{"id":20,"date":"2019-02-04T04:49:19","date_gmt":"2019-02-04T04:49:19","guid":{"rendered":"http:\/\/pb.ee.pw.edu.pl\/pb\/elap\/?post_type=chapter&p=20"},"modified":"2020-12-22T14:41:38","modified_gmt":"2020-12-22T14:41:38","slug":"exercise-4","status":"publish","type":"chapter","link":"http:\/\/pb.ee.pw.edu.pl\/pb\/elap\/chapter\/exercise-4\/","title":{"raw":"Exercise 4","rendered":"Exercise 4"},"content":{"raw":"
\r\n

SWITCHING OFF DC WITH USE OF TRANSISTOR<\/strong><\/p>\r\n\r\n<\/div>\r\n

Introduction<\/h2>\r\n
Semiconductor connectors are characterized by important advantages in relation to the contact switches. The switching process is implemented in them by changing the\u00a0 conductivity of the semiconductor connectors, and not, as is the case in the contact switches, by mechanical joining or separation of the contacts. When switching off, there is no problem to quench the arc, and then to rebuild the electrical strength of the pit channel. When switching on, there is no problem in welding or melting the contacts. The reliability of electronic components is much easier to achieve than in the case of complex mechanical devices, such as contact switches. They are also characterized by far greater, virtually unlimited, connection durability and connection speed. The electronic structure of semiconductor switches is a strong motivation for replacing them with contact switches wherever possible, i.e. where the maximum overvoltage and overcurrent values do not exceed the permissible values for semiconductor connectors.<\/div>\r\n
<\/div>\r\n
An important disadvantage of semiconductor switches, which prevents such a procedure, is their high inter-clamp resistance, which exceeds the magnitude of inter-clamp resistance of the contact switches, and leads to an increased release of thermal energy during the conductive current through the switch. This energy should not only be dissipate, but, above all, the customer has to pay for it. In addition, semiconductor switches are not in the state of open galvanic break in the circuit and require complex means to protect them against the consequences of switching processes, for which they are very sensitive. Semiconductor switches are currently only installed in such cases in which the benefits of their electronic construction outweigh the losses resulting from their high clamping resistance and the costs of necessary circuits and additional devices.<\/div>\r\n
<\/div>\r\n
An example of a system in which semiconductor switches are used, is a 3 kV direct current network (in Poland it is the PKP electric traction network), HVDC network (200 - 1100 kV), generators excitation (10 - 30 kV), or alternative sources energy (from several to several thousand volts). In this network, direct current creates difficult switching conditions for contact switches, due to the lack of a natural transition of the current through a zero value. It results in the necessity of forcing the passage of current through zero, which at high voltage is an extremely thankless task. Under such conditions, the contact switches are successfully replaced by semiconductor switches.<\/div>\r\n
<\/div>\r\n

