power density.<\/h5>\n<p>Vector product of phasors of electric field strength and conjugate of magnetic field strength at arbitrary point in space builds the phasor of the Poynting\u2019s vector<\/p>\n<p>[latex]\\overrightarrow{\\bf{S}} = \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^*  \\label{Poynting_1}\\tag{4.72}[\/latex]<\/p>\n<p>Surface integral of the phasor of the Poynting\u2019s vector over any closed surface [latex]s[\/latex], oriented outwards versus solid enclosing some antennas gives the phasor of total apparent power [latex]{\\bf{P}}_{app}[\/latex] on this surface <span class=\"citation\" data-cites=\"Hammond_2\">[@Hammond_2]<\/span><\/p>\n<p>[latex]{\\bf{P}}_{app}=\\oiint_s \\left( \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^* \\right) \\cdot \\overrightarrow{ds}  \\label{Power_app} \\tag{4.73}[\/latex]<\/p>\n<p>crucial is that [latex]\\overrightarrow{ds}[\/latex] is the surface versor oriented outward versus the solid with antennas.<\/p>\n<p>If phasors in Eq.\u00a0(4.72) are scaled with RMS values then Poynting\u2019s vector [latex]\\overrightarrow{\\bf{S}}[\/latex] gives density per surface unit of apparent power at the point of interest<a href=\"#fn10\" class=\"footnote-ref\" id=\"fnref10\" role=\"doc-noteref\"><sup>10<\/sup><\/a>. Sense of the Pointing\u2019s vector shows direction of power transportation in case of electric antennas and oposite direction in case of magnetic antennas. In the near and reactive field zone the imaginary part of the Poynting\u2019s vector represents density of reactive power traveling back and forth between the antenna and the point of interest and real part represents density of real power radiated out of the antenna.<\/p>\n<figure><img src=\"Propagation\/Figures\/Poynting.png\" id=\"Poynting\" alt=\"\" \/><\/p>\n<figure id=\"attachment_193\" aria-describedby=\"caption-attachment-193\" style=\"width: 742px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-193\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-31.png\" width=\"742\" height=\"319\" alt=\"\" \/><figcaption id=\"caption-attachment-193\" class=\"wp-caption-text\">Figure 4.32: The triple field vectors on the equiphase surface by electric antenna a), by magnetic antenna b) .<\/figcaption><\/figure>\n<\/figure>\n<p>In the far field zone the imaginary part of the Poynting\u2019s vector decays. It is actually attribute of the far field zone. There is no phase shift between fields\u2019 strengths. Magnitude of the Poynting\u2019s vector represents solely density of radiated power.<\/p>\n<p>Each antenna, even geometrically complex is seen, from the observation point located at the wavefront, as the point radiator. Therefore three features of the dipoles\u2019 fields in the far field zone can be extended to antennas with any shape. Two of them are listed below:<\/p>\n<ul>\n<li>the equiphase surface is the sphere,<\/li>\n<li>the field strength decreases in the far field zone reciprocally proportional to the distance.<\/li>\n<\/ul>\n<p>For formulation of the third one, additional definition is necessary. It will be introduced later in this paragraph.<\/p>\n<p>Let us imagine wavefront of an antenna with electric field strength polarized in the [latex]\\varphi = const[\/latex] plane and magnetic field strength polarized in the [latex]z = const[\/latex] plane, green coloured in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> a). The triple of the vectors can be oriented as shown there. The magnitude of the Poynting\u2019s vector yields<\/p>\n<p>[latex]S_r(r,\\theta) = E_\\theta(r,\\theta) H_\\varphi(r,\\theta) = \\frac{{E_\\theta^2(r,\\theta)}}{Z_0} = Z_0 {H_\\varphi^2(r,\\theta)}  \\label{Poynting_2}\\tag{4.74}[\/latex]<\/p>\n<p>Let us imagine wavefront of an antenna with magnetic field strength polarized in the [latex]\\varphi = const[\/latex] plane and electric field strength polarized in the [latex]z = const[\/latex] plane, green coloured in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> b). The triple of the vectors can be oriented as shown there. The magnitude of the Poynting\u2019s vector yields<\/p>\n<p>[latex]S_r(r,\\theta) = E_\\varphi(r,\\theta) H_\\theta(r,\\theta) = \\frac{{E_\\varphi^2(r,\\theta)}}{Z_0} = Z_0 {H_\\theta^2(r,\\theta)}  \\label{Poynting_3}\\tag{4.75}[\/latex]<\/p>\n<p>If the antenna in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> a) is elementary electric dipole with strengths expressed with Eqs.\u00a0(4.57) and (4.58) then<\/p>\n<p>[latex]S^{(E)}_r(r,\\theta) = \\frac{p_z^2 \\beta_0^2 Z_0 } {16 \\pi^2 } \\cdot \\frac{\\sin^2{\\theta}}{r^2}  \\label{Poynting_4}\\tag{4.76}[\/latex]<\/p>\n<p>Total power radiated by the electric dipole is equal to the surface integral of the Poynting vector Eq.\u00a0(4.73) and can be calculated over equiphase sphere with arbitrary radius [latex]r[\/latex].<\/p>\n<p>[latex]P_{rad}^{(E)}=\\unicode{x222F}_s S_r^{(E)} \\overrightarrow{1}_r \\cdot \\overrightarrow{ds} =  \\unicode{x222F}_s S_r^{(E)} r^2 \\sin{\\theta} d\\theta d\\varphi =[\/latex]<\/p>\n<p>[latex]=\\frac{p_z^2 \\beta_0^2 Z_0 } {16 \\pi^2 }  {\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sin^3{\\theta} d\\theta} {\\int_{0}^{2 \\pi} d\\varphi} = \\nonumber[\/latex]<\/p>\n<p>[latex]= \\frac{p_z^2 \\beta_0^2 Z_0 } {6 \\pi }  \\label{Power_rad} \\tag{4.77}[\/latex]<\/p>\n<p>If the antenna in Fig.\u00a0<a href=\"#Poynting\" data-reference-type=\"ref\" data-reference=\"Poynting\">4.32<\/a> b) is elementary magnetic dipole with strengths expressed with Eqs.\u00a0(4.59) and (4.60) then<\/p>\n<p>[latex]S^{(H)}_r(r,\\theta) = -\\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {16 \\pi^2 Z_0 } \\cdot \\frac{\\sin^2{\\theta}}{r^2}  \\label{Poynting_5}\\tag{4.78}[\/latex]<\/p>\n<p>Total power radiated by the electric dipole is equal to the surface integral of the Poynting vector Eq.\u00a0(4.73) and can be calculated over equiphase sphere with arbitrary radius [latex]r[\/latex].<\/p>\n<p>[latex]P_{rad}^{(H)}=\\unicode{x222F}_s S_r^{(H)} \\left( -\\overrightarrow{1}_r \\right) \\cdot \\overrightarrow{ds} =  \\unicode{x222F}_s S_r r^2 \\sin{\\theta} d\\theta d\\varphi = \\nonumber[\/latex]<\/p>\n<p>[latex]=-\\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {16 \\pi^2 Z_0 }  {\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sin^3{\\theta} d\\theta} {\\int_{0}^{2 \\pi} d\\varphi} = \\nonumber[\/latex]<\/p>\n<p>[latex]= \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 }{6 \\pi Z_0 }  \\label{Power_rad_H} \\tag{4.79}[\/latex]<\/p>\n<p>In Eqs.