ENEE 206

March 28, 2000



Laboratory 8 - Nonideal Passive Components

A. Lab Goals

In this lab you will quantify the nonideal properties of resistors, capacitors, and inductors and examine the phase and amplitude relationships between sources and passive components in simple AC circuits.

B. Background Reading

Review sedction 4.2 and read sections 9.2-9.4 (up to Example 9.3 on page 356) in (M/L).

C. Definitions

D. Laboratory Equipment

There is no new equipment required for the lab. oscilloscopes, DMM's, and LC meters will be used. in the course of the ecperiment.

E. New Hardware

The inductors used in this lab are encased in plastic to protect them from damage. The braided wires are soldered to solid wires at the end so that the inductors can be connected to the breadboard. The inductors are to be passed out by the TA at the beginning of the lab and returned to the TA at the end of the lab.

F. Circuit Analysis

The low frequency models for a nonideal inductor and capacitor are given in Fig. 8.1a and b, respectively. The inductor resistance RL comes from the finite conductivity of the wire that comprises the inductor. The cpacitor conductance Gc comes from the nonzero conductance of the dielectric material that separates the two capacitor plates. Capacitors typically perform more closely to the ideal model than inductors do; in this lab you will quantify this idea. At high frequencies, all inductors, capacitors, and resistores display the characteristics of each other. A typical model is shown in Fig. 8.1c.
The relative values of the three components depend greatly on the type of construction. As you might expect from the model, all components can exhibit a self-resonant effect at certain frequencies and only behaves as expected at frequencies well below self-resonance. Special fabrication techniues and configurations are required to produce components that work at high frequencies. While this is a very interesting subject for radio frequencty (RF) circuits - it is beyond the scope of this course.

Let's see how these nonideal properties affect system performance. Consider the ideal RL circuit shown in Fig. 8.2a. The normalized voltage across the inductor as a function of frequency is given by:

Vo/Vin = jwL/(R + jwL).

The magnitude of the voltage ratio is given by:

|Vo/Vin| = wL/[R2 + (wL)2]1/2 ,

and the phase angle is given by:

< Vo/Vin = tan-1(R/wL).

Note that at low frequencies, the output voltage is nearly zero. As the frequency increases, the relative output voltage increases until it approaches the input voltage (in magnitude and phase) at high frequencies. A circuit that displays this type of frequency response is called a high-pass filter. The frequency above which the output voltage is "nearly" equal to the source voltage is called the corrner frequency. It is defined as the frequency where

|Vo/Vin| = 2-1/2,

which occurs for this circuit when

wL = R/L.

PSpice simulation



How do the frequency characteristics change when the nonideal properties of the inductor are taken into consideration? The RL circuit with the nonideal inductor is shown in Fig 8.2b. The voltage divider equation is:

Vo/Vin = (jwL + RL)/(jwL + RL + R).

If we define the quality factor Q = wLL/RL, then

Vo/Vin = (1 + jQw/wL)/(R/RL + 1 + jQw/wL),

so

|Vo/Vin| = [1 + (Qw/wL)2]1/2/[(1 + R/RL)2 + (Qw/wL)2]1/2,

and

< Vo/Vin = tan-1(Qw/wL) - tan-1[Qw/wL)/(1 + R/RL)].

At high frequency, the ideal and nonideal circuit equations approach the same result. However, at low frequency the output voltage is

|Vo/Vin| -> RL/(RL + R) ~ RL/R

for a good inductor. At other frequencies, the nonideal equations can be used with the measured results to double-check the component values and vice-versa. However, the sensitivity of the calculation to experimental measurement error must be considered to ascertain the acuracy of our results. For example, if you know R and L, you could measure the angle between Vo/Vin to estimate RL. But if you can only measure the phase angle to an accuracy of 5o, what is the resulting uncertainty in RL? And what frequency is that uncertainty minimized? Note that the relative voltage magnitude can also be used to estimate parameters and the optimal frequency for that is likely to be different than that for the angle measurement. Here is one final word of caution. The nonideal properties may be a function of frequency! for example, the resistance will increase at high frequencies due to the skin effect, which is an effect whereby the current at the center of a conductor decreases with increasing frequency due to Maxwell's equations. The frequencies where this effect is evident depend on material composition and component size but are generally very high.

