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][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][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!
Experimental Procedures
1)Select a resistor and capacitor so that the crossover frequency from equation [3] is a few thousand hertz. You should measure the resistance of the resistor with the ohmmeter, but use the listed value for the capacitance. Assume ±10% for the capacitance unless otherwise specified.
2)Set the function generator to a low frequency.
3)Using the cables, the resistor, the capacitor, and the function generator, make a series RC circuit. The negative side of the function generator should be connected to the negative side of the capacitor.
4)Use three multi-meters configured as AC voltmeters to measureΔVr, ΔVc, and E0for an extremely largerange of frequencies in an approximately geometric series.The frequency range should be large enough to measure several large potential differences (>0.8* E0), several small potential differences (<0.2* E0), andseveral potential differences in the intermediate range for both the capacitor and resistor. If you do not use an extremely large range of frequencies as defined above, then you will not receive full credit on your report.
5)Calculate the theoretical potential differences across the capacitor and resistorfor the measured frequencies using equations [1] and [2]. You do not need to include error propagation in the theoretical potential difference.
6)Plot the experimental and theoretical potential difference (vertical axis) versus the frequency on a single graph for both the resistor and the capacitor. 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, then right click on the x-axis and select “format axis”. Show your graph to your instructor.
7)Compare the theoretical and experimental plots. In this case, since you do not have error propagation, there will be some subjectivity in the comparison and you are allowed to use the word “close” in your report.
8)Estimate the crossover frequency from the graph and compare this to the theoretical value from equation [3].
9)Repeat steps 1 through 9 with a different resistor or capacitor.