\u00a0(4.76) and (4.78) is apparent that the surface power density in the far field zone of elementary radiators decreases reciprocally proportional to the distance [latex]r[\/latex] in square. This is the third feature valid also for realised antennas with any shape and size.<\/p>\n<h5 id=\"radiation-pattern.\">Radiation pattern.<\/h5>\n<p>Ability of antennas to radiate electromagnetic energy in different directions is portrayed with radiation pattern. It is defined for electric, magnetic field strength as well as for radiated power in far field zone. The radiation pattern is a function built over equiphase surface. Its value represents vector magnitude or power density. For electric\/magnetic antenna oriented along z-axis, as shown in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b)\/Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0c) electric\/magnetic field has only [latex]\\theta[\/latex] component dependent on all co-ordinates [latex]r[\/latex], [latex]\\theta[\/latex] and [latex]\\varphi[\/latex]. Patterns normalized with maximal value are objective measures for comparison of antennas\u2019 performance. They are independent on distance [latex]r[\/latex]<\/p>\n<p>[latex]F^{(E)}(\\theta,\\varphi) = \\frac{E_\\theta(r,\\theta,\\varphi)}{ E_{\\theta_{max}}} \\hspace{1.5cm}  F^{(H)}(\\theta,\\varphi) = \\frac{H_\\theta(r,\\theta,\\varphi)}{ H_{\\theta_{max}}}  \\label{F_EH}\\tag{4.80}[\/latex]<\/p>\n<p>Obviously normalized pattern of power radiation yields, see Eq.\u00a0( 4.74)<\/p>\n<p>[latex]F_P^{(E)}(\\theta,\\varphi)=\\frac{S_r^{(E)}(r,\\theta,\\varphi)}{ S_{r_{max}}^{(E)}} = \\left[ F^{(E)} (\\theta,\\varphi) \\right]^2[\/latex]<\/p>\n<p>[latex]\\nonumber[\/latex]<\/p>\n<p>[latex]F_P^{(H)}(\\theta,\\varphi)= \\frac{S_r^{(H)}(r,\\theta,\\varphi)}{ S_{r_{max}}^{(H)}} = \\left[ F^{(H)} (\\theta,\\varphi) \\right]^2  \\label{F_S} \\tag{4.82}[\/latex]<\/p>\n<p>Radiation pattern of field strength of the electric and magnetic dipole according to Eq.\u00a0(4.57) and Eq.\u00a0( 4.60) are identical<\/p>\n<p>[latex]F^{(E)}(\\theta) = F^{(H)}(\\theta) = F(\\theta) = \\sin{(\\theta)}  \\label{F_pm}\\tag{4.84}[\/latex]<\/p>\n<figure><img src=\"Propagation\/Figures\/Pattern_E.png\" id=\"Pattern_E\" alt=\"\" \/><\/p>\n<figure id=\"attachment_194\" aria-describedby=\"caption-attachment-194\" style=\"width: 733px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-194\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-32.png\" width=\"733\" height=\"288\" alt=\"\" \/><figcaption id=\"caption-attachment-194\" class=\"wp-caption-text\">Figure 4.33: Radiation pattern of electric\/magnetic field strength round the electric\/magnetic dipole.<\/figcaption><\/figure>\n<\/figure>\n<p>and of surface power density<\/p>\n<p>[latex]F^{(E)}_P(\\theta) = F^{(H)}_P(\\theta) = F_P(\\theta) = \\sin^2{(\\theta)}  \\label{F_pma}\\tag{4.84}[\/latex]<\/p>\n<figure><img src=\"Propagation\/Figures\/Pattern_S.png\" id=\"Pattern_S\" alt=\"\" \/><\/p>\n<figure id=\"attachment_195\" aria-describedby=\"caption-attachment-195\" style=\"width: 771px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-195\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-33.png\" width=\"771\" height=\"260\" alt=\"\" \/><figcaption id=\"caption-attachment-195\" class=\"wp-caption-text\">Figure 4.34: Radiation pattern of surface power density round the electric\/magnetic dipole.<\/figcaption><\/figure>\n<\/figure>\n<p>Radiation pattern of field strength of the electric and magnetic dipole in the spherical system of coordinates [latex]r, \\theta, \\varphi[\/latex] is function of one variable [latex]\\theta[\/latex]. In the system of Cartesian coordinates [latex]x, y, z[\/latex] it is function of all variables and can be depicted as a map of colours on the surface of the sphere with radius [latex]r=1[\/latex], as shown in Fig.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0a). From the engineer point of view such picture is useless.<\/p>\n<p>Antennas\u2019 engineers are accustomed with mapping the one variable pattern [latex]F(\\theta)[\/latex] into the two variables pattern [latex]F(r,\\theta)[\/latex] in which [latex]r = \\sin{(\\theta)}[\/latex]. Such surface is the torus (doughnut) without hole in centre with circular cross-section in the plane [latex]\\theta = 90^{\\circ}[\/latex] with radius [latex]r = 1[\/latex] and two excentrically placed touching circles with radii equal [latex]0.5[\/latex]. Distance between the origin of the Cartesian system of coordinates and the point at the pattern\u2019s surface is equal to the value of the field strength by that elevation [latex]\\theta[\/latex], see Fig.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0b). From such portrait radiation ability can be directly determined quantitatively.<\/p>\n<p>The same concerns the pattern of the surface power density which is identical for both dipoles. In the spherical system of coordinates [latex]r, \\theta, \\varphi[\/latex] it is function of one variable [latex]\\theta[\/latex]. In the system of Cartesian coordinates [latex]x, y, z[\/latex] it is function of all variables and can be depicted as a map of colours on the surface of the sphere with radius [latex]r=1[\/latex], as shown in Fig.\u00a0<a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0a). Usually it is mapped into the two variables pattern [latex]F(r,\\theta)[\/latex] in which [latex]r = \\sin^2{(\\theta)}[\/latex]. Such surface is the vertically squeezed torus (doughnut) without hole in centre with circular cross-section in the plane [latex]\\theta = 90^{\\circ}[\/latex] with radius [latex]r = 1[\/latex] and two excentrically placed touching, deformed circles. Distance between the origin of the Cartesian system of coordinates and the point at the pattern\u2019s surface is equal to the value of the surface power density by that elevation [latex]\\theta[\/latex], see Fig.\u00a0<a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0b).<\/p>\n<p>Solid representing the two variables radiation pattern [latex]F(r,\\theta)[\/latex] is called the lobe. The electric as well as magnetic dipole has only one lobe which is the torus. The realized antennas have so called main lobe roundabout the desired radiation direction and usually more than one side lobe which is undesired, side effect causing waste of radiated power.