Transistors IGBT<\/h2>\r\n
The\u00a0 Insulated\u00a0 Gate\u00a0 Bipolar\u00a0 Transistor\u00a0 (IGBT)\u00a0 is\u00a0 a\u00a0 minority-carrier\u00a0 device\u00a0 with\u00a0 high input impedance and large bipolar current-carrying capability. Many designers view IGBT as a device with MOS input characteristics and bipolar output characteristic that is a\u00a0 voltage-controlled\u00a0 bipolar\u00a0 device.\u00a0 To\u00a0 make\u00a0 use\u00a0 of\u00a0 the\u00a0 advantages\u00a0 of\u00a0 both\u00a0 Power MOSFET and BJT, the IGBT has been introduced. It\u2019s a functional integration of Power MOSFET and BJT devices in monolithic form. It combines the best attributes of both to achieve optimal device characteristics.<\/div>\r\n
The IGBT is suitable for many applications in power electronics, especially in Pulse Width Modulated (PWM) servo and three-phase drives requiring high dynamic range control and low noise. It also can be used in Uninterruptible Power Supplies (UPS), Switched-Mode Power Supplies (SMPS), and other power circuits requiring high switch repetition rates. IGBT improves dynamic performance and efficiency and reduced the level of audible noise. It is equally suitable in resonant-mode converter circuits. Optimized IGBT is available for both low conduction loss and low switching loss. The main advantages of IGBT over a Power MOSFET and a BJT are:<\/div>\r\n
    \r\n \t
  • It has a very low on-state voltage drop due to conductivity modulation and has superior on-state current density. So smaller chip size is possible and the cost can be reduced.<\/li>\r\n \t
  • Low driving power and a simple drive circuit due to the input MOS gate structure. It can be easily controlled as compared to current controlled devices (thyristor, BJT) in high voltage and high current applications.<\/li>\r\n \t
  • Wide SOA. It has superior current conduction capability compared with the bipolar transistor. It also has excellent forward and reverse blocking capabilities.<\/li>\r\n<\/ul>\r\n
    The main drawbacks are:<\/div>\r\n
      \r\n \t
    • Switching speed is inferior to that of a Power MOSFET and superior to that of a BJT. The collector current tailing due to the minority carrier causes the turn-off speed to be slow.<\/li>\r\n \t
    • There is a possibility of latchup due to the internal PNPN thyristor structure.<\/li>\r\n<\/ul>\r\n
      The basic structure of IGBT was shown on Fig. 1.<\/div>\r\n
      <\/div>\r\n
      \r\n\r\n[caption id=\"attachment_29\" align=\"alignnone\" width=\"605\"]\"\" Fig. 1. The basic structure of IGBT[\/caption]\r\n\r\n<\/div>\r\n
      <\/div>\r\n
      During the experiment the IGBT model PM600HSA120 (Fig. 2) will be used.<\/div>\r\n
      <\/div>\r\n
      \r\n\r\n[caption id=\"attachment_30\" align=\"alignnone\" width=\"278\"]\"\" Fig. 2. The IGBT model PM600HSA120[\/caption]\r\n\r\n<\/div>\r\n
      <\/div>\r\n
      The main nominal parameters of the IGBT model PM600HSA120:<\/div>\r\n
        \r\n \t
      • max voltage \u2013 1200 V<\/li>\r\n \t
      • max current \u2013 600 A<\/li>\r\n \t
      • max frequency \u2013 20 kHz<\/li>\r\n \t
      • max switching off time \u2013 3,5 \uf06ds<\/li>\r\n \t
      • switching time \u2013 0,5-3,5 \uf06ds<\/li>\r\n \t
      • max power of losses \u2013 3470 W<\/li>\r\n \t
      • max temperatur\u0119 \u2013 150\u2070C<\/li>\r\n \t
      • overload current \u2013 740-1000 A<\/li>\r\n \t
      • short-circuit current \u2013 higher than 1000 A<\/li>\r\n \t
      • time delay of overload signal \u2013 5 \uf06ds<\/li>\r\n<\/ul>\r\n

        The laboratory setup<\/h2>\r\n
        The laboratory setup scheme is shown on Fig. 3.<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_31\" align=\"alignnone\" width=\"440\"]\"\" Fig. 3. The laboratory setup scheme[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n

        Simulation model<\/h2>\r\n
        The Simulink model of a switching process with use of IGBT is shown on Fig. 43.<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_32\" align=\"alignnone\" width=\"585\"]\"\" Fig. 4. Simulink model of switching process with use of IGBT[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        The model was built from the blocks:<\/div>\r\n
        1) \u2018Repeating sequence\u2019 \u2013 Simulink\/Sources<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        2) DC Voltage Source<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        3) Capacitor \u2013 Series RLC branch<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        4) Diode<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        5) IGBT<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        6) Load RL \u2013 Series RLC branch<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        7) Sink block parameters \u2013 To workspace<\/div>\r\n
        \r\n

