The University of Hong Kong
Department of Physics
Experimental Physics Laboratory
PHYS2255 Introductory Electricity and Magnetism
2255-2 LABORATORY MANUAL
Experiment 2: The A.C. Circuitry
This experiment illustrates the main properties of a simple resistance-capacitance (RC)
circuit, including the charge and discharge curves of capacitor through a resistor.
I.
RC circuit Charge and Discharge : Theoretical Results
A.
Charging to final voltage Vo (Constant voltage Vo is applied to the capacitor)
V
R
Vo
I
Vo
C
-1
1/2 Vo
t 1/2
(1-e )Vo = 0.632 Vo
2
t
At t 0 , the charge at the capacitor q(0)=0. After the switch is closed,
Vo V R VC i t R qt  C , where the current i t dqt  dt , so we get
𝑑𝑞(𝑡)
1
𝑉𝑜
+
𝑞(𝑡) =
(1)
𝑑𝑡
𝑅𝐶
𝑅
By solving (1) using the above initial condition, the charge and the voltage on the capacitor
can be obtained as:
𝑞(𝑡)
(2)
𝑞(𝑡) = 𝑉𝑜 𝐶(1 − 𝑒 −𝑡/𝜏 ),
𝑉𝐶 (𝑡) =
= 𝑉𝑜 (1 − 𝑒 −𝑡/𝜏 ),
𝐶
where 𝜏 = 𝑅𝐶 is called the time constant of the circuit. When 𝑉𝐶 (𝑡) reaches 𝑉𝑜 /2, 𝑡1/2 = 𝜏 ln 2.
B.
Discharge from initial voltage V1 (“Charge relaxation”)
When t 0 , the initial charge and voltage on the capacitor are q(0)=Q1, and V1 Q1 C ,
respectively. Now closing the switch of the circuit, then VR VC 0 , i.e.,
dqt  1
qt 0

dt
RC
By solving (3) using the initial condition, the voltage across the capacitor is
VC t q t  C V1e t / 
1
(3)
(4)
V
R
I
V1
C
-1
V1 e = 0.358 V 1
t 1/2
II.
2
t
RC Circuit Charge and Discharge : Experimental Details
In this experiment, we aim to test both the charging and discharging for short time
constant (order of milliseconds) RC circuits using a signal generator and a CRO. First please
check the internal resistance Ri of your signal generator using the method described in
part (H) of Laboratory 1 on both square and sine waves at frequency 800Hz. Set the Output
Multiplier to (10). Record your checked result at Output Adjust 5 (i.e. the peak-to-peak
amplitude of the trace) on the report sheet.
(A)
Displaying charge/discharge curves
Setup the following circuit, initially taking R = 10k, C = 0.1F. Note the use of a
double-pole switch to allow you to disconnect from the RC circuit and connect the signal
generator to the CRO. Be sure you understand the switch connections: they can be a little
confusing at first. Set the signal generator to square-wave.
2
1
1
2
Switc
hS
fig.1
O
Signal
Generator
CRO
O
If we apply a square wave to the RC circuit, during the positive half-cycle the
capacitor C will charge up according to equation (2); and during the negative half-cycle it
will discharge according to equation (4).
Choose a signal generator frequency such that the half-period of the square wave,
T 2 , equals to time constant  of the circuit. With the switch S in position 1, measure T 2
using the CRO time scale. Record the period T and hence the frequency f (=1/T).
Meanwhile, also record the nominal frequency read from the signal generator
2
Now turn the switch S to position 2 and adjust the VARIABLE setting. Change R by
a factor of at least 4 to make  T 2 , and then change R again to make  T 2 . You
should get displays as shown below:
(i)
1
T ~
2
e.g. R = 10k
(B)
(ii)
1
T >>
(iii)
1
T <<
2
e.g. R ~ 35k
2
e.g. R ~ 2k
Measurements of time-constant, 
It is clear that only in case (ii) above is the majority of the charge or discharge curve
observed. From such setting one can measure t1 2 as a fraction of T 2 , both from the charge
and the discharge cases, and deduce .