<\/p>\n<p>Dipoles and antennas presented up to now are oriented along z-axis of the global system of coordinates i.e.\u00a0vertically. For presentation of radiation pattern at the plane, as it is usually the case in the antennas\u2019data sheets, it is more convenient to use system of coordinates tied to the antenna. Then the cross-section including only electric field component or magnetic field component is called E-plane or H-plane respectively. For the dipole in Figs.\u00a0<a href=\"#Pattern_E\" data-reference-type=\"ref\" data-reference=\"Pattern_E\">4.33<\/a>\u00a0b) and <a href=\"#Pattern_S\" data-reference-type=\"ref\" data-reference=\"Pattern_S\">4.34<\/a>\u00a0b), the presented cross-section is the E-plane if it is electric dipole and H-plane if it is magnetic dipole. The cross-section [latex]\\theta = 90^{\\circ}[\/latex] would be H-plane for electric dipole and E-plane for magnetic dipole.<\/p>\n<h5 id=\"directive-gain-and-directivity.\">Directive gain and directivity.<\/h5>\n<p>Compared is antenna under interest with isotropic antenna radiating the same power. Directive gain [latex]D(r,\\theta)[\/latex] is ratio of the surface radiated power of the first and the second. Directivity is maximal value of the directive gain. Both parameters gives the measure of squeezing the power beam referred to the omnidirectional radiation.<\/p>\n<p>Let us derive these parameters for the electric and magnetic dipole as shown in Fig.\u00a0<a href=\"#xyz\" data-reference-type=\"ref\" data-reference=\"xyz\">4.20<\/a>\u00a0b) and c). Division of total radiated power of the electric dipole Eq.\u00a0(4.77) and magnetic dipole Eq.\u00a0(4.79) by the solid angle of the sphere [latex]4\\pi r^2[\/latex] gives surface power density of the isotropic antenna<\/p>\n<p>[latex]S_i(r) = \\frac{p_z^2 \\beta_0^2 Z_0 } {24 \\pi^2 r^2 } \\hspace{2cm} S_i(r) = \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 } {24 \\pi^2 Z_0 r^2 }  \\label{Poynting_i}\\tag{4.85}[\/latex]<\/p>\n<p>According to the definition, the directive gain is fraction of surface power density of the dipole and density of isotropic radiator Eq.\u00a0(4.85). The numerator in this fraction is Eq.\u00a0(4.76) by the electric dipole and Eq.\u00a0(4.78) by the magnetic dipole<\/p>\n<p>[latex]D(\\theta) = \\frac{3 } {2 } \\sin^2{\\theta}  \\label{Directivity_1}\\tag{4.86}[\/latex]<\/p>\n<p>and directivity<\/p>\n<p>[latex]D_{max} = D(90^{\\circ}) = \\frac{3 } {2 }  \\label{Directivity_2}\\tag{4.87}[\/latex]<\/p>\n<p>In the [latex]dB[\/latex] scale [latex]D_{max} \\approx 1.8[\/latex]\u00a0[latex]dB_i[\/latex]<a href=\"#fn11\" class=\"footnote-ref\" id=\"fnref11\" role=\"doc-noteref\"><sup>11<\/sup><\/a>. In the direction of the maximal radiation, the dipole radiates 50% or [latex]1.8[\/latex]\u00a0[latex]dB_i[\/latex] more power then isotropic radiator with the same totally radiated power.<\/p>\n<h5 id=\"energetic-gain\">(Energetic) gain,<\/h5>\n<p>usually named simply gain is the directivity diminished by energetic efficiency of the antenna [latex]\\eta[\/latex] defined as ratio of total power radiated by the antenna [latex]P_{rad}[\/latex] and power at the antenna feeding port [latex]P_{in}[\/latex]<\/p>\n<p>[latex]G_i = \\eta D_{max} = \\frac{P_{rad}}{P_{in}} D_{max}  \\label{Gain}\\tag{4.88}[\/latex]<\/p>\n<p>By interpretation of power at the antenna feeding port [latex]P_{in}[\/latex] ambiguity can creep in. If net power<a href=\"#fn12\" class=\"footnote-ref\" id=\"fnref12\" role=\"doc-noteref\"><sup>12<\/sup><\/a> is introduced as the input power [latex]P_{in} = P_{fwr} - P_{rev}[\/latex] then the gain is preceded with the adjective absolute. If forward power sent from the signal source is introduced as the input power [latex]P_{in} = P_{fwr}[\/latex] then the gain is preceded with the adjective realised. By absolute gain, unlike by realised gain mismatching at the feeding point is taken into account. Be prudent by evaluation of catalogue data of an antenna in respect of its gain.<\/p>\n<p>More practical meaning has the gain [latex]G_d[\/latex] referred to the half wave dipole. Since the gain of the half wave dipole referred to the isotropic antenna is [latex]1.64[\/latex] in the linear scale and [latex]2.15[\/latex]\u00a0[latex]dB_i[\/latex], see <span class=\"citation\" data-cites=\"Clayton\">[@Clayton]<\/span><\/p>\n<p>[latex]G_d=\\frac{1}{1.64} G_i[\/latex]<\/p>\n<p>[latex]& \\nonumber[\/latex]<\/p>\n<p>[latex]G_{{dB}_d}=G_{{dB}_i} - 2.15dB_i  \\label{G_d} \\tag{4.90}[\/latex]<\/p>\n<h5 id=\"equivalent-isotropically-radiated-power-eirp\">Equivalent isotropically radiated power EIRP<\/h5>\n<p>is the input power at the terminal of the isotropic radiator necessary for radiation with the surface power density equal to the maximum of the magnitude of the surface power density [latex]S(r)[\/latex] of the considered antenna in the far field zone. It is product of this value and solid angle of the sphere<\/p>\n<p>[latex]EIRP = 4 \\pi r^2 |S(r)|_{max}  \\label{EIRP}\\tag{4.91}[\/latex]<\/p>\n<p>The EIRP does not depend on the distance [latex]r[\/latex]. Indeed the EIRP is explicitly proportional to the distance [latex]r[\/latex] in square but reciprocal proportionality to distance [latex]r[\/latex] in square is implied in the [latex]|S(r)|_{max}[\/latex] due to attribute of the far field zone.<\/p>\n<p>For the electric dipole calculation is based on Eq.\u00a0(4.76) and for magnetic dipole on Eq.\u00a0(4.78) with setting [latex]\\theta = 90^{\\circ}[\/latex]<\/p>\n<p>[latex]EIRP^{(E)} = \\frac{p_z^2 \\beta_0^2 Z_0 }{4 \\pi } \\hspace {1.5cm} EIRP^{(H)} = \\frac{m_z^2 \\omega^2 \\mu_0^2 \\beta_0^2 }{4 \\pi Z_0 }  \\label{EIRP_EH}\\tag{4.92}[\/latex]<\/p>\n<p>Let us derive the measurement procedure of the EIRP for an electric antenna. The set up is established in the full anechoic chamber in which walls, ceiling and floor are lined with the absorbers marked with pyramids in Fig.\u00a0<a href=\"#EIRP_Fig\" data-reference-type=\"ref\" data-reference=\"EIRP_Fig\">4.35<\/a>, simulating reflection free space. The investigated antenna [latex]T_x[\/latex] and receiving antenna [latex]R_x[\/latex] are placed at the same height above the floor in order to match their elevations with [latex]\\theta = 90^{\\circ}[\/latex]. The investigated antenna must be rotated about the vertical axis to the azimuth maximizing signal [latex]U_R[\/latex] in the measurement receiver and fixed in this position.