## Lab 8: AC Circuits

This lab investigates properties of various electrical circuits when an alternating potential difference is applied.

### I. Addition of Voltages. Phasor Diagrams

1. Connect the circuit shown below using a resistor R of about l000 ohms in series with a large capacitor.
2. Using the multimeter, measure the rms voltages across R, across C, and across the series combination of R and C. Using your known value of R, calculate the rms current in the circuit. Then determine the impedance (in this case assumed to be a perfect reactance) of the capacitor and, finally, evaluate the capacitance C.

3. Connect the next circuit shown using a resistor R of about l000 ohms in series with an inductor.

The inductor coil is designated (L,r) to emphasize that it has both inductance L and resistance r. Use the multimeter to measure the rms voltages across R, across the coil, and across the series combination. Considering that R is accurately known, calculate the rms current and then determine the coil impedance. Using the following expression representing the law of cosines, evaluate the coil phase angle:

Then determine values for Vr, VL, and evaluate r and L for the coil using vector methods.

II. Voltage Phase Relations

Set up the following series circuit:

Use the oscilloscope to explain qualitatively the phase differences between the various components. First look at the two voltages simultaneously using the CHOP mode of the oscilloscope for the following five connections for Ground, X Input, and Y Input, respectively:

1. b, c, d
2. c, b, d
3. b, a, c
4. d, e, c
5. d, e, b

What phase difference do you see in each case? Imagine what the Lissajous pattern would look like, keeping in mind what the X and Y connections will be. Now observe the Lissajous pattern to confirm your prediction.

### III. Frequency Dependence of Voltage and Phase

The phase difference between two AC voltages can be studied by using a dual beam oscilloscope and comparing the time when each voltage peaks. Take the small wooden box with the knob on top and hook it up as shown below. The diagram below is drawn so that you can see what each circuit contains.

A selector switch allows you to connect either R, C, L, LC series or LC parallel as the "second" element in the circuit, in series with the top resistor that is always in the circuit. Once one of these has been selected and hooked into the circuit, vary the frequency of the signal generator from about 20 to 200 Hz and observe the corresponding variations in both:

a) the magnitude of the observed CURRENT (the voltage across R is proportional to the current), and

b) the PHASE DIFFERENCE between the total voltage supplied by the generator and the circuit current.

In interpreting the phase difference, remember that the two beams of the scope are looking at opposite sides of the output of the oscillator. Thus, when there is no real phase difference, one beam should be at its maximum when the other is at its minimum. Check the pattern with the selector switch at R to see what this means.

What you should see is roughly as follows. You should be able to explain the forms of these graphs.  For the circuit elements we are using, the frequency scale for these graphs will run from 100-1000 Hz.

For the circuits that have series and parallel combinations, also evaluate the RESONANT FREQUENCY from the scope and compare the results with the calculated value from the theoretical formula.

### Internet Resources

You can find some LRC circuit labs on the internet. For an advanced treatment, point your browser at: LCR Circuits Lab.