Physics 241 Lab: RC Circuits – AC Source
http://bohr.physics.arizona.edu/~leone/ua/ua_spring_2010/phys241lab.html
Name:____________________________
Section 1:
1.1.
Last week you studied RC circuits, examining the exponential time dependence of the capacitor
voltage as you charged and discharged the capacitor with a constant source voltage. To do this you used
a square wave with a DC offset. Today you will examine the behavior of a capacitor when a sinusoidal
source voltage is applied: Vsource (T)  Vsource sin( D t) , where D is called the driving frequency of the
amplitude
circuit (note that it is an angular frequency).
The capacitor voltage will no longer exhibit exponential time behavior. Instead the capacitor
voltage will oscillate sinusoidally with the same frequency as the source driving frequency. This can be
proven by writing the differential equation for the circuit, finding its solution, and checking the solution.
However, this requires knowledge of solving inhomogenous differential equations.
Instead, the most useful results of that calculation are provided: the time dependent voltages
across each component. Thus, you are not required to be able to derive the solutions to the AC-driven
RC circuit, but you must understand and be able to use these results.
Each component of the sinusoidally driven RC circuit has a sinusoidally varying voltage across it,
but each peaks at a different time determined by a phase shift. The solutions for the time dependent
voltages of each component are given by the equations:

Vsource (t)  Vsource sin( D t   )
amplitude


R 
VR (t)   Vsource sin( D t)
Z  amplitude
 

 
VC (t)   C Vsource sin D t  
 Z  amplitude 
2 
There are several new parameters to discuss. First notice that the source voltage is now written
with a source phase shift , the capacitor voltage has a phase shift of –/2, and the resistor voltage has no
phase shift. What this means in practice is that we will use the resistor voltage as a reference for all other
components in the circuit: i.e. we will measure the phases of each component in relation to what is
happening inside the resistor. This is because the resistor is Ohmic and can always provide the time
dependent current via Ohm’s law, which can often be useful to know.
1.2.
The Source Voltage Equation: Vsource (t)  Vsource sin( D t   )
amplitude
The source voltage equation is straightforward. It oscillates sinusoidally, i.e. it is a sine function
of time. The maximum voltage applied across the whole circuit is Vsource . The source oscillates with an
amplitude
angular driving frequency  D  2f D (which you will set later with your function generator). The source
 
voltage is phase shifted from the resistor voltage by an amount   arctan C  where XC is the reactive
 R 
1
capacitance of the capacitor given by the equation C 
(more on this later). Note that this x-like
 DC
variables are really the capital Greek letter Chi (pronounced kai).
If you look at the equation for resistor voltage, you will see no phase shift. Again, what this
means is that we measure all phases in relation to the resistor not the source. The resistor will have its
maximum voltage at a different time than when the source voltage is maximum.
Imagine a sinusoidally driven RC circuit. If the source voltage has an amplitude of
Vsource
amplitude
=1.8 volts, a linear driving frequency fD=555 Hz, a resistance R=150 , and a capacitance C=1.5x10-5 F,
find the phase shift of the source voltage compared to the resistor. Your work and answer:
R 
The Resistor Voltage Equation: VR (t)   Vsource sin( D t)
Z  amplitude
The resistor voltage oscillates sinusoidally without a phase shift while R is simply the resistance.
Z is the impedance of the whole circuit. Z acts like the “total resistance” of the circuit. Z is measured in
1.3.
SI units of Ohms and is given by the equation Z  R 2   L  C  .
This definition has new stuff, too. XL and XC are like the “resistances” of the inductor and
capacitor, respectively. We won’t study inductors until later in the semester, but it is easier to memorize
the complete equation. Since we don’t have an inductor (coil) in the circuit, you can set this to zero. So
we have Z  R 2  C2 . C is called the reactive capacitance and is measured in Ohms.
Now examine the resistor equation as VR ( t )  Vresistor sin( D t ) . The maximum and minimum
2
amplitude
R
Vsource .
Z amplitude
amplitude
Imagine a sinusoidally driven RC circuit. If the capacitance is increased, explain what happens to the
amplitude of the resistor voltage? Your explanation:
voltage would oscillate across the resistor is Vresistor 
If the frequency is increased what happens to the amplitude of the resistor voltage? Your answer:
Explain what happens to the current through the circuit if the resistor voltage amplitude decreases?
Your explanation:
Explain what happens to the power lost through heating the resistor if the resistor voltage amplitude
decreases? (Remember that PR=IRVR.) Your explanation:
Imagine a sinusoidally driven RC circuit with source voltage amplitude VS, resistance R, and capacitance
C. Explain whether the resistor will become hotter if you increase the driving frequency? Use the
concept that Z = total impedance of the circuit. Your answer and explanation:
1.4.
The Capacitor Voltage Equation:
 

 
VC ( t )   C Vsource sin D t  
 Z  amplitude 
2 
The capacitor voltage oscillates sinusoidally and lags behind the resistor voltage by 90o. The
reactive capacitance C is like the resistance of the capacitor and is measured in SI units of Ohms.
The “resistance” of the capacitor is related to the capacitance of the capacitor and the driving
1
can be derived from the differential equation modeling the
frequency. This relationship C 
 DC
circuit, but you must memorize it. The larger the capacitance, the less “resistance” in the capacitor. But
just as importantly if the driving frequency is increased, the “resistance” of the capacitor decreases. This
is why a capacitor is often used as a high pass filter in electronics: the capacitor has less resistance to
more quickly oscillating currents. BE SURE TO REMEMBER THIS DURING TODAY’S LAB!

