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 resistor
Q = charge on one plate of a capacitor
C = capacitance of the capacitor
The potential difference across the power supply varies with time; ε = Vmaxsin(ωt). The current is dQ/dt or ΔQ/Δt. This sets up a differential equation as follows:
Vmaxsin(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 impedance and reactance were developed to greatly simplify the analysis. For a capacitor, the capacitive reactance, Xc, is 1/(ωC), where ω=2πf.The equivalent impedance of the series circuit in question can be calculated with complex arithmetic (beyond the scope of this course) or taken from the text:
Z = √[R2 + Xc2]
Z = √[R2 + 1/(ωC)2]
Then, the AC version of Ohm’s law can be used to calculate the rms current:
Vinput = IZ for the power supply
I = Vinput/Z
I = Vinput/√[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:
Vc = IXc
Vc = Vinput/√[R2+1/(ωC)2]*1/ωC
Vc = Vinput/√[(ωRC)2+1][1]
Vr = IR
Vr= Vinput/√[R2 + 1/(ωC)2]*R
Vr= Vinput/√[1+1/(ωRC)2][2]
These are the equations 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 thesound card.
1)Disconnect the cable from the back of your computer speakers. Do not disconnect the cable from your computer.
2)Set the volume on your computer to maximum.
3)Download and install Audacity freeware from the following web site:
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 = 0.1/(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 and resistor.
13)Repeat steps 11 and 12 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 differences across the capacitor 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 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 x-axis and select “format axis”.
16)Compare the theoretical and experimentalplots. 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.