ELEC 477 - Bucknell University



ELEC 225L Circuit Theory I Laboratory Fall 2010

Lab #5: Resonant Circuits in the Real World

Introduction

Circuits consisting of DC sources and resistors can accomplish many important tasks, but the really interesting applications make use of capacitors, inductors, time-varying (especially sinusoidal) sources, and a few other components. Using these devices, a talented engineer can build signal sources, amplifiers with tailor-made frequency responses, filters, and many other useful circuits. One key to making these kinds of circuits operate properly is a good understanding of the frequency-dependent behavior of capacitors and inductors, but almost equally important is the realization that there is no such thing as a pure resistor, inductor, or capacitor. All electronic components are imperfect, and it is especially difficult to fabricate inductors that do not exhibit significant internal resistance. In this lab exercise you will explore some of the realistic properties of a simple series combination of a capacitor and an inductor driven by a function generator.

Theoretical Background

The circuit we will study is shown in Figure 1. The voltage source Vth and the resistor Rth form a Thévenin equivalent representation of a function generator. As shown in Figure 1a, you will be connecting a capacitor (several different values) and an inductor to the function generator and measuring various voltage values. Almost all inductors are fabricated by winding one or more turns of wire around some type of support structure, called a coil form. Sometimes the coil form is an insulating material such as ceramic or plastic, and sometimes it is made of ferrite. The latter material is used if a high inductance value is desired in a compact size. The disadvantages of using ferrite coil forms are their weight and their nonlinear properties in certain cases.

Some inductors, especially those used in low-frequency (say, a few kHz or lower) applications, require many turns of wire, and the wire has to be thin in order to save space. A long piece of thin wire can have significant resistance that can cause unintended effects not anticipated by the circuit designer. The wire resistance is a distributed quantity, meaning that it is spread evenly along the length of the wire (assuming the wire has a uniform diameter). The inductance of the coil is also distributed. Thus, the coil’s inductance and its wire resistance are intermingled along the length of the coil. As shown in Figure 1b, the inductance and resistance are often modeled as separate, distinct circuit elements. This is a matter of convenience, and it aids clarity in the circuit representation. However, it is not possible to measure the voltage across L and Rw separately. A measurement of the voltage cross a coil yields vRL, the voltage across the series combination of L and Rw.

The wire resistance of commercially manufactured inductors is often listed along with the inductance value in data sheets. The resistance value is normally accompanied by the frequency at which it was measured because the resistance is highly frequency dependent for reasons you will learn in ELEC 390. Briefly, the wire resistance varies because of a phenomenon called the skin effect, in which a sinusoidal current flowing through a wire confines itself to the layer of metal right at the surface of the wire. The higher the frequency, the thinner the layer in which the current flows, and therefore the higher the wire resistance.

[pic]

Figure 1. Series LC circuit driven by a Thévenin equivalent sinusoidal source.

a) Ideal representation. b) More realistic representation that models the wire resistance in the inductor as Rw. The dashed line indicates that the inductance and resistance of the coil are intermingled, and therefore the voltages vL and vR across each cannot be measured separately. Only voltage vRL can be measured for the inductor.

Capacitors and resistors also have resistance in the leads connected to them, and the leads exhibit inductance as well. However, these quantities are often too small to be of much consequence and are frequently ignored. One important exception is in the selection of capacitors used to smooth current spikes in digital circuits. The switching of logic gates from one state to another usually causes brief demands for relatively large amounts of current. Capacitors connected across the power supply leads store charge for those moments of high demand. The capacitor discharges into the gate while the current from the power supply builds up to the level needed. These types of capacitors, often called bypass or decoupling capacitors, must have low equivalent resistance in order to supply the sudden demand for current quickly.

One of our goals in this exercise will be to measure the wire resistance of a coil at several different operating frequencies. We can’t do that directly with the equipment we have in the lab, but we can determine it indirectly. To start, we note that the total impedance Zeq of the capacitor and real inductor in series is given by

[pic],

where ZC is the impedance of the capacitor (assumed to be ideal in this case), ZL is the impedance of the inductive part of the coil, and ZR is the wire resistance of the coil. Using the voltage divider formula, the phasor voltage VRLC across the capacitor and inductor combination is

[pic],

where Vth is the phasor representation of the Thévenin equivalent voltage of the function generator, and Zth is the Thévenin equivalent impedance (which is equal to Rth = 50 Ω in this case).

Recall that the impedance of a capacitor has a negative imaginary value and that the impedance of an inductor has a positive imaginary value. As frequency increases, the magnitude of the capacitor’s impedance falls, and the magnitude of the inductor’s impedance rises, since

[pic] and [pic].

There is a single frequency where the two impedances exactly cancel, that is, where

[pic].

This quantity can only be zero if

[pic].

This frequency is sometimes labeled ωo, and the corresponding cyclic frequency fo is given by

[pic].

At the resonant frequency, ZC = −ZL, and the voltage across the LC combination becomes

[pic].

If we knew the exact values of L and C, we could determine the resonant frequency, tune the function generator to that frequency, and then we could determine ZR easily by solving the voltage divider formula above for ZR in terms of the known quantities Vth, VRLC, and Zth. We don’t actually know the phase of the Thévenin equivalent voltage relative to VRLC, but we can infer it knowing that at resonance Vth and VRLC must be in phase because they are related by the real ratio of resistor values given above.

In practice, we do not know the exact values of L and C. The tolerances of the component values are likely to be 20% or more, and we do not have the equipment to measure capacitance and inductance accurately. We need another way to determine whether or not the operating frequency of the function generator is equal to the resonant frequency of the LC combination. If we tune the function generator, we need some way to determine when we have arrived at the resonant frequency. Figure 2 suggests one way to do this. If the operating frequency is exactly equal to the resonant frequency of the LC combination, the impedances of the capacitor and inductor exactly cancel, and the resulting equivalent circuit is shown in Figure 2a.

