LC

LCCircuits0

0

t

V

V

C

L

t

t

UB

UE

Today...• A little review

• Oscillating voltage and current

• Qualitative descriptions:• LC circuits (ideal inductor)

• LC circuits (L with finite R)

• Quantitative descriptions:• LC circuits (ideal inductor)

• Frequency of oscillations

• Energy conservation?Text Reference: Chapter 31.1, 31.3, and 31.5

Example: 31.4

Review of Voltage DropsAcross Circuit Elements

IdtQV

C C Voltage determined by

integral of current and capacitance

C

I(t)

2

2

dI d QV L L

dt dtVoltage determined by derivative of current and inductance

L

I(t)

What’s Next?• Why and how do oscillations occur

in circuits containing capacitors and inductors?

• naturally occurring, not driven for now

• stored energy

• capacitive <-> inductive

Where are we going?• Oscillating circuits

• radio, TV, cell phone, ultrasound, clocks, computers, GPS

Oscillating Current and Voltage

Q. What does mean??osint

A. It is an A.C. voltage source. Output voltage appears at the terminals and is sinusoidal in

time with an angular frequency .

osint R

I(t)

ωtR

εI(t) o sin

Oscillating circuits have both AC voltage and current.

Simple for resistors, but...

Energy in the Electric and Magnetic Fields

21

2U LI

2

magnetic0

1

2

Bu

… energy density ...

Energy stored in an inductor …. B

Energy stored in a capacitor ...

21

2U CV

2electric 0

1

2u E… energy density ...

+++ +++

- - - - - -E

LC Circuits

• Consider the RC and LC series circuits shown:

• Suppose that the circuits are formed at t=0 with the capacitor charged to value Q.There is a qualitative difference in the time development of the currents produced in these two cases. Why??

• Consider from point of view of energy!

• In the RC circuit, any current developed will cause energy to be dissipated in the resistor.

• In the LC circuit, there is NO mechanism for energy dissipation; energy can be stored both in the capacitor and the inductor!

LCC R++++

- - - -

++++

- - - -

RC/LC Circuits

RC:

current decays exponentially

C R

-It

0

0

I

Q+++

- - -

LC

LC:

current oscillates

I

0

0 t

I

Q+++

- - -

LC Oscillations(qualitative)

LC+ +

- -

0I

0QQ

LC+ +

- -

0I

0QQ

LC

0II

0Q

LC

0II

0Q

Alternate way to draw:

L

C

V=0

VC

VL

VC+VL = 0

VC = -VL

LC Oscillations(qualitative)

0

I

Q0

t

0

dIdt

t

0

VC

0

VL1

These voltages are opposite, since the

cap and ind are traversed in “opposite”

directions

Lecture 18, Act 1• At t=0, the capacitor in the LC circuit

shown has a total charge Q0. At t = t1, the capacitor is uncharged.

– What is the value of Vab=Vb-Va, the voltage across the inductor at time t1?

(a) Vab < 0 (b) Vab = 0 (c) Vab > 0

(a) UL1 < UC1 (b) UL1 = UC1 (c) UL1 > UC1

– What is the relation between UL1, the energy stored in the inductor at t=t1, and UC1, the energy stored in the capacitor at t=t1?

1B

1A

LC

LC

+ +

- -Q =0Q Q= 0

t=0 t=t 1

a

b

Lecture 18, Act 1• At t=0, the capacitor in the LC circuit

shown has a total charge Q0. At t = t1, the capacitor is uncharged.

– What is the value of Vab=Vb-Va, the voltage across the inductor at time t1?

(a) Vab < 0 (b) Vab = 0 (c) Vab > 0

1A

• Vab is the voltage across the inductor, but it is also (minus) the voltage across the capacitor!

• Since the charge on the capacitor is zero, the voltage across the capacitor is zero!

LC

LC

+ +

- -Q =0Q Q= 0

t=0 t=t 1

a

b

Lecture 18, Act 1• At t=0, the capacitor in the LC

circuit shown has a total charge Q0. At t = t1, the capacitor is uncharged.

(a) UL1 < UC1 (b) UL1 = UC1 (c) UL1 > UC1

1B

• At t=t1, the charge on the capacitor is zero.

02

21

1 C

QUC 0

22

1 202

11 C

QLIU L

• At t=t1, the current is a maximum.

