Oscillations in an LC Circuit
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Transcript Oscillations in an LC Circuit
Oscillations in an LC Circuit
AP Physics C
Montwood High School
R. Casao
• When a charged capacitor is connected to an
inductor as shown in the figure and the switch is
then closed, oscillations will occur in the current
and charge on the capacitor.
• If the resistance of the
circuit is zero, no energy is
dissipated as joule heat and
the oscillations will persist.
• The resistance of the circuit
will be ignored.
• Assume that the capacitor has an initial charge Q
and that the switch is closed at t = 0 s.
• It is easiest to describe what happens in terms of energy.
• When the capacitor is fully charged, the total energy U in
the circuit is stored in the electric field of the capacitor
and is equal to
Qm 2
U
2 C
• At this time, the current is zero and there is no energy
stored in the inductor.
• As the capacitor begins to
discharge, the energy stored
in its electric field decreases.
• The circuit behavior is
analogous to an oscillating
mass-spring system.
• At the same time, the current increases and some energy
is now stored in the magnetic field of the inductor.
• Energy is transferred from the
electric field of the capacitor to the
magnetic field of the inductor.
• When the capacitor is fully
discharged, it stores no energy.
• At this time, the current reaches
its maximum value and all of the energy is now stored in
the inductor.
• The process then repeats in the reverse direction.
• The energy continues to transfer between the inductor
and capacitor indefinitely, corresponding to oscillations
in the current and the charge.
• The potential energy stored in a stretches spring,
0.5·k·x2, is analogous to the potential energy
stored in the capacitor,
Qm 2
U
2 C
• The kinetic energy of the moving mass, 0.5·m·v2,
is analogous to the energy stored in the inductor,
0.5·L·I2, which requires the presence of moving
charges.
• All of the energy is stored as potential energy in the capacitor at
t = 0 s because I = 0 A.
• All of the energy is stored as kinetic energy in the inductor,
0.5·L·Im2, where Im is the maximum current.
• In figure a, all of the energy is stored as electric potential energy
in the capacitor at t = 0 s.
• In figure b, which is one-fourth of a period later, all of the energy
is stored as magnetic energy 0.5·L·Imax2 in the inductor.
• In figure c, the energy in the LC circuit is stored completely in the
capacitor, with the polarity of the plates now opposite to what it
was in figure a.
• In figure d, all of the energy is stored as magnetic energy
0.5·L·Imax2 in the inductor.
• In figure e, the system has returned to the initial position,
completing one oscillation.
• At intermediate points, part of the energy is potential
energy and part is kinetic energy.
• At some time t after the switch is closed and the
capacitor has a charge Q and the current is I.
– Both the capacitor and the inductor store energy, but the sum
of the two energies must equal the total initial energy U stored
in the fully charged capacitor at time t = 0 s.
– Since the circuit resistance is zero, no energy is
dissipated as joule heat and the total energy must
remain constant over time.
U UC
– Therefore:
2
Q
2
UL
0.5 L I
2 C
dU
0
dt
• Differentiating the energy equation with respect to time
and noting that Q and I vary with time:
Q2
2
d
0.5 L I
2 C
dU
0
dt
dt
dU
dt
dU
dt
dU
dt
dU
dt
Q2
d
d 0.5 L I 2
2 C
0
dt
dt
2
2
1 dQ
dI
0.5 L
0
2 C dt
dt
1
21 dQ
21 dI
2 Q
0.5 L 2 I
0
2 C
dt
dt
Q dQ
dI
L I
0
C dt
dt
• We can reduce the equation to a differential equation of
one variable by using the following relationships:
dQ
dI d 2Q
I
and
dt
dt
dt 2
dU Q dQ
dI
L I
0
dt
C dt
dt
dU Q
d 2Q
I L I
0
2
dt
C
dt
Q
d 2Q
I L I 2
C
dt
2
2
Q
d Q
d Q
1
L
Q
C
C L
dt 2
dt 2
• We can solve for the function Q by noting that the
equation is of the same form as that of the mass-spring
system (simple harmonic oscillator):
d 2 x k
2
x
x
2
m
dt
– where k is the spring constant, m is the mass, and k m
– The solution for the equation has the general form
x A cos t
– where ω is the angular frequency of the simple harmonic
motion, A is the amplitude of the motion (the maximum value
of x), and is the phase constant; the values of A and depend
on the initial conditions.