[pic]

Figure 2. a) Equivalent circuit representation at the resonant frequency of the LC combination. b) Equivalent circuit representation when the operating frequency is below resonance. The capacitor symbol merely indicates that the total impedance of the RLC circuit has a negative imaginary part; the capacitance is not equal to C in the circuit. c) Equivalent circuit representation when the operating frequency is above resonance. The inductor symbol indicates that the total impedance of the RLC circuit has a positive imaginary part; the inductance is not equal to L in the circuit. The elements Vth and Rth together represent the function generator.

However, if the operating frequency is below resonance, the magnitude of the capacitor’s impedance will be greater than that of the inductor, yielding a net negative imaginary part. This state can be modeled as shown in Figure 2b. The capacitor in Figure 2b is not equal to C in the original circuit; it simply represents the net negative imaginary impedance in series with Rth and Rw. If the operating frequency is above resonance, the combined impedance of the capacitor and inductor has a positive imaginary value, and the net result can be modeled as an inductor as shown in Figure 2c.

We have no way of knowing what the net capacitance or inductance is at any particular non-resonant frequency; however, we can measure VRLC. The resistances Rth and Rw do not vary significantly with frequency. (The wire resistance is frequency dependent, but it changes very little over any given narrow range of frequencies.) If a net capacitance or a net inductance is present because the operating frequency is not at resonance, then the magnitude of VRLC will be greater than its value at resonance because the magnitude of the impedance across which VRLC is measured is larger relative to Rth than it is at resonance. (The impedance across which VRLC is measured is simply Rw at resonance.) Thus, we can find resonance by tuning the frequency of the function generator until |VRLC| has its minimum value.

Experimental Procedure

• Read the “Theoretical Background” section above. It will save you a lot of time as you complete the tasks listed below.

• Construct the circuit shown in Figure 3 using a 10-mH that will be provided to you. Obtain four capacitors with values that will resonate with the inductor at frequencies in the neighborhood of 100 Hz, 1 kHz, 10 kHz, and 100 kHz. In your report, show how you determined the capacitor values. Begin with the capacitor that yields resonance near 100 Hz. Adjust the function generator so that it produces a sinusoidal waveform with a Thévenin equivalent voltage of 10 Vpp or so. (Remember that the amplitude displayed on the generator is not the Thévenin equivalent voltage.)

[pic]

Figure 3. Series LC circuit driven by a function generator.

• Monitor the voltage VRLC using channel 1 of the oscilloscope. Find the resonant frequency for your particular LC combination, and record it. Then determine the wire resistance of the inductor assuming that Rth = 50 Ω and that Vth has the magnitude inferred by the function generator’s display. (It is not equal to the displayed value.)

• Repeat the previous step for the other three resonant frequencies using the capacitor values you calculated above. Record the wire resistance data obtained in each case.

• Demonstrate to the instructor or TA the voltage measurement and wire resistance calculation for one of the resonant frequencies.

• In your report, plot the measured wire resistance Rw as a function of frequency on a semi-log graph (linear for resistance; log for frequency).

• Use channel 2 of the oscilloscope to display the voltage VRL across the coil at one of the resonant frequencies. Capture the screen image for this case. At least one full cycle of the waveform should be displayed, and the vertical scale (V/div setting) should be adjusted appropriately so that the amplitude is clearly readable. Provide in your report any important information not included already on the screen image, such as the frequency of operation.

• Without changing the triggering controls or horizontal scale (time/div setting) from the previous measurement, use the differential measurement feature of the oscilloscope to measure the voltage VC across the capacitor. The voltage scale for channels 1 and 2 should be the same to obtain an unambiguous reading. Capture the screen image of the differential waveform, and record any important missing information.

• In your report, explain in your own words why you need to use the differential measurement feature to measure VC. That is, why can’t you measure VC directly by placing the test leads across the capacitor?

• Demonstrate to the instructor or TA your differential measurement of VC.

• In your report, comment on the phase relationship between VC and VRL. Are they nearly in phase or out of phase? If they are not exactly in or out of phase, what might account for that result?

Grading

Each group is expected to submit their results and the answers to the questions posed in this handout in a clear, concise, and professional narrative. The report does not have to be word-processed, but it should be easy to follow and as neat and well organized as possible with the context for each question included. The report should be written as if you were describing to another professor what you did in lab. Any writing issues specifically identified in the “Lab Report Guidelines” (available on the lab web site) that appear in this report or the remaining reports will be assessed significant grade penalties. The report is due at noon on Thursday, Nov. 4 in Dana 301 (the EE main office). Each group member will receive the same grade, which will be determined as follows:

30% Report – Completeness and technical accuracy, with all questions answered

10% Report – Organization, neatness, and style (professionalism)

10% Report – Spelling, grammar, and punctuation

10% Quality and completeness of wire resistance vs. frequency plot

20% First demonstration (wire resistance calculation)

20% Second demonstration (differential measurement)

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(c)

+

vRLC

−

Rw

Rth = 50 Ω

+

vRL

−

+

vRLC

−

Vth

+−

+

vRLC

−

Rw

Rth = 50 Ω

Vth

+−

+

vRLC

−

Rw

Rth = 50 Ω

Vth

+−

(b)

+

vC

−

+

vL

−

Rth = 50 Ω

(a)

L

Vth

(AC)

C

+−

+

vRLC

椷楺榰槤槥

−

+

vC

−

(b)

Rth = 50 Ω

+

−

(a)

Vth

+

VRLC

−

L

and

Rw

TEC of function generator

C

+

VC −

+

VRL −

Rw

Rth = 50 Ω

L

Vth

(AC)

C

+−

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