{"id":121,"date":"2020-08-13T11:26:23","date_gmt":"2020-08-13T10:26:23","guid":{"rendered":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/?post_type=chapter&p=121"},"modified":"2020-08-13T13:21:44","modified_gmt":"2020-08-13T12:21:44","slug":"radio-frequency-electromagnetic-disturbances","status":"publish","type":"chapter","link":"http:\/\/pb.ee.pw.edu.pl\/pb\/cemc\/chapter\/radio-frequency-electromagnetic-disturbances\/","title":{"raw":"Radio frequency electromagnetic disturbances","rendered":"Radio frequency electromagnetic disturbances"},"content":{"raw":"

Long-term disturbances<\/h2>\r\n

Electric discharge in gases<\/h3>\r\nElectrons in metal are free to move around within it. Some of these electrons will have sufficient velocity to escape from the surface of the material. However, when they leave, they produce an electric field which pulls them back to the surface. If an external electric field is present with sufficient field strength, it can overcome the force that normally returns the electron to the surface. The electrons are therefore, removed from the surface and set free.\r\n\r\nAs electrons finally bombard the anode, the material of anode is heated and may vaporise. In general, either the anode or cathode may vaporise first, depending on the rates at which heat is delivered to and removed form the two contacts. Vaporized metal forms plasma between the electrodes [@Ott]<\/span> which displaces ions as current\u2019s carrier. Transition from ions\u2019 to plasma\u2019s current takes place in time less than a nanosecond [@Ott]<\/span>.\r\n\r\nThe filed strength required for initiation of this phenomenon is approximately $latex 500kV\/m$ according to [@Ott]<\/span> and $latex 10 GV\/m$ according to [@Clayton]<\/span>. In Fig.3.2<\/a> it is represented with the line with $latex 100 MV\/m$ (black continuous line on the left, with black dashed line continuation on the right).\r\n\r\nSuch big field strength can be generated by small distance of electrodes. Therefore this phenomenon is called short arc. By transition from ions\u2019 to plasma\u2019s current voltage across electrodes drops significantly. To sustain arc minimum arcing voltage and current is required.\r\n\r\nAfter the short arc is initiated the voltage across as well as current through the contact must exceed minimum arcing (sustaining) values. They are dependent on electrodes\u2019 material, as gathered in Tab.3.1<\/a>, [@Ott]<\/span>.\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Table 3.1: Minimal parameters for sustaining metal-vapor discharge. (* 69% gold, 25% silver, 6% platinum)<\/caption>\r\n
Material<\/th>\r\nMinimum arcing voltage $latex [V]$<\/th>\r\nMinimum arcing current $latex [mA]$<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Silver<\/td>\r\n12<\/td>\r\n400<\/td>\r\n<\/tr>\r\n
Gold<\/td>\r\n15<\/td>\r\n400<\/td>\r\n<\/tr>\r\n
Gold alloy*<\/td>\r\n9<\/td>\r\n400<\/td>\r\n<\/tr>\r\n
Palladium<\/td>\r\n16<\/td>\r\n800<\/td>\r\n<\/tr>\r\n
Platinum<\/td>\r\n17.5<\/td>\r\n700<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nShort arc can occur even in vacuum, since it does not require the presence of gas. Therefore another name metal-vapor arc. The short arc forms the basis for instance by cutting metals in electro erosion machine tools or in arc welding tools.\r\n\r\nIf distance between electrodes is to big for initiating short arch discharge, then discharge follows other way which is common also for natural lightning by thunderstorm. Namely, three discharge regimes can be distinguished: dark, glow and long arc. This phenomenon is illustrated in Fig.\u00a03.1<\/a> as voltage versus current. Consciously lacks scale of voltage axis in Fig.\u00a03.1<\/a> because its quantification depends on type of gas, electrodes\u2019 distance and electrodes\u2019 shape.\r\n
\"\"\r\n\r\n[caption id=\"attachment_127\" align=\"aligncenter\" width=\"690\"]\"\" Figure 3.1: Regimes of discharge in gas.[\/caption]<\/figure>\r\nIn the region from $latex 1$ to $latex 2$ the background ionisation due to cosmic rays or other sources of ionising radiation takes place. There exist small amount of free electrons and positively charged atoms in gas. Applied voltage accelerates ions. Number of ions does not increase. Some of them, but not all reaches attracting electrode. In point $latex 3$ stream of ions reaching attracting electrode saturates. Range between point $latex 3$ and $latex 4$ means passage to avalanche process. Energy of free electrons is enough for liberating other electrons from electrically neutral atoms by striking them. In consequence number of free electrons and positively charged atoms increases. This is accompanied with emission of electromagnetic energy in form of radio-frequency electromagnetic waves. This is not self sustained process called partial discharge or Ramsauer-Townsend discharge named for its discoverers. Its intensity is proportional to energy of electrons i.e.\u00a0to the applied voltage. It can cause buzzing on radio receivers. Between $latex 4$ and $latex B$ emission of electromagnetic energy shifts to visible light. It is called corona discharge which can be observed by high voltage overhead power lines. The same nature has St.\u00a0Elmo\u2019s fire observable by sailors on top of masts by thunderstorms.\r\n\r\nThere is hysteresis between $latex 4$ and $latex G$. By increased voltage passage follows points $latex 4$, $latex B$ and $latex G$, by decreased $latex G$, $latex G'$ and $latex 4$. Between $latex G$ and $latex 5$ the gas ionization becomes self-sustained.\r\n
