EXPERIMENT 5 Series Resonance in AC Circuits Figure 1

Physics 112 Laboratory Exp. 5 – Series Resonance in AC Circuits (2004 Fall)
EXPERIMENT 5
Series Resonance in AC Circuits
Reference: Serway & Faughn, College Physics (6th. ed.), Chapter 21, Sections 21.4-6
Purpose of this Experiment:
A. To experimentally find the resonant frequency of an RLC circuit, and to compare it with the
value predicted using the inductance and capacitance in the circuit.
B. To study the behavior of the current, the inductive reactance, and the capacitive reactance in
the vicinity of resonance.
The calculations and answers to each of the Questions (below) should be included in
your Lab Report, as well as your conclusions There is a data sheet at the end of this document
(remember to get it signed by the T.A. before you leave the Laboratory).
Overview & Theory:
Figure 1 shows a resistor, a capacitor, and an inductor in series, connected to an AC source.
R
L
C
a
b
AC Source (oscillator) of
peak voltage
Vmax
Figure 1
RLC Circuit
The peak voltage across each element is given by:
V R = i max R (1)
V C = i max X C , where X C is the capacitive reactance (2)
V L = i max X L , where X L is the inductive reactance (3)
For the phase relations between these voltages, see Fig. 21.9 in your text.
The peak voltage of the AC source is given by:
V max = i max Z (4)
where Z is the impedance of the circuit in ohms (Ω).
Z is given by:
Z
( X − X ) 2
2
= R +
L C
(5)
where X L = 2πfL, XC = 1/(2πfC), and R is the resistance in the circuit.
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Physics 112 Laboratory Exp. 5 – Series Resonance in AC Circuits (2004 Fall)
Note, in Equations (1)-(4) the maximum values of the AC currents and voltages are used rather
than the effective values I and V. Since I = 0.707 i max and V = 0.707 v max , we are free to use either set
of values. Since it is easier to obtain i max and V max from the screen of the oscilloscope, we use the
maximum values here.
(X L - X C ) is the equivalent reactance between points a and b in Figure 1. This part of the circuit
consists of an inductor and a capacitor in series. The minus sign of X C indicates that the voltage across
the inductor is 180° out of phase with the voltage across the capacitor. The maximum voltage V L+C
between points a and b is given by:
V L+C = i max (X L -X C ) (6)
There is a specific frequency, called the resonant frequency, f res , at which X L = X C . The theoretical
value of f res is obtained by setting
leading to
X L = 2πf res L equal to X
f res
C
1
=
2π
f
res
C
1 1
= (7)
2π
LC
At resonance, according to Equation (5) and Figure 2, Z = R. Therefore, i max = V max /R and the
current in the circuit has its maximum value. Also, Equation (6) predicts that V L+C = 0 since X L - X C
= 0. In practice, V L+C is a minimum rather than zero at resonance because of the small amount of
resistance between points a and b due to the resistance of the inductance coil.
In this experiment, you will find f res for a particular RLC circuit by noting where V L+C is a
minimum. This experimental value of f res will then be compared to the theoretical value given by
Equation (7).
Apparatus:
Fixed-value inductor, capacitance box, resistance box, AC ammeter, oscilloscope, signal
generator.
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Physics 112 Laboratory Exp. 5 – Series Resonance in AC Circuits (2004 Fall)
Procedure:
1. Set up the circuit shown in Figure 1. Choose a capacitance which, when combined with the
given inductance, will give a resonant frequency somewhere between 1,000 and 10,000 Hz.
Use Equation (7) to find an approximate value of C corresponding to whatever f res you choose.
Then choose a capacitance close to that. Record the values of C, L, and R in the Datasheet.
2. Connect the CRO across points a and b in the circuit of Figure 1. (This will enable you to
measure V L+C .) Sweep through a range of frequencies below and above f res . Notice that V L+C is a
minimum at f res . It’s not zero as Equation (6) predicts because there is a non-negligible
amount of resistance in the inductor.
3. Place an AC ammeter in the circuit to measure the current.
4. Set the frequency to something below resonance and record the value of V L+C from off the CRO,
and the current from the AC ammeter. Recall from Experiment 4 that you likely will need to
measure the frequency with the CRO.
5. Measure the inductive reactance X L . To do this, measure the voltage across the inductor, V L ,
with the oscilloscope and use Equation 3. Since the AC ammeter measures effective current,
the maximum current must be calculated by multiplying by 2 .
6. Next measure the capacitive reactance, X C , using Equation 2.
7. Repeat procedures 4 through 6 for nine additional frequencies (four more below and five above
resonance). One of your frequencies should be as close to resonance as you can get. The
frequencies above and below should show the full range of changes in current.
Graph One:
Plot V L+C vs. frequency, f, for the ten frequencies at which you measured V L+C .
Question One:
Find the experimental value of f res from this plot. In other words, by hand of
otherwise draw a best fit line between the points. Remember the parabolic
nature of the curve.
Question Two:
Calculate the theoretical value of f res from Equation (7).
Question Three:
Calculate the percentage difference between the experimental and theoretical
values of f res .
Graph Two:
Plot the values of X L and X C obtained above as a function of frequency.
Immediately below, plot the current as a function of frequency.
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Physics 112 Laboratory Exp. 5 – Series Resonance in AC Circuits (2004 Fall)
Question Four:
Compare the following three results from your data:
a. The frequency at which V L+C is a minimum.
b. The frequency at which X L = X C .
c. The frequency at which the current is a maximum.
Question Five:
From your results suggest alternative definitions of the resonant frequency of
an RLC series circuit.
Final Conclusions
At the end of your Lab Report, cover the following topics in separate, concise paragraphs
1. The Experimental Results, including the measurement uncertainties ('errors').
2. The Sources contributing to the Uncertainty in the final result(s), including
estimates for the magnitudes and sources of systematic and random errors.
3. Summary of what you have leant from this experiment, and how it relates to the
Objectives and Theory behind the experiment.
END OF EXPERIMENT
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Physics 112 Laboratory Exp. 5 – Series Resonance in AC Circuits (2004 Fall)
DATA SHEET #1
Name:(Last/First)
Date
T.A. signature
Section
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