A phasor is a complex number having a magnitude and a phase angle. The magnitude of phasor voltages and currents can be measured directly with the DMM. However the phase angle of a phasor is always taken relative to some standard; it represents the phase shift of the sinusoidal current or voltage in question, with respect to some reference sinusoidal current or voltage. In the circuit below we will take the reference quantity to be the current, and we will measure the phase shift of various voltages with respect to this current. Actually, since voltages are more convenient to deal with than currents we will use the voltage -VR/R, which is equal to i.
Figure 1
Measuring the magnitude and relative phase shift of some voltage V with respect to the reference current i involves an initial setup and a somewhat tricky measurement, detailed on the following pages.
- Measure R accurately so that from VR, you can calculate
i accurately.
- Connect VR as the CH2 input and V as the CH1 input.
- Set the SOURCE mode to CH1 (or CH2 - whichever provides the most stable
display.)
- Use the INVERT option for the CH2 display. The reason for this is that
VR = -Ri, so that reversing the polarity of
VR results in a signal which is Ri, i.e. has the same
polarity as i. If R = 1000 ohms then the signal being displayed is i
in milliamp units.
- Set the function generator's DC OFFSET to zero and the function type
to sinusoid.
- Adjust the vertical positions of the traces for CH1 and CH2 so that
they are accurately centered on the scope face, with the coupling
mode set to GND. This step is important, and you might want to check
the centering from time to time, for example if you change the vertical
sensitivity, since an error in vertical position will result in an error
in the phase measurement. During all measurements in this experiment, the
coupling should be set to their AC positions for both channels.
- Select the frequency of interest. Display CH2 only. Uncalibrate the
time axis (by the TIME/DIV dial), and adjust it, the horizontal position
knob, and the CH2 sensitivity until you get a single sinusoid covering 7.2 centimeters per cycle. Adjust the horizontal position until this sinusoid
starts at a convenient position. At this point the horizontal axis of the
scope face is calibrated, at 360 degrees per 7.2 cm, or 50 degrees per
centimeter, or 10 degrees per small division on the time axis; zero degrees
corresponds to the position[s] at which the sinusoid crosses the axis
going positive.
Measuring the phase angle and magnitude of v - Apply v to the CH1 input and VR to CH2. Adjust the
controls until you get a good picture. Be sure (by momentarily GND-ing CH1)
that the CH1 trace is vertically centered.
- Read off the phase angle of V with respect to VR
from the horizontal scale which has been calibrated above at 50 degrees/cm.
- Obtain the magnitude of v with the DMM. Together with the phase
angle from VR, this determines the phasor V.
When the frequency of the sinusoid is changed you must return to step g and start over, recalibrating the horizontal scale.
For Z (NL in the Figure), use the capacitor provided by your instructor, and let V be the voltage VZ across Z. Measure the magnitude and phase angle. Repeat the measurement at 10 frequencies provided by your instructor.
- Present your results in three forms:
- as a table;
- as a graph of Z in the complex plane showing the points Z(w) for 10 different values of w, each point labeled with the value of w; and
- as a graph of |Z(w)| vs w. Estimate the value of C from
the theoretical |Z(w)| = 1/wC and put it in the table. Expect
some deviation from theoretical since capacitors also involve inherent
resistance, primarily leakage resistance in parallel with the C. Calculate
the average value of C from the 10 estimates. This value will be used in
a later section.
Next repeat the investigation of the last paragraph, this time using an inductor provided by your instructor. (In the tabular representation, the last column will be "L".) Your results will reflect the fact that practical inductors really consist of a resistor and an inductor in series; the RL is inherent resistance of the wire from which the inductor is wound.
From your investigation, deduce the values of RL and L for each of the 10 trials, and average them. These values will be used later.
This part investigates an impedance which exhibits series resonance. The impedance consists of the capacitor you measured earlier, in series with the practical inductor you measured earlier.
Display VR on CH2 and V on CH1 as you have done before. Adjust the frequency of the signal generator until you find the frequency at which V has a zero phase angle, i.e. the impedance Z(w) is purely real. This is the observed series resonant frequency of the circuit. Call it wo. Now calculate the resonant frequency, using the theoretical formula wo = 1/(LC)1/2 and using the values of L and C which you determined earlier in part II above. Compare with the observed value.
Investigate the magnitude and phase angle of vZ, with respect to i, at frequencies in the vicinity of wo. On the scope you should be able to see a very dramatic change of magnitude and angle of vZ in this vicinity. Observe and record your results in the same 3 forms explained earlier, but without the last column (C or L) of the earlier tables. Take sufficient data in the vicinity of the resonant frequency to allow you to draw a good graph of |Z(w)|; draw the graph as you take the data. Then on the same axes with your experimentally derived plot of Z(w), draw the plot which would be expected on a theoretical basis, and comment.
Last modified: Mon Nov 12 10:49:07 2001