The RC series circuit is shown in Fig. 8.3a with an ideal capacitor and in Fig. 8.3b with a nondeal capacitor. The circuit functions as a low-pass filter, with Vo ~ Vin at low frequencies and Vo -> 0 as the frequency becomes large. For an ideal capacitor, the corner frequency where |Vo|/|Vin| = 1/21/2 is

wc = 1/RC.

When we include the conductance we find Vo/Vin = (1 + RGc + jwRC)-1,

so

|Vo/Vin| = [(1 + RGc)2 + (wRC)2]-1/2 and < Vo/Vin = -tan-1(wRC/(1 + RGc).

These equations can be used with oscilloscope measurements to verify component values; for good capacitors you should find Gc to be very small.

As the next example, consider the RLC series circuit shown in Fig. 8.4. While both a nonideal inductor and nonideal capacitor are shown, we will simplify the analysis by assuminng Gc = 0. The 51 W resistor is included to protect the function generator, which will be used to generate the input signal. We are interested, bowever, in measuring the inductor voltage relative to the voltage across the LC combination:

Vo/V1 = (jwL + RL)/[jwL(1 - 1/w2LC) + RL].

The resonant frequency when RL = 0 is

wR = (LC)-1/2.

At the resonant frequency, Vo/V1 is infinite. However, when RL is not equal to 0, the relative output voltage at the ideal resonant frequency becomes:

|Vo/V1| = [1 + (RL/RL)2]1/2 = (1 + Q2)1/2 for Q = wRL/RL.

This quality factor can be quite large, so the ratio of voltges can be quite large. Let's take as an example the following parametes: L = 1 mH, RL = 10 W, C = 1 nF, Vin = 10 V, and R = 51 W. The resonant frequency is wR = 106. The quality factor is Q = 106 x 10-3/10 = 100. The ratio of the voltage magnitudes is |Vo/V1| ~ 100, but to finalize the result we need the ratio |V1/Vin|. The circuit in Fig. 8.5 shows the general result, but at resonance there is a simplification because the capacitance and inductance "cancel" each other out and Z = RL. Thus,

|V1/Vin| = RL/(51 + RL) = 10/61 and |Vo/Vin| = |Vo/V1||V1/Vin| = 1000/61 = 16.

Notice that the voltage cross the inductyor is higher than the input voltage by a factor of 16! This amplification factor is not too high, but could be much higher if the quality factor is larger or if the 51 W resistor were omitted. Without the 51 W resistor, for example, the inductor voltage wuold be 1000 volts!

The quality factor is also a measure of the relative "bandwidth" of the resonance in the output voltage. For x = w/wr, the relative output voltage can be rewritten:
Vo/V1 = (x - j/Q)/[(x + 1)(x - 1)/x - j/Q].

The bandwidth is defined to be the frequency range where the output voltage is at least 2-1/2 times as large as the maximum value. The frequencies where the relative voltage is down by 21/2 are called the 3 dB frequencies. If Q is large, then the two 3 dB points are symmetric about wr and can be written as w+/- = wr +/- Dw, where Dw << wr. Let x+/- = 1 +/- Dw/wr, so x ~ 1, and we can simplify the expression for the output voltage to Vo/V1 ~ 1/[2(x - 1) - j/Q]. In that case the relative bandwidth is just BW = 2Dw/wr = x+ - x-. But |Vo/V1| ~ 1/{2-1/2[(x - s)2 - 1/(4Q)2)]}, so x+/- = 1 +/- 1/(2Q), and the relative bandwidth BW = 1/Q.

As a final example, consider the ideal pararllel LC circuit shown in Fig. 8.6. The 51 W rsistor is in series with the LC combination to protect the function generator at low and high frequencies. The voltage divider ratio is:

Vo/Vin = 1/{1 + jwRC[1 - 1/(w2LC)]}.

The resonant frequency is still given by wR = (LC)-1/2, however, the voltage ratio simply goes to one at resonance (the parallel LC impdedance is infinite) and the ratio of the magnitude is less thasn 1 everywhere else.

