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Chapter 7 – Response of First-Order RL and RC Circuits
Study Guide
Objectives:
1. Be able to determine the natural response of both RL and RC circuits.
2. Be able to determine the step response of both RL and RC circuits.
3. Know how to analyze circuits with sequential switching.
4. Be able to analyze op amp circuits containing resistors and a single capacitor.
Mastering the Objectives:
1. Read the Introduction, Section 7.1, and Section 7.2.
a) Define the following
 Natural response;
 Step response;
 First-order circuit;
 Time constant.
b) The process of determining the natural response of an RL circuit can be broken
down into the following steps:
i.
Determine the initial value of the current in the inductor. This value may
be given to you in the problem statement. More often, you will need to
draw the circuit for t < 0 and use simple dc circuit analysis to find the
inductor current. Remember that in the circuit for t < 0 the inductor is
replaced by a short circuit and the only other components are resistors.
You need to find the current through the short circuit. Call this initial
inductor current Io.
ii. Determine the time constant  = L/R. To do this, draw the circuit for t > 0
and find the equivalent resistance seen by the inductor, using series and
parallel combinations of resistors if necessary.
iii. Write the expression for the current in the inductor: i(t) = Ioe-t/, t > 0.
iv.
Determine any other quantities of interest using the expression for i(t).
You may need Ohm’s law, current division, v = Ldi/dt, or other circuitanalysis techniques.
c) Practice the technique for determining the inductor current in an RL circuit on the
following:
i.
In Example 7.1, draw the circuit for t < 0 and calculate Io. Then draw the
circuit for t > 0 and calculate the time constant. Write the expression for
the inductor current.
ii. In the circuit for Assessment Problem 7.1, draw the circuit for t < 0 and
calculate Io. Then draw the circuit for t > 0 and calculate the time
constant. Use this analysis to complete Assessment Problem 7.1.
iii. In the circuit for Assessment Problem 7.2, draw the circuit for t < 0 and
calculate Io. Then draw the circuit for t > 0 and calculate the time
constant. Use this analysis to complete Assessment Problem 7.2.
iv.
In the circuit for Chapter Problem 7.1, draw the circuit for t < 0 and
calculate Io. Then draw the circuit for t > 0 and calculate the time
constant. Use this analysis to complete Chapter Problem 7.1.
d) Read Chapter Problem 7.2. What is a make-before-break switch? When must
you use a make-before-break switch in a circuit?
e) The process for determining the natural response of an RC circuit is analogous to
determining the natural response of an RL circuit. To summarize,
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i.
Determine the initial value of the voltage drop across the capacitor. This
value may be given to you in the problem statement. More often, you will
need to draw the circuit for t < 0 and use simple dc circuit analysis to find
the capacitor voltage. Remember that in the circuit for t < 0 the capacitor
is replaced by an open circuit and the only other components are resistors.
You need to find the voltage drop across the open circuit. Call this initial
capacitor voltage Vo.
ii. Determine the time constant  = RC. To do this, draw the circuit for t > 0
and find the equivalent resistance seen by the capacitor, using series and
parallel combinations of resistors if necessary.
iii. Write the expression for the voltage across the capacitor: v(t) = Voe-t/, t >
0.
iv.
Determine any other quantities of interest using the expression for v(t).
You may need Ohm’s law, current division, i = Cdv/dt, or other circuitanalysis techniques.
f) Practice the technique for determining the capacitor voltage for an RC circuit on
the following:
i.
In Example 7.3, draw the circuit for t < 0 and calculate Vo. Then draw the
circuit for t > 0 and calculate the time constant. Write the expression for
the capacitor voltage.
ii. In the circuit for Assessment Problem 7.3, draw the circuit for t < 0 and
calculate Vo. Then draw the circuit for t > 0 and calculate the time
constant. Use this analysis to complete Assessment Problem 7.3.
iii. In the circuit for Chapter Problem 7.22, draw the circuit for t < 0 and
calculate Vo. Then draw the circuit for t > 0 and calculate the time
constant. Use this analysis to complete Chapter Problem 7.22.