        \"\"<\/p>\r\n\r\n<\/div>\r\n

        <\/div>\r\n
        Matlab script to work with Simulink was added in the end of this instruction.<\/div>\r\n
        During the exercise both the voltage and current are observed on oscilloscope. On Fig. 5 the voltage and current versus time was shown. On Fig. 6 the voltage and current versus time was shown with markers.<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_40\" align=\"alignnone\" width=\"368\"]\"\" Fig. 5. Voltage and current versus time[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_41\" align=\"alignnone\" width=\"605\"]\"\" Fig. 6. Voltage and current versus time with markers[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        <\/div>\r\n
        On Fig. 6 the four switching processes are shown and can be described:<\/div>\r\n
        1) Process Z \u2013 switching on the load<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_42\" align=\"alignnone\" width=\"517\"]\"\" Fig. 7. Equivalent circuit and the current flow during the switching on[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        Current equation:<\/div>\r\n
        $$L\\frac{di}{dt}+Ri={{E}_{o}}$$, from zero initial conditions<\/div>\r\n
        Current transform:<\/div>\r\n
        $$I\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( Ls+R \\right)}$$<\/div>\r\n
        2) Process A - switching off the transistor: from cutting off current in transistor to the moment of zero voltage on switching capacitance<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_43\" align=\"alignnone\" width=\"519\"]\"\" Fig. 8. Equivalent circuit and the current flow during switching off the transistor[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        Current and voltage equations:<\/div>\r\n
        $$L\\frac{di}{dt}+Ri+u={{E}_{o}}$$<\/div>\r\n
        $$C\\frac{du}{dt}=i $$<\/div>\r\n
        for initial conditions: $$i\\left( 0 \\right)={{I}_{A0}}=\\frac{{{E}_{o}}}{R};u\\left( 0 \\right)=0$$,<\/div>\r\n
        Equations are valid up to the moment $${{t}_{Ae}}:u\\left( {{t}_{Ae}} \\right)=0$$<\/div>\r\n
        <\/div>\r\n
        Voltage transform:<\/div>\r\n
        $$U\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( LC{{s}^{2}}+RCs+1 \\right)}+\\frac{L{{I}_{A0}}}{LC{{s}^{2}}+RCs+1}$$<\/div>\r\n
        3) Process B \u2013 dissipation of magnetic energy: from the moment of zero voltage on switching capacitance (diode D starts conducting) up to the moment of termination of conduction through the diode D<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_44\" align=\"alignnone\" width=\"519\"]\"\" Fig. 9. Equivalent circuit and the current flow during dissipation of magnetic energy[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        Current equation:<\/div>\r\n
        $$L\\frac{di}{dt}+Ri={{E}_{o}}$$, przy $$I\\left( 0 \\right)={{I}_{Ae}}$$ - the current in the inductance at the end of the A process<\/div>\r\n
        Equation are valid up to the moment $${{t}_{Be}}:i\\left( {{t}_{Be}} \\right)=0$$<\/div>\r\n
        Current transform:<\/div>\r\n
        $$I\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( Ls+R \\right)}+\\frac{L{{I}_{Ae}}}{Ls+R}$$<\/div>\r\n
        4) Process C - reconstruction of the source voltage on the connector terminals: from the moment of termination of the conduction through the diode D to the component decay on the recovery voltage<\/div>\r\n
        <\/div>\r\n
        \r\n\r\n[caption id=\"attachment_45\" align=\"alignnone\" width=\"582\"]\"\" Fig. 10. Equivalent circuit and current flow during reconstruction of the source voltage on the connector terminals[\/caption]\r\n\r\n<\/div>\r\n
        <\/div>\r\n
        \r\n\r\nCurrent and voltage equations:\r\n\r\n<\/div>\r\n
        $$L\\frac{di}{dt}+Ri+u={{E}_{o}}$$<\/div>\r\n
        $$C\\frac{du}{dt}=i $$<\/div>\r\n
        for zero initial conditions Voltage transform:<\/div>\r\n
        $$U\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( LC{{s}^{2}}+RCs+1 \\right)}$$<\/div>\r\n
        <\/div>\r\n

        MATLAB program for simulation<\/h2>\r\n
        clear all;<\/code>\r\nclose all;<\/code>\r\nclcsim('igbt_instr'); % name of the Simulink scheme<\/code>\r\n\u00a0 t = u.time;<\/code>\r\n\u00a0 u = u.signals.values;<\/code>\r\n\u00a0 i = i.signals.values;% basic oscillograms<\/code>\r\n\u00a0 clf;<\/code>\r\n\u00a0 plot(t,u,'r',t,10*i,'k');grid;<\/code><\/div>\r\n ","rendered":"
        \n

        SWITCHING OFF DC WITH USE OF TRANSISTOR<\/strong><\/p>\n<\/div>\n

        Introduction<\/h2>\n
        Semiconductor connectors are characterized by important advantages in relation to the contact switches. The switching process is implemented in them by changing the\u00a0 conductivity of the semiconductor connectors, and not, as is the case in the contact switches, by mechanical joining or separation of the contacts. When switching off, there is no problem to quench the arc, and then to rebuild the electrical strength of the pit channel. When switching on, there is no problem in welding or melting the contacts. The reliability of electronic components is much easier to achieve than in the case of complex mechanical devices, such as contact switches. They are also characterized by far greater, virtually unlimited, connection durability and connection speed. The electronic structure of semiconductor switches is a strong motivation for replacing them with contact switches wherever possible, i.e. where the maximum overvoltage and overcurrent values do not exceed the permissible values for semiconductor connectors.<\/div>\n
        <\/div>\n
        An important disadvantage of semiconductor switches, which prevents such a procedure, is their high inter-clamp resistance, which exceeds the magnitude of inter-clamp resistance of the contact switches, and leads to an increased release of thermal energy during the conductive current through the switch. This energy should not only be dissipate, but, above all, the customer has to pay for it. In addition, semiconductor switches are not in the state of open galvanic break in the circuit and require complex means to protect them against the consequences of switching processes, for which they are very sensitive. Semiconductor switches are currently only installed in such cases in which the benefits of their electronic construction outweigh the losses resulting from their high clamping resistance and the costs of necessary circuits and additional devices.<\/div>\n
        <\/div>\n
        An example of a system in which semiconductor switches are used, is a 3 kV direct current network (in Poland it is the PKP electric traction network), HVDC network (200 – 1100 kV), generators excitation (10 – 30 kV), or alternative sources energy (from several to several thousand volts). In this network, direct current creates difficult switching conditions for contact switches, due to the lack of a natural transition of the current through a zero value. It results in the necessity of forcing the passage of current through zero, which at high voltage is an extremely thankless task. Under such conditions, the contact switches are successfully replaced by semiconductor switches.<\/div>\n
        <\/div>\n

