Physics 184
#8A RLC Circuits: Free Oscillations
Goals
In this lab we investigate the properties of a series RLC circuit. Such circuits are interesting, not only for
there widespread application in electrical devices, but also for their mathematical solutions which
simulate other physical systems, including damped mechanical oscillators.
Reading
A discussion of Inductors can be found in Sec. 30.2. RLC circuits are discussed in Young and Freedman,
Sec. 30.6 in the 12th ed. The preceding two chapters discuss RL and LC circuits.
Theory
In Lab. 3 we studied Ohm’s law, which states that current i flowing through resistor R results in a
potential difference
VR = iR .
(1)
In Lab. 4 we learned that a capacitor C stores charge Q = CV. Because it takes time to charge or discharge
a capacitor, the current through a capacitor circuit changes exponentially with time. As a result, the
charge stored on the capacitor can be expressed as a time integral of the current: Q = ∫idt, so that
(2)
VC = ∫idt /C .
In Labs. 5 and 6 we studied and utilized a consequence of Faraday’s law, that a changing current in one
coil induces an emf in a second coil according to
E2 = − M
di1
(3)
dt
where M is the mutual inductance. However, the same effect is also realized in a single isolated coil: A
current in the coil establishes a magnetic field—and hence a magnetic flux—through the coil. If the
current changes, the flux changes, and hence an emf will be induced in the same coil. In analogy with Eq.
(3), this emf can be expressed as
E =− L
di
dt
(4)
where L is the self-inductance—or simply the inductance—of the coil. The unit for inductance is the same
as for mutual inductance, i.e., the Henry. The negative sign in Eq. (4) implies that the direction of the
induced emf is always such as to oppose the change in the current. For example, if the current through
the coil were to increase, the coil would produce an emf or voltage that would
reduce the magnitude of the current increase. Note that the induced emf does
not oppose the current itself; rather, it opposes, and tends to reduce, the
current change. Think of an inductor simply as a coil that slows down the
R
response of a circuit to changes by generating a changing magnetic flux.
The voltage decrease VL across an inductor will always be
di
VL = L
.
(5)
dt
Consider the series circuit shown in Fig. 1, with the capacitor uncharged when
the switch is closed at t = 0. Kirchoff’s law states that—at all times—the total
voltage summed around the circuit is zero, i.e.:
V − V R − V L − VC = 0
di 1
⇒ V − Ri − L − ∫ i dt = 0.
dt C
1
(6)
L
V
C
Fig. 1. Series RLC
circuit.
Physics 184
Differentiating with respect to time t:
di
d 2i i
+L
+ =0
dt
dt 2 C
R
⇒
d 2i
dt 2
+
R di
1
⋅ +
⋅ i =0
L dt LC
(7)
This second-order differential equation has the same form as the equation that describes the oscillation of
a pendulum in air, where the mechanical energy transforms back and forth between kinetic and
gravitational potential energy, but is gradually reduced or damped by air resistance. In the RLC circuit,
the electromagnetic energy oscillates between the electric field of the capacitor and the magnetic field of
the inductor, but is slowly dissipated by the resistor. Thus the RLC series circuit is also an example of a
damped oscillator. The current and its derivatives are analogous to the position, velocity, and acceleration
of the mechanical oscillator.
The solution to Eq. (7) has different forms depending on the relative values of R, L, and C. In each case
the presence of the inductor demands that i = 0 at t = 0; the capacitor requires i → 0 at t → ∞. The three
most important solutions are:
Underdamped Oscillator:
For small R the solution is:
2
R << 4L/C
i(t) = A1 exp(–Rt/2L) · sin(ω0 t)
where
2
ω0 ≈ 1/ (LC)
(8)
(9)
1/2
and A1 is a constant, approximately proportional to (C/L) . The form of this solution shows that the
current oscillates at angular frequency ω0 but with an exponentially-decaying amplitude characterized by
time constant τ = 2L/R, as illustrated in Fig. 2(a).
2
Critically damped Oscillator: R = 4L/C
2
As R increases more energy is dissipated in the resistor, and the oscillation ceases. When R = 4L/C
i(t) = A2 t exp(–Rt/2L) .
(10)
For t << 2L/R the current i(t) is proportional to t , but, as indicated in Fig. 2(b), it ends up by approaching
zero exponentially with time constant τ = 2L/R, as for the underdamped oscillator.
2
R >> 4L/C
i(t) = A3 [ exp(–t/RC) – exp(–Rt/L) ] .
(11)
Initially the current increases from zero, but at large times the second exponential goes to zero much
faster than the first, so that current ends up dying away with long time constant τ = RC, as shown in Fig.
2(b).
Overdamped Oscillator:
A mechanical analogy to the damped oscillator is a swinging two-way door, such as to a restaurant
kitchen. An underdamped door swings back and forth with decreasing amplitude. An overdamped door
takes for ever to close. A critically damped door is just right, and closes relatively quickly, without
oscillating.
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Physics 184
(a)
(b)
1.0
R = 50 Ω
L = 25 mH
C = 0.1μF
0.8
0.6
Overdamping
2 2
R /R0 = 50
.8
Relative current
0.4
Relative current
L = 25 mH; C = 0.1 μF
1.0
0.2
0.0
-0.2
-0.4
-0.6
.6
Critical damping
2
2
R = R0 = 4L/C
= 1.00 kΩ
.4
.2
-0.8
-1.0
0
1
2
3
Time (ms)
4
5
.0
.0
.1
.2
Time (ms)
.3
.4
Fig. 2. (a) Underdamped oscillations of series RLC circuit. (b) Overdamped and critically damped oscillations.