<\/p>\n<figure><img src=\"Propagation\/Figures\/EIRP.png\" id=\"EIRP_Fig\" alt=\"\" \/><\/p>\n<figure id=\"attachment_196\" aria-describedby=\"caption-attachment-196\" style=\"width: 522px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-196\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-34.png\" width=\"522\" height=\"291\" alt=\"\" \/><figcaption id=\"caption-attachment-196\" class=\"wp-caption-text\">Figure 4.35: Ascertainment of the EIRP in the full anechoic chamber.<\/figcaption><\/figure>\n<\/figure>\n<p>Applied must be Eq.\u00a0(4.74) for surface power density in the direction of maximal radiation i.e.\u00a0by elevation [latex]\\theta = 90^{\\circ}[\/latex] and Eq.\u00a0(<a href=\"#F_A\" data-reference-type=\"ref\" data-reference=\"F_A\">4.31<\/a>) for antenna factor of an electric antenna<\/p>\n<p>[latex]S_r(r,90^{\\circ}) = \\frac{E_{\\theta}^2(r,90^{\\circ})}{Z_0} = \\frac{\\left( AF^{(E)} \\right)^2}{Z_0} U_A^2(r)  \\label{EIRP_2}\\tag{4.93}[\/latex]<\/p>\n<p>For EIRP this formula must be multiplied by the solid angle of the sphere<\/p>\n<p>[latex]EIRP = \\frac{4 \\pi r^2}{Z_0} \\left( AF^{(E)} \\right)^2 U_A^2(r)  \\label{EIRP_3}\\tag{4.94}[\/latex]<\/p>\n<p>Requested unit for the EIRP is [latex]mW[\/latex] and for voltage at the antenna terminal [latex]\\mu V[\/latex]. Suitable formula yields<\/p>\n<p>[latex]EIRP \\left[ mW\\right] = \\frac{4 \\pi r^2 \\left[m^2 \\right]}{377 [\\Omega]} \\left( AF^{(E)} \\right)^2 \\left[ \\frac{1}{m^2}\\right] U_A^2(r) \\left[(\\mu V)^2 \\right] 10^{-9}  \\label{EIRP_4}\\tag{4.95}[\/latex]<\/p>\n<p>Both sides of equation are logarithmised and multiplied by 10<\/p>\n<p>[latex]EIRP_{_{dB(mW)}}(r)= 10\\log{ \\left( \\frac{4 \\pi r^2 \\left[ m^2 \\right] }{377 [\\Omega]} \\right)} - 90_{_{dB}}+ 20\\log{ \\left\\{ AF^{(E)}\\left[ \\frac{1}{m} \\right] \\right\\}} + \\nonumber[\/latex]<\/p>\n<p>[latex]\\nonumber[\/latex]<\/p>\n<p>[latex]& +A_{C_{dB}} + 20\\log{ \\left\\{ U_R(r) \\left[ \\mu V \\right] \\right\\} }  \\label{EIRP_5} \\tag{4.96}[\/latex]<\/p>\n<p>Voltage at the feeding terminal of the receiving antenna [latex]U_{A_{dB(\\mu V)}}[\/latex] is greater than voltage measured by the receiver [latex]U_{R_{dB(\\mu V)}}[\/latex] about attenuation [latex]A_{C_{dB}}[\/latex] of the measurement path (cables, feed through) [latex]U_{A_{dB(\\mu V)}} (r) = A_{C_{dB}} + U_{R_{dB(\\mu V)}} (r)[\/latex]<\/p>\n<p>[latex]EIRP_{_{dB(mW)}}(r)= \\left[10\\log{ \\left( \\frac{4 \\pi r^2 }{377} \\right)} - 90_{_{dB}} \\right]+ AF_{_{dB(1\/m)}}^{(E)} + \\nonumber[\/latex]<\/p>\n<p>[latex]\\nonumber[\/latex]<\/p>\n<p>[latex]& +A_{C_{dB}} + U_{R_{dB(\\mu V)}} (r)  \\label{EIRP_6} \\tag{4.98}[\/latex]<\/p>\n<p>The EIRP is independent on distance [latex]r[\/latex]. Indeed there is radius in square in the numerator in Eqs. from (4.94) to (4.98) but measurement is done in the far field zone in which surface power density decreases reciprocally proportional to the distance in square. Consequently the same concerns the voltage at the antenna terminal. By increasing distance e.g.\u00a0three times the term in square bracket in Eq.\u00a0(4.98) increases about 9.5\u00a0dB but in the same time the voltage at the antenna terminal [latex]U_{A_{dB(\\mu V)}}[\/latex] and consequently at the receiver input [latex]U_{R_{dB(\\mu V)}}[\/latex] decreases about the same amount.<\/p>\n<p>Typical distance for performing the measurements of the EIRP is ([latex]r= 3[\/latex]\u00a0[latex]m[\/latex]). By doing so correction factor in the square brackets is [latex]-95.2_{{dB}}[\/latex].<\/p>\n<h5 id=\"radiation-resistance\">Radiation resistance<\/h5>\n<p>Total radiated power is represented in the equivalent circuit of the antenna with radiation resistance [latex]R_{rad}[\/latex] through which current [latex]I[\/latex] from the source feeding antenna is driven [latex]R_{rad}I^2[\/latex].<\/p>\n<p>Let us calculate this resistance for the electric dipole. Total radiated power is ruled with Eq.\u00a0(4.77). Replacing the magnitude of dipole moment with the product of current magnitude [latex]I[\/latex] and infinitesimal segment [latex]l[\/latex] Eq.\u00a0(4.41), moreover with implementing the wavelength [latex]\\lambda_0[\/latex] instead of the phase constant [latex]\\beta_0[\/latex], according to Eq.\u00a0(4.50) yields<\/p>\n<p>[latex]R_{rad}^{(E)} = \\frac{2 \\pi Z_0}{ 3} \\left(\\frac{l}{\\lambda_0} \\right)^2  \\label{R_rad}\\tag{4.99}[\/latex]<\/p>\n<p>In vacuum and approximately in air this resistance is [latex]R_{rad}^{(E)} \\approx 789.57 \\left( \\frac{l}{\\lambda_0} \\right)^2 [\\Omega][\/latex] if the dipole length [latex]l[\/latex] and the wavelength [latex]\\lambda_0[\/latex] have the same units.<\/p>\n<p>In feasible short dipole relation [latex]l\/\\lambda_0[\/latex] and consequently radiation resistance is very small. Therefore radiation ability is very weak<a href=\"#fn13\" class=\"footnote-ref\" id=\"fnref13\" role=\"doc-noteref\"><sup>13<\/sup><\/a>.<\/p>\n<p>For example, for a length ([latex]l = 1[\/latex]\u00a0cm) and a frequency of 300 MHz ([latex]\\lambda_0 = 1[\/latex]\u00a0m), the radiation resistance is [latex]R_{rad}^{(E)} \\approx 79 [m\\Omega][\/latex] In order to radiate 1\u00a0W of power, required is current of 3.6 A!<\/p>\n<p>If the frequency is changed to 3 MHz ([latex]\\lambda_0 = 100[\/latex]\u00a0m), the radiation resistance is [latex]R_{rad}^{(E)} \\approx 7.9 \\left[ \\mu \\Omega \\right][\/latex] and the current required to radiate 1\u00a0W is 356 A!<\/p>\n<p>Moreover resistance [latex]R_{rad}[\/latex] is serially circuited with the element representing antenna outlet to the lossless space. Depending on electrical length of the dipole [latex]l\/\\lambda_0[\/latex] it can be capacitance, shortcircuit or inductance but for a short dipole it is huge capacitance limiting current driven through radiation resistance. It is the second factor contributing to very small radiation efficiency of the Hertzian as well as short dipole.<\/p>\n<p>Let us proceed now to the radiation resistance of magnetic dipole. Total radiated power is ruled with Eq.\u00a0(4.79). Replacing the magnitude of dipole moment with the product of current magnitude [latex]I[\/latex] and infinitesimal loop area [latex]\\pi a^2[\/latex] Eq.\u00a0(4.42), moreover with implementing the wavelength [latex]\\lambda_0[\/latex] instead of the phase constant [latex]\\beta_0[\/latex], according to Eq.