It is left as an exercise to calculate the quality factor of this circuit and to look at the voltage divider ratio when the inductor and capacitor are not ideal, i.e. when RL and Gc can't be neglected.

Helpful Hints

  1. Don't forget the 51 W resistor in the LC circuits. You will burn out the function generator without it and that is very costly to repair.
  2. Because the voltages in LC circuits can be much higher than the source voltage, DO NOT TOUCH any part of the circuit while the function gerrator is on.
  3. There are several different types of "grounds" that are referred to in circuit analysis. Perhaps the most fundamental is "earth ground" which refers to the (voltage) potential of the earth. The earth ground is the ultimate reference and the found connection in wall outlets (the round hole) is tied to it. The "neutral" wire (the bigger of the two slots in an outlet) is in principle connected to earth ground at some point in the building, but because that wire normally carries current and the (copper) wires have finite resistance, the potential in the lab of the neutral wire is usually not exactly zero. The ground wire is not designed to normallyt carry current and is essentially at earth ground. "Chassis ground" referes to the potential of the case of a piece of electronic equipment. If the power cord for the equipment has 3 prongs (hence the ground wire), the chassis ground is tyopically connected to earth ground. "Circuit ground" sometimes refers to the node with the greatest connectivity, but usually referes to the point in a circuit that has the "negative" side of AC sourdes or DC supplies connected to it. However, most sources and powler supplies can "float", i.e.,neither side is connected to ground, so there is no guarantee that circuit ground is the same thing as earth ground or even chassis graound. It is generally best to make some point in the circuit a true earth ground; otherwisle there is a risk of receiveing an electrical shock. There MUST only be one "earth ground" in the circuit. The ground wire on an oscilloscope probe is connected to chassis ground, so care must be taken when measuring voltages across specific components. For example, if you want to measure the voltage across an inductor in an RLC series circuit, you should build the circuit with the inductor connected to circuit ground. If you also want to measure the voltage across another component that does not have one side connected to circuit ground, you must use two probes and use the math functions on the oscilloscope to subtract the signals.
  4. To help with the PSpice simulations of the analog circuits, you can download a PSpice esample called lab7.sch from the web page. The example has an RLC series circuit which is driven by a "VSIN" source. The simulation is set up to do both an AC frequency sweep AND a transient analysis. Take a careful look at the VSIN attributes and the setup menus to see how the analyses should be run. Additional information can be found in the 204 book and in the PSpice references. If when you click on the link with the right mouse button and select "save link as" from the menu. If your calculations and PSpice don't agree, try increasing the accuracy of the PSpice simulation. In the "analysis setup" window, click on "options" (upper right box). Scroll through the right box until you see the option "RELTOL" with the value 0.001. Make it somewhat smaller (a good value to try is 0.0001), then click the "Accept" button and then the "O.K." button.
  5. The key results from a calculation of the behavior of an RLC circuit (with a nonideal inductor) are given velow.
WARNING!
In this lab there can be large voltages across the capacitors and indudctors. Do not touch any components whil the function generator is on (as always)! Set the function generator output to 2 V peak-to-peak and leave it there for all measurementws!!!

Laboratory 8 Description - Nonideal passive components

Objective:

To quantify the nonideal properties of resistors, capacitors, and inductors and to examine the phase and ampolitude relationships betwqeen sources and passive components in simple AC circuits.

Available Hardware:

Resistors: 51 W, 220 W
Capacitors: 1.8 nF, 56 nF, 4.71.0mF, 4.7mF

Pre-lab preparation:

  1. Calculate the 3 dB frequencies for the series RC combination of the 51 W resistor with all 3 capacitors.
  2. Calculate the resonant frequencies of all 9 possible series combination of the 220 W resistor with all 3 inductors.
  3. Calculate the resonant frequencies of all 9 possible series combinations of inductors and capacitors.
  4. Assume you have a series RLC circuit with R = 220 W, C = 1.8 nF, and L = 4.7 mH. Assume the function generator is connected to the circuit with a peak-to-peak voltage of 2 V.
      Calculate the maximum voltage across the inductor and the frequency at which the voltage occurs.
    1. Use PSpice to simulate this RLC circuit and plot the inductor voltage within two decades of the frequency where the maximum voltage is obtained.

