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Physics Online
AC Circuits
An AC power supply connected to a resistor and capacitor in series can be analyzed using
Kirchhoff’s voltage law:
ε = Δvr + Δvc
ε = instantaneous potential difference across power supply
Δvr = instantaneous potential difference across resistor
Δvc = instantaneous potential difference across capacitor
An AC potential difference is sinusoidal, so the potential difference across the power supply
can be specified. Ohm’s law and the definition of capacitance can be used to specify the
potential differences across the resistor and capacitor respectively.
E0sin(ωt) = iR + q/C
E0 = maximum potential difference across power supply
ω = angular frequency of power supply
i = instantaneous current
R = resistance of resistor
q= instantaneous charge on one plate of a capacitor
C = capacitance of the capacitor
The definition of current can be placed into the equation to make the equation have only
one unknown, charge:
E0sin(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 reactance and impedance were
developed to greatly simplify the analysis. The reactance is a property of the capacitor that
functions like resistance. The impedance is a property of the entire circuit which functions
like equivalent resistance does in a DC circuit.
Xc = 1/(ωC)
Xc = capacitive reactance
Z = √[R2 + Xc2] (only for this particular circuit)
Z = √[R2 + 1/(ωC)2]
Z = impedance
The AC version of Ohm’s law can be used to determine the current:
E0 = IZ
I = E0/Z = E0/√[R2 + 1/(ωC)2]
This current will be the current for all circuit elements. Ohm’s law can be used to calculate
the potential difference across the individual capacitor and resistor:
ΔVc = IXc
ΔVc = E0/sqrt[R2+1/(ωC)2]*1/ωC
ΔVc = E0/sqrt[(ωRC)2+1]
ΔVc = E0/sqrt[(2πfRC)2+1]
ΔVr = IR
ΔVr = E0/sqrt[R2 + 1/(ωC)2]*R
ΔVr = E0/sqrt[1+1/(ωRC)2]
ΔVr = E0/sqrt[1+1/(2πfRC)2]
[1]
[2]
The crossover frequency, fc, is defined as the frequency for which the potential differences
across the resistor and capacitor are the same.
ΔVr = ΔVc
E0/sqrt[1+1/(2πfcRC)2] = E0/sqrt[(2πfcRC)2+1]
fc = 1/(2πRC)
[3]
These are the equations you will test. Physics is fun!
Equipment You Procure
 digital camera
 computer with external speakers with a cable that detaches from the speakers OR
disposable set of headphones
Equipment From Kits
 2 digital multi-meters
 clips and wires (do not use nichrome, the bare wires, unless you want to start a fire)
 resistors
Experimental Procedures
AC RC circuit
Note: Please be very careful in this lab because if you connect one output directly to the
other, then you could damage your sound card.
1) Disconnect the cable from the back of your computer speakers. Do not disconnect
the cable from your computer. If your cable does not disconnect from the back of
your speakers (or you do not have external speakers), then cut off one ear bud from
a headphone and strip off the insulation. Be sure to keep the exposed wires
separated throughout the experiment.
2) Set the 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 insulation.
6) Click on the play button (the green triangle) and check if you obtain a measurable
potential difference (it should be at least 0.4 V). Record this value (Vinput) 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 = 0.05/(2πRC).
9) Connect the capacitor, resistor, and output of the computer in a single loop.
10) Connect the voltmeters 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
differences across the capacitor
V
V
and resistor.
13) Repeat steps 11 and 12 for
approximately 50% more than
your just measured frequency
until the frequency is at least 400
times your initial frequency.
14) Calculate the theoretical
potential differences across the capacitor and resistor for the measured frequencies
using equations [1] and [2]. You do not need to include error propagation for the
theoretical potential differences.
15) Plot the experimental and theoretical potential differences (vertical axis) versus the
frequency 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, right click on the xaxis and select “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 through 16 with a different resistor.