\r\n\r\n[caption id=\"attachment_128\" align=\"aligncenter\" width=\"511\"]\"\" Figure 3.2: Variation of voltage threshold for different discharge regimes in gas.[\/caption]<\/figure>\r\nBy $latex B$ amount of ions is such that gas isolation ability ends. Gas starts to glow. This is called breakdown. Paschen found that the breakdown voltage $latex U_B$ is dependent on the product of electrode separations $latex d$ and pressure $latex p$ as\r\n\r\n$latex U_B(p \\cdot d) = \\frac{K_1 \\cdot p \\cdot d}{K_2 + \\ln{(p \\cdot d)}}\r\n\\label{Paschen_law}\\tag{3.16} $\r\n\r\nwhere $latex K_1$ and $latex K_2$ are constants that depend on gas. $latex \\ln{(p \\cdot d)}$ is normalised with product of units in which pressure and distance are expressed.\r\n\r\nThe Paschen curve is shown in Fig.3.2<\/a> for air in standard atmospheric pressure as red dashed curve on the left side and solid red curve continuation on the right. Air pressure $latex p$ and distance $latex d$ are taken in $latex Atm$ and $latex m$ respectively. $latex K_1 = 43.6\\cdot 10^6 \/ (Atm \\cdot m)$ and $latex K_2 = 12.8$ are applied in Fig.3.2<\/a>. Minimal breakdown voltage in air by standard atmospheric pressure is approximately $latex U_B^{min} \\approx 320 V$ and occurs at electrodes\u2019 separation $latex d^{min} \\approx 7.6 \\mu m$.\r\n\r\nJumping to the self-sustained glow discharge situated on the right to point $latex G$ is possible, provide the current available from the external circuit exceeds minimum glow discharge sustaining current $latex I_G$. Voltage across electrodes drops to\r\n\r\n$latex U_G(d) = 280 + 1000 \\cdot d\r\n\\label{Glow_sust}\\tag{3.17} $\r\n\r\n$latex U_G(d)$ is presented with blue dashed curve in Fig. 3.2<\/a>.\r\n\r\nVoltage drop by transition from $latex B$ to $latex G$ is relatively small. In air by atmospheric pressure the breakdown voltage is of order $latex U_B = 320V$ and glow voltage of order $latex U_G = 280V$. Minimum glow sustaining current is quite variable. Its representative range is $latex I_G \\approx 1 - 100 mA$ [@Clayton]<\/span> Glow regime is utilised in operation of electrical gas lighting equipment.\r\n\r\nThe voltage across the electrodes $latex U_G(d)$ for currents between $latex G$ and $latex 5$ is primarily determined by the dark region between the cathode and the beginning of the glow region, so called the cathode fall region. As the current increases, the dimension of the glow region increases toward anode, but the voltage across the gap remains $latex U_G(d)$. In consequence the fall region shrinks and by crossing the threshold field strength for short arc initiation the arcing process starts. State is reached where the heating causes vaporization of the contact metal (point $latex P$) resulting in a rapid drop in contact voltage, which marks the beginning of the arc discharge region where in the volume of ionsied gas channels filled with plasma are initiated, point $latex A$ [@Clayton]<\/span>.\r\n\r\nIn long arc regime building of channel is completed. Plasma can freely flow between electrodes building conducting bridge. Plasma regime is accompanied with intensive emission of whole spectrum of visible light.\r\n\r\nIt must be reemphasized that, in order to sustain a glow \/ arc discharge, the voltage across and current through the contact that are available from the external circuit must exceed $latex U_G$, $latex I_G$ and $latex U_A$, $latex I_A$ respectively. They are the most left points on the plot in Fig. 3.2<\/a>.\r\n\r\nActually, plasma current is initiated by long arc discharge identically as by short arc discharge. As it is explained before, voltage fall between electrodes is only in dark, not illuminating segments. In addition by increased current glowing region expands and dark segments shrinks. It means that field strength in dark regions increases leading to heating electrode materials, vaporisation and formation of plasma.\r\n

Signals with Amplitude Modulation<\/h3>\r\nElectronic nowadays is predominately digital. It uses variety of modulations for signal transmission. However experience shows, that electronic is mostly susceptible on amplitude modulation AM. That\u2019s why application of AM by most radio frequency EMC immunity tests should be not amazing. Therefore approach to this theme here.\r\n\r\nTwo waves are combined in AM. One is modulating signal $latex u_m(t)= m A \\cos(2 \\pi f_m t)$ in which message is coded and carrier $latex u_c(t)= A \\cos(2 \\pi f_0 t)$ responsible for message transportation. $latex m$ is called modulation depth or modulation index.\r\n
