UNIVERSITY OF MASSACHUSETTS DARTMOUTH
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
ECE 201
CIRCUIT THEORY I
MEASUREMENT OF GAIN AND PHASE
INTRODUCTION
In general, when sinusoidal voltages are applied to resistive circuits, all of the voltages and currents will be sinusoidal in wave shape (differing only in amplitudes), and “in phase” (the phase difference between them will be equal to zero degrees).
This is not the case when the circuits contain any energy storage, or reactive elements, such as inductors and capacitors. The voltages and currents are still sinusoidal in wave shape, but they differ in their amplitudes (or magnitudes) and/or phase differences.
Consider the RC circuit shown here in Figure 1.
Figure 1. An RC circuit driven by a sinusoidal voltage source.
A plot of the source voltage and the voltage across the capacitor are shown in Figure 2 on the next page.
Figure 2. The source and capacitor voltages for the circuit of Figure 1.
Using the source voltage waveform as a reference, we can observe that the capacitor voltage waveform is a sinusoid of the same frequency as the source voltage, has a peak amplitude of 0.707 volts, and is “phase shifted” to the right of the source voltage.
First we will determine the amount of phase shift between the two sinusoidal waveforms. Let’s take a look at just the waveform of the input voltage. One cycle of that voltage is contained within 10 horizontal divisions, or, we could say that there is 360°/10, or 36°/division worth of phase.
In order to measure the phase difference between the waveforms, we need to do a comparison of the phase differences between corresponding points on both waveforms. We could choose to compare phase difference between the positive peaks, the negative peaks, the positive (or negative) zero-crossings, etc. In this case, it’s easy to compare the phase difference between the zero-crossings. It looks as if there is a phase difference of 1.25 divisions between the zero crossings. We can compute the phase difference in degrees as
(1.25 div)(36°/div) = 45° phase difference
Next, we have to determine whether there is a phase “lead” or “lag”. In order to do this, we need to establish a reference point. Usually it makes sense to use the source voltage as the reference. Doing this, we see that the zero-crossing of the source voltage occurs before the zero-crossing of the capacitor voltage. Therefore, the capacitor voltage “lags” behind the source voltage. We could also say that the source voltage “leads”, or, is ahead of, the capacitor voltage.
PRELIMINARY WORK
PROBLEM #1
Consider the voltages shown below in Figure 3.
Figure 3. Voltages for Preliminary Problem #1.
Using the source voltage as the reference, determine the phase difference between the source and output voltages. Is the output voltage leading or lagging the source voltage?
PROBLEM #2
Construct the circuit of Figure 4 in MultiSim. Set the frequency of the function generator to 1 kHz and observe the input and output voltages on the oscilloscope. Comment on the amplitude and phase of the output voltage with the input voltage as a reference.
Figure 4. RC circuit from Figure 1 with the resistor and capacitor interchanged.
Using the Bode Plotter in MultiSim7, run linear frequency response plots of the gain and phase for the frequency range from 10 Hz to 50 kHz. Comment on whether this is a “low-pass” or a “high-pass” circuit.
LABORATORY PROCEDURE / RESULTS
1. Construct the circuit shown here in Figure 5.
Figure 5. RC Circuit driven by a Function Generator
In order to obtain a 0.16 μF capacitor, use a parallel combination of 0.1 μF, 0.05 μF, and
0.01 μF capacitors. Set the function generator to provide an output voltage of 1 volt peak at a frequency of 1 kHz.
2. Keeping the input voltage constant at 2 volts peak-to-peak, measure the peak-to-peak voltage across the capacitor and the phase difference between the capacitor and input voltages for frequencies of 100 Hz, 500 Hz, 1 kHz, 2 kHz, and 10 kHz.
In your lab notebook, arrange your data in a table that presents the “gain” (the ratio of the output and input voltages) and the phase shift in degrees between the output and input voltages. Use the phase of the input voltage as your reference.
Use the floppy drive option to save a copy of your results for a frequency of 1 kHz to be included in your report.
3. Construct separate plots for the gain and phase shift as a function of the frequency
over the range from 100 Hz to 10 kHz. Comment on the performance of the circuit as a function of frequency. Would you call this circuit a “low-pass” or a “high-pass” circuit? How is the capacitor behaving as a function of frequency? Check your experimentally obtained results with those obtained using the Bode Plotter in MultiSim7.