Michael Lin
Tuesday Section
Partners: Josh Narciso, Bryant Rolfe
Due Date: 04/24/07
AC Circuits
Michael Lin
The objective of this experiment was to study the time and frequency-dependent
behavior of an alternating current (AC) circuit driven by a sinusoidal voltage. By
scanning through a range of frequencies and recording the peak to peak voltages, the
resonance frequencies for circuits of three different resistances were recorded. The
phase differences between the driving voltage and the resistor, capacitor, and inductor
were measured using the digital oscilloscope. The observed frequency of the RLC circuit
was 3644.24 +/- 62.83 Hz. The inductance of the large copper coil was calculated to be
0.02527 +/- 0.0027 H. Both of these measured values were within 1σ of their respected
actual values. The measured lead of the inductor to the driving voltage was 87.78 +/-
5.00°, and the lag of the capacitor to the driving voltage was -75.19 +/- 0.02°, showing
statistically significant discrepancies from the predicted +90° lead and -90° lag.
INTRODUCTION
An alternating current (AC) circuit is one that contains the usual elements of resistors, capacitors,
etc. but is driven by a voltage that varies in time. For an AC circuit the voltage is sinusoidal, and
therefore the current in each circuit element will also be sinusoidal; however, the current may be out of
phase with the voltage across that element.
V(t) = V
max
sinωt
I(t) = I
max
sinωt
The voltage and current in a resistor have the same time dependence based on Ohm’s Law, and
therefore are in phase with each other. For an inductor, the current running through the device will be
out of phase with the voltage across of it. Through Faraday’s Law, it can be proven that the voltage
leads the current in an inductor by a phase shift of +90°. The quantity X
L
= ωL is known as the
reactance of the inductor, and is obtained at the point of maximum voltage through the inductor. The
relationship between the reactance and the resistance in Ohm’s Law serves to explain how an inductor
works. For a capacitor, the voltage and current are also out of phase, but now the voltage lags behind
the current by a phase shift of -90°.By observing the maximum voltage through the capacitor, the
quantity X
C
= 1/ωC is obtained, and is described as the reactance of the capacitor. The difference
between X
C and
X
L
is that unlike X
L,
X
C
decreases as a function of ω. In essence, the capacitor is a mirror
image of the inductor, resisting the flow of current when the driving frequency is small, but barely
resisting the flow when the driving frequency is high.
Because the voltages across each circuit element are sinusoidal, the total voltage across an
overall RLC combination must be sinusoidal and have some phase. The overall phase shift of the RLC
combination depends on the reactance of the capacitors and inductors. Because the both the reactance
for the capacitor and for the inductor are functions of the driving frequency ω, the total phase of the