\"\"\r\n\r\n[caption id=\"attachment_129\" align=\"aligncenter\" width=\"782\"]\"\" Figure 3.3: Functional block diagram of Double Sideband - Full Carrier DSB-FC amplitude modulation.[\/caption]<\/figure>\r\nThe simplest way of AM, in fact applied in EMC immunity tests is called Double Sideband - Full Carrier DSB-FC. Its principle is explained in Fig.3.3<\/a>. The modulating signal $latex u_m(t)$ and amplitude of carrier $latex A$ are conveyed to the adder. Sum of both signals along with carrier signal are conveyed to multiplier (mixer). Modulated signal at the output of mixer yields\r\n\r\n$latex \\begin{array}{rcl}\r\nu(t) & = & A \\,\\sin(2 \\pi f_0 t) + mA \\, \\cos(2 \\pi f_m t) \\, \\sin(2 \\pi f_0 t) = \\\\\r\n& = & \\frac{m}{2} \\, A \\, \\sin\\left[2 \\pi (f_0 - f_m) t \\right] + A \\, \\sin (2 \\pi f_0 t) + \\frac{m}{2} \\, A \\, \\sin\\left[2 \\pi (f_0 + f_m) t \\right]\r\n\\end{array}\r\n\\label{AM-DSB}\\tag{3.18} $\r\n
\"\"\r\n\r\n[caption id=\"attachment_130\" align=\"aligncenter\" width=\"661\"]\"\" Figure 3.4: Amplitude modulation with DSB-FC: in the time domain a), in the frequency domain b).[\/caption]<\/figure>\r\nIt can be interpreted as the modulated signal (green line in Fig.3.4<\/a>a)) had been superposition of the pure carrier and carrier with amplitude alternated according to modulating signal, see middle part of Eq.(3.18). Envelope of the instantaneous amplitude of the modulated signal is brown line in Fig.3.4<\/a>a). In the frequency domain the DSB-FC is represented with three spectral lines, see the right part of Eq.(3.18) and Fig.3.4<\/a>b).\r\n\r\nObviously RMS of AM DSB-FC signal yields $latex U_{RMS} = \\frac{ A }{\\sqrt{2}} \\, \\sqrt{1 + \\frac{m^2}{2}}\r\n\\label{AM_RMS}\\tag{3.19} $\r\n\r\nand consequently power relation of modulated signal $latex P_m+P_c$ and carrier signal $latex P_c$\r\n\r\n$latex \\frac{P_m + P_c} {P_c} = 1 + \\frac{ m^2}{2 }\r\n\\label{AM_Power}\\tag{3.20} $\r\n\r\nMaximal instantaneous amplitude of the modulated signal is bigger than amplitude of carrier signal by $latex (1+m)$ or about $latex (1+m)dB$ in floating dB. Minimal instantaneous amplitude of the modulated signal is smaller than amplitude of carrier signal by $latex (1-m)$ or about $latex (1-m)dB$ in floating dB.\r\n\r\nMaximal instantaneous power of the modulated signal is bigger than power of carrier signal by $latex (1+m)^2$ or about $latex (1+m)dB$ in floating dB. Minimal instantaneous power of the modulated signal is smaller than power of carrier signal by $latex (1-m)^2$ or about $latex (1-m)dB$ in floating dB.\r\n\r\nModulation depth can be easily calculated from the maximal and minimal peak-to-peak spans $latex U_{p-p}^{max}$ and $latex U_{p-p}^{min}$ taken from the signal observed in the time domain as shown in Fig.3.4<\/a>a)\r\n\r\n$latex \\begin{array}{rclclrl}\r\n\\frac{U_{p-p}^{max}}{U_{p-p}^{min}} & = & \\frac{ 1+m}{1-m} & \\Rightarrow & m & = & \\frac{U_{p-p}^{max} - U_{p-p}^{min}}{U_{p-p}^{max} + U_{p-p}^{min}}\r\n\\end{array}\r\n\\label{AM_m}\\tag{3.21} $\r\n
\"\"\r\n\r\n[caption id=\"attachment_131\" align=\"aligncenter\" width=\"518\"]\"\" Figure 3.5: Functional block diagram of Double Sideband - Suppressed Carrier DSB-SC amplitude modulation.[\/caption]<\/figure>\r\nPrinciple of Double Sideband - Suppressed Carrier DSB-SC is explained in Fig.3.5<\/a>. The modulating signal $latex u_m(t)$ and carrier signal $latex u_c(t)$ are conveyed to multiplier. Modulated signal at the output of it yields\r\n
\"\"\r\n\r\n[caption id=\"attachment_132\" align=\"aligncenter\" width=\"676\"]\"\" Figure 3.6: Amplitude modulation with DSB-SC: in the time domain a), in the frequency domain b).[\/caption]<\/figure>\r\n$latex \\begin{array}{rcl}\r\nu(t) & = & mA^2 \\, \\cos(2 \\pi f_m t) \\sin(2 \\pi f_0 t) = \\\\\r\n& = & \\frac{mA^2}{2} \\, \\sin \\left[ 2 \\pi (f_0 - f_r) t \\right] + \\frac{mA^2}{2} \\, \\sin \\left[ 2 \\pi (f_0 + f_r) t \\right]\r\n\\end{array}\r\n\\label{Eq_AM_DSB-SC}\\tag{3.22} $\r\n\r\nIt can be interpreted as the carrier signal (green line in Fig.3.6<\/a>a)) would have amplitude alternated according to modulating signal, see middle part of Eq.(3.22). Envelope of the instantaneous amplitude of the modulated signal is brown line in Fig.3.6<\/a>a). In the frequency domain the DSB-SC is represented with two spectral lines, see the right part of Eq.(3.22b) and Fig.3.6<\/a>b).\r\n\r\nThere is another amplitude modulation called Single Sideband - Suppressed Carrier SSB-SC. It is realized similarly to DSB-SC with only one difference. Namely low pass or high pass filter is mounted behind the mixer in Fig.3.5<\/a>. It suppresses either upper or lower spectral line by frequency $latex f_0+f_m$ or $latex f_0 - f_m$ respectively.\r\n

