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

• 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.

2

2mQUC

• 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, 2

2mQUC

• 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·Im

2, 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·Imax

2 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·Imax

2 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.

– Therefore:

• Differentiating the energy equation with respect to time and noting that Q and I vary with time:

220.5

2C LQ

U U U L IC

0dUdt

220.5

20

Qd L I

CdUdt dt

2

2

2 2

2 1 2 1

0.520

10.5 0

21

2 0.5 2 02

0

Qd

d L ICdUdt dt dt

dU dQ dIL

dt C dt dtdU dQ dI

Q L Idt C dt dtdU Q dQ dI

L Idt C dt dt

• We can reduce the equation to a differential equation of one variable by using the following relationships:

2

2

2

2

2 2

2 2

0

0

1

dU Q dQ dIL I

dt C dt dt

dU Q d QI L I

dt C dt

Q d QI L I

C dt

Q d Q d QL Q

C C Ldt dt

2

2

dQ dI d QI and

dt dt dt

• 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):

– where k is the spring constant, m is the mass, and

– The solution for the equation has the general form

– 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.

22

2

d x kx x

mdt

km

cosx A t

• Writing in the same form as the

differential equation of the simple harmonic oscillator, the solution is:

– where Qm is the maximum charge of the capacitor and the angular frequency ω is given by:

– The angular frequency of the oscillation depends on the inductance and capacitance of the circuit.

• Since Q varies periodically, the current also varies periodically.

2

2

1d QQ

C Ldt

cosmQ Q t

1

L C

• Differentiating the harmonic oscillator equation for the LC oscillator with respect to time:

cos

cos

cos

sin

m

m

m

m

Q Q t

d Q tdQdt dt

d tdQI I Q

dt dtd t

I Q tdt

• 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.

sin

sin 0

sin

m

m

m

d t dI Q t

dt dt

dtI Q t

dt

I Q t

sin

0 sin 0

0 sin

m

m

m

I Q t

Q

Q

• The phase constant = 0.

• The time variation of Q and I are given by:

– 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.

1

0 sin

0 sin sin 0

0

mQ

m

cos

sin

I sin

m

m

Q Q t

I Q t

I t

– 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:

220.5

2C LQ

U U U L IC

m

cos

I sin

mQ Q t

I t

• Total energy:

• 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.

22

22

22 2 2

0.52

cos0.5 sin

2

cos 0.5 sin2

C L

mm

mm

QU U U L I

C

Q tU L I t

C

QU t L I t

C

• When the energy stored in the capacitor has its maximum value, , the energy stored in the inductor is zero.

• When the energy stored in the inductor has its maximum value, 0.5·L·Im

2, the energy stored in the capacitor is zero.

• The sum of the UC + UL is a

constant and equal to the total

energy .

2

2mQ

C

2

2mQ

C

• Since the maximum energy stored in the capacitor (when I = 0) must equal the maximum energy stored in the inductor (when Q = 0),

• Substituting this into the total energy equation:

220.5

2mQ L IC

22 2 2

2 22 2

22 2

cos 0.5 sin2

cos sin2 2

cos sin2

mm

m m

m

QU t L I t

C

Q QU t t

C C

QU t t

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.

22 2

2 2

2

cos sin2

cos sin 1

2

m

m

QU t t

C

t t

QU

C

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:

• 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

6

1 12 2

1 1

2 2 0.00281 9 10

1 10

ffL C L C

fL C H x F

f x Hz

12 109 10 12 1.08 10mQ C V x F V x C

• Maximum current is related to the maximum charge:

• Determine the charge and current as functions of time.

m

6 10m

4m

I 2

I 2 1 10 1.08 10

I 6.79 10

m mQ f Q

x Hz x C

x A

6 6

10 6

4 6m

2 2 1 10 2 10

cos 1.08 10 cos 2 10

I sin 6.79 10 sin 2 10

m

f x Hz x Hz

Q Q t x C x Hz t

I t x A x Hz t

Energy Oscillations in the LC Circuit and the Mass-Spring System

(harmonic oscillator)