        Transistors IGBT<\/h2>\n
        The\u00a0 Insulated\u00a0 Gate\u00a0 Bipolar\u00a0 Transistor\u00a0 (IGBT)\u00a0 is\u00a0 a\u00a0 minority-carrier\u00a0 device\u00a0 with\u00a0 high input impedance and large bipolar current-carrying capability. Many designers view IGBT as a device with MOS input characteristics and bipolar output characteristic that is a\u00a0 voltage-controlled\u00a0 bipolar\u00a0 device.\u00a0 To\u00a0 make\u00a0 use\u00a0 of\u00a0 the\u00a0 advantages\u00a0 of\u00a0 both\u00a0 Power MOSFET and BJT, the IGBT has been introduced. It\u2019s a functional integration of Power MOSFET and BJT devices in monolithic form. It combines the best attributes of both to achieve optimal device characteristics.<\/div>\n
        The IGBT is suitable for many applications in power electronics, especially in Pulse Width Modulated (PWM) servo and three-phase drives requiring high dynamic range control and low noise. It also can be used in Uninterruptible Power Supplies (UPS), Switched-Mode Power Supplies (SMPS), and other power circuits requiring high switch repetition rates. IGBT improves dynamic performance and efficiency and reduced the level of audible noise. It is equally suitable in resonant-mode converter circuits. Optimized IGBT is available for both low conduction loss and low switching loss. The main advantages of IGBT over a Power MOSFET and a BJT are:<\/div>\n
          \n
        • It has a very low on-state voltage drop due to conductivity modulation and has superior on-state current density. So smaller chip size is possible and the cost can be reduced.<\/li>\n
        • Low driving power and a simple drive circuit due to the input MOS gate structure. It can be easily controlled as compared to current controlled devices (thyristor, BJT) in high voltage and high current applications.<\/li>\n
        • Wide SOA. It has superior current conduction capability compared with the bipolar transistor. It also has excellent forward and reverse blocking capabilities.<\/li>\n<\/ul>\n
          The main drawbacks are:<\/div>\n
            \n
          • Switching speed is inferior to that of a Power MOSFET and superior to that of a BJT. The collector current tailing due to the minority carrier causes the turn-off speed to be slow.<\/li>\n
          • There is a possibility of latchup due to the internal PNPN thyristor structure.<\/li>\n<\/ul>\n
            The basic structure of IGBT was shown on Fig. 1.<\/div>\n
            <\/div>\n
            \n
            \"\"
            Fig. 1. The basic structure of IGBT<\/figcaption><\/figure>\n<\/div>\n
            <\/div>\n
            During the experiment the IGBT model PM600HSA120 (Fig. 2) will be used.<\/div>\n
            <\/div>\n
            \n
            \"\"
            Fig. 2. The IGBT model PM600HSA120<\/figcaption><\/figure>\n<\/div>\n
            <\/div>\n
            The main nominal parameters of the IGBT model PM600HSA120:<\/div>\n
              \n
            • max voltage \u2013 1200 V<\/li>\n
            • max current \u2013 600 A<\/li>\n
            • max frequency \u2013 20 kHz<\/li>\n
            • max switching off time \u2013 3,5 \uf06ds<\/li>\n
            • switching time \u2013 0,5-3,5 \uf06ds<\/li>\n
            • max power of losses \u2013 3470 W<\/li>\n
            • max temperatur\u0119 \u2013 150\u2070C<\/li>\n
            • overload current \u2013 740-1000 A<\/li>\n
            • short-circuit current \u2013 higher than 1000 A<\/li>\n
            • time delay of overload signal \u2013 5 \uf06ds<\/li>\n<\/ul>\n

              The laboratory setup<\/h2>\n
              The laboratory setup scheme is shown on Fig. 3.<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 3. The laboratory setup scheme<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n