Sequence of toggled DC voltage<\/h2>\r\nIn today\u2019s electronic there are two important branches based on toggling DC voltage. In power electronic, semiconductor valves operating in ON\/OFF state are core of appliances such as: AC\/AC inverters described in Chapter 3.1<\/a>, switched mode power supplies SMPS, DC\/DC converters, power factor correctors PFC. They are beneficial due to smaller losses and weight and they guarantee flexible, rigid and robust energy sources. In digital, binary electronic logical state 0 and 1 are realized with VOLTAGE\/VOLTAGE-LOSS state of the gates.\r\n\r\nIt is of primary interest to know frequency representation of toggled DC voltage in order to estimate hazards due to side effects of operation of electronic devices.\r\n

Rectangular waveform<\/h3>\r\nNeglecting the rise and fall time of pulses leads to simplification of toggling which in the time domain is represented with the rectangular waveform as shown in Fig. 3.7<\/a> a). Instant $latex t=0$ in Fig. 3.7<\/a> a) coincides with the half of the pulse width $latex tw$, therefore coefficients $latex b_n$ in the Fourier series expansion disappears.\r\n
\r\n\r\n[caption id=\"attachment_133\" align=\"aligncenter\" width=\"716\"]\"\" Figure 3.7: Rectangular waveform with the duty cycle of 0.23 in the time domain a) and its selected Fourier expansions b).[\/caption]<\/figure>\r\nCoefficient $latex a_0 = A t_w\/T$ is average value of the waveform. Remaining expansion coefficients $latex a_n$ are as follows\r\n\r\n$latex a_n = \\frac{ 2A}{T} \\int_{-\\frac{t_w}{2}}^{\\frac{t_w}{2}} \\cos( n\\omega t) dt = 2A \\frac{t_w}{T} \\frac{\\sin\\left(n\\pi \\frac{t_w}{T} \\right)}{n\\pi \\frac{t_w}{T}}\r\n\\label{Rect_an}\\tag{3.23} $\r\n\r\nThe pulse width expressed as a ration of the period $latex t_w\/T$, present in Eq.(3.23) is called duty cycle.\r\n\r\nFourier expansion of the rectangular waveform is as follows\r\n\r\n$latex u(t) = A \\frac{tw}{T} + 2A \\frac{t_w}{T} \\sum_{n=1}^{\\infty} \\left[ \\frac{\\sin\\left(n\\pi \\frac{t_w}{T} \\right)}{n\\pi \\frac{t_w}{T}} \\cos \\left(2 \\pi n f_1 t \\right) \\right]\r\n\\label{u(t)}\\tag{3.24} $\r\n\r\nSelected Fourier expansions for $latex n = 1$, $latex n = 1, 2, ...,5$ and $latex n = 1, 2, ...,10$ of the rectangle waveform from Fig. 3.7<\/a> a) are presented in Fig. 3.7<\/a> b).\r\n\r\nCoefficients $latex a_n$ are values of continuous function $latex a(f)$ by spot frequencies $latex f_n = n f_1$, where $latex f_1 = 1\/T$ is fundamental harmonic of the waveform\r\n\r\n$latex a(f) = 2A\\frac{ t_w}{T} \\frac{\\sin\\left(\\pi t_w f \\right)}{\\pi t_w f}\r\n\\label{Envelope1}\\tag{3.25} $\r\n\r\nEqs. from (3.23) to (3.25) encompasses term $latex \\sin x\/x$ which is called cardinal sine function $latex sinc$ and is typical by frequency representation of pulses.\r\n\r\n$latex \\mathop{\\mathrm{sinc}}(x) = \\frac{\\sin x}{x}\r\n\\label{Sinc}\\tag{3.26} $\r\n\r\nFunction $latex sinc$ shown in Fig.3.8<\/a> is continuous, symmetric, periodic function with period $latex 2\\pi$ and value 1 by argument equal to zero, $latex \\mathop{\\mathrm{sinc}}(0) = 1$. It decays to 0 by x approaching plus and minus infinity $latex \\lim_{x \\to \\pm \\infty} \\left[\\mathop{\\mathrm{sinc}}(x) \\right] = 0$.\r\n
\"\"\r\n\r\n[caption id=\"attachment_134\" align=\"aligncenter\" width=\"522\"]\"\" Figure 3.8: Cardinal sine function.[\/caption]<\/figure>\r\nIn respect of the harmonics, the lowest frequency of function Eq.(3.25) which must be considered is $latex f_1$. Absolute function imposed on $latex a(f)$ of the waveform from Fig.3.7<\/a>a) in the frequency range from $latex f_1$ to infinity is represented with black continuous curve in Fig. 3.9<\/a>a). It is the envelope of absolute values of the Fourier coefficients $latex \\left|a_n \\right|$ for variability of the duty cycles [@Schnorren]<\/span>. Its downward extrapolation is represented with black dashed line. Notable is its value by $latex f=0$, $latex a(0) = 2A\\frac{ t_w}{T}$. It is double value of the $latex a_0$ of the Fourier expansion. Distinctive are zero cross frequencies which falls by multiple of inverse of the pulse width$latex 1\/t_w$. Notable is the number of harmonics between consecutive zero crossing. It oscillates by reciprocal of the duty cycle. It is four or five for duty cycle equal to $latex 0.23$ as shown in Fig. 3.7<\/a> and Fig. 3.9<\/a>.\r\n
