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					          E245 Transient RL/RC and Thevenin Solution Example
                   Thevenin Equivalent Circuits and RC/RL Transient Circuits

The solutions to an earlier homework are provided here to illustrate how to solve general DC
circuit problems. Included is a circuit problem solved using Kirchoff's loop equations, along
with instructions on how to use MatLab to solve the resulting simultaneous equations.



       NOTE: THIS HOMEWORK: In problem 1, you will be reducing a circuit
       with a load resistance to a Thevenin equivalent circuit (Thevenin voltage source
       in series with Thevenin resistance) driving that load resistance. In problems 2
       and 3, you will be replacing that load resistance with first a capacitor and then an
       inductor. To solve problems 2 and 3, you will need to solve problem 1 first.
       If unable to solve problem 1, assume that the Thevenin voltage is 9 volts and
       that the Thevenin resistance is 3 ohms (these values are not the correct solution
       to problem 1)




Contents:

PROBLEM 1: THEVENIN EQUIVALENT CIRCUIT (40 POINTS) ..................................................................2

PROBLEM 2: RC CIRCUIT TRANSIENT ANALYSIS (30 POINTS) ................................................................3

PROBLEM 3: RL CIRCUIT TRANSIENT ANALYSIS (30 POINTS) .................................................................5
Problem 1: Thevenin Equivalent Circuit (40 points)
Figure 1a shows a circuit with a voltage source Vin, five resistors R1 through R5, and a "load"
resistance Rload that will be added to the circuit. The resistor and voltage source values are
given in Figure 1d.


               R1            R4                           R1               R4
                                                                                      VA

                                     RLoad
                R2                                             R2                          RLOAD
      Vin                   R5                      Vin                   R5

              R3                                          R3                          VB




                      (a)                                           (b)

                                                                     Resistance Values
                        RTh ev                                       R1 = 3 ohms
                                              VA
                                                                     R2 = 10 ohms
                                                                     R3 = 7 ohms
                                                                     R4 = 3 ohms
             VTh ev                              RLOAD
                                                                     R5 = 2 ohms
                                                                     Rload = 5 ohms

                                                                     Vin = 12 volts
                                               VB
                            (c )                                            (d)


                                              Figure 1
The Thevenin equivalent circuit concept allows the circuitry "seen" by the load resistance to be
replaced by a single voltage source, the Thevenin voltage VThe v , in series with a single resistance,
the Thevenin resistance RThe v . Figure 1b shows the approach used. The load resistance has been
taken out of the circuit and connected to the circuit shown in the gray-shaded area. Your task
here is to analyze the circuit in the gray-shaded area to determine the Thevenin voltage and the
Thevenin resistance. Figure 1c shows the general form of the Thevenin equivalent circuit, along
with the connections that can be made to the external load resistor.


1A: The Thevenin voltage is the voltage appearing across the terminals VA and VB with the
    load resistor removed. This is the so-called "open circuit output voltage." Using Kirchoff's
    loop equations (sum of voltages around a loop equals zero) or node equations (sum of
     currents into nodes equals zero, with one node labeled ground), determine the open circuit
     output voltage Vope n circuit  VA  VB . Remember that the load resistance is NOT connected.
     This voltage is the Thevenin voltage VThe v  Vope n circuit.
1B: The Thevenin resistance is the resistance between the nodes labeled VA and VB with the
    load resistance NOT connected. To calculate the resistance, you need to apply the
    technique discussed in class to remove the voltage source V1 (replace with either an open
    circuit or a short circuit). Show a picture of the circuit in the gray-shaded circuit of Figure
    1b with the voltage source V1 removed and show all of the steps used to determine the
    equivalent resistance appearing between the VA and VB nodes.
1C; Using the results from 1A and 1B above, show a picture of the Thevenin equivalent circuit
    with the load resistance connected. What is the output voltage Vout  VA  VB for a load
    resistance Rload = 10 ohms. Note that once you have determined the Thevenin circuit, it is
    easy to determine the output voltage for different load resistors.



Problem 2: RC Circuit Transient Analysis (30 Points)
Figure 2a shows the circuit in Figure 1 with the load resistor replaced by a capacitor with
capacitance 5 microFarads. The values of R1, R2, ... R5 are the same as in Problem 1. For the
problem here, the input voltage changes at time t = 0 from having been 3 volts (for a long time
before t = 0) to a value of 6 volts (remaining for a long time after t = 0).


                  R1              R4
                                                        V i n (t)


                                                          6 volt
                                       C
                      R2
      Vin (t)                     R5
                                                                    3 volt
                 R3

                                                                (0,0)                time t

                       (a)                                          (b)

                                             Figure 2


In this problem, you will determine the transient response of the voltage VC (t) across the
capacitor and the current IC (t) through the capacitor.
2A. Using the results from Problem 1, redraw the circuit in Figure 2a using the Thevenin
    equivalent circuit. Given the voltages Vin (t  0) and Vin (t  0) shown in Figure 2b, what
    are the Thevenin voltages VThev(t  0) and VThev(t  0) before and after t = 0.
    2B. Next, you will determine the voltage VC (t) across the capacitor and the current IC (t)
        through the capacitor before the input voltage changes For DC voltages, is a capacitor
        replaced by a short circuit (wire) or open circuit (no connection)? Look at the symbol for
        the capacitor to see whether the two sides are connected. Show the drawing of the circuit
        you obtained in 2A above for DC voltages with the capacitor replaced by a short or open
        circuit according to your decision.
    2C. Given your answers in 2A and 2B above, what are the values of VC (t  0) and IC (t  0)
        before the input voltage changes. The voltage across the capacitor just before t = 0 is
        VC (t  0 ) .
    2D. At t = 0, the input voltage switches abruptly from 3 volt to 6 volts, with a corresponding
        abrupt change of the Thevenin voltage. Specify whether each of the following can change
        abruptly or must remain initially unchanged ("sticks") at the time immediately after
        Vin (t) changes.
              Capacitor C.
                 Voltage across C?
                 Current through C?
              Thevenin Resistor R The v .
                 Voltage across R The v ?
                  Current through R The v ?
             Based on above, can the current around the circuit (Thevenin with load capacitor)
              change instantaneously?
                                                                                                    