 
If we rewrite the capacitor equation as VC ( t )  Vcapacitor sin D t  , the capacitor voltage

2 
amplitude
C
amplitude is given by Vcapacitor 
Vsource . That means that the ratio of the capacitive reactance and the
Z amplitude
amplitude
total circuit impedance times the source amplitude gives the amplitude of the voltage across the capacitor.
In a previous equation, you found that the resistor voltage amplitude increases when the frequency
is increased. Since the voltage across the resistor and capacitor must add to the voltage across the source,
if the resistor voltage amplitude increases, then the capacitor voltage must decrease. Therefore, as you
increase the driving frequency, the resistor voltage amplitude increases while the capacitor voltage
amplitude decreases. (Not a question)
Section 2:
2.1.
Work though an example before beginning. Remember the equations below as you work.
R 
 

 
Vsource ( t )  Vsource sin( D t   )
VR ( t )   Vsource sin( D t )
VC ( t )   C Vsource sin D t  
Z  amplitude
 Z  amplitude 
2 
amplitude
-7
If your circuit has Vsource  2 Volts , R  10,000  , C  1x10 Farads, and  D  1,500 radians/sec find
amplitude
the following values with correct units. Your answers:
XC =
Z=
=
VR,amplitude. =
VC,amplitude. =
Now examine VR,amplitude + VC,amplitude =
Your answer to this previous question adds to more than Vsource amplitude!!! No, you didn’t make a
mistake. Since the voltages are out of phase, their maximums do not add together at the same
time.
Now let’s try and visualize this circuit’s behavior:
Write the functions for VS ( t ) , VR ( t ) and VC ( t ) using the numerical solutions to the previous questions.
Quickly sketch VR ( t ) and VC ( t ) on the oscilloscope screen below using a graphing calculator. Don’t
worry about providing the scale of the time axis. Then sketch VR (t) + VC (t) onto the screen using a
dotted line. This should equal the function VS ( t ) so check it using your graphing calculator.
Section 3:
3.1. Now you will set up the sinusoidally driven RC
circuit with R  10,000  , and C  1x10 -7 Farads. Set your
function generator to create a sin wave with a voltage
amplitude of a nice round number like 3 Volts. You may
want to adjust your frequency later, but start at about 400
Hz. Set up a middle ground to view the voltage across both
the resistor and the capacitor simultaneously making sure
to invert the correct channel (a necessary step when using
a middle ground). Make a sketch on the oscilloscope screen
below.
Label the signals VR (t) and VC (t) on your sketch
Explain which signal is phase shifted to lag by 90o. Your explanation:
Find the amplitudes of each signal by measuring the peak-to-peak voltage of each signal. Your
observation:
Use the labeled values to determine the impedance of your circuit for this driving frequency.
Remember Z  R 2   C2 . . Your work and answer:


Use your previous answer to determine what the signal amplitudes should and then compare these
predicted (calculated) amplitudes to your measured amplitudes in the other previous question (they
should be close). Your work and answers:
Find the frequencies f of each signal using oscilloscope measurements. Your observation:
Use your answers to the previous questions to write equations for VR ( t ) , VC ( t ) and VS ( t ) entirely with
numerical values (no free parameters). (Don’t forget the phase shift.) Your solutions:
3.2. Set your oscilloscope to plot VR (t) on the x-axis and VC (t) on the y-axis (an XY plot). Sketch the
result on the oscilloscope screen below. Your sketch:
In an XY plot, if the signal on the y-axis oscillates twice as fast as the signal on the x-axis and the signals
are 90o out of phase, then sketch what will appear on the oscilloscope screen below. Your sketch:
Section 4:
1
by observing a sinusoidally driven RC circuit using
 DC
many different driving frequencies. Use the same circuit set up as in the previous part of the lab. As you
increase the driving frequency, the amplitude of the resistor voltage will increase because the total circuit
R
impedance is decreasing, i.e. Vresistor  Vsource (work through this logic!). Meanwhile, as the driving
Z amplitude
amplitude
frequency increases, the capacitor amplitude decreases. This makes sense because the resistor and the
capacitor are the only two components in the circuit other than the source. Since the voltages across both
must add up to the source voltage, if the voltage amplitude of one increases, then the other must decrease.
Therefore, there must be some specific driving frequency when the amplitude of the resistor
voltage is the same as the capacitor voltage: Vresistor  Vcapacitor for a specific D. Substitute
4.1.
Next you will test the relationship C 
amplitude
amplitude
R


