General Physics Experiment Handout
RC Circuits
Purpose
The purpose of this exercise is to use an oscilloscope and a function generator to observe the
charging and discharging periods of RC circuits, as well as the effects of the capacitance of the capacitor
in the circuit on the time constant.
Principles
Assume the capacitor in the circuit (shown in Figure 1) is initially electrically neutral. When
Switch S is thrown to position a, a circuit consisting of a battery, a resistor, and a capacitor is formed. A
charge begins to flow in the circuit, and the capacitor begins to charge as the charges of opposite signs
accumulate on the two plates.
However, if the switch is thrown to position b, this new circuit contains only a capacitor and a
resistor. The charges of opposite signs accumulated in the capacitor now discharge by flowing through
the wires and resistor, and become neutralized.
The following is the quantitative analysis of the charging and discharging periods:
Figure 1. RC circuit (Halliday: Fundamentals of Physics, 8th ed.）
A.
Charging:
At the instant S is closed, we set time t = 0. Because the circuit is closed, the potential difference
V0 of the battery drives the charges to accumulate in the plates of capacitor C until the potential
difference across the capacitor matches V0. Applying the loop rule to this circuit gives
V0 − iR −
where i ( i =
q
=0
C
(1)
dq
) is the current, and q represents the charges stored in the capacitor. Both i and q are
dt
instantaneous values that depend on time. Formula (1) shows that i and q are given by
q = CV0 (1 − e
V −t
i = 0 e RC
R
−t
RC
)
(2)
(3)
Figures 2(a) and 2(b) show the plots of the values of i and q versus t.
At t = 0, the charge on and the potential difference across the capacitor are both zero. At the instant
S is closed, i is determined by V0 and R, giving i =
V0
. As time passes, charges continuously accumulate
R
on the capacitor across which the potential difference is also increased. An increased potential difference
reduces the current in the circuit.
After an interval, the potential difference across the capacitor is increased to V0, matching that of
the battery. The current at this time approaches zero, but the charge on the capacitor reaches a
maximum value.
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General Physics Experiment Handout
12
10
) 8
C
(= 6
q4
2
0
8
)6
A
4
m
(
i2
0
0 2 4 6 8 10 12
t (ms)
0 2 4 6 8 10 12
t (ms)
(a)
(b)
Figure 2. RC circuit charging curves. (a) Capacitor charge versus time; (b) Current versus time. R =
2000 Ω、C = 1 µF、V0 = 10 V。
notes for Fig. 2.
B. Discharging
After a certain interval, throw the switch to position b. A closed series circuit is formed by a
battery and capacitor. Charges accumulated on the capacitor flow through the resistor and become
neutralized.
Assume t = 0 at the instant S is thrown to position b. Applying the loop rule to this circuit gives
iR +
q
=0
C
(4)
from which we obtain i and q as functions of time:
−t
q = CV0 e RC
V −t
i = − 0 e RC
R
(5)
(6)
Figures 3(a) and 3(b) show the plots of i and q versus time respectively.
At t = 0, the charge on and the potential difference across the capacitor are both zero. At the instant
S is closed, i is determined by V0 and R, giving i =
V0
. As time passes, charges continuously accumulate
R
on the capacitor across which the potential difference is increased. An increased potential difference
reduces the current in the circuit.
12
10
) 8
C
( 6
q4
2
0
0
)-2
A
-4
m
(
i -6
-8
0 2 4 6 8 10 12
t (ms)
(a)
0 2 4 6 8 10 12
t (ms)
(b)
Figure 2. RC circuit discharging curves. (a) Capacitor charge versus time; (b) Current versus time. R =
2000 Ω、C = 1 µF、V0 = 10 V。
C.
Time constant
In an RC circuit, the rates of capacitor charging and discharging are characterized according to the
values of C and R. Formula (2) shows the charges as a function of time for a capacitor being charged:
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General Physics Experiment Handout
(
q ( t ) = CV0 1 − e
−t
RC
)
Charges on a capacitor are determined according to the term e −
t
RC
. A higher RC increases the
interval for q ( t ) to reach a certain value; likewise, a smaller RC shortens the interval for q ( t ) to
reach the same value, resulting in a faster charge.
Charges as a function of time, during the discharge, are shown as Formula (5): q = CV e
0
−t
RC
.A
higher RC increases the interval for q ( t ) to release a certain amount of charges; likewise, a smaller RC
shortens the interval for q ( t ) to release the same amount of charges, resulting in a faster discharge.
