Notes on Experiment #9

Theorems of Linear Networks

Week 9 Lab Sessions (3/11 and 3/13)

Be sure to print a copy of experiment #9 and bring it with you to your lab session. There will not be any copies available in the lab.

Prepare for this experiment!

If you prepare, you can finish in 90 minutes. If you do not prepare, you will not finish even half of this experiment. So, do your preliminary work. Set up data tables and graphs before you come to lab.

Bring cm X cm graph paper

Measure the Resistors First!

The resistors must be accurate in this experiment. Discard any with an error greater than 5%. Ask your lab instructor for a replacement.

The resistor values should be:

Procedure

We will do the experiment almost "as is" in the experiment. The discussion below gives a bit more detail about the procedures of this experiment.

Part 1: Maximum Power Transfer Theorem

We will do this part twice. The first time through we will use a pure DC source. See Figure 1. The second time through we will use a pure AC source. See Figure 2.

For each case above we will measure and record VL for ten different test values of RL in the range 0.1RS to 10RS. This, of course, will require you to know the value of RS. It is very important to include RL = RS as the center test value of set of RL. So use this set of RL:

RL = {.1RS, .3RS, .5RS, .7RS, .9RS, RS, 2RS, 5RS, 8RS, and 10RS}

You will then calculate the power absorbed by RL:
PABS_RL = (VRL)2/RL for each value of RL. Use your data to plot PABS_RL as a function of RL.

To begin each case you will measure VOC, the "open-circuit" voltage. See Figure 3. This is the case when RL = infinity. i.e. there is no RL connected. Note that VOC = VS. Then connect a variable resistor as RL and adjust RL until the voltage VL becomes exactly 0.5VOC. When VL = 0.5VOC then we know that RL is exactly equal to RS. (See circuit analysis below.) So, we have just experimentally found RS! Use this value of RS to determine the test values required as explained above and measure the voltages VL as explained above.

Part 1A: DC Case

Build the circuit using these discreet values:

Now get the data for the various RL and plot the power curve.

Part 1B: AC Case

The circuit is the Function Generator! RS and VS are inside the function generator. DO NOT INCLUDE AN EXTERNAL RS!!!

Set VS = 5 Volts RMS (Pure AC. The DC = 0.) To set this just use the DMM to measure the AC voltage at the terminals of the function generator and adjust the amplitude control until the AC (RMS) meter reads 5.00 Volts. Now connect the resistor decade box as RL and follow the above procedures to determine the value of the internal RS of the function generator. Now get the data for the various RL and plot the power curve.

Answer these questions:

  1. Does RL = RS when VL = 0.5VOC?
  2. Does RL = RS when the maximum power is being delivered to RL?

Part 2: Linearity

Part 2A: DC Point by Point Plot (The hard way)

  1. Set up the circuit in Figure 4. Use a DC supply for VS.
  2. Measure VO for these values of VS:
    VS = { -4, -2, -1, 0, 1, 2, and 4} Volts.
  3. Plot VO as a function of VS. Connect the points to get a continuous relation. Is the relation linear?
  4. Verify that the slope VO /VS is the same value as calculated in your circuit analysis.

Part 2B: Automatic Plotting (The easy way)

  1. Set up the circuit in Figure 5. Use the function generator for VS.
  2. Connect the scope as indicated in Figure 5.
  3. Scope Setup
    1. Put the scope in "X-Y" mode.
    2. Set both channels to GND and position the "dot" to center screen.
    3. Now set both channels to 1 Volt/DIV
  4. Function Generator Setup:
    1. Turn DC to Off
    2. Use a sinusoidal waveform
    3. Set AC amplitude to maximum
    4. Set frequency to a "low" value ~60 to 120 Hz (whatever frequency give the best or "cleanest" image)
  5. You should now see a continuous plot of VO as a function of VS. Sketch it. Is the relation linear?
  6. Verify that the slope VO /VS is the same value as calculated in your circuit analysis.

Are the plots from the above two methods the same? Which method was easier?

Part 3: Superposition

  1. Set up the circuit in Figure 6.
  2. Use the DMM to accurately set:
    1. VS1 = 5.00 Volts.
    2. VS2 = 4.00 Volts.
  3. Now verify that superposition holds for V1 and I2. This requires that you show that:
    1. V1|(VS1 = 5, VS2 = 0) + V1|(VS1 = 0, VS2 = 4) = V1|(VS1 = 5, VS2 = 4)

      and

    2. I2|(VS1 = 5, VS2 = 0) + I2|(VS1 = 0, VS2 = 4) = I2|(VS1 = 5, VS2 = 4)
  4. HINT:After setting the sources, the best way to go back to Zero Volts (as is needed during data taking) is to remove the cables from a voltage source terminals and connect the cables together. You will have the Zero Volts required. Then, when you need the non-zero value again, just plug the cables back into the source. That way you do not waste time re-setting the source voltages.
  5. So, fill in a data table like the one below and verify that the addition of rows one and two is equivalent to row three for each column.
Superposition Data Table

Set up appropriate data tables and plots for all the expected data for each part.
You will then compare this data to the calculated values from your circuit analysis and do error analysis for each part.

Circuit Analysis

Note: An arrow through a resister is the circuit symbol for a variable resister. Your Lab instructor will show you how to use the POWER RESISTOR DECADE BOX as a variable resistor.

Part 1A: DC Case
  • RS = 3.3K, and
  • VS = 8 Volts DC

Figure 1.

Part 1B: AC Case
  • RS = 50 Ohms, and
  • VS = 5 Volts AC (RMS)

Figure 2.

For each circuit above the "open circuit voltage" VOC is the value of VL when RL is infinite. Note that in that case

VOC = VS. See Figure 3.

Figure 3.

Note that in Figures 1 and 2 if RL = RS then

VL = 0.5VS = 0.5VOC.

Which can be found easily by voltage division.

Also, when we have the above conditions, RL is absorbing the maximum power that the circuit is able to deliver. See pages 143-145 in your text for a proof.

Part 2: DC Point-by-Point Plot

For the circuit in Figure 4. find the ratio of VO /VS. You can do this using by successive voltage division of VS. Note that this ratio is a constant now matter what the value of VS. Show all of your work.
  • Part 2 Elements:

    R1 = 3.3K

    R2 = 6.8K

    R3 = 4.7K

    R4 = 10K

Figure 4.

VS = { -4, -2, -1, 0, 1, 2, and 4 volts}

Part 2: AC Continuous Plot

The circuit in Figure 5. shows how to connect the oscilloscope to easily verify linearity.

Figure 5.

Part 3: Superposition

Use the principle of superposition to find V1 and I2 for the circuit in Figure 6. Show all of your work.
  • Part 3 Elements:

    R1 = 3.3K

    R2 = 6.8K

    R3 = 4.7K

    R4 = 10K

Figure 6.

VS1 = 5 volts.
VS2 = 4 volts.

Have fun.

Last modified: Sun Mar 2 11:58:16 2008