Note :
You can spread out the waveform to show only slightly over T 2 if you wish.
The ratio of t1 2 to T 2 is of course independent of the VARIABLE setting.
Now measure the time-constant in cases (i) and (ii) and record the results on the
report sheet. You should be able to get similar display as cases (i), (ii) and (iii) with different
input frequencies. In fact any combination of f, C, and R giving the same product
CR 2fCR (or the same ratio /T ) should result in the same display.
(C)
The "Differentiating" Circuit
Now return to your original frequency chosen in part (A) and R = 10k. Interchange
C and R in the circuit of Fig.1 and observe the shape of the waveform (i.e. the variation
in time of the potential difference) across R. Repeat for larger and smaller values of R and
sketch the various waveforms on the report sheet. You should find that with a small enough
R (and hence ) you get alternately positive and negative sharp spikes corresponding to each
sharp rise and fall of the square wave input voltage. Measure the height of the voltage spikes
Vp. You should find that Vp corresponds to Vs from the previous measurements.
For CR<<1, i.e., for small time-constant, the output waveform is approximately the
derivative of the input waveform. Hence this circuit is called a "differentiating" circuit. Note
that for the circuit of section (B) with CR>>1, the output waveform is approximately the
integral of the input waveform. That circuit is therefore called an "integrating" circuit.
3
(D)
Voltage across the Capacitor
(Please study the course Lecture Notes Chapter 10, or Chapter 29 of the textbook
Physics for Scientists and Engineers by Tipler & Mosca for theories on phasors and AC
circuits) From the phasor diagram of Fig. 2 below, it is readily shown that the voltage across
the capacitor Vco is given by Vco=VoX/Z, where X=1/C and Z 2 R 2 X 2 , so that
Vco 
Vo
(5)
( RC) 2 1
In this equation Vco and Vo may be both peak-to-peak voltage (more convenient when
using the CRO) or both rms (more convenient if using a voltmeter).
The phase difference between V and Vc is given by
tan  R X CR
(6)
The negative sign means that Vc lags behind V. Note that the instantaneous values of V is
given by Vo sin t , with  being negative.
R
RI
Signal
Generator
C
CRO
VR
-JI
wc
oI
Vc
V
Fig. 2
Take C=0.01F, R=20k, and choose a value of f such that CR is about 1, and
set up the circuit of Figure 1 with the signal generator on sine wave and set to your
chosen frequency. Record the values. According to the chosen values of f, R and C,
calculate the expected value of phase difference o with equation(6). Repeat this calculation
for a lower and a higher frequency (e.g. f/10, 10f). Complete Table 1 in the Report.
The phase difference  (representing the time difference between corresponding
maxima of V and Vc) can be measured directly in at least three different ways. Each of these
illustrates a new technique of using to CRO which you will need to learn and try out first. In
this experiment two ways are to be tried. In each case it is useful first to be able to see
qualitatively the difference between
(i)
a low frequency (e.g. f/10) for which ~0
(ii)
your chosen frequency f for which ~ /4 (one-eighth of a period)
(iii)
a higher frequency (e.g. 10f) for which ~ /2
Now you will try the first way. In this method, please use CRO to measure Vo by
switching S to position 1 according to the circuit shown in Figure 1. Then measure Vco by
switching S to position 2. The phase difference measured this way is denoted by 1.
Complete Table 2 in the Report.
4
(E)
Measure Phase Difference using External Triggering
If we use internal triggering, the waveforms of V and Vc start at the same point in their
own cycle and so no phase difference can be detected. If we instead use the external
triggering, the phase difference of V and Vo can be measured from the waveform shown on
the screen.
Connect the signal generator sine-wave output (V) to the EXT TRIG (EXT HOR)
input terminal, set the SOURCE to EXT, the COUPLING to AC, the SLOPE to +, and the
Level lock to "Lock position". Then when we switch Vc using switch S set to 2, the sweep
is still triggered by V, so the whole waveform is shifted, as shown below in Figure 3(ii),
which shows that Vc lags V, in this case by an angle of /4.