To characterize the charge and discharge in various RC circuits, the time constant of an RC circuit
is defined as
(6)
τ = RC
After charging for an interval τ , the charges on the capacitor are
q ( t ) = CV0 (1 − e−1 ) ≅ 0.63CV0
Observe the potential difference across the capacitor:
V (t ) =
q (t )
C
= V0 (1 − e−1 ) ≅ 0.63V0
When discharging for an interval τ , the charges on the capacitor are
q ( t ) = CV0 e −1 ≅ CV0 0.37
Observe the potential difference across the capacitor:
V (t ) =
q (t )
12
10
) 8
V
( 6
c
V4
2
0
C
= V0 ( e−1 ) ≅ 0.37V0
τ
0 2 4 6 8 10 12
t (ms)
(a)
12
10
) 8
V
( 6
c
V4
2
0
τ
0 2 4 6 8 10 12
t (ms)
(b)
Figure 2. Plots of charges versus the time constant for a charging and discharging RC circuit. (a) During
the charge; (b) During the discharge. R = 2000 Ω、C = 1 µF、V0 = 10 V
Experimental instruments
1.
2.
3.
4.
5.
Oscilloscope
Function generator
Breadboard and cables
Three resistors and three capacitors
Signal cable and T-connector
Notes
※ The shells of the female BNC connectors on the oscilloscope and on the function generator used in
the lab are connected to the cases of the two devices and to the ground terminal of the power cords.
Therefore, to prevent observing the wrong signals in any circuit, ensure that all ground terminals
(black alligator clips) of all devices’ signal cables are connected to the same position in the circuit.
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General Physics Experiment Handout
Procedures
A.
Oscilloscope and function generator setup
1. Input the calibration signal (CAL, an output of a Vp-p = 2V and 1-kHz signal) of the oscilloscope
to CH1.
2. Adjust oscilloscope settings until the waveform of the calibration signal can be clearly observed.
3. Measure the period and Vp-p of the observed waveform, and ensure that the results are identical
to the original calibration signal.
4. After calibrating CH1, input the CAL signal to CH2 and repeat Procedures 2 to 3. Ensure CH2
is properly calibrated.
5. Generate a square wave using the function generator, and send the wave to CH1 of the
oscilloscope. Ensure that the function generator is working properly!
※ The cable used for connecting the CAL might have a BNC connector on the one end and two
alligator clips on the other end. Wave signals should be transmitted through the red alligator clip.
We suggest testing whether the black alligator clip must connect to the BNC shells!
B.
RC circuit configuration.
A power source, a resistor R, and a capacitor C are connected in series in an RC circuit.
Because the shells of the female BNC connectors are connected to the wall outlets as well as to the
cases of the oscilloscope and the function generator, the grounding of the black alligator clips
should be considered, and relevant adjustments should be performed when connecting RC circuits
to the devices.
Figure 3 shows an example circuit for a potential difference (VC) measurement across the
capacitor. Ensure that the two black alligator clips of the oscilloscope and the function generator are
connected to the same wire.
Figure 3. Schematic diagram of the measurement of VC in a
RC circuit using an oscilloscope
Figure 3 shows an example circuit for a potential difference (VR) measurement across the
resistor. Again, the two black alligator clips of the oscilloscope and the function generator should
be connected to the same wire.
Figure 4. Schematic diagram of the measurement of VR in a RC circuit using an
oscilloscope
The function generator generates a square wave VS. The maximum of the square wave charges
the RC circuit, and the minimum discharges the circuit.
During the experiment, the square wave VS should be split by a t-connector and should be
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General Physics Experiment Handout
transmitted to the circuit and any of the CH inputs of the oscilloscope. Set the oscilloscope to Dual
mode to simultaneously observe VS and the output signal versus time.
C.
Effects of changes in the charging and discharging periods
1. Determine a resistor R and a capacitor C.
2. Estimate the value of the time constant τ in this circuit composed of R and C.
3. Set the period T of VS equal to 30τ, and record the waveforms of VS, VC, and VR.
4. Set T equal to 10τ, and record the waveforms of VS, VC, and VR.
5. Set T equal to 0.5τ, and record the waveforms of VS, VC, and VR.
D.
Relationship between the time constant τ and the capacitor C.
1. Set up the RC circuit shown in Figures 3 and 4, and measure the time constant τ using the
oscilloscope. Compare the result to theoretic values.
☉ Choose either VC or VR as the signal for Procedure D.
☉ Choose an appropriate frequency VS during the charge and discharge. The frequency is
appropriate only if a complete charging or discharging cycle can be observed on the
oscilloscope.
2. Repeat Procedure D-1 with two different resistors R.
3. Let R be the horizontal axis and τ the vertical axis, and plot a data graph. Determine the linear
equation of the trend line.
Discussion questions
1.
Explain why we use a square wave to charge and discharge the circuits.
2.
Describe the process of determining the time constant based on the trace of the oscilloscope.
3.
By using the linear equation of the trend line derived from the R-τ data graph in Procedure D, can
we determine the equivalent resistance for the function generator and the oscilloscope?
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