Physics 184
#8B RLC Circuits: Forced Oscillations and Resonance
Goal
To investigate resonance in a series RLC circuit that is driven by a sinusoidal voltage source.
This is analogous to resonance in mechanical oscillating systems with a driving force.
Reading
Resonance in RLC circuits is discussed in Young and Freedman, Sec. 31.5 in the 12th ed. Preceding
chapters discuss the impedance (the ac analog of resistance) of capacitors and inductors.
Theory
In Lab. 8a we saw how, if the resistance is relatively small, a series RLC circuit can be stimulated into
damped oscillations by means a sudden voltage change. The effect is analogous to a pendulum set into
oscillation by a single push. Eventually the mechanical energy of the pendulum dissipates away–mainly
due to air resistance–and the oscillations cease.
However, the oscillations can be sustained if the pendulum is pushed periodically. The largest oscillations
are obtained when the frequency of the pushes exactly matches the natural frequency of the pendulum.
This is called resonance.
We will now study this resonance condition by driving the series RLC circuit with a periodic sinusoidal
voltage source. The response of an RLC circuit to a sinusoidal driving voltage Vs cos( ωt) is
i (t )=
Vs cos(ωt )
1 ⎞
⎛
R 2 + ⎜ ωL −
⎟
ωC ⎠
⎝
2
= I 0 cos(ωt )
(1)
where ω is the angular frequency of the source and I0 is the amplitude of the sinusoidal current. Equation
(1) is simply a general form of Ohm’s law for ac circuits, with the denominator representing the ac
impedance. Note that the impedance contributions from the inductor and capacitance are frequency
dependent. The maximum current amplitude—corresponding to resonance—occurs at the frequency that
minimizes the impedance of the series circuit, i.e., for ω = ω0 :
V
1
1
I 0, max = s when ω 0 L −
= 0 ⇒ ω0 =
.
R
ω0C
LC
(2)
The frequency ω0 = 2πf0 is the natural frequency of the series RLC circuit. The current in the circuit
progressively decreases as f departs from f0, and the resultant bell-shaped resonance curve is characterized
by its maximum height and width. The width of the response curve is often expressed as the frequency
difference Δf between the two points on the curve–one each side of the maximum–where the current is
half its maximum value. The quantity Δf is often called the Full-Width at Half-Maximum (FWHM).
For a series RLC circuit the FWHM is related to R and L by:
Δω
Δf
R
.
=
= 3a = 3
ω0
f0
ω0L
(3)
This is easy to show in the approximation that ω ≈ ω0 . The parameter a = R/(ω0 L) characterizes not only
the width of the resonance, but also the magnitude. As a decreases, the resonance curve becomes
narrower and higher.
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Physics 184
Scope
Measurements - REVISED
C
The circuit for this experiment is almost the same as employed for Lab 8a.
The difference is that rather than switching the RLC combination on and
off using a low-frequency square wave, the circuit is to be powered by a
sinusoidal wave of variable frequency f of the order of 1 kHz.
L
First connect the four circuit elements: R0 L, C, and the function generator in
series, as shown. Any wire connected to the function generator ground (black
terminal) should be black. Any wire connected to the function generator output
signal (red terminal) should be red.
Scope
Now connect the ground to the channel 1 oscilloscope ground (black terminal)
with a black wire. Connect the function generator output signal to the scope
channel 1 input (red terminal) with a red wire. Connect the non-ground side of
R0 to the scope channel 2 input with a red wire.
As in past experiments, use the Measure menu to set the digital oscilloscope to
report the channel 1 frequency and the peak-to-peak voltages on channels 1 & 2.
Ratios of peak-to-peak voltages are equal to the ratios of the amplitudes.
R0
Scope
ground
Fig. 4. Circuit for studying
forced oscillations.
The measurements will determine the amplitude I0 of the current i(t) as a function of frequency and the
resistance R0. The current amplitude I0 is determined by measuring the voltage across R0.
As indicated by Eq. (1), the impedance of the circuit —and hence the voltage amplitude delivered by
the function generator— changes with frequency f. This makes it difficult and time-consuming to keep
VS constant, and we will not attempt to do that. However, the ratio of the current amplitude to the
source voltage amplitude has the same functional form as the current amplitude for constant VS.
The analysis of resonance in R-L-C circuits will be done by studying how the ratio
I 0 VS = VR R0 VS varies with frequency, and with the value of R0..
1. Record the inductance L, inductance resistance RL, and capacitance C. Calculate the expected resonant
frequency f0, and the total circuit resistance R that would result in critical damping. (See expt. 8a.)
2. These measurements are aimed at studying the maximum current and FWHM of the resonance shape
for various values of R0. Start with a value of R0 that ensures that the circuit is underdamped. A value
of 50 Ω or 100 Ω is suggested.
3. With the output of the function generator on oscilloscope channel 1, and the voltage across resistor R0
on channel 2, find the resonant frequency by tuning the frequency to obtain maximum VR. You will
notice that these two signals are in phase only at the resonant frequency. This observation of the phase
is the most sensitive way to determine the resonant frequency.
4. Change the frequency so that the peak-to-peak VR is about 80% of the maximum value. Record the
frequency and the peak-to-peak values of VR and VS.
5. Repeat this for frequencies at which peak-to-peak VR values are about 60%, 40%, and 20% of the
maximum value. This requires a set of frequencies above resonance, and another set below.
6. Double the value of R0, and repeat steps 3 through 5. Double R0 again, and repeat steps 3 through 5.
7. Data tables and suggestions for analysis of the data in terms of Equation 3 are on the report sheet.
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Physics 184
#8B Laboratory Report Sheet
RLC Circuits: Forced Oscillations and Resonance
Name: _______________________________ Partner: _____________________________
Lab Section: __________________________
Date: _____________________________
1. Parameter values:
RL = _____________ Ω
L =_____________ H
Predicted f =
0
1
2π
C =_____________ μF
1
= ______________ kHz
LC
2. Experimental resonant frequency and resonance width:
Take data as indicated in the instructions, and plot three resonances curves on a single graph with a
frequency axis extending from about 500 Hz to 2500 Hz. Partners can share measurement, calculating,
and graphing duties. Measure the FWHM from the graphs.
R0 = _____________ Ω
Freq.
(kHz)
VR0
VS
R0 = _____________ Ω
V R R0
Freq.
VS
(kHz)
f0
VR0
VS
V R R0
VS
f0
FWHM = fHI – fLO = ___________Hz
FWHM = fHI – fLO = ___________Hz
Δω = 2 π (fHI – fLO) = ___________s-1
Δω = 2 π (fHI – fLO) = ___________s-1
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Physics 184
R0 = _____________ Ω
Freq.
VR0
(kHz)
VS
R0 = _____________ Ω
V R R0
Freq.
VS
(kHz)
F0
VR0
VS
V R R0
VS
f0
FWHM = fHI – fLO = ___________Hz
FWHM = fHI – fLO = ___________Hz
Δω = 2 π (fHI – fLO) = ___________s-1
Δω = 2 π (fHI – fLO) = ___________s-1
3. Final Data Analysis
• Plot a graph of Δω as a function of R0. The R axis should include about 100 Ω to the left of zero
(negative R). Draw the best straight line through the data. How much additional resistance would
have to be added to each data point to make the line pass through the origin? Compare this with RL.
• Calculate the slope of the straight line, and calculate L from the slope.
R
According to Equation 3, Δω = 3 . Comment on your result.
L
• Calculate the product of the maximum current and Δω for each of the 3 sets of data.
Compare these three products.
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