Download ELEC 225L Circuit Theory I Laboratory Fall 2010

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.
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Rth = 50 
Rth = 50 
+
C
Vth
(AC)
+
−
vRLC
L
+
+
vC
−
+
vL
−
C
Vth
(AC)
+
−
−
vRLC
L
vRL
Rw
−
−
(a)
+
vC
−
+
(b)
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
Z eq  Z C  Z L  Z R ,
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
VRLC  Vth
ZC  Z L  Z R
,
Z C  Z L  Z R  Z th
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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  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
ZC 
1
jC
and
Z L  jL .
There is a single frequency where the two impedances exactly cancel, that is, where
0  ZC  Z L 
1
j
1 

 jL  
 jL  j  L 
.
jC
C
C 

This quantity can only be zero if
L 
1
1
 0  L 
C
C
  2 LC  1   2 
1
LC
 
1
LC
.
This frequency is sometimes labeled o, and the corresponding cyclic frequency fo is given by
fo 
o
2

fo 
1
2 LC
.
At the resonant frequency, ZC = ZL, and the voltage across the LC combination becomes
VRLC  Vth
Rw
ZR
.
 Vth
Z R  Z th
Rw  Rth
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.
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Rth = 50 
Rth = 50 
Rth = 50 
+
+
+
+
−
vRLC
Vth
+
−
Rw
−
(a)
+
−
vRLC
Vth
Rw
vRLC
Vth
Rw
−
(b)
−
(c)
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 nonresonant 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
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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.)
Rth = 50 
+
Vth
+
−
VRLC
−
C
L
and
Rw
+
VC
−
+
VRL
−
TEC of function generator
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  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?
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
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 wordprocessed, 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%
10%
10%
10%
20%
20%
Report – Completeness and technical accuracy, with all questions answered
Report – Organization, neatness, and style (professionalism)
Report – Spelling, grammar, and punctuation
Quality and completeness of wire resistance vs. frequency plot
First demonstration (wire resistance calculation)
Second demonstration (differential measurement)
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