    2E. Given your value of VC (t  0 ) just before t = 0, what is the value of the voltage VC (t  0 )
        just after t = 0.
    2F. Next consider the values of these voltages and the current as time extends to infinity. At
        that time, far away, the conditions are again DC conditions but now with an input voltage
                                                                                                 
        Vin (t  0)  6 volts. Using the same means by which you obtained the voltage VC (t  0 )
        across the capacitor just before the input voltage changed, what is the voltage VC (t  )
        across the capacitor long after the input voltage has changed.
 2G. The transient response will include an exponential term of the form exp(t /  ) where  is
       the time constant of the circuit. What is the equation for and value of this time constant for
       the RC circuit in Figure 2a.
    2H. What is the general form of the equation for the voltage VC (t  0) across the capacitor for
                                                        
        times after t = 0, given the values of VC (t  0 ) and VC (t  ) and the value of the time
        constant  for this RC circuits.
    2I.   What is the equation relating the current IC (t) through the capacitor to the voltage VC (t) ?
          Using this equation and your solution to 2H above, what is the equation for the current
          IC (t  0) ?
2J.   Roughly plot the voltage VC (t) showing its value before the input voltage switches, its
      values just after the input voltage switches, and its transitions to its values at time t ->
      infinity.



Problem 3: RL Circuit Transient Analysis (30 Points)
This problem repeats the questions for the RC circuit in Problem 2 above, but for an RL circuit
shown in Figure 3a. The resistors and voltage are the same as in problem 2. The value of the
inductor is L = 10 millihenries. As in Problem 2, the input voltage changes at time t = 0 from
having been 3 volts (for a long time before t = 0) to a value of 6 volts (remaining for a long time
after t = 0).


                                                        V i n (t)
                      R1              R4

                                                            6 volt
                                             L
                         R2
         Vin (t)                      R5
                                                                     3 volt
                    R3
                                                                (0,0)                   time t
                              (a)                                    (b)

                                                 Figure 3


In this problem, you will determine the transient response of the current IL (t ) across the inductor
and the voltage VL (t) across the inductor.
3A. Using the results from Problem 1, redraw the circuit in Figure 3a using the Thevenin
    equivalent circuit. Given the voltages Vin (t  0) and Vin (t  0) shown in Figure 2b, what
    are the Thevenin voltages VThev(t  0) and VThev(t  0) before and after t = 0.
3B. Next, you will determine the voltage VL (t) across the inductor and the current IL (t )
    through the inductor before the input voltage changes For DC voltages, is an inductor
    replaced by a short circuit (wire) or open circuit (no connection)? Look at the symbol for
    the inductor and remember that it is a coil of wire. Show the drawing of the circuit you
    obtained in 3A above for DC voltages with the inductor replaced by a short or open circuit
    according to your decision.
3C. Given your answers in 3A and 3B above, what are the values of VL (t  0) and IL (t  0)
    before the input voltage changes. I will represent the current through the inductor just
                             
    before t = 0 as IL (t  0 ) .
    3D. At t = 0, the input voltage switches abruptly from 3 volt to 6 volts, with a corresponding
        abrupt change of the Thevenin voltage. Specify whether each of the following can change
        abruptly or must remain initially unchanged ("sticks") at the time immediately after
        Vin (t) changes.
               Inductor L.
                  Voltage across L?
                  Current through L?
               Thevenin Resistor R The v .
                  Voltage across R The v ?
                   Current through R The v ?
             Based on above, can the current around the circuit (Thevenin with load inductor)
              change instantaneously?
                                                                                                    
    3E. Given your value of IL (t  0 ) just before t = 0, what is the value of the current IL (t  0 )
        just after t = 0.
    3F. Next consider the values of these voltages and the current as time extends to infinity. At
        that time, far away, the conditions are again DC conditions but now with an input voltage
        Vin (t  0)  6 volts. Using the same means by which you obtained the current IL (t  0  )
        through the inductor just before the input voltage changed, what is the current IL (t  )
        through the inductor long after the input voltage has changed.
 3G. The transient response will include an exponential term of the form exp(t /  ) where  is
       the time constant of the circuit. What is the equation for and value of this time constant for
       the RL circuit in Figure 3a?
    3H. What is the general form of the equation for the current IL (t  0) across the capacitor for
                                                        
        times after t = 0, given the values of IL (t  0 ) and IL (t  ) and the value of the time
        constant  for this RL circuits.
    3I.   What is the equation relating the voltage VL (t) across the inductor to the current IL (t )
          through the inductor? Using this equation and your solution to 3H above, what is the
          equation for the voltage VL (t  0)
    3J.   Roughly plot the current IL (t ) showing its value before the input voltage switches, its
          values just after the input voltage switches, and its transitions to its values at time t ->
          infinity.

				
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