Alternating Current in RC Circuits

Physics 241 Lab – Matt Leone
Week 8: Alternating Current in RC Circuits
Leone@physics.arizona.edu (email preferred), PAS 376, o. 520-621-6819
Office Hours: M & W 11:00-11:50, or by appointment. Consultation Room (PAS 372): F 12:00-12:50
http://bohr.physics.arizona.edu/~leone/phy241/phys241lab.html
General Comments:
• Today we study ALTERNATING CURRENT charging and discharging of capacitors.
• You will practice using your oscilloscope in a test-like environment.
Lab 8 – Summary:
There are three key concepts you should know about an RC circuit driven with a sine source.
1. The capacitor voltage lags behind the resistor voltage by 90 o (phase shift).
2. The capacitor is described by a capacitive reactance X C that acts like a resistance.
3. A sinusoidally driven circuit is described by an overall IMPEDANCE, Z that acts
like the total resistance of the whole circuit.
Lab 8 – Matt’s discussion on board: now on the paper!
We now study the RC circuit driven by a sinusoidal alternating voltage source:
For an advanced course, you could write a differential equation for this circuit and solve it to
study its behavior mathematically. We will only discuss how to use the solution in this lab.
Each component of the above circuit has a sinusoidally varying voltage across it, though each
peaks at a different time. The voltage of each is given by the equations:
• V source
(t) = V MAX
sin("t + #)
"
• V R
(t) = R %
$ ' V
# Z &
MAX
sin((t)
!
#
• V C
(t) = " &
C
%
$ Z '
( V sin #
)t * + &
%
MAX $ 2 (
'
where ! there are several new parameters to discuss. Let’s discuss each equation in detail. Incidentally,
the current is also oscillating with the same frequency. The oscillating current in the resistor is often
used as a starting point to teach $ this material (but not here because I think it confuses the big idea).
!
%
Also, in case you’re curious " = arctan # L
$ # C
(
' *.
& R )
!
1

!
V source
(t) = V MAX
sin("t + #)
The source voltage equation is straightforward. It oscillates sinusoidally, i.e. it is a sine
function of time. It determines the maximum voltage across the whole circuit with V MAX . It oscillates
with an angular frequency " = 2#f .
Notice that it also has a phase shift φ. Now why on earth would we describe the source voltage
with a phase shift? If you look below at the resistor equation, you will see no phase shift. 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.
!
"
V R
(t) = R %
$ ' V
# Z &
MAX
sin((t)
The resistor voltage equation includes some new stuff. It also oscillates sinusoidally and
without a phase shift. R is the resistance.
What is new is Z. Z is the impedance of the whole circuit. It is given by the equation
( ) 2 .
Z = R 2 + " L
# " C
Uh oh, this definition has new stuff, too. Now X L and X C are like the resistances of the inductor and
capacitor in the circuit. Since we don’t have an inductor (coil) in the circuit, you can set this to zero.
So we have Z = R 2 + " 2 C
.
!
Now what if I rewrote the resistor equation as V R
(t) = V R,MAX
sin("t). What maximum and
minimum voltage would oscillate across the resistor? V R,MAX
= R
!
Z V MAX
. That means that the ratio of
the resistor resistance and the total circuit resistance (impedance) times the source amplitude gives you
the amplitude of the voltage just across the resistor.
!
!
#
V C
(t) = " &
C
%
$ Z '
( V sin #
)t * + &
%
MAX $ 2 (
'
!
The capacitor voltage oscillates sinusoidally and lags the voltage across the resistor by 90 o .
$
Knowledge of trigonometry allows you to also write V C
(t) = " # '
C
& ) V
% Z (
MAX
cos (*t) since sine shifted left
by 90 o is the negative cosine. This is sometimes useful for graphing.
The reactive capacitance is like the resistance of the capacitor. More precisely, " C
= 1
#C .
!
Yikes! The “resistance” of the capacitor is related to the capacitance of the capacitor. The larger the
capacitance, the less “resistance” in the capacitor. But just as importantly X C is also related to the
frequency of the driving voltage. If the frequency is increased, the “resistance” of the capacitor
decreases. This is why a capacitor is often used as a high pass filter. The capacitor ! has less resistance
to more quickly oscillating currents. BE SURE TO REMEMBER THIS DURING TODAY’S LAB!
%
Now what if I rewrote the capacitor equation as V C
(t) = V C,MAX
sin "t # $ (
' *. What maximum
& 2 )
and minimum voltage would oscillate across the capacitor? V C,MAX
= " C
Z V MAX
. That means that the
ratio of the capacitive reactance and the total circuit resistance (impedance) times the source amplitude
gives you the amplitude of the voltage just across
!
the capacitor.
!
2

Lab 8 – Predictions – reminder: your solutions on these pages are always worth points.
!
Lets work though an example before we begin. Remember the equations,
V source
(t) = V MAX
sin("t + #)
"
V R
(t) = R %
#
$ ' V
# Z &
MAX
sin((t) V C
(t) = " &
C
%
$ Z '
#
)t * + &
%
MAX $ 2 (
'
If your circuit has V MAX
= 2 Volts, R =10,000 " , C =1x10 -7 Farads, and " =1,500 radians/sec find
the following values with correct units
1. X C .
!
!
!
!
!
!
2. Z
3. V R,MAX .
4. V C,MAX .
5. Now examine V R,MAX + V C,MAX .
They add to more than V MAX !!! 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 visualize this.
6. Write the functions for V R
(t) and
V C
(t).
7. Plot V R
(t) and V C
(t) on the o-scope screen below. Be sure to label units and hash marks
! !
correctly for full credit.
!
!
8. On your graph, sketch the addition of the V R
(t) and V C
(t). This is V source
(t).
3
!
!
!