1
1
1
0.5
0.5
0.5
1
2
3
4
5
1
6
-0.5
-0.5
-1
-1
1
1
0.5
0.5
1
2
3
-0.5
4
5
6
2
3
4
5
6
1
4
5
6
d
1
0.5
1
2
3
4
5
1
6
2
3
4
5
6
-0.5
-1
-1
Fig. 3(i)
3
-1

-0.5
-1
2
-0.5
Fig. 3(ii)
Fig. 3(iii)
Try this : the phase difference is easily seen from the relative shift  in the positions
of the peak. Change to a much lower frequency such as f/10 and to a much higher
one such as 10f in turn, and see whether you can confirm the effects shown in the
figure above i.e. ~0 and ~/2 respectively. If the period corresponds to a distance d,
then the phase difference can be determined from  2 d .
Now, measure  by this method for the three frequencies used in the first
method in. section (D) above. The phase difference measured by this way is denoted
by 2. Complete Table 3 in the Report. Switch the SOURCE back to CH1 or CH2 and
disconnect the EXT TRIG terminal.
Note : We can also use the CH1 and CH2 terminals to show both waveforms
simultaneously. The method is : connect the signal generator Output (V) to the CH1(X)
terminal, and the Vc to the CH2(Y) terminal. All the above switches are set to the original.
Set the VERT MODE to DUAL and switch S to 2. Adjust the  position so that both
waveforms overlap. The phase difference can be easily found.
5
(F)
The X input terminal, and Lissajou’s Figures
One may wish to apply one’s own signal to the X plate instead of the internal sweep
waveform normally used to give a time base. To do this switch the Time/Div to X-Y
EXT HOR and VARIABLE to clockwise. And set the SOURCE to EXT.
Use the 6.3V 50Hz supply to the Y-input, and the signal generator O/P with
Output Range set at 1V to the EXT TRRIG input terminal at the upper part of the
CRO, and adjust Volt/Div and output multiplier to get the amplitudes of the two
signals roughly equal.
Caution : Note that the signal applied to EXT TRIG should not exceed 2V peak-to-peak.
Another method is : to apply the 6.3V 50Hz supply to the Y input CH1(X),
and the signal generator O/P to the CH2(Y). Set the Time/Div to X-Y EXT HOR, the
SOURCE to CH1 (X-Y), the VERT MODE to CH2 (X-Y). The same results can thus be
obtained.
Now adjust the signal generator frequency in turn to 25Hz, 50Hz, 75Hz, 100Hz, etc.
When the frequencies are in an exact ratio like 2:1, 3:2, etc, the pattern becomes stable.
The followings are some examples.
Ratio(1:2)
Ratio(2:1)
Ratio(2:3)
Ratio(1:1)
The number of antinodes (maxima) on each side indicates the frequency ratio.
This provides a method of adjusting a frequency to be in some exact ratio to a
standard or given one; and also a method of calibrating the Signal Generator dial at a
number of spot frequencies.
Tabulate the signal generator nominal and measured frequencies corresponding
to 25Hz, 50Hz, 75Hz, 100Hz and any others you wish, assuming the mains supply (and
hence the 6.3V transformer output) is exactly 50Hz while the nominal reading on the
signal generator is unreliable. Complete Table 4 in the Report.
6
Yet another method to try phase shift, (You may also want to give it a try!)
Apart from the methods described in parts (D) and (E) to find the phase difference ,
it can be done in the following ways:
(1) switch off the time base.
(2) connect Vc across the CRO input terminals (i.e. to the Y-plates)
(3) connect the signal generator output V directly to the EXT TRIG input terminal, or another
method as part (E).
If the amplitude of the two signals are the same, a diagram like the following can be
obtained. And the phase can be determined from sin  B A .
A=2y1
B=2
y sin
1
The basic theory of this method is described here. The x and y coordinates are given
by x  x1 cos t , y y1 cos t , where is the frequency of the Vc and signal from
signal generator. The distance B is just twice the y displacement at time when x 0 , such
as t   2 . At this time, y  y1 cos 2  y1 sin . So B 2 y1 sin  , as shown in
the figure. We also have A 2 y1 , and thus sin  can be obtained.
7