\u00a0(4.50) and relation [latex]\\omega = 2 \\pi v\/\\lambda_0[\/latex] between angular frequency, wave velocity [latex]v[\/latex] and wavelength yields [latex]\\omega \\beta_0 = \\frac{4 \\pi^2}{\\lambda_0^2} v[\/latex]<\/p>\n<p>Finally<\/p>\n<p>[latex]R_{rad}^{(H)} = \\frac{8 \\pi^3 \\mu_0^2 v^2}{ 3 Z_0} \\left( \\frac{\\pi a^2}{\\lambda_0^2} \\right)^2  \\label{R_rad_H}\\tag{4.104}[\/latex]<\/p>\n<p>Value of the wave speed in vacuum is given in Eq.(<a href=\"#v_0\" data-reference-type=\"ref\" data-reference=\"v_0\">[v_0]<\/a>) therefore this resistance in air is approximately [latex]R_{rad}^{(H)} \\approx 31'170.91 \\left( \\frac{\\pi a^2}{\\lambda_0^2} \\right)^2 [\\Omega][\/latex]<\/p>\n<p>if radius [latex]a[\/latex] and wavelength [latex]\\lambda_0[\/latex] have the same units.<\/p>\n<p>Consider a loop of radius ([latex]a= 1[\/latex]\u00a0cm). At 300 MHz the wavelength ([latex]\\lambda_0 = 1[\/latex]\u00a0m) and the radiation resistance is [latex]R_{rad}^{(H)} \\approx 3.08 \\mu \\Omega[\/latex] In order to radiate 1\u00a0W, the loop requires a current of 18\u00a0A!<\/p>\n<p>At 3 MHz the wavelength ([latex]\\lambda_0 = 100[\/latex]\u00a0m) and the radiation resistance is [latex]R_{rad}^{(H)} \\approx 30.8 p \\Omega[\/latex] and current required to radiate 1 W is 180\u00a0kA!<\/p>\n<p>Both elementary dipoles radiate very weakly but their comparison shows that magnetic dipole is overwhelmingly worse radiator than electric. In practice they are scarcely met as unintentional antennas.<\/p>\n<h5 id=\"half-power-beamwidth-hpbw\">Half power beamWidth HPBW<\/h5>\n<p>called also 3 dB beamwidth is an angle span, either in azimuth or in elevation between two directions encompassing set of directions with the surface power density decaying not more then to the half of the maximum. In other words not reduced about more then 3 dB.<\/p>\n<figure><img src=\"Propagation\/Figures\/HPBW.png\" id=\"HPBW\" alt=\"\" \/><\/p>\n<figure id=\"attachment_197\" aria-describedby=\"caption-attachment-197\" style=\"width: 551px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-197\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-35.png\" width=\"551\" height=\"264\" alt=\"\" \/><figcaption id=\"caption-attachment-197\" class=\"wp-caption-text\">Figure 4.36: Illustration of the half power beamwidth of electric and magnetic dipole.<\/figcaption><\/figure>\n<\/figure>\n<p>The electrical dipole is omnidirectional in respect to azimuth [latex]\\varphi[\/latex], therefore determining the HPBW on the plane [latex]\\theta = 0^{\\circ}[\/latex] is groundless. For the elevation<a href=\"#fn14\" class=\"footnote-ref\" id=\"fnref14\" role=\"doc-noteref\"><sup>14<\/sup><\/a> the upper bound is [latex]\\theta = 45^{\\circ}[\/latex] and lower bound [latex]\\theta = 135^{\\circ}[\/latex] therefore [latex]HPBW = 90^{\\circ}[\/latex], see Fig.\u00a0(<a href=\"#HPBW\" data-reference-type=\"ref\" data-reference=\"HPBW\">4.36<\/a>).<\/p>\n<p>HPBW expresses spreading of the power beam. The narrower HPBW, the better concentration of radiated power roundabout desired direction.<\/p>\n<h4 id=\"unintentional-antennas\">Unintentional antennas<\/h4>\n<p>It is mentioned frequently, that any cable can be either transmitting or receiving antenna.<\/p>\n<p>By the measurement of radiated emission of the table top equipments in the Semi Anechoic Chamber SAC, the EUT must be placed on the table with 80\u00a0cm height. The mains cord should be stretched towards the table edge, dropped down towards the floor and plugged in the supply socket under the metal floor. Typical arrangement of the mains cord is shown in Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0a). Such cable is a monopole antenna while the EUT case is its ground reference. If the common mode current is driven in the cord and the cord length is matched with the wavelength, then the cable radiate and it is very often the case.<\/p>\n<figure><img src=\"Propagation\/Figures\/Cable_as_antenna.png\" id=\"Cable_antenna\" alt=\"\" \/><\/p>\n<figure id=\"attachment_198\" aria-describedby=\"caption-attachment-198\" style=\"width: 756px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-198\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-36.png\" width=\"756\" height=\"420\" alt=\"\" \/><figcaption id=\"caption-attachment-198\" class=\"wp-caption-text\">Figure 4.37: Examples of the mains cable as an unintentional antenna: monopole antenna a), loop antenna b) excess cable folded to and forth c).<\/figcaption><\/figure>\n<\/figure>\n<p>In Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0b) excess cable is arranged as it should not be. It is wound in few turns building the loop antenna. If the common mode current is driven in the cord and the loop diameter is matched with the wavelength, then it can radiate. Excess cable should be folded to and forth in order to form a low inductive bundle as shown in Fig.\u00a0<a href=\"#Cable_antenna\" data-reference-type=\"ref\" data-reference=\"Cable_antenna\">4.37<\/a>\u00a0c).<\/p>\n<p>Let\u2019s consider the metal case with the primary plane wave inside, propagating towards the wall with vector\u2019s orientation of: electric field [latex]\\overrightarrow{E}_p[\/latex], magnetic field [latex]\\overrightarrow{H}_p[\/latex] and Poynting vector [latex]\\overrightarrow{S}_p[\/latex] as shown in the side projection in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b). By continuous metal wall of the case, surface current would be induced on the internal surface of the wall. The slot enables the current to leak outside. It flows on the external surface of the wall, in surrounding of the slot as shown with reddish region in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a). Black arrows on the edges of the slot shows direction of the driven current. It is accompanied with electric field in the vicinity of the slot oriented as vector [latex]\\overrightarrow{E}[\/latex] in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b). Imagine that the edges of the slot build two dipoles shortcircuited at the top of the arms. Such entity can hardly radiate due to cancellation in the far field zone. But as shown with the reddish region in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a) current is driven also on the external surface around the edges.