\"\"\r\n\r\n[caption id=\"attachment_135\" align=\"aligncenter\" width=\"750\"]\"\" Figure 3.9: Frequency spectrum of the waveform from Fig. 3.7a) in linear scale a) and in double logarithmic scale b).[\/caption]<\/figure>\r\nVery practical is representation of the frequency spectrum from Fig. 3.9<\/a>a) in the double logarithmic scale as shown in Fig. 3.9<\/a>b). The envelope is as follows\r\n\r\n$latex |U(f)| [dB] = 20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} + 20 \\log{\\left|\\frac{\\sin\\left(\\pi t_w f \\right)}{\\pi t_w f } \\right|}\r\n\\label{Envelope2}\\tag{3.27} $\r\n\r\nFor upper bound of the spectrum, $latex \\left|\\mathop{\\mathrm{sinc}}(\\pi t_w f )\\right|$ is set to one by low frequencies and $latex \\left|\\sin(\\pi t_w f )\\right|$ is set to one by high frequencies yielding\r\n\r\n$latex |U_p(f)|= \\left\\{ \\begin{array} {ll}\r\n20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} & \\text{for}~~f < \\frac{1}{\\pi t_w } \\\\\r\n20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} - 20 \\log{\\left( \\pi t_w f \\right)} & \\text{for}~~f > \\frac{1}{\\pi t_w } \r\n\\end{array}\r\n\\right.\r\n\\label{Bounds}\\tag{3.33} $\r\n\r\nIn consequence the upper bound is composed of two segments of the strait line, horizontal by low frequencies and declined with the slope $latex -20dB\/decade$ by high frequencies, see blue line in Fig.3.9<\/a>b). The corner frequency depends on the pulse width $latex 1\/(\\pi t_w)$.\r\n

Rectangular modulation, radar pulsing<\/h3>\r\n

Trapezoidal waveform<\/h3>\r\nConsidered is the case with equal rise and fall time $latex t_r = t_f$. Assumption of $latex t_w = t_{50\\%}$ simplifies expression for coefficient $latex a_0 = A \\frac{t_w}{T}$ which is trapeze area divided by the period $latex T$ i.e.\u00a0it is average value of the waveform.\r\n
\"\"\r\n\r\n[caption id=\"attachment_136\" align=\"aligncenter\" width=\"870\"]\"\" Figure 3.10: Trapezoidal waveform with the duty cycle of 0.42 in the time domain a) and its selected Fourier expansions b).[\/caption]<\/figure>\r\nInstant $latex t=0$ in Fig. 3.10<\/a> a) coincides with the half of the pulse width $latex tw$, therefore coefficients $latex b_n$ in the Fourier series expansion disappears. Remaining expansion coefficients $latex a_n$ are as follows\r\n\r\n$latex a_n = \\frac{ 2A}{T} \\left\\{ \\int_{-\\frac{t_w}{2} - t_r}^{\\frac{-t_w}{2}} \\left[ \\frac{1}{t_r} \\left(t + t_r + \\frac{t_w}{2} \\right)\r\n\\cos(n\\omega t) \\right] dt + \\right. \\nonumber $\r\n\r\n$latex + \\int_{-\\frac{t_w}{2}}^{\\frac{t_w}{2}} \\cos(n\\omega t) dt + \\nonumber $\r\n\r\n$latex + \\left. \\int_{\\frac{t_w}{2}}^{\\frac{t_w}{2} +t_r} \\left[ \\frac{1}{t_r} \\left(-t + t_r + \\frac{t_w}{2} \\right) \\cos(n\\omega t) \\right] dt \\right\\} = \\nonumber $\r\n\r\n$latex = 2A \\frac{t_w}{T} \\frac{\\sin\\left(n\\pi \\frac{t_w}{T} \\right)}{n\\pi\\frac{t_w}{T}}\r\n\\frac{\\sin\\left(n\\pi \\frac{t_r}{T} \\right)}{n\\pi\\frac{t_r}{T}}\r\n\\label{Trap_an} \\tag{3.29} $\r\n\r\nThe $latex 50\\%$ pulse width expressed as a ration of the period $latex t_w\/T$, present in Eq.(3.29) is also called duty cycle.\r\n\r\nFourier expansion of the trapezoidal waveform is as follows\r\n\r\n$latex u(t) = A \\frac{tw}{T} + 2A \\frac{t_w}{T} \\sum_{n=1}^{\\infty} \\left[ \\frac{\\sin\\left(n\\pi \\frac{t_w}{T} \\right)}{n\\pi \\frac{t_w}{T}}\r\n\\frac{\\sin\\left(n\\pi \\frac{t_r}{T} \\right)}{n\\pi \\frac{t_r}{T}}\r\n\\cos \\left(2 \\pi n f_1 t \\right) \\right]\r\n\\label{ut(t)}\\tag{3.30} $\r\n\r\nSelected Fourier expansions for $latex n = 1$, $latex n = 1, 2, ...,5$ and $latex n = 1, 2, ...,10$ of the trapezoidal waveform from Fig. 3.10<\/a> a) are presented in Fig. 3.10<\/a> b).\r\n\r\nCoefficients $latex a_n$ are values of continuous function $latex a(f)$ by spot frequencies $latex f_n = n f_1$, where $latex f_1 = 1\/T$ is fundamental harmonic of the waveform\r\n\r\n$latex a(f) = 2A\\frac{ t_w}{T} \\frac{\\sin\\left(\\pi t_w f \\right)}{\\pi t_w f} \\frac{\\sin\\left(\\pi t_r f \\right)}{\\pi t_r f}\r\n\\label{TEnvelope1}\\tag{3.31} $\r\n\r\nEq.(3.31) encompasses product of two $latex sinc$ functions $latex \\mathop{\\mathrm{sinc}}\\left(\\pi t_w f \\right)$ and $latex \\mathop{\\mathrm{sinc}}\\left(\\pi t_r f \\right)$.\r\n\r\nIn respect of the harmonics, the lowest frequency of function Eq.(3.31) which must be considered is $latex f_1$. Absolute function imposed on $latex a(f)$ of the waveform from Fig.3.10<\/a>a) in the frequency range from $latex f_1$ to infinity is represented with black continuous curve in Fig. 3.11<\/a>a). It is the envelope of absolute values of the Fourier coefficients $latex \\left|a_n \\right|$ for variability of the duty cycles. Its downward extrapolation is represented with black dashed line. Notable is its value by $latex f=0$, $latex a(0) = 2A\\frac{ t_w}{T}$. It is double value of the $latex a_0$ of the Fourier expansion.\r\n\r\nVery practical is representation of the frequency spectrum from Fig. 3.11<\/a>a) in the double logarithmic scale as shown in Fig. 3.11<\/a>b). The envelope is as follows\r\n\r\n$latex |U(f)| [dB] = 20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} + 20 \\log{\\left|\\frac{\\sin\\left(\\pi t_w f \\right)}{\\pi t_w f } \\right|} +\r\n20 \\log{\\left|\\frac{\\sin\\left(\\pi t_r f \\right)}{\\pi t_r f } \\right|}\r\n\\label{TEnvelope2}\\tag{3.32} $\r\n\r\nFor upper bound of the spectrum, $latex \\left|\\mathop{\\mathrm{sinc}}(\\pi t_w f )\\right|$ and $latex \\left|\\mathop{\\mathrm{sinc}}(\\pi t_r f )\\right|$ are set to one by low frequencies, and $latex \\left|\\sin(\\pi t_w f )\\right|$ and $latex \\left|\\sin(\\pi t_r f )\\right|$ are set to one by high frequencies. In the intermediate frequencies $latex \\left|\\mathop{\\mathrm{sinc}}(\\pi t_r f )\\right|$ is set to one and $latex \\left|\\sin(\\pi t_w f )\\right|$ is set to one yielding\r\n\r\n$latex |U_p(f)|= \\left\\{ \\begin{array} {ll}\r\n20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} & for~~f < \\frac{1}{\\pi t_w } \\\\\r\n20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} - 20 \\log{\\left( \\pi t_w f \\right)} & for~~\\frac{1}{\\pi t_w } < f < \\frac{1}{\\pi t_r } \\\\\r\n20 \\log{\\left( 2A\\frac{ t_w}{T}\\right)} - 20 \\log{\\left( \\pi t_w f \\right)} - 20 \\log{\\left( \\pi t_r f \\right)} & for~~f > \\frac{1}{\\pi t_r $ }\r\n\\end{array}\r\n\\right.\r\n\\label{Bounds}\\tag{3.33} $\r\n