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Physics 184
Measurements
All measurements in this lab will be conducted using the circuit shown in
Fig. 3, in which the dc voltage source and switch of the Fig. 1 circuit are
replaced with a low frequency square wave derived from a function
generator. This automates the procedure of continually opening and closing
the switch, and allows us to easily display the voltages on an oscilloscope.
The voltage across R0 monitors the current in the circuit.
C
L
(i) Underdamped oscillations
1. Drive the circuit with a 20 Hz square wave and start with C = 0.1 μF and
R0 = 100 Ω, values that are low enough to ensure that the circuit is
underdamped. Your first task is to observe how the resonant frequency
increases with increasing C. The angular frequency ω0 of the decaying
oscillations can be deduced from measurements of the period of the
scope trace. Compare the result with the value predicted by Eq. (9).
Repeat for two other values of C, one below and the other above 0.1 μF.
V0 cos(ωt)
Scope
R0
Scope
ground
2. Now examine the exponential decay of the oscillations. Using C = 0.1 μF
measure the amplitude of the voltage across resistor R0 —proportional to
Fig. 3. Circuit used
throughout this lab.
the current i—at successive oscillation maxima. On semi-log graph paper
plot the voltage versus time t. A straight line should result if the
amplitude reduces exponentially, as suggested by Eq. (8). The time constant τ characterizing the decay
is equal to the time required for the voltage to drop to 1/e = 0.37 = 37% of the amplitude of the first
maxima.
Repeat the measurement of the time constant τ using another value of R0 between10 and 100 Ω.
Note: Because we expect τ = 2L/R, the measured time constant τ should be inversely dependent on R.
A graph of R0 as a function of 1/τ will form a straight line, but it will not pass through the origin,
because we have neglected the resistance of the inductor and the internal resistance of the function
generator.
(ii) Critically damped oscillations
1. With C = 0.1 μF, increase R0 until the circuit becomes critically damped, i.e., the oscillations cease
and the voltage across the resistor decreases rapidly to zero. It is not easy to do this precisely. Perhaps
the best way is to gradually increase R0 until the voltage does not discernibly dip below zero.
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2. For critical damping R = 4L/C. Use the values of L and C to predict Rc , the critical value of R.
3. Another way of getting information on Rc is from the time constant of the decay at large time. Adjust
the oscilloscope trace to examine the large t section of the oscillation well beyond the voltage peak,
and from this determine the time constant τ using the 37% rule. As in the underdamped case, we
expect τ = 2L/R. Now, R can be calculated from the observed time constant and the inductance L.
4. Comment on the above three values of R.
(ii) Overdamped oscillations
1. Increase R further by about a factor of 10, sufficient to ensure overdamping. Use the oscilloscope to
determine the time constant τ at large time t. For under- and critically damped oscillations τ depends
on R and L, but for overdamped oscillations we expect τ = RC.
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Physics 184
#8A Laboratory Report Sheet
RLC Circuits: Free Oscillations
Name: _______________________________ Partner: _____________________________
Lab Section: __________________________
Date: _____________________________
A. Preliminaries:
Inductor resistance = ……………………… Ω
Inductor = ……………………… mH
B. Underdamped oscillations:
Note: ωT = 2π
1. Oscillation frequencies
R0
(Ω)
C
(μF)
Osc. period T
(ms)
Oscillation ω
(s−1)
1 / LC
(s−1)
Difference
(%)
100
100
0.1
100
20
0.1
2. Oscillation decay time constants
Take t = 0 to be at first maximum in the oscillating signal.
The other maxima will be at t = T, t = 2T, etc., which are t/T = 1, t/T = 2, etc.
Record voltage maxima for each of two resistance values, R0.
Use C = 0.1 μF.
R0 (Ω) T (ms)
100
t/T
0
1
2
3
4
τ (ms)
Ampl. (mV)
Ampl. (mV)
• (a) Using one sheet of semi-log graph paper, plot the oscillation amplitude as a function of time
for each R0, using the ratio t/T as the time axis, and
draw the “best fit” line through the data points. A straight line indicates an exponential decrease.
(b) For each of the 3 lines determine the value of τ as the time at which the amplitude graph has
decreased to 1/e = 0.368 of the initial value.
(c) Express this time, τ, in seconds, or in ms. (Multiply by the period T.) Enter τ and 1/τ in the table.
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Physics 184
C. Critically damped oscillations: (Please show any calculations)
1. Adjust R0 to obtain critical damping of the oscillations.
Value of resistor required for critical damping R0 = ……………… Ω
2. L = ________________________
Estimated uncertainty = ±…………… Ω.
C = _________________________
Predicted value of Rc from nominal L and C values: Rc = ………………… Ω
3. Critical damping time constant
Measured time constant at critical damping: τ = ……………… ms
Predicted value of R from τ and nominal L value: R = ………………… Ω
4. Comment on the above values.
D. Overdamped oscillations:
R0 = ……………… Ω (see page 4 of instructions)
Ratio R2 / (4L/C) = ……………………… >> 1
Measured time constant for overdamping: τ = ……………… ms
E. Observation:
Note that the time constant for critical damping is the smallest observed in the experiment.
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