<\/p>\n<figure><img src=\"Propagation\/Figures\/Slot_antenna.png\" id=\"Slot_antenna\" alt=\"\" \/><\/p>\n<figure id=\"attachment_199\" aria-describedby=\"caption-attachment-199\" style=\"width: 668px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-199\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-37.png\" width=\"668\" height=\"181\" alt=\"\" \/><figcaption id=\"caption-attachment-199\" class=\"wp-caption-text\">Figure 4.38: Half wavelength slot antenna.<\/figcaption><\/figure>\n<\/figure>\n<p>Therefore the secondary wave oriented as shown with the vectors of: electric field [latex]\\overrightarrow{E}_s[\/latex], magnetic field [latex]\\overrightarrow{H}_s[\/latex] and Poynting vector [latex]\\overrightarrow{S}_s[\/latex] in the side projection in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0b) propagates from the case, particularly by matching the slot length. Example is [latex]\\lambda\/2[\/latex] matching shown in Fig.\u00a0<a href=\"#Slot_antenna\" data-reference-type=\"ref\" data-reference=\"Slot_antenna\">4.38<\/a>\u00a0a). Memorize that orientation of the propagated field is dual to the electric dipole. Horizontally oriented slot propagates vertically oriented electric field.<\/p>\n<figure><img src=\"Propagation\/Figures\/Slot_antenna_2.png\" id=\"Slot_antenna_2\" alt=\"\" \/><\/p>\n<figure id=\"attachment_200\" aria-describedby=\"caption-attachment-200\" style=\"width: 740px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-200\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-38.png\" width=\"740\" height=\"244\" alt=\"\" \/><figcaption id=\"caption-attachment-200\" class=\"wp-caption-text\">Figure 4.39: Gap by point wise fixing of two pieces of the case a), window in a case as the aperture antenna b).<\/figcaption><\/figure>\n<\/figure>\n<p>By point wise techniques of joining parts of the metal cases with screws, nuts or spot welding, gaps between places tied together, marked with yellowish area in Fig.\u00a0<a href=\"#Slot_antenna_2\" data-reference-type=\"ref\" data-reference=\"Slot_antenna_2\">4.39<\/a>\u00a0a) are formed. Things like incertitude of joined edges and deformations caused by excessed contact pressing force are contribution factors of it. In order to reduce effect of secondary radiation from such gaps distances between fixed spots must ensure length mismatching or\/and EMC gaskets obstructing electromagnetically the gaps must be applied.<\/p>\n<p>Windows and openings in cases are indispensable. They serves for mounting displays, cameras, switches, buttons, as sight hole or for ventilation and manipulation of the arrangement. Due to the same rules as by the gaps in the metal case the surface currents flow on the external surface of the wall as shown with the reddish area in Fig.\u00a0<a href=\"#Slot_antenna_2\" data-reference-type=\"ref\" data-reference=\"Slot_antenna_2\">4.39<\/a>\u00a0b). The window can build aperture antenna enabling the secondary wave to radiate out of the case.<\/p>\n<p>There is alway maximal frequency called cut off, up to which the window is not capable to radiate. It is associated with the biggest wavelength which can be radiated. Above the cut off frequency the wavelength for the rectangular window with sides [latex]ab[\/latex] and the circular window with the radius [latex]r[\/latex] are approximately ruled with following inequalities<\/p>\n<p>[latex]\\begin{array} {cc}  \\lambda_{\\framebox[0.15in]{}} < 2 \\sqrt{\\pi a b} &#38; \\hspace{2cm} \\lambda_{\\bigcirc} < 2 \\pi^{\\frac{3}{2}} r \\approx 11.14 r  \\end{array}  \\label{cut_off_1}\\tag{4.108}[\/latex]\n\nInequality for the circular window is convergent with matching of the circumference of the magnetic dipole to the [latex]\\lambda\/2[\/latex] wavelength [latex]\\lambda \\approx 12.57 r[\/latex] as illustrated in Fig.\u00a0<a href=\"#mPol_dipol\" data-reference-type=\"ref\" data-reference=\"mPol_dipol\">4.29<\/a>. It is also in line with condition of not being electrically small i.e. bigger than [latex]0.1 \\lambda[\/latex], see subsection <a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a>.<\/p>\n<h3 id=\"wave-guides\">Guided waves<\/h3>\n<p>So far in the chapter <a href=\"#wave_coupling\" data-reference-type=\"ref\" data-reference=\"wave_coupling\">4.4<\/a> we have been discussing conveying energy, signals and disturbances with radiation. By this art of transportation both field vectors [latex]\\overrightarrow{E}[\/latex] as well as [latex]\\overrightarrow{H}[\/latex] are perpendicular to the direction of propagation. This type of waves is called TEM (transverse electric magnetic) mode. The same TEM mode is present in transmission lines elaborated by galvanic coupling in chapter <a href=\"#galvanic_coupling\" data-reference-type=\"ref\" data-reference=\"galvanic_coupling\">4.2<\/a> as well as by electric and magnetic coupling in chapter <a href=\"#LC_coupling\" data-reference-type=\"ref\" data-reference=\"LC_coupling\">4.3<\/a>. Transportation passed along the line and fields are perpendicular to it<a href=\"#fn15\" class=\"footnote-ref\" id=\"fnref15\" role=\"doc-noteref\"><sup>15<\/sup><\/a>. Therefore TEM mode is often referred to as the transmission line mode.<\/p>\n<figure><img src=\"Propagation\/Figures\/WaveGuide.png\" id=\"waveguide\" alt=\"\" \/><\/p>\n<figure id=\"attachment_201\" aria-describedby=\"caption-attachment-201\" style=\"width: 451px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-201\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-39.png\" width=\"451\" height=\"296\" alt=\"\" \/><figcaption id=\"caption-attachment-201\" class=\"wp-caption-text\">Figure 4.40: A rectangle waveguide.<\/figcaption><\/figure>\n<\/figure>\n<p>Let us consider tubular structure with rectangular cross-section, built of perfectly conducting material [latex]\\sigma = \\infty[\/latex] filled with lossless dielectric [latex]\\sigma = 0[\/latex] with dielectric permittivity [latex]\\epsilon_0[\/latex] and magnetic permeability [latex]\\mu_0[\/latex] as shown in Fig.\u00a0<a href=\"#waveguide\" data-reference-type=\"ref\" data-reference=\"waveguide\">4.40<\/a>. Since [latex]\\overrightarrow{E}[\/latex] must be always perpendicular and [latex]\\overrightarrow{H}[\/latex] always tangential to the boundary, it is not possible to excite TEM in such structure.