Experiment6: Response of First Order RL and RC
Circuits
1
Objectives
In this experiment the natural and step responses of RL and RC circuits are examined. The use of computer
controlled equipment is also introduced here.
2
Introduction and Test Circuits
Inductors and capacitors have the ability to store energy. It is important to determine the voltages and
currents that arise in circuits composed by resistors, and either inductors or capacitors, when energy is
released or acquired by the inductor or capacitor as a consequence of an abrupt change in the DC voltage or
current in the circuit. The description of the voltages and currents in this type of circuits is done in terms
of differential equations of first order.
2.1
Natural response
The currents and voltages that arise when the energy stored in an inductor or capacitor is suddenly released
to the resistors in the circuit are referred to as the natural response of the circuit. The behavior of these
currents and voltages depends only on the nature of the circuit, and not on external sources of excitation.
2.1.1
Natural response of an RL circuit
In an RL circuit, the natural response is described in terms of the voltage and current at the terminals of
the resistor when the external source of power stops delivering energy to the circuit. The expressions for the
current and voltage across the resistor are:
i(t) = Io e−t/τ , t ≥ 0
(6-1)
v(t) = Io Re−t/τ , t ≥ 0
(6-2)
where Io is the initial current through the inductor before the power source goes off and the inductor starts
releasing energy to the circuit.
The symbol τ represents the time constant of the circuit,
τ = L/R
(6-3)
which determines the rate at which the current or voltage decays to zero.
An RL circuit is shown in Fig. 6-1. Here, Vs provides a square signal with a DC offset voltage such that the
bottom part of the waveform is aligned with the zero volts level. Rs is the internal resistance of the voltage
source, hence when the square wave takes the value of zero volts (and can be viewed as a short circuit) the
energy of the inductor L is released through the combination of R1 and Rs . In order to observe the natural
response of the circuit, the period T of the square wave must be long enough to allow the complete charge
and discharge of the inductor. Usually T = 20τ is appropriate for this purpose.
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Summarizing, the natural response of an RL circuit is calculated by (1) finding the initial current Io through
the inductor, (2) finding the time constant of the circuit (Eq. 6-3), and (3) using Eq. 6-1 to generate i(t).
2.1.2
Natural response of an RC circuit
The natural response of an RC circuit is analogous to that of an RL circuit. The expressions for the current
and voltage across the resistor are:
(6-4)
v(t) = Vo e−t/τ , t ≥ 0
Vo −t/τ
i(t) =
e
, t≥0
(6-5)
R
τ = RC
(6-6)
where Vo is the initial voltage across the (fully charged) capacitor before the power source is switched off and
the capacitor starts releasing energy to the circuit. An RC circuit is shown in Fig. 6-2. The same concepts
discussed for the circuit of Fig. 6-1 are applied here.
Summarizing, the natural response of an RC circuit is calculated by (1) finding the initial voltage Vo across
the capacitor, (2) finding the time constant of the circuit (Eq. 6-6), and (3) using Eq. 6-4 to generate v(t).
2.2
Step response
The response of a circuit to the sudden application of a constant voltage or current source is referred to as
the step response of the circuit. This case presents the opposite conditions of the natural response. Now, in
RL or RC circuits, the inductor or capacitor (assumed to be completely discharged) begins acquiring energy
after a sudden application of an external power source. The voltages and currents that arise in the circuit
under these conditions are discussed next.
2.2.1
Step response of an RL circuit
In an RL circuit the initial conditions to determine the step response are assumed to be Io =0. The expressions
for the current in the circuit and the voltage across the inductor after the voltage source is applied are:
i(t) =
Vs
(1 − e−t/τ )
R
v(t) = Vs e−t/τ
(6-7)
(6-8)
Notice that Eq. 6-7 indicates that the current increases from zero to a final value of Vs /R at a rate determined
by the time constant τ = L/R.
The circuit of Fig. 6-1 can be used to determine the step response providing that the period of the square
wave is long enough to allow the complete charge and discharge of the inductor in successive cycles. For this
circuit, R = Rs + R1 .
2.2.2
Step response of an RC circuit
In an RC circuit (see Fig. 6-3) the initial voltage across the capacitor is assumed to be Vo =0. The expressions
for the current and voltage in the capacitor after the current source is applied are:
v(t) = Is R(1 − e−t/τ )
31
(6-9)
i(t) = Is e−t/τ
(6-10)
Notice that Eq. 6-9 indicates that the voltage increases from zero to a final value of Is R at a rate determined
by the time constant τ = RC.
By applying a source transformation to the circuit of Fig. 6-3, the circuit of Fig. 6-2 can be used to determine
the step response provided that Vs = Is R, R = Rs + R2 , and the period of the square wave is long enough
to allow the complete charge and discharge of the capacitor in successive cycles.
3
Preparation
In preparation for the lab do the following activities:
1. Natural response of an RL circuit. Consider the circuit shown in Fig. 6-1 and the component values of
Table 6-1. Determine Io , R, τ and the expression for the current i(t) through the inductor. Generate
a computer graph of i(t) versus time in the interval from 0 to 7τ seconds in increments of 7τ /100
seconds.
2. Natural response of an RC circuit. Consider the circuit shown in Fig. 6-2 and the component values of
Table 6-2. Determine Vo , R, τ and the expression for the voltage v(t) across the capacitor. Generate
a computer graph of v(t) versus time in the interval from 0 to 7τ seconds in increments of 7τ /100
seconds.
3. Step response of an RL circuit. Consider the circuit shown in Fig. 6-1 and the component values of
Table 6-3. Determine R, τ and the expression for the current i(t) through the inductor. Generate a
computer graph of i(t) versus time in the interval from 0 to 7τ seconds in increments of 7τ /100 seconds.