\"\"\r\n\r\n[caption id=\"attachment_137\" align=\"aligncenter\" width=\"744\"]\"\" Figure 3.11: Frequency spectrum from Fig. 3.10 b) in double logarithmic scale.[\/caption]<\/figure>\r\nIn consequence the upper bound is composed of three segments of the strait line, horizontal by low frequencies, declined with the slope $latex -20dB\/decade$ by intermediate frequencies and declined with the slope $latex -40dB\/decade$ by high frequencies, see blue line in Fig.3.11<\/a>b). The corner frequencies depends on the pulse width $latex 1\/(\\pi t_w)$ and the pulse rise time $latex 1\/(\\pi t_r)$.\r\n\r\nZero cross frequencies marked in and Fig. 3.11<\/a>a) falls by multiple of inverse of the pulse width $latex 1\/t_w$ in the low and intermediate frequencies because term $latex \\left|\\mathop{\\mathrm{sinc}}(\\pi t_r f )\\right|$ is negligible. The number of harmonics between consecutive zero crossing in that range oscillates by reciprocal of the duty cycle. It is two or three for duty cycle equal to $latex 0.417$ as shown in Fig. 3.10<\/a> and Fig. 3.11<\/a>.\r\n\r\nThe frequency bandwidth $latex BW$ of the trapezoidal waveform defined as follows\r\n\r\n$latex BW = \\frac{1}{t_r}\r\n\\label{BW}\\tag{3.34} $\r\n\r\nis marked in Fig. 3.11<\/a>b). It seems to be very conservative, but if the pulse width approaches the rise time, such bandwidth ensures removal of the spectral lines which approximately are only 27 dB below the fundamental harmonic.\r\n

Damped oscillatory waveform<\/h3>\r\nVery ofter happens, that toggled DC voltage is followed with ringing (overshoot\/undershoot) due to parasitic inductances and capacitances, for instance in power cable connecting motor with the inverter or in PCB signal path. Analysis of the series RLC circuit supplied with rectangularly switched ON and OFF electromotive force, as shown in Fig.3.12<\/a> gives good insight into how this phenomenon is mirrored in the frequency domain.\r\n
\"\"\r\n\r\n[caption id=\"attachment_138\" align=\"aligncenter\" width=\"623\"]\"\" Figure 3.12: Ringing (undershoot\/overshoot) in series RLC circuit supplied with rectangularly switched ON and OFF electromotive force.[\/caption]\r\n\r\n
<\/figcaption><\/figure>\r\n
\"\"\r\n\r\n[caption id=\"attachment_139\" align=\"aligncenter\" width=\"741\"]\"\" Figure 3.13: Piece of period from Fig.3.12, damped oscillatory by: EMF ON a), EMF OFF b).[\/caption]<\/figure>\r\nPattern of damped oscillatory in the period shown in Fig.3.12<\/a> is as follows\r\n\r\n$latex u(t)= \\left\\{ \\begin{array} {ll}\r\nA - Ke^{-\\alpha \\left( t+\\frac{t_w}{2}\\right)} \\sin\\left[ 2\\pi f_r (t+\\frac{t_w}{2}) + \\varphi \\right] & \\text{for}~~\\frac{-t_w}{2 } < t < \\frac{t_w}{2 } \\\\\r\nKe^{-\\alpha \\left( t-\\frac{t_w}{2}\\right)} \\sin\\left[ 2\\pi f_r (t-\\frac{t_w}{2}) + \\varphi \\right] & \\text{for}~~\\frac{t_w}{2 } < t < T - \\frac{t_w}{2 } \r\n\\end{array}\r\n\\right.\r\n\\label{Ringing_t1}\\tag{3.35} $ \r\n\r\nwhere $latex A$ is amplitude of the EMF, $latex t_w$ is pulse width and $latex T$ pulse period as shown in Fig.3.12<\/a>.\r\n\r\nThe RLC circuit has resonance frequency $latex f_0$. By this frequency its impedance is shrunk to resistance $latex R$. Damping coefficient $latex \\alpha$ depends on resistance $latex R$ and inductance $latex L$.\r\n\r\n$latex \\begin{array} {lcr}\r\nf_0 = \\frac{1}{2\\pi} \\sqrt{LC} & ; & \\alpha = \\frac{R}{2L}\r\n\\label{f0alfa}\r\n\\end{array}\\tag{3.36} $\r\n\r\nDamped oscillatory frequency $latex f_d$ is combined with resonance frequency $latex f_0$ and damping coefficient $latex \\alpha$. After excitation (step change of EMF in this case) extinguishes, the circuit exhibits damped oscillations with frequency $latex f_d$, see Fig.3.13<\/a>. That\u2019s why $latex f_d$ is called also natural frequency. In that figure the rise time $latex t_r$ defined as time interval between transient initiation and peak value of the first overshoot is shown.\r\n\r\n$latex K$ depends on amplitude $latex A$, resonance frequency $latex f_0$ and natural frequency $latex f_d$. $latex \\varphi$ is the phase angle of damped oscillations.\r\n\r\n$latex \\begin{array} {lcccr}\r\nf_d = \\frac{1}{2\\pi} \\sqrt{4 \\pi^2 f_0^2 - \\alpha^2} & ; & K = \\frac{f_0}{f_d} A & ; & \\varphi = \\arctan\\left( \\frac{2\\pi f_d}{\\alpha} \\right)\r\n\\label{frKfi}\r\n\\end{array}\\tag{3.37} $\r\n