2. Read Sections 7.3 and 7.4.
a) Determining the step response of an RL or RC circuit is very similar to
determining the natural response, but one additional task is required – finding the
final value of the variable of interest. That is, you must find the value of the
variable as t  . The steps are as follows:
i.
Find the initial value from the circuit for t < 0;
ii. Find the final value from the circuit for t  ;
iii. Find the time constant from the circuit for t > 0;
iv.
Write the step-response expression using the initial value, the final value,
and the time constant.
Practice these steps on the following circuits:
i.
Figure 7.19;
ii. The circuit described in Assessment Problem 7.5;
iii. Figure P7.34;
iv.
Figure 7.22;
v.
Figure 7.22, but now assume that the switch is at position 2 for t < 0 and
switches to position 1 at t = 0;
vi.
Figure P7.47, where the behavior of the sources is described in the
problem statement.
b) What is the final value of the inductor current in a first-order RL circuit that has a
natural response, not a step response? (See Fig. 7.4). What is the final value of
the capacitor voltage in a first-order RC circuit that has a natural response, not a
step response? (See Fig. 7.11). Show that for these final values, the equation for
the step response (Eq. 7.35 for the RL circuit, Eq. 7.51 for the RC circuit) reduces
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to the equation for the natural response (Eq. 7.6 for the RL circuit, Eq. 7.22 for the
RC circuit).
c) In a circuit with a switch whose position changes at t = 0, what is the difference
between t = 0- and t = 0+? If this circuit contains an inductor, what is true about
the inductor current at t = 0- and t = 0+? Is this also true of the inductor voltage?
Why or why not? Suppose a circuit with a switch whose position changes at t = 0
has a capacitor. Make observations about the capacitor voltage and current at t =
0- and t = 0+ similar to those made about the circuit with the inductor.
3. Read Section 7.5.
a) Define sequential switching.
b) To solve a sequential switching problem, we follow the solution technique
developed for the natural and step response of the first-order RL and RC circuits
for the first switching time, t1. That is, we use one circuit for t < t1 to determine
the initial value of the current in the inductor or the voltage drop across the
capacitor. We use a second circuit for t > t1 to determine the final value and the
time constant. Note that if the circuit does not have an independent source for t >
t1 the final value is 0. Then we write the equation for the inductor current or the
capacitor voltage using the initial value, the final value, and the time constant. We
then evaluate the equation at t = t2 to determine the initial value of the variable for
the next time interval. We repeat the process by drawing the circuit for t > t 2 to
determine the final value and the time constant for the next time interval and write
the equation for inductor current or capacitor voltage using the initial value, final
value, and time constant for the new interval. If there is a third time interval
starting at t3 we evaluate the last expression at t3, draw the circuit for t > t3 and
repeat the process. Review Example 7.12 and draw the circuits for t < 0, 0  t <
15 ms, and t  15 ms to assist you in understanding this example.
c) Solve Assessment Problem 7.7 and Chapter Problem 7.70. Be sure to draw
circuits for the relevant time intervals to assist you.
4. Read Section 7.6.
a) Define the term “unbounded response.”
b) How could you tell that a response is unbounded by looking at the equation for a
voltage or a current?
c) Why is a circuit with only an independent source incapable of exhibiting an
unbounded response?
d) What is unusual about the Thevenin-equivalent resistance in a circuit with an
unbounded response?
e) If a real circuit has an unbounded response, what normally happens to the circuit
after some time has elapsed?
5. Read Section 7.7 (assuming you have studied Chapter 5).
a) The circuit in Fig. 7.40 is an integrating amplifier. Define “integrating.” Define
“amplifier.” The circuit might more accurately be called an inverting integrating
amplifier – why?
b) What determines the time interval for which the circuit in Fig. 7.40 behaves as an
inverting amplifier? What happens after that time interval?
c) If you want the circuit in Fig. 7.40 to integrate but not amplify, what must be true
about the values of Rs and Cf?
d) Solve Assessment Problems 7.9 and 7.10.
Assessing Your Mastery:
Review the Objectives for this unit. Once you are satisfied that you have achieved these
Objectives, take the Chapter 7 Test.