Notes on Experiment #12

The report for experiment 12 is due at the beginning of your next lab session. So, you have extra time to write the report. So, the report better be perfect!

Phasors and Sinusoidal Analysis

Be sure to print a copy of experiment #12 and bring it with you to your lab session. There will not be any copies available in the lab.

We will do experiment #12 AS IS. Follow the instructions in the experiment as given. The value of R in the circuit setup will be 100 Ohms and not the 1K shown in the experiment. Also, NL is the imdedance Z. This is the only change for this experiment.

PREPARE FOR THIS EXPERIMENT!

You will take 75 data values during this experiment! That takes time! So, you must come to the lab prepared! Read about and know about the setup and measuring techniques for this experiment. If you are not prepared you will not finish this experiment. But, as with all the other experiment, if you prepare in advance you will finish early.

There is no circuit analysis for this experiment so nothing to do there. Prepare the purpose, theory, and procedure as usual.

You _must_ set up three separate data tables using the headings given in the lab manual. The first two tables will have the following frequencies (in Hz) down the first column: 100, 200, 400, 600, 800, 1000, 1500, 2000, 2500, 3000. The frequencies for the third table will be found experimentally so you cannot list them until you get some data. The right most column will have the heading "C" for table 1 and "L" for table 2. Table 3 does not need that last column.

All of the above preparation must be submitted to your lab instructor at the beginning of the lab session for scoring and will be returned to you after you set up the first circuit.

The lab report will not be due at the end of the lab session. It will be collected at the _beginning_ of your next lab session. This is because the data evaluation and plotting will need more time then we have during the lab session. So, you have an extra week to write up this report. Given that much time your reports are expected to be perfect! ;)

What You Will Do in Lab

For this experiment you will be solving the mystery of the unknown elements. You will be given a capacitor and inductor of unknown values. By indirectly measuring the current-voltage characteristics you will determine the element values. Since the measuring techniques used are not very accurate (due to visual estimations) we will take many data samples and get an average value of the data. In the past the average values of the data have given elements values within 3-5% accuracy!

During the lab session for each impedance you will measure:
• Voltage |V_Z|;
• Voltage |V_R|; and
• Phase Angle Ø_Z.

What You Will Do at Home

After completing the experiment you will take your data home and:
• Fill in the remaining columns of the table;
• Calculate the average value of C and L from data of tables 1 and 2 respectively:
• Calculate F_{O_Calc} and compare it to F_{O_Exp} (See How it works below);
• Plot data:
  For each table, plot at each frequency:
  • The complex impedance vector in the complex plane; and
  • |Z| = F(w) (Connect the points to get an estimated continuous function.)
• Write your conclusion.

How it works

|Z| / Ø_Z = |V_Z| / Ø_V / |I_Z| / Ø_I

The Magnitude of Z

At each frequency in the table you will be directly measuring the RMS magnitude of the voltage across your elements and indirectly measuring the RMS magnitude of the current through your elements using the DMM. The ratio of these magnitudes gives the magnitude of the Complex Impedance of your element. SO,

|Z| = |V_Z| / |I_Z|

You will use a 100 Ohm resistor (NOTE CHANGE FROM LAB MANUAL 1K) in series with Z to indirectly measure the magnitude of I_Z. Since R is in series with Z they have the same current. If we measure |V_R| then:

|I_Z| = |V_R| / R [Be sure you measure R so you know its exact value.]

The Phase Angle of Z

Also at each frequency, you will directly measure the Phase Angle difference between the sinusoidal voltage and current using the oscilloscope (scaled to degrees using the technique explained in your lab manual. READ IT! KNOW IT!) This phase angle difference is the phase angle of the complex impedance of your element since:

Ø_Z = Ø_V - Ø_I

On the scope we will position I_Z so that Ø_I = 0. That way Ø_Z = Ø_V.

• Lead: If the voltage "leads" the current (voltage has a t-axis cross over point to the left of the current t-axis cross over point) then the angle is the distance in degrees between the two signals and has a positive sign.
• Lag: If the voltage "lags" the current (voltage has a t-axis cross over point to the right of the current t-axis cross over point) then the angle is the distance in degrees between the two signals and has a negative sign.

Evaluating Z

Z = a ± Jb

The value of b will be used to determine the value of the unknown element.

Z_C = 0 - J/wC

So from your data:

b = 1/wC so,

C = 1/wb

Calculate the value of C using the value of b at each frequency in table 1. Then get the average value of the ten values of C. Call it C_AVG.

The inductor you will be given is made of very long piece (10 yds?) of very fine wire wrapped around a metal core. This wire will have a resistance R_L of about 40 to 80 Ohms depending on your actual inductor. We cannot remove that resistance. So your actual "practical" inductor will behave like a resister in series with an ideal inductor so:

Z_{L_prac} = R_L + JwL

So from your data:

a = R_L and

b = wL so,

L = b/w

Calculate the value of L using the value of b at each frequency in table 2. Then get the average value of the ten values of L. Call it L_AVG. Note that the average value of data a is the average value of R_L.

Please Note: In the drawing of the practical inductor in your lab manual the ideal R is R_L and NOT the 100 Ohm resister in the circuit setup used to find |I_Z|.

Z_LC = Z_C + Z_{L_prac} = 1/JwC + ( R_L + JwL). So,

Z_LC = R_L + J(wL - 1/wC)

Notice that at just the right frequency w_o (called the resonant frequency):

w_oL - 1/w_oC = 0

At w_o Z_LC = R_L + J0 is a pure Real number. So the phase angle is zero. It is easy to show that:

w_o = 1/[(LC)^1/2]

Define: F_{O_Calc} = 1/{2(pi)[(L_AVG *C_AVG)^1/2]} as the calculated resonant frequency in Hertz.

You will find the experimental resonant frequency F_{O_Exp} by adjusting the frequency control dial on the function generator until the voltage and current images on the scope display cross the t-axis together everywhere. This means the phase angle is zero. Note that for your L-C combinations F_{O_Exp} will be in the range of 1000 to 6500 Hz. For table 3 use frequencies based on F_{O_Exp} as follows:

• F_{O_Exp} - 1000 Hz
• F_{O_Exp} - 500 Hz
• F_{O_Exp}
• F_{O_Exp} + 500 Hz
• F_{O_Exp} + 1000 Hz

You must fill in table 3. So, you will need the voltage and phase angles for each of the above calculated frequencies. Don't forget to get that data.

Setup Tips

• Run the function gen. amplitude at max output.
• Ch 2 must be set negative. (Pull Invert)
• The taller the images on the scope the more easily the angles can be measured. The images do not need to be the same height.
• You must alway reset the 7.2/cycle each time you change the frequency.
• Check that the baseline (line you see in GND mode) for both channels on the scope are in the same position - dead center on the screen - before each measurement.
• All of the phase angles for the capacitor should be -90 degrees. The cap. behaves in an ideal manner. For some of you the angle may drop a few degrees at frequencies above 1500 Hz. i.e you may get -87 or -82 but never -91 or higher(in magnitude.)
• The phase angle for the practical inductor at 100Hz is NOT zero. It is a very small angle in the range of about 2 to 12 degrees. Zero degrees will skew your data.

OK. That was a lot. But once you get into performing the experiment it will go quickly since it is so repetitive in nature.

Its my favorite experiment and quite instructive.

Have fun.