R
Vresistor  Vsource and Vcapacitor  C Vsource and you get C Vsource  Vsource for a specific D.
Z amplitude
Z amplitude
Z amplitude Z amplitude
amplitude
amplitude
The first method for finding the capacitance of an unknown capacitor makes use of the
previous equation. Adjust the driving frequency of your circuit until the capacito voltage amplitude and
1

R
the resistor voltage amplitude are equal. Then use C 
and C Vsource  Vsource for the specific
 DC
Z amplitude Z amplitude
D to find the capacitance. Obtain an accurate measurement for R using a DMM. Your observations,
work and answer for C determined experimentally:
4.2. The second method for finding an unknown capacitance is more involved. The voltage amplitudes
of the sinusoidally driven RC are:
R

Vresistor  Vsource and Vcapacitor  C Vsource .
Z amplitude
Z amplitude
amplitude
amplitude
 C  Vsource 
amplitude 



Vcapacitor
Vcapacitor
Z

  C
amplitude
. Therefore,  C  R amplitude .
Dividing these two equations gives


R  Vsource  R
Vresistor
Vresistor
amplitude 

amplitude
amplitude


Z


1
and experimentally determine C for your solenoid, simply
In order to test the relationship  C 
 driveC
combine the last two equations and rearrange:
V
1 resistor
amplitude
 C drive .
R Vcapacitor
amplitude
V
1 resistor
amplitude
Therefore if you graph
vs.  drive , you should obtain a linear graph with a slope equal to C.
R Vcapacitor
amplitude
Find C by collecting data for multiple driving frequencies, making a graph and finding the slope.
Make your observations and graph now. Then write your work and result for C:
Section 5: Test Yourselves - Each lab partner should take a turn doing this section.
Quickly set up a working circuit that simultaneously uses a random capacitor and a 1000  resistor in
series powered by a sinusoidal source voltage on your function generator. Then make the necessary
measurements to determine the capacitance of the capacitor. Be sure your experimentally determined
measurements give the correct capacitance. Your lab partners can give you verbal feedback, but only you
are allowed to touch the equipment. Record your results below:
Section 6: (Open-ended question / creative lab design)
Make a capacitor from the square cardboard pieces covered in conductive aluminum foil.
Sandwich a non-foil square of cardboard between the foiled boards, and be sure your makeshift capacitor
is not shorted out by accident. Measure the capacitance of your homemade capacitor. The equation for
 A
the capacitance of two parallel plates is given by C  o . Use this equation to report the dielectric
d
constant  of the sandwiched cardboard between the plates with correct units.
Note:
2
C
o  8.85 x1012
.
N  m2
At the following prompts, design an experiment to determine the capacitance of your cardboard
capacitor and the dielectric constant of the cardboard. Then implement your experiment and record your
observations. You may “cheat” by talking to other groups for ideas, but not “cheat” by already knowing
the answer or looking it up.
Your planned experiment, sketch of actual implementation and any theoretical calculations:
Your observations:
Your explanations & conclusion:
Report Guidelines: Write a separate section using the labels and instructions provided below. You may
add diagrams and equations by hand to your final printout. However, images, text or equations
plagiarized from the internet are not allowed!
 Title – A catchy title worth zero points so make it fun.
 Goals – Write a 3-4 sentence paragraph stating the experimental goals of the lab (the big picture).
Do NOT state the learning goals (keep it scientific). [~1-point]
 Concepts & Equations – [~5-points] Be sure to write a separate paragraph to explain each of the
following concepts.
o Compare and contrast how to find the capacitance of a capacitor using a DC source
(square wave) versus a sinusoidal source.
o Discuss at length the three time dependent voltage equations that describe the AC-driven
RC circuit. Be sure to explain:
o impedance
o reactive capacitance
o phase shifts
o Discuss how to find the amplitude of the current through the resistor and what
combination of parameters gives this value.
 Procedure & Results – Write a 2-4 sentence paragraph for each section of the lab describing
what you did and what you found. Save any interpretation of your results for the conclusion. [~4points]
 Conclusion – Write at least three paragraphs where you analyze and interpret the results you
observed or measured based upon your previous discussion of concepts and equations. It is all
right to sound repetitive since it is important to get your scientific points across to your reader.
Write a separate paragraph analyzing and interpreting your results from your open-ended
experiment. Do NOT write personal statements or feeling about the learning process (keep it
scientific). [~5-points]
 Graphs – All graphs must be neatly hand-drawn during class, fill an entire sheet of graph paper,
include a title, labeled axes, units on the axes, and the calculated line of best fit if applicable. [~5points]
o The graph from section 4.2.
 Worksheet – thoroughly completed in class and signed by your TA. [~5-points.]