RC CIRCUITS
th
Reference: Halliday, Resnick & Krane, 4 Ed., Chapter 33.
1 Introduction:
In this experiment the rates at which capacitors in series with resistors can be charged
and discharged are measured directly with a Picoscope digital oscilloscope (scope) and the
RC time constants are found. To make the measurements, the output voltage of a function
generator which is set to generate square waves is applied to the RC circuit. The scope is
used to measure directly the time variation of the voltage across the capacitor while it is
charging and discharging. From these measurements, the time constants of the RC circuits
can easily be determined.
2 Equipment:
Square-wave function generator, Picoscope computer interface, digital multimeter,
“2R-2C” circuit board, wires for circuit hookup.
3 Physics Introduction
3.1 Charging a capacitor
When a constant voltage (potential difference) εo is applied at time t =0 to a series RC
circuit (a series combination of a resistor and a capacitor) it takes time to charge the
capacitor. If the capacitor is initially uncharged, the charge Q on the capacitor grows with
time as:
−t/τ
Q(t)= Qo {1 −e }
and the current I in the circuit decreases with time as:
−t/τ
I(t)=Ioe .
Here Qo= εoC is the final equilibrium charge on the capacitor which is approached at large
time t and Io= εo/R is the current at t = 0. The constant τ is called the (capacitive) time
constant for the circuit and has the value τ=RC. Since the voltage across a capacitor is always
V = Q/C, the voltage across the capacitor in the circuit varies with time as
−t/τ
V(t)=εo{1 −e }.
3.2 Discharging a capacitor
If the applied voltage is suddenly reduced to zero at a later time (a new t = 0) when
the charge on the capacitor is Q1,the charge on the capacitor and the voltage across the
capacitor decay (decrease) with time as
Q(t) = Q1e
−t/τ
and V(t)= V1e
−t/τ
−t/τ
=(Q1/C)e ,
and the current (opposite direction from the original current) in the circuit decreases as:
−t/τ
I(t) = I1e
with I1= Q1/RC.
A square-wave function generator will be used to apply and remove a potential
difference Vo=
εo across the
RC circuit under study. The function generator switches the
voltage repeatedly from V =0 to V =Vo =
εo
and back at the frequency ν of the square wave
as shown in the top graph. The period τo of the wave is 1/ν. Since the voltage across C varies
-t/τ
-1
as V(t)= Vo{1 − e }, it starts at V =0 at t = 0 and rises to V = Vo(1 − e ) ≈ 0.632Vo at t = τ.
The time constant τ may be determined directly by measuring how long it takes for
the voltage to rise from V = 0 to V = 0.632Vo. This is done by observing the voltage V as a
function of time with the Picoscope. The trace on the scope screen represents a plot of V as a
function of t. Since you will be interested in the shape of the curve, i.e. its time dependence,
rather than the actual magnitude of the voltage (which depends on the circuit parameters and
the applied voltage), it will not be necessary to measure V in volts. Instead, you can simply
measure V directly on the screen of the Picoscope as the vertical displacement (in scale
divisions on the scope screen) of the scope trace from the baseline (which corresponds to
zero volts, or ground). It is necessary to know the number of vertical scale divisions on the
screen corresponding to Vo. To determine this, you must apply the constant voltage to the RC
circuit for a time long enough for the charge to approach its maximum value.
The time constant τ can also be determined by measuring how long it takes for the
voltage to drop from V = Vo to V = 0.368Vo. This is done by observing the voltage V as a
function of time with the Picoscope.
IMPORTANT: Setting up the Picoscope
Connect the square wave function generator to the scope (600  to red and
Ground to black of input). Turn on the function generator and make sure it is set for
square wave output. Go through the following steps to initialize the Picoscope.
(If the scope freezes at any time close it and reopen it and follow pt. 2 again)
1. Make sure the Computer is running windows. If it is not, then hold down the
option button and restart the machine. Keep the option button depressed till
you reach a screen asking you to choose between windows and Mac.
2. On the Windows Desktop you will find the “Picoscope” icon (with the picture
of a wave). Open Picoscope, maximize it to fill the screen and check the
following:
a. The “Trigger”(bottom right) should be set to “Auto”.
b. Go to “Settings → Options” and choose “Current(Filtered) in the
Display Mode box”
c. Set the “Time Base” (upper right) to 50 s.
Preliminary observations
The top graph in the figure below shows what the square-wave output Va of the
function generator should look like. Try changing the frequency and amplitude of
the square wave and see how the pattern changes. Experiment with the scope
controls to see how they affect the display. Try to adjust the Picoscope “time base”
(top right) so that you see about one complete wave.
The two lower graphs in the figure show Vc (the voltage across the capacitor C in an
RC circuit) versus t for two cases: Case (1) for RC »1/3 (to/2), and Case (2) for RC <<
to/2 . Make sure you reset the controls to the suggested settings before proceeding
with your measurements.
The pattern of traces on the oscilloscope should look as follows:
MEASUREMENTS:
A. TIME CONSTANT FOR SMALL R &C
On the 2R-2C board, select the smaller resistance as RE and the smaller capacitance
as C. Record the capacitance as marked on the board, and measure and record the
value of RE. Connect the output of the function generator in series with RE and C
and connect the scope across C as shown in the schematic diagram. Leave the TTL
output connected to the scope external trigger input.
· Warning: the scope and function generator have internal grounds of their own. If
the circuit is to function properly the Ground terminal of the function generator
and the grounded scope input (black) lead must be connected to the same point
in the circuit(marked V = 0 in the diagram).
· Note that the function generator has an output impedance (effective resistance) RI
of about 600. This acts as an additional resistance in the series RC circuit so the
total resistance is RT = RE + RI and the time constant is  = RTC. Assume that RI =
600.