<\/p>\n<p>Let us assume that in such structure propagates wave with three component of phasor\u2019s electric field: two cross-sectional [latex]{\\bf{E}}_x(x,y,z)[\/latex], [latex]{\\bf{E}}_y(x,y,z)[\/latex] and longitudinal [latex]{\\bf{E}}_z(x,y,z)[\/latex]. The last is governed with the partial differential equation called the wave equation<\/p>\n<p>[latex]\\nabla^2 {\\bf{E}}_z + \\omega^2 \\mu_0 \\epsilon_0 {\\bf{E}}_z = 0  \\label{Ez_1}\\tag{4.109}[\/latex]<\/p>\n<p>This equation can be solved by the method of separation of variables [latex]{\\bf{E}}_z(x,y,z) = {\\bf{X}}(x) {\\bf{Y}}(y) {\\bf{Z}}(z)[\/latex]<\/p>\n<p>[latex]\\frac{{\\bf{X}}^{''}}{{\\bf{X}}} + \\frac{{\\bf{Y}}^{''}}{{\\bf{Y}}} + \\frac{{\\bf{Z}}^{''}}{{\\bf{Z}}} + \\beta_0^2 = 0  \\label{Ez_2}\\tag{4.110}[\/latex]<\/p>\n<p>where [latex]\\beta_0[\/latex] is phase constant as defined in Eq.\u00a0(4.50).<\/p>\n<p>Eq.\u00a0(4.110) can be split into three independent ordinary differential equations<\/p>\n<p>[latex]\\label{X}  {\\bf{X}}^{''} - {\\bf\\gamma}_x^2 {\\bf{X}} = 0 \\tag{4.111}[\/latex]<\/p>\n<p>[latex]\\label{Y}  {\\bf{Y}}^{''} - {\\bf\\gamma}_y^2 {\\bf{Y}} = 0 \\tag{4.112}[\/latex]<\/p>\n<p>[latex]\\label{Z}  {\\bf{Z}}^{''} - {\\bf\\gamma}_z^2 {\\bf{Z}} = 0 \\tag{4.113}[\/latex]<\/p>\n<p>along with dispersion equation<\/p>\n<p>[latex]{\\bf\\gamma}_x^2 + {\\bf\\gamma}_y^2 + {\\bf\\gamma}_z^2 + \\beta_0^2 = 0  \\label{Ez_4}\\tag{4.114}[\/latex]<\/p>\n<p>where [latex]{\\bf\\gamma}_x[\/latex], [latex]{\\bf\\gamma}_y[\/latex] and [latex]{\\bf\\gamma}_z[\/latex] are separations\u2019 constants.<\/p>\n<p>Solution of Eq.\u00a0(4.111) is following linear combination of waves [latex]{\\bf{X}}(x) = A e^{{\\bf\\gamma}_x x} + B e^{-{\\bf\\gamma}_x x}[\/latex] traveling forth and back along x axis. [latex]{\\bf\\gamma}_x x = \\alpha_x + j\\beta_x[\/latex] can be a complex quantity. As [latex]E_z[\/latex] is tangential to the walls of the waveguide but tangential component of electric field must be zero on perfectly conducting metal wall, [latex]{\\bf{X}}(x)[\/latex] must disappear by [latex]x =0[\/latex] and [latex]x= a[\/latex]. This is possible only if [latex]\\alpha_x = 0[\/latex]. Consequently magnitude [latex]X(x)[\/latex] is a standing wave with the nodes on walls<\/p>\n<p>[latex]X_m(x) = (A_m - B_m) \\sin{(\\beta_{x,m}}x) = X_{0,m} \\sin{(\\beta_{x,m} x)}[\/latex]<\/p>\n<p>[latex]\\beta_{x,m} = m \\frac{\\pi}{a}  \\label{beta_x}\\tag{4.117}[\/latex]<\/p>\n<p>where [latex]m[\/latex] is integer number.<\/p>\n<p>Identical procedure applied to Eq.\u00a0(4.112) yields [latex]Y_n(y) = Y_{0,n} \\sin{(\\beta_{y,n} y)}[\/latex] along with<\/p>\n<p>[latex]\\beta_{y,n} = n \\frac{\\pi}{b}  \\label{beta_y}\\tag{4.119}[\/latex]<\/p>\n<p>where [latex]n[\/latex] is integer number.<\/p>\n<p>Solution of Eq.\u00a0(4.113) is following linear combination of waves [latex]{\\bf{Z}}(z) = C e^{{j\\gamma}_z z} + D e^{{-j\\gamma}_z z}[\/latex] traveling back [latex]C e^{{j\\gamma}_z z }[\/latex] and forth [latex]D e^{{-j\\gamma}_z z}[\/latex] along the z-axis. The waveguide is infinite in respect to z-axis. Therefore physical meaning has only one wave.<\/p>\n<p>The source of field is placed somewhere in the waveguide. From that location on, two waves propagate: one forwards another backwards in respect to z-axis. Assuming that source of field is placed by [latex]z<0[\/latex] so that by [latex]z=0[\/latex] regular propagation takes place, then by [latex]z \\ge 0[\/latex] propagates only\n\n[latex]{\\bf{Z}}_{mn}(z) = D_{mn} e^{-{\\bf\\gamma}_{z,mn} z}  \\label{Z_2}\\tag{4.121}[\/latex]\n\nwhere\n\n[latex]{\\bf\\gamma}_{z,mn}^2 = - \\beta_0^2 - \\left( {\\bf\\gamma}_{x,mn}^2 + {\\bf\\gamma}_{y,mn}^2 \\right) = \\beta_{x,mn}^2 + \\beta_{y,mn}^2 - \\beta_0^2  \\label{gamma_z}\\tag{4.125}[\/latex]\n\n[latex]{\\bf\\gamma}_{z,mn}[\/latex] can be real, imaginary or can have zero value by the break point which corresponds to the cut-off frequency [latex]f_{c,mn}[\/latex]\n\n[latex]f_{c,mn} = \\frac{v_0}{2} \\sqrt{\\left( \\frac{m}{a} \\right)^2 + \\left( \\frac{n}{b}\\right)^2}  \\label{gamma_zz}\\tag{4.123}[\/latex]\n\nwhere [latex]v_0[\/latex] is wave velocity in vacuum and indexes [latex]mn[\/latex] corresponds to Eqs.\u00a0(4.117) and (4.119).\n\nFinally separation constant [latex]{\\bf{\\gamma}}_{z,mn}[\/latex] which is identical with propagation constant in z direction is\n\n[latex]{\\bf{\\gamma}}_{z,mn} = \\left\\{ \\begin{array} {ll}  \\frac{\\omega}{v_0}\\sqrt{ \\left( \\frac{f_{c,mn}}{f} \\right)^2 - 1} &#38;  \\text{for}~~f < f_{c,mn}\\\\  \\\\  j\\frac{\\omega}{v_0} \\sqrt{ 1 - \\left( \\frac{f_{c,mn}}{f} \\right)^2 } &#38;  \\text{for}~~f > f_{c,mn}  \\end{array}  \\right.  \\label{gamma_z2}\\tag{4.124}[\/latex]<\/p>\n<p>and z component of the phasor of electric field<\/p>\n<p>[latex]{\\bf{E}}_{z,mn}(x,y,z) = E_{z,mn}(0) \\sin{\\left(\\frac{ m\\pi }{a}x \\right)} \\sin{\\left(\\frac{n \\pi }{b}y \\right)} e^{-\\gamma_{z,mn} z}  \\label{gamma_z}\\tag{4.125}[\/latex]<\/p>\n<p>[latex]E_{z,mn}(0)[\/latex] is product of [latex]X_{m}0[\/latex], [latex]Y_{n}0[\/latex] and [latex]D_{mn}[\/latex]. For unique solution the value [latex]E_{z,mn}(0)[\/latex] must be known.<\/p>\n<p>Derivation of remaining field components can be found for instance in <span class=\"citation\" data-cites=\"Hammond_2\">[@Hammond_2]<\/span>.<\/p>\n<p>Eq.\u00a0(4.125) can be rearranged with introduction of phase constant by propagation in free space with cut-off frequency [latex]\\beta_0(f_{c,mn})[\/latex]<\/p>\n<p>[latex]{\\bf{\\gamma}}_{z,mn} = \\left\\{ \\begin{array} {ll}  \\beta_0({f_{c,mn}}) \\sqrt{ 1 - \\left( \\frac{f}{f_{c,mn}} \\right)^2 } &  \\text{for}~~\\frac{f}{f_{c,mn}} < 1\\\\  \\\\  j\\beta_0({f_{c,mn}}) \\sqrt{ \\left( \\frac{f}{f_{c,mn}} \\right)^2 - 1} &#38;  \\text{for}~~\\frac{f}{f_{c,mn}} > 1  \\end{array}  \\right.  \\label{gamma_z3}\\tag{4.126}[\/latex]<\/p>\n<p>Dependence of the module of the propagation constant [latex]|\\gamma_z|[\/latex] versus frequency related to the cut-off frequency [latex]f\/f_{c,mn}[\/latex] is shown in Fig.<a href=\"#gamma_plot\" data-reference-type=\"ref\" data-reference=\"gamma_plot\">4.41<\/a>.<\/p>\n<figure><img src=\"Propagation\/Figures\/Gamma_z.png\" id=\"gamma_plot\" alt=\"\" \/><\/p>\n<figure id=\"attachment_202\" aria-describedby=\"caption-attachment-202\" style=\"width: 485px\" class=\"wp-caption aligncenter\"><img class=\"wp-image-202\" src=\"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-content\/uploads\/sites\/12\/2020\/08\/Pasted-into-Propagation-of-disturbances-40.png\" width=\"485\" height=\"303\" alt=\"\" \/><figcaption id=\"caption-attachment-202\" class=\"wp-caption-text\">Figure 4.41: Frequency dependence of the module of the propagation constant [latex]\\gamma_z[\/latex].