CIRCUITS LABORATORY NOTES
Resonance
Parts List
resuistors
capacitors
inductors
820 Ω, 1 kΩ, 10 kΩ
1nF
10 mH
Objective
The objective of this experiment is to examine experimentaly various aspects of series and parallel resonant
circuits.
'Series' Resonance
A series 'resonant' circuit consists in general of
the series combination of a idealized resistor, an
idealized inductor, and an idealized capacitor as
illustarted to the right. In practice a 'real' inductor
invariably has an associated wiring resistance of
significance, and in relating the idealized circuit
model to a practical realization that should be
taken into account. The meaning conveyed by the
adjective 'resonant' will appear
in the discussion to follow. However one inference to be taken is that excitation of the circuit will
be sinusoidal. For that reason it will be useful to compute the input impedance of the resonant circuit:For
reasons to appear Qo is called the 'quality' factor of the circuit, and ωo is called the resonant frequency.
Note that inductive reactance increases w increases from 0 to ∞, while the capacitive reactance
decreases from ∞ to 0. Hence there is a
frequency, it is ωo, at which the two cancel and
the input impedance is simply resistive. For
frequencies greater than ωo the inductive
reactance dominates the capacitive reactance, and
vice versa at frequencies lower than ωo. It
follows also that the magnitude of the impedance
Z(ω) is a minimum at the resonant frequency ωo.
The figure to the left is a sketch of |Z(ω)|/R vs
radian frequncy ω. Two points other than the
resonance point are noted; these provide a
measure of how fast the curve widens as a
function of frequency ( called the 'bandwidth',
normalized to ωo in this case). The two points often are referred to colloquially as the '3 db' or '.707'
points. and correspond to frequncies at which the real and imaginary parts of Z have equal magnitude, i.e.,
Experiment #10 Resonance V2
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Copyright 1997 M H Miller
This forms two quadratic equations, with four roots for w. However only the two positive roots have
physical meaning. Thus
where ∆ω3db is the difference between the positive roots, i.e., the 'bandwidth' between the roots as
indicated on the curve. Note that the higher the 'quality' factor Qo the narrower the bandwidwidth, i.e., the
'sharper' the resonance. Quality factors range from about 5 or so using general purpose inductors up to
many thousands using specially built devices.
Experiment #1
Assemble the series resonant circuit drawn to the
left; the signal source is the Function Generator.
Set the function generator to produce a constant
10 volt p-p sinusoidal voltage. This voltage is to
be kept constant for this experiment, and should
be checked during the course of the experiment to
assure that an inadvertant change was not made.
Use the oscilloscope to measure the amplitude of
the voltage drop from A to B at various frequencies. Knowing the fixed Function Generator voltage, the
measured voltage drop from A to B, and the (measured) value of the nominal 1kΩ resistance the source
current amplitude can be calculated. From these data the magnitude of Z(w) can be calculated.
Make sufficient measurements to plot the magnitude of the impedance as a function of frequency
adequately; the frequency range of interest is from about 10 kHz to about 100 kHz. However the objective
is to obtain an experimental curve corresponding to the sketch of |Z(w)| vs w on the preceding page, and
you should plot your data to assure adequacy, particularly in the vicinity of resonance. Compare the
experimentally determined resonant frequency with the theoretically expected value.
A 'real' inductor has a circuit behavior which is that of a resistor inseparably in series with an inductance;
the resistor accounts for the resistance of the wire with which the inductor is wound. Often this resistance
is a significant part of the resonant behavior. Determine the 3 db bandwidth, and from this calculate an
experimental value for the circuit Q. Use this value of Q to calculate the wire resistance of the inductance,
and compare to the measured value.
Display the voltage A-B concurrently with the voltage drop across the capacitor, and determine the ratio of
the capacitor voltage to VAB. From this ratio calculate a value for Q and compare to the other calculated
values, and the theoretical value.
'Parallel' Resonance
A parallel resonant circuit is drawn to the right;
the admittance looking into this circuit is
We can save some effort in analysing this circuit
by recognizing the formal equivalence of the equation for Y(ω) with that for Z(ω) for the series resonant
circuit. The equations transform one into the other by making the variable changes
Y(ω)<->Z(ω)
G<->R, and
jωL<->jωC.
Hence all the relationships derived for series resonance translate to the corresponding expressions for the
parallel resonant circuit on making the variable substitutions noted. For example for series resonance Q is
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Copyright 1997 M H Miller
defined as ωoL/R. For parallel resonance ωoL-> ωoC, and R->G, and so Q is defined as ωoC/G. Since
ωo2LC is 1 in both cases this means that Q = R/woL for parallel resonance (R = 1/G).
A parallel resonant circuit assembled from discrete components is not entirely realistic in that it neglects the
wiring resistance of the inductor. If the inductive branch is replaced by the series R-L branch as drawn to
the left (
)
the admittance expression becomes
and the resonant frequency ωo ( defined as the frequency at which Y(ω) is real) is given by
ωo2LC = 1 - R2C/L.
Experiment #2
Assemble the parallel resonant circuit drawn on the right. Plot on the same graph the admittance Y(w) for
G = 0S (open-circuit), and also for G = 10-4S. The frequency range of interest will be from (roughly)
10 kHz to about 100 kHz. From the G=0S data
estimate the value of R (coil resistance) from the
resonant frequency. (Note that at resonance the
admittnace when G = 0S is RC/L.) Estimate R
from the data of G = 10-4S also and compare
with the other values. Estimate the value of
(woL/R)2 from the data and comment on how
significant the coil resistance is in this case.
Experiment #3
The circuit diagram to the left is that used for
experiment #2( with G = 0), and is reused in the
following experiment. This time adjust the
Function Generator to output a square wave with
a nominal repetition frequency of 50 kHz.
The periodic square wave can be represented by a Fourier series consisting of a sum of sinusoids with
frequencies which are integer multiples of 50 kHz:
Observe the signal input and the voltage Vo concurrently. Explain your observations.
Experiment #10 Resonance V2
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Copyright 1997 M H Miller