4. Step response of an RC circuit. Consider the circuit shown in Fig. 6-2 and the component values of
Table 6-4. Determine R, Is (in terms of Vs and R by applying a source transformation), τ and the
expression for the voltage v(t) across the capacitor. Generate a computer graph of v(t) versus time in
the interval from 0 to 7τ seconds in increments of 7τ /100 seconds.
4
Procedure
This part of the experiment requires assembling the circuits discussed in the previous section and measuring data from all of them. Refer to the appendices regarding the use of the equipment and breadboard.
The use of a computer to control the equipment is introduced here. It requires running a program known
as a VI (virtual instrument) from the computer desktop, entering the desired parameters and collecting data.
Equipment settings
• Function generator. Type of waveform: Square wave. Amplitude=5 V. Offset=2.5 V. Frequency:
determined by circuit parameters. Notice that the actual amplitude of the signal is 10 V.
• Oscilloscope. External trigger: from the SYNC out terminal of the function generator. Channel 2:
disconnected (disconnect the probe from channel 2).
• Computer. Enter the computer network password at start-up. Load channel1.vi (a shortcut is located
in the desktop). A floppy disk is required to collect data.
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The procedure necessary to determine the circuit responses experimentally is described next. Measure all
the components and report their actual values in the corresponding entries of Tables 6-1 through 6-4.
1. Natural response of an RL circuit.
• Using the component values of Table 6-1, assemble the circuit shown in Fig. 6-1. To avoid
ground conflicts with the equipment, connect the inductor to the positive terminal of the function
generator and the resistor to the negative terminal (it is a series circuit, so the order of connection
is irrelevant).
• Set the frequency of the function generator to about 0.05/τ Hz.
• Connect the oscilloscope probe of channel 1 across the resistor, this will give a scaled version (by a
factor of R1 ) of the current through the inductor. Press Autoscale . Press Main/Delayed and
select Lft under Time Ref from the soft menu in the oscilloscope screen. Press Slope/Coupling
and select the second slope in the soft menu (going from high to low). Adjust the Time/Div knob
to the closest value to τ . Notice that the amount of time per division is shown on top of the
m
oscilloscope display. For example, 20.0 s / means the time span displayed in the screen is (20
ms)×10 equal to 200 ms. At this point the natural response of the RL circuit should be displayed
in a form that is comparable with the plot created in the Preparation section.
• In the front panel of the virtual instrument channel1, enter a suitable data filename, i.e., A:\RL.dat.
Use always drive A to store the datafiles. Move the switch to ON to allow the recording of the
data. Run the virtual instrument by pressing Ctrl-R in the computer keyboard. Print a hard
copy of the computer screen by pressing Ctrl-P.
2. Natural response of an RC circuit.
(a) Repeat the above procedure to obtain the natural response of the RC circuit with the following
modifications. Use the circuit of Fig. 6-2 and the component values of Table 6-2. Connect the oscilloscope probe across the capacitor. Use a different filename to store the data.
(b) Keeping the equipment settings unchanged, connect the resistor R2 in parallel with the capacitor.
Observe the waveform decreasing the time span by a factor of 20. Record and print this response using
a different filename.
3. Step response of an RL circuit. Repeat the procedure of (1) to obtain the step response of an RL circuit
with the following modifications. Use the circuit of Fig. 6-1 and the component values of Table 6-3.
Select the first slope (going from low to high). Use a different filename to store the data.
4. Step response of an RC circuit. Repeat the procedure of (1) to obtain the step response of an RC circuit
with the following modifications. Use the circuit of Fig. 6-2 and the component values of Table 6-4.
Connect the oscilloscope probe across the capacitor. Select the first slope (going from low to high).
Use a different filename to store the data.
5
Analysis
This section is intended for the analysis and comparison of the experimental and theoretical results. Answer
all the questions.
1. Estimate the time constant for all the circuits by using two coordinates of their natural (or step)
responses (from the data files) during the discharge (or charge) and the following expressions, for the
natural response:
t2 − t1
(6-11)
τ=  ‘
ln VV12
33
Parameter
Vs
Rs
R1
L
τ
Unit
V
Ω
Ω
mH
µs
Theor
10
50
270
0.64
Exper
%Error
Table 6-1: RL circuit values to determine the natural response.
Parameter
Vs
Rs
R2
C
τ
Unit
V
Ω
Ω
µF
µs
Theor
10
50
1000
0.1
Exper
%Error
Table 6-2: RC circuit values to determine the natural response.
for the step response:
τ=
t2 − t1
 ‘
ln VV21
(6-12)
Complete the entries in Tables 6-1 through 6-4. Compare the estimated time constants to the theoretical
time constants for each circuit.
2. Compare the theoretical and experimental plots of all the responses. Explain any differences between
them.
3. Estimate τ from the response in 2(b) of the Procedure section. To explain the discrepancy with the
result of part 2(a) show a schematic of the equivalent circuit when the function generator output is
zero. Calculate (analytically) the time constant for this circuit and compare it with the estimated
value.
Parameter
Vs
Rs
R1
L
τ
Unit
V
Ω
Ω
µH
ns
Theor
10
50
220
56
Exper
%Error
Table 6-3: RL circuit values to determine the step response.
34
Parameter
Vs
Rs
R2
C
τ
Unit
V
Ω
Ω
µF
µs
Theor
10
50
470
0.01
Exper
%Error
Table 6-4: RC circuit values to determine the step response.
Vs
Rs
R1
L
Figure 6-1: RL circuit.
Vs
Rs
R2
C
Figure 6-2: RC circuit.
t=0
Is
+
v
-
R
i
C
Figure 6-3: RC circuit for step response.
35