AC Circuits
An AC power supply connected to a resistor and capacitor in series can be analyzed using
Kirchoff’s voltage law:
V
= IR + Q/C
V =
potential difference
across entire circuit
I = current through entire circuit
R = resistance of resi
stor
Q = charge on one plate of a capacitor
C = capacitance of the capacitor
The
potential difference
across the power supply varies with time; ε = V
max
sin(ωt). The
current is dQ/dt o
r
ΔQ/Δt. This sets up a differential equation as follows:
V
max
sin(ωt) = dQ/dt*R + Q/C
This is a difficult differential equation to solve, and things only get worse with
additional
circuit elements. Instead of this approach, the concepts of imped
a
nce and reactance were
developed to greatly simplify the analysis. For a capacitor, the
capacitive
reactance, X
c
, is
1/(ωC)
, where ω=2πf.
The
equivalent
impedance of the
series
ci
rcuit in question can be
calculated with complex arithmetic (beyond the scope of this course) or taken from the text:
Z =
√
[R
2
+ X
c
2
]
Z =
√
[R
2
+ 1/(ωC)
2
]
Then,
the AC
version of Ohm’s law can be used to calculate the
rms
current:
V
input
= IZ for t
he power supply
I =
V
input
/Z
I =
V
input
/
√
[R
2
+ 1/(ωC)
2
]
This current will be the curren
t for all circuit elements.
Ohm’s law
can be used to
calculate
the
potential difference
across the
individual
capacitor
and resistor
:
V
c
=
IX
c
V
c
=
V
input
/
√
[R
2
+
1
/(ωC)
2
]*
1/ωC
V
c
=
V
input
/
√
[(ωRC)
2
+
1]
[1]
V
r
= IR
V
r
= V
input
/
√
[R
2
+ 1/(ωC)
2
]*R
V
r
=
V
input
/
√
[1+1/(ωRC)
2
]
[2]
These are
the equation
s
we will investigate as a test of this type of circuit analysis
.
Physics is
fun!
Experimental Procedures
AC RC
circuit
Note: Please be very careful in this lab because if you connect one output directly to the
other, you could damage
the
sound card.
1)
Disconnect the cable from the back of your computer speakers.
Do not disconnect
the cable from your computer.
2)
Set th
e volume on your computer to maximum.
3)
Download and install Audacity freeware from the following web site:
http://audacity.sourceforge.net/
4)
In the Audacity program, click on “Generate” and “Tone”. It should default to
Waveform “Sine” and amplitude of 1. For
frequency, choose 100 Hz. Hit “enter”.
5)
Configure a DMM as a voltmeter at the lowest setting for AC
potential difference
and connect it to the
cable from your computer
(or the wires from your
headphones)
.
There should be a small piece of insulation on the
cable separating the
+ and

. Place a probe on each side of the in
sulation.
6)
Click on the play button (the green triangle) and check if you o
btain a measurable
potential difference
(it should be at least 0.4 V).
Record this value
(V
input
)
and press
the stop
button on Audacity.
7)
Select a capacitor and resistor so that their time constant (R*C) is approximately 10

3
s.
8)
Calculate the
initial
frequency
using f = 0.1/(2πRC
).
9)
Connect the capacitor, resistor, and output of
the
computer
in a single loop.
10)
Connect the
voltmeter
s
so that
they
measure the
potential difference
across the
capacitor
and resistor
.
See figure at right.
11)
In the Audacity program, click on “Generate” and “Tone”. For frequency, choose
your calculated value. Hit “enter”.
12)
Measure the
potential differ
ence
s
across
the
capacitor
and resistor
.
13)
Repeat steps
1
1
and 1
2
for approximately
50% more than
your just measured
frequency until the
frequency is at least 100 times your initial frequency
.
14)
Calculate the theoretical
potential difference
s
across the capaci
tor
and resistor
for the
measured frequencies
using equations [1] and [2]
. You do not need to include an
error analysis in the
theoretical
potential difference
.
15)
Plot the experimental and theoretical
potential difference
s
(vertical axis) versus the
frequenc
y
on a single graph
. Use a logarithmic scale for the horizontal axis and a
linear scale for the vertical axis.
To use a logarithmic scale, make your graph
according to the general lab instructions. Once you have a graph, r
ight click on the
x

axis and selec
t “format axis”
.
16)
Compare the
theoretical and experimental
plots
.
In this case, since you do not have a
complete error analysis, there will be some subjectivity in the comparison and you
are allowed to use the word “close” in your report.
17)
Repeat steps
7
thr
ough
1
6
with a
different
resistor.
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