<\/figcaption><\/figure>\n<\/figure>\n<p>Below the cut-off frequency (yellowish region) the propagation constant is real [latex]\\gamma_z = \\alpha_z[\/latex]. The wave is attenuated. Above the cut-off frequency (greenish region) the propagation constant is imaginary [latex]\\gamma_z = j\\beta_z[\/latex]. The wave is able to propagate. The propagation constant in this region approaches asymptote [latex]\\beta_{z,mn} = \\beta_0 \\left(f_{c,mn} \\right) \\frac{f}{f_{c,mn}}[\/latex]<\/p>\n<p>It is blue strait line in Fig.<a href=\"#gamma_plot\" data-reference-type=\"ref\" data-reference=\"gamma_plot\">4.41<\/a>.<\/p>\n<p>The type of fields discussed up to now are so called E-mode named equivalently TM-mode. Notice that the lowest mode of this type of fields can be [latex]E_{11}[\/latex] ([latex]TM_{11}[\/latex]) due to constraint on tangential component of E field on the walls.<\/p>\n<p>Similarly [latex]H_z[\/latex] component for the H mode (TE) mode can be derived<\/p>\n<p>[latex]{\\bf{H}}_{z,mn}(x,y,z) = H_{z,mn}(0) \\cos{\\left(\\frac{ m\\pi }{a}x \\right)} \\cos{\\left(\\frac{n \\pi }{b}y \\right)} e^{-\\gamma_{z,mn} z}  \\label{gamma_z4}\\tag{4.128}[\/latex]<\/p>\n<p>Dependence in x and y direction is cosinusoidal since H field must be tangential on the metal walls i.e. [latex]\\frac{\\partial H_z}{\\partial n} = 0[\/latex]. Consequently the lowest mode that can be propagated can have one number equal zero i.e.\u00a0[latex]H_{10}[\/latex], [latex]H_{01}[\/latex] ([latex]TE_{10}[\/latex], [latex]TE_{01}[\/latex]).<\/p>\n<section class=\"footnotes\" role=\"doc-endnotes\">\n<hr \/>\n<ol>\n<li id=\"fn1\" role=\"doc-endnote\">Proximity effect between strains makes the exact analysis of the problem more complex but the tendency is unambiguous.<a href=\"#fnref1\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn2\" role=\"doc-endnote\">[latex]w \\gg t[\/latex]<a href=\"#fnref2\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn3\" role=\"doc-endnote\">If in a given instant the field strength at one point of the equiphase surface has e.g.\u00a0positive magnitude, then it also has at all remaining points.<a href=\"#fnref3\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn4\" role=\"doc-endnote\">[latex]{\\bf{E}}_r[\/latex] component by electric dipole Eq.\u00a0(4.43) and [latex]{\\bf{H}}_r[\/latex] component by magnetic dipole Eq.\u00a0(4.54) disappears due to lack of summand with the first power of radius [latex]r[\/latex].<a href=\"#fnref4\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn5\" role=\"doc-endnote\">Exactly cosinusoidal distribution is valid for lossless symmetrical transmission line with parallelly layouted forward and return conductors. Per unit length parameter [latex]LC[\/latex] of such line are constant. By straightened out conductors [latex]LC[\/latex] parameters depend on position along the arms.<a href=\"#fnref5\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn6\" role=\"doc-endnote\">By field probes the parasitic capacitances can be neglected due to small electrical size. It is not the case by antennas.<a href=\"#fnref6\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn7\" role=\"doc-endnote\">Common mode chokes are covered in chapter <a href=\"#Filters\" data-reference-type=\"ref\" data-reference=\"Filters\">[Filters]<\/a><a href=\"#fnref7\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn8\" role=\"doc-endnote\">By EMC measurements of radiated emission, antenna height must be scanned from 1\u00a0m to 4\u00a0m. Up to the height 2\u00a0m the capacitances are hardly equal.<a href=\"#fnref8\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn9\" role=\"doc-endnote\">It is a circle with the radius [latex]r = \\lambda\/(4\\pi)[\/latex].<a href=\"#fnref9\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn10\" role=\"doc-endnote\">By scaling with the magnitudes, power density is [latex]\\frac{1}{2} \\left( \\overrightarrow{\\bf{E}} \\times \\overrightarrow{\\bf{H}}^* \\right)[\/latex].<a href=\"#fnref10\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn11\" role=\"doc-endnote\">Index [latex]i[\/latex] points out the isotropic radiator as the reference.<a href=\"#fnref11\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn12\" role=\"doc-endnote\">Difference of power sent from the source and power reflected at the antenna feeding terminal by mismatching.<a href=\"#fnref12\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn13\" role=\"doc-endnote\">Notice that it is in line with definition of electrical dimension and with the rule about radiation inability of electrically small objects, see chapter\u00a0<a href=\"#El_size\" data-reference-type=\"ref\" data-reference=\"El_size\">[El_size]<\/a>.<a href=\"#fnref13\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn14\" role=\"doc-endnote\">Introduce [latex]F_P(\\theta) = 1\/2[\/latex] into equation on power radiation pattern of the dipole Eq.\u00a0(4.84).<a href=\"#fnref14\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<li id=\"fn15\" role=\"doc-endnote\">It is apart from small region next to the conductors\u2019 surfaces where the Poynting\u2019s vector is leant toward the inside of conductors where the wave energy is converted into heat.<a href=\"#fnref15\" class=\"footnote-back\" role=\"doc-backlink\">\u21a9\ufe0e<\/a><\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":1,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3,"_links":{"self":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapters\/159"}],"collection":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":4,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapters\/159\/revisions"}],"predecessor-version":[{"id":251,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapters\/159\/revisions\/251"}],"part":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapters\/159\/metadata\/"}],"wp:attachment":[{"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/wp\/v2\/media?parent=159"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/pressbooks\/v2\/chapter-type?post=159"},{"taxonomy":"contributor","embeddable":true,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/wp\/v2\/contributor?post=159"},{"taxonomy":"license","embeddable":true,"href":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/wp-json\/wp\/v2\/license?post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}