\"\"\r\n\r\n[caption id=\"attachment_140\" align=\"aligncenter\" width=\"769\"]\"\" Figure 3.14: Frequency spectrum of damped oscillatory from Fig.3.12: magnitude a), phase angle b).[\/caption]<\/figure>\r\nDamped oscillatory as shown in Fig.3.12<\/a> is overlapping of three time sequences: rectangle with amplitude A by EMF ON in the time interval $latex [-t_w\/2,t_w\/2]$ and zero by EMF OFF in the time interval $latex [t_w\/2,T - t_w\/2]$, up ringing in the time interval $latex [-t_w\/2,t_w\/2]$, see Fig.3.13<\/a>a) and down ringing in the time interval $latex [t_w\/2,T - t_w\/2]$ see Fig.3.13<\/a>b). Therefore coefficients of the Fourier expansion Eq.([an]<\/a>) and Eq.([bn]<\/a>) are sums of three integrals. Its complex representation with $latex C(f)$ as magnitude and phase angle is shown in Fig.3.14<\/a>.\r\n\r\nUpper bound of the spectrum, blue line in Fig.3.14<\/a>a) are in the lower frequency range the same as for rectangle toggling, i.e. segment of the strait horizontal line up to the corner frequency depending on the pulse width $latex 1\/(\\pi t_w)$ and line decreased with the slope $latex -20dB\/decade$ above the corner frequency, compare Fig.3.9<\/a>b).\r\n\r\nRemarkable in the spectrum is resonant region in a narrow frequency band. Quality factor and bandwidth of this region depends on $latex f_0$ and $latex \\alpha$. Maximum in this region is by ringing frequency $latex f_r$. In the case presented here ringing frequency $latex f_r$ is about forty times bigger than corner frequency and quality of resonance is moderate, therefore maximum of resonance is significantly below the horizontal upper bound. If corner frequency and $latex f_r$ are closer one to another and resonance has big quality, the resonant region can exceed the upper bound.\r\n\r\nAbove the resonant region upper bound decays with the slope approximately $latex \\sim (-60dB\/decade)$. Bandwidth frequency defined as usual $latex BW=1\/t_r$ is situated on this slope.\r\n
\"\"\r\n\r\n[caption id=\"attachment_141\" align=\"aligncenter\" width=\"506\"]\"\" Figure 3.15: Reconstruction of damped oscillatory from the frequency domain to the time domain.[\/caption]<\/figure>\r\nIn Fig.3.15<\/a> reconstruction of ringing from the frequency domain into the time domain with $latex N=500$ harmonics (red dotted line) together with the original (black line) ringing is shown. Added is reconstruction of the rectangle (blue line).\r\n

Single events - pulses<\/h2>\r\n

Spectral density of triangle pulse<\/h3>\r\n
\"\"\r\n\r\n[caption id=\"attachment_142\" align=\"aligncenter\" width=\"551\"]\"\" Figure 3.16: Triangle pulse with equal rise and fall time $latex tr=t_f$, a) and with bigger fall time $latex t_f>t_r$, b).[\/caption]<\/figure>\r\nThe Fourier integral of the triangle pulse with equal rise and fall time $latex t_r = t_f$, as shown in Fig.3.16<\/a>a) according to Eq.([Fourier_int]<\/a>) yields\r\n\r\n$latex u(f) = At_r \\mathop{\\mathrm{sinc}}^2\\left(\\pi t_r f \\right)\r\n\\label{Triangle_tr}\\tag{3.38} $\r\n\r\nThis spectrum density is a real function with value equal to the pulse area $latex At_r$ by zero frequency and has zero minima by multiples of the inverse of the pulse rise time $latex 1\/t_r$ as shown in Fig.3.17<\/a>a).\r\n
\"\"\r\n\r\n[caption id=\"attachment_143\" align=\"aligncenter\" width=\"662\"]\"\" Figure 3.17: Spectrum density of triangle pulse with equal rise and fall time in linear scale a) and in double logarithmic scale b).[\/caption]<\/figure>\r\nThe upper boundary of the spectrum density represented in double logarithmic scale is composed of horizontal line $latex 20 \\log\\left( At_r \\right)$, a slope $latex -40dB\/decade$ and the corner frequency by $latex f=1\/(\\pi t_r)$, see Fig.3.17<\/a>b).\r\n\r\n$latex u_p(f)= \\left\\{ \\begin{array} {ll}\r\n20 \\log\\left( At_r \\right) & \\mbox{ for f < \\frac{1}{\\pi t_r } }\\\\\r\n20 \\log\\left( At_r \\right) - 40 \\log{\\left( \\pi t_r f \\right)} & \\mbox{ for f > \\frac{1}{\\pi t_r } }\r\n\\end{array}\r\n\\right.\r\n\\label{Bounds_triangle}\\tag{3.39} $\r\n\r\nBandwidth estimation in the same way as for the trapezoidal wave form $latex BW=1\/t_r$ Eq.(3.4<\/a>), ensures removal of the spectral lines which are approximately 27 dB below the value of spectral density by zero frequency.\r\n\r\nThe Fourier integral of the triangle pulse with different rise and fall time $latex t_r \\neq t_f$, as shown in Fig.3.16<\/a>b) is complex function\r\n\r\n$latex \\begin{array} {lcl}\r\nRe[u(f)] &= \\frac{A}{2 \\pi^2 f^2} \\left[ \\frac{1}{t_r} \\sin^2(\\pi f t_r) + \\frac{1}{t_f} \\sin^2(\\pi f t_f) \\right] = \\\\\r\n & =\\frac{A t_r t_f}{2} \\left[ \\frac{1}{t_f} \\mathop{\\mathrm{sinc}}^2(\\pi f t_r) + \\frac{1}{t_r} \\mathop{\\mathrm{sinc}}^2(\\pi f t_f) \\right] \\\\\r\n Im[u(f)] & = \\frac{A}{2 \\pi^2 f^2} \\left[ \\frac{-1}{t_r} \\sin(\\pi f t_r) \\cos(\\pi f t_r) + \\frac{1}{t_f} \\sin(\\pi f t_f) \\cos(\\pi f t_f)\\right]\r\n\\end{array}\r\n\\label{Triangle_trf_f_Eq} \\tag{3.42} $\r\n\r\nIt is depicted in Fig.3.18<\/a>a) in single logarithmic scale in order to accentuate spectrum by high frequencies. This spectrum density is equal to area of the pulse $latex A(t_r+t_f)\/2$ by zero frequency since $latex \\mathop{\\mathrm{sinc}}^2(0)=1$.