Experimental Procedure:

During this experiment, be certain that you:
  • Ask the TA questions regarding any procedures about which you are uncertain.
  • Turn off all power supplies any time that you make any change to the circuit.
  • Arrange your circuit components neatly and in a logical order.
  • Compare your breadboards carefully with your circuit diagrams before applying power to the circuit.
  • Discharge ALL capacitors beffore using the capacitance-inductance analyzwe(LC meter) to measure their capacitance. NOTE: There is only one meter; you can measure the values at any time during the lab.
  • Complete the following tasks:
    1. Use the LC meter to measure the inductance and capacitance of all 6 components listed above in the pre-lab section.
    2. Measure the dc resistance of each resistor.
    3. Measure the dc resistance of aall 3 inductors.
    4. Connect the 220 W resistor and the 4.7 mH inductor in series. Measure the reltive apmpitude and phase of the inductor voltage at the 3 dB frequency. (Telative to the amplitude and phase of the voltage source) Measure also the relative amplitude and phase of the resistor volrtage (use waveform math).
    5. Repeat the preceding step for the other two inductors.
    6. For ONE of the RL circuits, store voltage traces for the source, the resistor, and the indujctor, and plot the 3 volttages AND the difference of the source and resistor voltages.
    7. Connect an RLC series circuit consisting of the 220 W resistor and the 1.8 nF capacitor and 4.7 mH inductor. Determine the resonant frequency and quality factor of the circuit. Measure the peak voltage on the inductor.
    8. Reat the prededing step for any 2 additional combinations of inductors and capacitors. Plot the voltage across the resistor and the voltage acxross the inductor for one of the combinations at resonance.
    9. Connect the 1.8 nF capacitor and the 4.7 mH inductor in parallel and connect that comgination in series with the 51 W resistor. Determine the resonant frequency and quality factor of the circuit. Measure the peak voltage on the inductor.

      Extra Credit

    10. Connect the tqo outer terminals of the potentiometer to the function generator via a seies resistor of 51 W. Measure the phase difference between the voltage accross the 51 W resistor and the potentiometer every decade from 1 kHz to 10 MHz.

    Post-lab analysis:

    Generate a lab report following the sample report available in Appendix A. Mention any difficulties encountered during the lab. Describe any results that were unexpected and try to account for the origin of these results(i.e. explain what happened). In ADDITION, answer the following questions:
    1. What are the relative phase differences begween the source voltage, the inductor voltages, and the resistor voltages for the 3 RL circuits tested? Are these values different from what youwould expectr? Why or why not?
    2. What are the amplitude differences between the soource voltage, the inductor voltrages, and the rsstor voltages for the 3 RL circuits tested? Are these values different from what you would expect? Why or why not?
    3. How do the relative voltage amplitudes and phases between the resistor and the inductor change if the frequency is 10 times lower than the 3 dB frequency? If the frequency is 10 times higher than the 3 dB frequency?
    4. What are the relative phase differences between the source voltage, the capacitor voltages, and the resistor voltages for the 3 RC circuits tested? Are these values different from what you would expect? Why or why not?
    5. What is the relative phase difference between the capacitor voltage, the inductor voltage, and the resistor voltage for the RLC circuit tested in step 9? Are these values different from what you would expect? Why or why not?
    6. Use the 3 dB measurements in steps 4-5 to estimate the values of inductance from the RL measurements for all 3 inductorrs. do the results agree with the LC meter?
    7. Use the 3dB measurements in step 7 to esstimate the values of capacitance from the RC measurements for all e capacitors. Do the results agree with the LC meter?
    8. Estimate the internal resistance of the inductorrs as a function of frequency from all abailable data. Interpret the results.
    9. Estimate the internal resistance of thecapacitors as a function of frequency from all abailable data. Interpret the results.
      Extra Credit:
    10. Estimate the internal inductance of the potentiometer.




    Laboratory 8 Pre-lab Questions - Nonideal passive components

    1. ....
    2. ....
    3. ....
    4. The quality factor of an RLC parallel circuit is .....




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