Adjust the frequency of the function generator to a value sufficiently
low that the voltage Vc approaches the value Vo asymptotically as
shown for Case (2) in the figure. If you use a frequency which is too
high, you will observe a waveform such as that shown for Case (1), and
it will be very difficult (if not impossible) to determine the time when
Vc = 0.632Vo .

Adjust the horizontal (time axis) scale of the display using the "time
base" (TOP RIGHT) controls on the Picoscope so that you can see at
least one full cycle for V as a function of time (as shown in the figure
above).

Print the graph on the Picoscope from the “File” menu on the top
toolbar.
Measure V(t): if you drag your mouse horizontally in the Picoscope graph, a
horizontal line will be created which will help you do this. Clicking close to this line
will make it move up or down. Position this line at the top of a crest and measure V 0.
Alternatively, you can adjust the “Amplitude” knob on the frequency generator to
get a desired voltage like 6V on the graph. Measure and record V and the time t in
scale divisions for about 10 values of t spread over the charging part of the cycle. Be
sure to include the time t at which V starts to rise from V = 0. Estimate the errors (in scale
divisions) in your measurements of both t and V. Be sure to record the time in seconds
corresponding to one scale division on the Picoscope.
B. Time Constant for Charging
 Locate the point where V is a minimum and is beginning to rise
sharply.
 Create a horizontal line in the picoscope where the voltage is a
minimum. Create a vertical line here by dragging the mouse
vertically.
 Record the x(time, t1) and y(voltage) coordinates that appear at the
top of the screen.
 Create a horizontal line at 0.632V0 of this rising curve. Try to make
it as exact as possible by noting the y coordinate readings at the top of
the screen and adjusting the position of the line.
 Create a vertical line where the horizontal line meets the curve.
Record the coordinates(t2, V2).
CAUTION: Clicking next to a line moves it so be very careful where
you click on the picoscope.
The time constant  is given by (t2- t1). Be sure to record the time in scale divisions and
the time in seconds corresponding to one scale division. Estimate the uncertainty  (in
scale divisions) of this value for .
C. Time Constant for discharging.
 Locate the point where V is a maximum and is beginning to fall
sharply.
 Create a horizontal line in the picoscope where the voltage is a
maximum. Create a vertical line here by dragging the mouse
vertically.
 Record the x(time, t1) and y(voltage) coordinates that appear at the
top of the screen.
 Create a horizontal line at 0.368V0 of this falling curve. Try to
make it as exact as possible by noting the y coordinate readings at the
top of the screen and adjusting the position of the line.
 Create a vertical line where the horizontal line meets the curve.
Record the coordinates(t2, V2).
 The time constant  is given by (t2- t1).
Estimate the uncertainty in this value for  as before.
TIME CONSTANT FOR LARGE R & LARGE C
 Reconnect the circuit using the larger of the two capacitances and the
larger of the two resistances



Adjust the frequency of the frequency generator(source) and the time
base of the Picoscope so that you get a graph similar to the previous
one.
Print the graph
Follow procedures A, B and C above to get the time constants for
charging and discharging for this circuit.
Analysis (Do this for both circuits):
Part A.
(1). Open the Graphical Analysis Program and enter values of t and V from your data
table. It is possible to plot the data so that the points should lie on a straight line . Since
we expect Vc = Vo{1 - e-t/}, we can rearrange this expression as (Vo - Vc)/Vo = e-t .
Then ln{(Vo - Vc)/Vo} = - t/ or ln{Vo/(Vo - Vc)} = t/. Thus plotting ln{Vo/(Vo - Vc)}
versus t should give a straight line with slope 1/ To do this, open a new column Z and
enter Z= ln (V0 / (V0 - Vc)) and plot Z versus t. Make a linear fit.
Print out the plot and slope value of the plot for each partner.
To determine  convert the slope to time in seconds. Compare this (probably more
accurate) value with the value you determined from the direct measurement in Part B
above.
(2). Using your measured value for RE and the given value for C, compute the total
resistance RT = RE + RI. Calculate the capacitive time constant  = RTC for this circuit from
your data. Calculate an approximate error for this value of  assuming a 5% error in C
and a 3% uncertainty in RT. Compare your measured and calculated values of . Do they
agree reasonably well when you consider your error estimates?
Parts B & C.
For the RC circuit used in parts B and C, convert your two measurements of one time
constant and the uncertainties  from times in scale divisions to times in seconds.
Compute the total resistance RT = RE + RI and the time constant t = RTC for this RC
circuit. Calculate an approximate error for this value of  assuming a 5% error in C and a
3% uncertainty in RT.
Questions for the report:
In your semilog plot of {Vo/(Vo - Vc)} versus t in part A(1), does your best line pass right
through the point (t = 0, {Vo/(Vo - Vc)} = 1)? Would you expect it to?
In parts B and C, do the time constants for charging and discharging the capacitor
which you measured using the Picoscope agree as expected when you consider your error
estimates? Does your calculated value of for this RC circuit agree within errors with
your measured values?