\r\n\r\nThe upper boundary of the absolute value of the spectrum density represented in double logarithmic scale can be established only by very low and very high frequencies. By very low frequencies imaginary part can be neglecter and the real part builds horizontal line $latex 20 \\log\\left[ A(t_r+t_f)\/2 \\right]$ as for zero frequency. It ends by corner frequency $latex 1\/(\\pi t_f)$. By very high frequencies which starts by another corner frequency $latex 1\/(\\pi t_r)$ again imaginary part can be neglecter and functions $latex \\sin^2(\\pi f t_r)$ and $latex \\sin^2(\\pi f t_f)$ in the real part can be replaced with 1. In consequence the upper bound decays with the slope $latex -40dB\/decade$, see Fig.3.18<\/a>b).\r\n\r\n$latex |u_p(f)|= \\left\\{ \\begin{array} {ll}\r\n20 \\log\\left( A \\frac{t_r + t_f}{2} \\right) & \\text{for}~~f < \\frac{1}{\\pi t_f } \\\\\r\n20 \\log\\left[ \\frac{A}{2 \\pi^2} \\left( \\frac{1}{t_r} + \\frac{1}{t_f} \\right)\\right] - 40 \\log{(f)} & \\text{for}~~f > \\frac{1}{\\pi t_r } \r\n\\end{array}\r\n\\right.\r\n\\label{Bounds_triangle_trf}\\tag{3.43} $\r\n
\"\"\r\n\r\n[caption id=\"attachment_144\" align=\"aligncenter\" width=\"746\"]\"\" Figure 3.18: Spectrum density of triangle pulse with different rise and fall time in single logarithmic scale a) and in double logarithmic scale b).[\/caption]<\/figure>\r\nBandwidth estimation in the same way as for the trapezoidal wave form $latex BW=1\/t_r$ Eq.(3.4<\/a>), is more conservative than for triangle pulse with equal rise and fall time.\r\n

Discharge of static electricity<\/h3>\r\nWhen two objects rub against each other, some charge (electrons) is transferred from material of one object to material of the other. When these two objects are separated thereafter, this charge may not return to the original one. If the two objects were originally neutral, they are now charged, one positively and the other negatively.\r\n\r\nThis method of generating static electricity is referred to as the triboelectric effect. Some materials readily absorb electrons while others tend to give them up easily. The triboelectric series is a listing of materials in order of their affinity for giving up electrons [@Ott]<\/span>.\r\n\r\nTable 3.2<\/a> is a typical triboelectric series. The materials at the top of the table easily give up electrons and therefore acquire a positive charge. The materials at the bottom of the table easily absorb electrons and therefore acquire a negative charge [@Ott]<\/span>.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Table 3.2: Triboelectric series<\/caption>\r\n
: {#Tri<\/th>\r\nboelectric se<\/th>\r\nries}<\/th>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
3.<\/td>\r\nAir<\/td>\r\n19.<\/td>\r\nSealing wax<\/td>\r\n<\/tr>\r\n
3.<\/td>\r\nHuman skin<\/td>\r\n20.<\/td>\r\nHard rubber<\/td>\r\n<\/tr>\r\n
3.<\/td>\r\nAsbestos<\/td>\r\n21.<\/td>\r\nMylar<\/td>\r\n<\/tr>\r\n
4.<\/td>\r\nRabbit fur<\/td>\r\n22.<\/td>\r\nEpoxy glass<\/td>\r\n<\/tr>\r\n
5.<\/td>\r\nGlass<\/td>\r\n23.<\/td>\r\nNickel, copper<\/td>\r\n<\/tr>\r\n
6.<\/td>\r\nHuman hair<\/td>\r\n24.<\/td>\r\nBrass, silver<\/td>\r\n<\/tr>\r\n
7.<\/td>\r\nMica<\/td>\r\n25.<\/td>\r\nGold, platinum<\/td>\r\n<\/tr>\r\n
8.<\/td>\r\nNylon<\/td>\r\n26.<\/td>\r\nPolystyrene foam<\/td>\r\n<\/tr>\r\n
9.<\/td>\r\nWool<\/td>\r\n27.<\/td>\r\nAcrylic rayon<\/td>\r\n<\/tr>\r\n
10.<\/td>\r\nFur<\/td>\r\n28.<\/td>\r\nOrlon<\/td>\r\n<\/tr>\r\n
11.<\/td>\r\nLead<\/td>\r\n29.<\/td>\r\nPolyester<\/td>\r\n<\/tr>\r\n
12.<\/td>\r\nSilk<\/td>\r\n30.<\/td>\r\nCelluloid<\/td>\r\n<\/tr>\r\n
13.<\/td>\r\nAluminium<\/td>\r\n31.<\/td>\r\nPolyurethane foam<\/td>\r\n<\/tr>\r\n
14.<\/td>\r\nPaper<\/td>\r\n32.<\/td>\r\nPolyethylene<\/td>\r\n<\/tr>\r\n
15.<\/td>\r\nCotton<\/td>\r\n33.<\/td>\r\nPolypropylene<\/td>\r\n<\/tr>\r\n
16.<\/td>\r\nWood<\/td>\r\n34.<\/td>\r\nPVC<\/td>\r\n<\/tr>\r\n
17.<\/td>\r\nSteel<\/td>\r\n35.<\/td>\r\nSilicon<\/td>\r\n<\/tr>\r\n
18.<\/td>\r\nAmber<\/td>\r\n36.<\/td>\r\nTeflon<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n
\r\n\r\nAmount of energy collected by triboelectric effect depends not only on the ordering of the materials in the series but also on the surface cleanliness, amount of rubbing, surface area in contact, smoothness of surface, the speed of separation and relative humidity [@Ott]<\/span>. By friction between the blades of the rotor of a helicopter or airfoil of airplane, between isolating wheels and ground floor voltage of static electricity can reach hundreds of kV.\r\n\r\nThe concern of this lecture notes is human discharge of static electricity which, depending on low\/high relative humidity can reach the level of [@Ott]<\/span>:\r\n