              Simulation model<\/h2>\n
              The Simulink model of a switching process with use of IGBT is shown on Fig. 43.<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 4. Simulink model of switching process with use of IGBT<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              The model was built from the blocks:<\/div>\n
              1) \u2018Repeating sequence\u2019 \u2013 Simulink\/Sources<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              2) DC Voltage Source<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              3) Capacitor \u2013 Series RLC branch<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              4) Diode<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              5) IGBT<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              6) Load RL \u2013 Series RLC branch<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              7) Sink block parameters \u2013 To workspace<\/div>\n
              \n

              \"\"<\/p>\n<\/div>\n

              <\/div>\n
              Matlab script to work with Simulink was added in the end of this instruction.<\/div>\n
              During the exercise both the voltage and current are observed on oscilloscope. On Fig. 5 the voltage and current versus time was shown. On Fig. 6 the voltage and current versus time was shown with markers.<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 5. Voltage and current versus time<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 6. Voltage and current versus time with markers<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              <\/div>\n
              On Fig. 6 the four switching processes are shown and can be described:<\/div>\n
              1) Process Z \u2013 switching on the load<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 7. Equivalent circuit and the current flow during the switching on<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              Current equation:<\/div>\n
              $$L\\frac{di}{dt}+Ri={{E}_{o}}$$, from zero initial conditions<\/div>\n
              Current transform:<\/div>\n
              $$I\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( Ls+R \\right)}$$<\/div>\n
              2) Process A – switching off the transistor: from cutting off current in transistor to the moment of zero voltage on switching capacitance<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 8. Equivalent circuit and the current flow during switching off the transistor<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              Current and voltage equations:<\/div>\n
              $$L\\frac{di}{dt}+Ri+u={{E}_{o}}$$<\/div>\n
              $$C\\frac{du}{dt}=i $$<\/div>\n
              for initial conditions: $$i\\left( 0 \\right)={{I}_{A0}}=\\frac{{{E}_{o}}}{R};u\\left( 0 \\right)=0$$,<\/div>\n
              Equations are valid up to the moment $${{t}_{Ae}}:u\\left( {{t}_{Ae}} \\right)=0$$<\/div>\n
              <\/div>\n
              Voltage transform:<\/div>\n
              $$U\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( LC{{s}^{2}}+RCs+1 \\right)}+\\frac{L{{I}_{A0}}}{LC{{s}^{2}}+RCs+1}$$<\/div>\n
              3) Process B \u2013 dissipation of magnetic energy: from the moment of zero voltage on switching capacitance (diode D starts conducting) up to the moment of termination of conduction through the diode D<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 9. Equivalent circuit and the current flow during dissipation of magnetic energy<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              Current equation:<\/div>\n
              $$L\\frac{di}{dt}+Ri={{E}_{o}}$$, przy $$I\\left( 0 \\right)={{I}_{Ae}}$$ – the current in the inductance at the end of the A process<\/div>\n
              Equation are valid up to the moment $${{t}_{Be}}:i\\left( {{t}_{Be}} \\right)=0$$<\/div>\n
              Current transform:<\/div>\n
              $$I\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( Ls+R \\right)}+\\frac{L{{I}_{Ae}}}{Ls+R}$$<\/div>\n
              4) Process C – reconstruction of the source voltage on the connector terminals: from the moment of termination of the conduction through the diode D to the component decay on the recovery voltage<\/div>\n
              <\/div>\n
              \n
              \"\"
              Fig. 10. Equivalent circuit and current flow during reconstruction of the source voltage on the connector terminals<\/figcaption><\/figure>\n<\/div>\n
              <\/div>\n
              \n

              Current and voltage equations:<\/p>\n<\/div>\n

              $$L\\frac{di}{dt}+Ri+u={{E}_{o}}$$<\/div>\n
              $$C\\frac{du}{dt}=i $$<\/div>\n
              for zero initial conditions Voltage transform:<\/div>\n
              $$U\\left( s \\right)=\\frac{{{E}_{o}}}{s\\left( LC{{s}^{2}}+RCs+1 \\right)}$$<\/div>\n
              <\/div>\n

              MATLAB program for simulation<\/h2>\n
              clear all;<\/code>
              \nclose all;<\/code>
              \nclcsim('igbt_instr'); % name of the Simulink scheme<\/code>
              \n\u00a0 t = u.time;<\/code>
              \n\u00a0 u = u.signals.values;<\/code>
              \n\u00a0 i = i.signals.values;% basic oscillograms<\/code>
              \n\u00a0 clf;<\/code>
              \n\u00a0 plot(t,u,'r',t,10*i,'k');grid;<\/code><\/div>\n

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