Lab 8 – On your own.
Experiment 1: Examining the phase shift between V R
(t) and
V C
(t).
a. Set your function generator to create a sin wave with a voltage amplitude of a nice round
number like 3 Volts. You may change your frequency later, but start at about 400 Hz..
! !
b. Set up the RC circuit with the sinusoidal driving function generator with R =10,000 " ,
and C =1x10 -7 Farads.
c. Set up a middle ground to view the voltage across both the resistor and the capacitor
simultaneously. Make a sketch on the o-scope screen below making
!
sure to invert the
!
correct channel for your graph (a necessary step when using a middle ground).
!
d. Label the signals V R
(t) and V C
(t) on your sketch and determine equations for V R
(t) and
V C
(t). Also explain which is phase shifted by 90 o . Write your equations and
explanation here:
!
!
!
4

Experiment 2: Testing the relationship " C
= 1 with many frequencies.
#C
a. Use the theoretical relationship " C
= 1 and the equations for voltage amplitutde on the resistor
#C
and capacitor, V C,MAX
= " C
!
Z V and
MAX
V R,MAX
= R Z V MAX
to calculate the driving angular frequency
necessary to have V R,MAX
= V
! C,MAX
. (You will want to take your data around this angular
frequency in the next parts of this experiment.) Report ω EQUAL here.
!
!
!
b. Set your function generator to create a sin wave with a voltage amplitude of a nice round number
like 3 Volts. You will be adjusting the frequency during the experiment. Use the same R and C
as in experiment 1.
c. Since
!
V C,MAX
= " C
Z V MAX and
V C,MAX
V R,MAX
= R Z V MAX
, dividing these equations gives
= " C
or after multiplying both sides by R, " C
= R #V C,MAX
.
V R,MAX
R V R,MAX
You can measure V!
C,MAX
and V R,MAX
on the oscilloscope, and you can measure R with a DMM.
Therefore, find X C for many different frequencies using the equation in the box above.
Remember that you can only be sure a DMM resistance measurement
!
!
is accurate by measuring
the resistor naked. Use the data table below:
! !
d. Plot X C vs. ω to support our claim that there is an inverse relationship between them. Be sure to
find ω from f before graphing. (Think about what the graph should look like if " C
= 1
#C ?)
GRAPH 1
e. Explain why your graph supports the relationship " C
= 1 . Write explanation ! here:
#C
!
5

Experiment 3: Finding C with a single frequency measurement.
a. Using the relationship from experiment 2,
" C
= R #V C,MAX
V R,MAX
You can use this relation for experimental determinations of C.
1
, you know that
"C = R # V C,MAX
V R,MAX .
b. Make a single measurement of ω, ! R, V R,MAX and V C,MAX using a ! frequency where V R,MAX
and V C,MAX are about the same and determine C. Report your value for C here:
c. Now perform an x-y measurement with the oscilloscope for V R
(t) versus
ω EQUAL and sketch your result on scope A. Be sure to label your axis.
V C
(t). Start with
d. Now lower the frequency so that V C
(t) is larger than V
! R
(t) and sketch your result on B
!
(another x-y measurement).
e. Now raise the frequency so that V
!
C
(t) is smaller than V
!
R
(t) and sketch your result on C
(another x-y measurement).
A B C
!
!
f. Explain why V C
(t) is larger than V R
(t) when the frequency is low enough. Write
explanation here:
Experiment 4:
!
Making and measuring
!
your own capacitor.
a. Make a capacitor from the square cardboard pieces covered in foil. Sandwich a non-foil
square with them, and be sure your makeshift capacitor is not shorted.
b. Measure the capacitance of your homemade capacitor using the technique from part 3. Be
sure to find a frequency where V R,MAX and V C,MAX are about the same. You may need to
use a different resistor. Report your homemade capacitor capacitance here:
5. CLEAN UP LAB STATION.
6

Lab 8 – Report Guidelines
I will be paying careful attention to your solutions in the above pages. (Ellipsis means you
should not need help here on what to write about).
1. Title - …
2. Goals – …
3. Theory –
• When an RC circuit is driven with a sinusoidal voltage source, how do the voltages
across the resistor and capacitor behave? What functions describe them?
• At any instant of time, the voltage across the resistor plus the voltage across the
capacitor should add to the voltage across the source. Sometimes the voltage across the
source is zero since it is alternating. At this particular instant of time, what must happen
with the voltages across the resistor and capacitor?
4. Procedure – No more than 3 sentences for each experiment please!!!
5. Results – Describe your results for each experiment with a separate short paragraph.
6. Conclusion – Write a short conclusion paragraph for each experiment and a final summarizing
paragraph.
7. Attachments including all the notes you took on the pages of this handout.
7