• Writing
d 2Q
dt 2
1
Q in the same form as the
C L
differential equation of the simple harmonic oscillator,
the solution is:
Q Qm cos t
– where Qm is the maximum charge of the capacitor and
the angular frequency ω is given by:
1
L C
– The angular frequency of the oscillation depends on
the inductance and capacitance of the circuit.
• Since Q varies periodically, the current also varies
periodically.
• Differentiating the harmonic oscillator equation for the
LC oscillator with respect to time:
Q Qm cos t
dQ d Qm cos t
dt
dt
d cos t
dQ
I
I Qm
dt
dt
d t
I Qm sin t
dt
d t d
I Qm sin t
dt
dt
dt
I Qm sin t
0
dt
I Qm sin t
• To determine the value of the phase angle , examine the
initial conditions. In the situation presented, when
t = 0 s, I = 0 A, and Q = Qm.
I Qm sin t
0 Qm sin 0
0 Qm sin
• The phase constant = 0.
0 Qm sin
0 sin
0
sin1 0
• The time variation of Q and I are given by:
Q Qm cos t
I Qm sin t
I Im sin t
– where Im = ω·Qm is the maximum
current in the circuit.
• Graphs of Q vs. t and I vs. t:
– The charge on the capacitor
oscillates between the extreme
values Qm and –Qm.
– The current oscillates between
Im and – Im.
– The current is 90º out of phase
with the charge.
– When the charge reaches an
extreme value, the current is 0;
when the charge is 0, the current
has an extreme value.
• Substituting the equations for
the oscillating LC circuit into
the energy equations:
U UC
Q2
UL
0.5 L I 2
2 C
Q Qm cos t
I Im sin t
• Total energy:
U UC
2
Q
UL
0.5 L I 2
2 C
Qm cos t
2
U
2 C
0.5 L I m sin t
2
Qm
U
cos2 t 0.5 L I m 2 sin2 t
2 C
• The equation shows that the energy of the system
continuously oscillates between energy stored in the
electric field of the capacitor and energy stored in the
magnetic field of the inductor.
2
• When the energy stored in the capacitor has its
maximum value, Qm 2 , the energy stored in the inductor
2 C
is zero.
• When the energy stored in the inductor has its maximum
value, 0.5·L·Im2, the energy stored in the capacitor is
zero.
• The sum of the UC + UL is a
constant and equal to the total
Qm 2
energy
.
2 C
• Since the maximum energy stored in the capacitor (when
I = 0) must equal the maximum energy stored in the
inductor (when Q = 0), Qm 2
2
2 C
0.5 L I
• Substituting this into the total energy equation:
Qm 2
2
2
2
U
cos t 0.5 L I m sin t
2 C
2
2
Qm
Qm
2
U
cos t
sin2 t
2 C
2 C
2
Qm
U
cos2 t sin2 t
2 C
2
Qm
2
2
U
cos t sin t
2 C
cos
2
t sin t 1
2
2
Qm
U
2 C
• The total energy U remains constant only if the energy
losses are neglected.
• In actual circuits, there will always be some resistance
and so energy will be lost in the form of heat.
• Even when the energy losses due to wire resistance are
neglected, energy will also be lost in the form of
electromagnetic waves radiated by the circuit.
An Oscillatory LC Circuit
• An LC circuit has an inductance of 2.81 mH and a
capacitance of 9 pF. The capacitor is initially charged
with a 12 V battery when the switch S1 is open and
switch S2 is closed.
• S1 is then closed at the same time
that S2 is opened so that the
capacitor is shorted across the
the inductor.
• Find the frequency of the
oscillation.
• Frequency for an LC circuit:
f
1
L C
1
2 L C
2 f
2 f
1
L C
1
2 0.00281 H 9 x 1012 F
f 1x 106 Hz
• What are the maximum values of charge on the capacitor
and current in the circuit?
– Initial charge on the capacitor equals the maximum charge:
12
Qm C V 9 x 10
10
F 12V 1.08 x 10
C
• Maximum current is related to the maximum charge:
Im Qm 2 f Qm
Im 2 1 x 106 Hz 1.08 x 10 10 C
Im 6.79 x 104 A
• Determine the charge and current as functions of time.
2 f 2 1 x 106 Hz 2 x 106 Hz
A sin 2 x 10
Hz t
Q Qm cos t 1.08 x 1010 C cos 2 x 106 Hz t
I Im sin t 6.79 x 10 4
6
Energy Oscillations in the LC
Circuit and the Mass-Spring System
(harmonic oscillator)