İzmir University of Economics
EEE 205 Fundamentals of Electrical Circuits Lab
EXPERIMENT 9
Step and Sinusiodal Responses of RC Circuits
A. Background
A.1. Step Response of RC Circuits
Consider the circuit given below in Fig. 9.1. The initial condition of the
capacitor voltage is given on the right.
R
+
VS
VC
C
VS(t) = VP
VC(0) = VC0
Fig. 9.1. A First Order RC Circuit
The differential equation that describes the behaviour of the capacitor voltage
is as follows:
The input is assumed a step function of amplitude VP as shown below.
VP
VS(t), volts
t, msec






Fig. 9.2. A First Order RC Circuit
Using several different approaches, the solution to the capacitor voltage may
be as obtained as
where  = RC
9-1
A detailed derivation can be found in Supplementary Notes – Part 1:
http://homes.ieu.edu.tr/~maskar/CE205/2010_11/CE205_2010_Supplement-P1.pdf
For the initial condition VC(0) = VC0 = 0 V, the input and output voltage
waveforms are plotted in Fig. 9.3.
VP
VS(t)
VC(t)

t, msec
Fig. 9.3. Input and Output Waveforms in a First Order RC Circuit
Now assume a periodic waveform with
> 5the output will also be periodic
since charging and discharging up to the final value is achieved at the end of
the high and low voltage durations.
VS(t), volts
VP




T

t, sec
Fig. 9.4. Periodic Input Waveforms
The input and output waveforms may be obtained using the similar approach
given in Supplementary Notes – Part 1:
http://homes.ieu.edu.tr/~maskar/CE205/2010_11/CE205_2010_Supplement-P1.pdf
The charging period, initial value
and during discharging,
VC0 = 0 V and
=
VP, then
-
VC0 = VP V and
=
0 V, then
-
Substituting t’ = t-T/2, it is obtained that
9-2
The plot of
VS and VC are given in Fig. 9.3.
VS(t)
VP
VC(t)
t, msec
Fig. 9.5. Periodic Input and Output Waveforms
A.2. Sinusoidal Response of RC Circuits
Assume a sinusoidal voltage is applied to the input of an RC circuit (Fig. 9.6).
R
+
VS
C
VC
-
Fig. 9.6. A First Order RC Circuit with Sinusoidal Input
The capacitor voltage is obtained as (see Class Notes)
-
where
-
For very large t, the first exponential term (called transient part) goes to zero.
Therefore for very large t, the output observed is
So for a sinusoidal input, the output is also a sinusoidal. However its amplitude
and phase has been changed. The amplitude is decreased by the factor
The phase of the output waveform is shifted by an angle
-
For
VSP = 10 V,
9-3
-
The input and output voltage waveforms are plotted in Fig. 9.7.
12
10
8
6
4
2
VS(t)
0
-2 0
45
90
135
180
225
270
315
360
405
VC(t)
-4
-6
-8
-10
-12
Fig. 9.7. Input and Output Waveforms at a First Order RC Circuit with Sinusoidal Input
9-4
B. Preliminary Work
1. Consider the circuit given below in Fig. 9.8. The input is a periodic 1 kHz
square wave as given below in Fig. 9.9.
R
R = 8.2 k
+
VS
C = 10 nF
VC
C
VC(0) = 0 V
Fig.9.8
5
vS(t), volts
t, msec












Fig.9.9
i. Determine the time constant .(
ii. Determine the charging behaviour of the output.
iii. Determine the discharging behaviour of the output.
9-5
iv. Plot the output on Fig. 9.10.
vS(t), volts
5
t, msec












Fig.9.10
2. Consider the circuit given below in Fig. 9.11. The input is a periodic
sinusoidal wave given in Fig. 9.12.
R
R = 15 k
+
VS
C
C = 10 nF
VC
Fig.9.11
12
10
8
6
4
2
0
-2 0
VS(t)
45
90
135
180
225
270
315
360
405
-4
-6
-8
-10
-12
Fig.9.12
i. Determine the frequency
at which
RC = 1(Note that
=2f0)?
9-6
-
ii. What is then
iii. What is then
?
?
iv. Plot VC(t) on Fig. 9.13.
12
10
8
6
4
2
0
-2 0
VS(t)
45
90
135
180
225
270
315
360
405
-4
-6
-8
-10
-12
Fig.9.13
9-7
C. Experimental Work
1. Construct the circuit given in Fig. 9.14.
i. Apply a square wave to the input. Adjust the amplitude to 5 V and
frequency to 1 kHz.
R
+
VS
C
VC
R = 8.2 k
C = 10 nF
Fig.9.14
ii. On the oscilloscope, measure both input and the output capacitor
voltage. Plot your waveforms on Fig. 9.15.
Time division:
………… ……./div
T = ……………….
f = ………………..
Channel 1 scale:
………… ……./div
Vp-p = ……………
Channel 2 scale:
………… ……./div
Vp-p = ……………
Fig.9.15
iii. Can you measure the time constant?
9-8
2. Construct the circuit given in Fig. 9.16.
i. Apply a sinusoidal input. Adjust the amplitude to 10 V. Adjust the
frequency so that phase difference between input and output is 450.
(f0 is close to 1 kHZ).
R
R = 15 k
+
VS
C
C = 10 nF
VC
Fig.9.16
ii. On the oscilloscope, measure both input and the output capacitor
voltage. Plot your waveforms on Fig. 9.17.
Time division:
………… ……./div
T = ……………….
f = ………………..
Channel 1 scale:
………… ……./div
Vp-p = ……………
Channel 2 scale:
………… ……./div
Vp-p = ……………
 = ……………….
Fig.9.17
9-9