LC

LC

+ +

- -Q =0Q Q= 0

t=0 t=t 1

a

b– What is the relation between UL1,

the energy stored in the inductor at t=t1, and UC1, the energy stored in the capacitor at t=t1?

LC Oscillations(L with finite R)

• If L has finite R, then– energy will be dissipated in R.

– the oscillations will become damped.

R = 0

Q

0

t t

0

Q

R 0

LC Oscillations(quantitative, but only for R=0)

• Guess solution: (just harmonic oscillator!)

where , Q0 determined from initial conditions

• Procedure: differentiate above form for Q and substitute into

loop equation to find .

• Note: Dimensional analysis

LC+ +

- -

I

Q

• What is the oscillation frequency ω0?

• Begin with the loop rule:

02

2

C

Q

dt

QdL

)cos( 00 tQQremember:

02

2

d xm kxdt

01 LC

LC Oscillations(quantitative)

• General solution:

)cos( 00 tQQLC

+ +

- -

02

2

C

Q

dt

QdL

• Differentiate:)sin( 000 tQ

dt

dQ

)cos( 00202

2

tQdt

Qd

• Substitute into loop eqn:

0)cos(1

)cos( 000020 tQ

CtQL 0

120

CL

Therefore,

LC

10

LCL

C

m

k 1/10

which we could have determinedfrom the mass on a spring result:

2

Lecture 18, Act 2• At t=0 the capacitor has charge Q0; the resulting

oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.

– What is the relation between 0 and 2, the frequency of oscillations when the initial charge = 2Q0?

(a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 20

(a) I = I (b) I = 2I (c) I = 4I

– What is the relation between I0 and I2, the maximum current in the circuit when the initial charge = 2Q0?

2B

2A

LC

+ +

- -Q Q 0

t=0

Lecture 18, Act 2• At t=0 the capacitor has charge Q0; the resulting

oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.

– What is the relation between 0 and 2, the frequency of oscillations when the initial charge = 2Q0?

(a) 2 = 1/2 0 (b) 2 = 0 (c) 2 = 20

2A

• Q0 determines the amplitude of the oscillations (initial condition)

• The frequency of the oscillations is determined by the circuit parameters (L, C), just as the frequency of oscillations of a mass on a spring was determined by the physical parameters (k, m)!

LC

+ +

- -Q Q 0

t=0

Lecture 18, Act 2• At t=0 the capacitor has charge Q0; the resulting

oscillations have frequency 0. The maximum current in the circuit during these oscillations has value I0.

– What is the relation between I0 and I2, the maximum current in the circuit when the initial charge = 2Q0?

(a) I2 = I0 (b) I2 = 2I0 (c) I2 = 4I0

2B

• The initial charge determines the total energy in the circuit: U0 = Q0

2/2C• The maximum current occurs when Q=0!• At this time, all the energy is in the inductor: U = 1/2 LIo

2

• Therefore, doubling the initial charge quadruples the total energy.• To quadruple the total energy, the max current must double!

LC

+ +

- -Q Q 0

t=0

LC OscillationsEnergy Check

• The other unknowns ( Q0, ) are found from the initial conditions. E.g., in our original example we assumed initial values for the charge (Qi) and current (0). For these values: Q0 = Qi, = 0.

• Question: Does this solution conserve energy?

)(cos2

1)(

2

1)( 0

220

2

tQCC

tQtU E

)(sin2

1)(

2

1)( 0

220

20

2 tQLtLitUB

• Oscillation frequency has been found from the loop equation. LC

10

UE

t0

Energy Check

UB

0t

Energy in Capacitor

)(cos2

1)( 0

220 tQ

CtUE

Energy in Inductor

)(sin2

1)( 0

220

20 tQLtUB

LC

10

)(sin2

1)( 0

220 tQ

CtUB

C

QtUtU BE 2)()(

20Therefore,

3

Lecture 18, Act 3• At t=0 the current flowing through the circuit is 1/2

of its maximum value.

– Which of the following plots best represents UB, the energy stored in the inductor as a function of time?

3ALC

+ +

- -

I

Q

– Which of the following is a possible value for the phase , when the charge on the capacitor is described by: Q(t) = Q0cos(t + )

3B

(a) (b) (c)

00

UB

time

00

UB

time

00

UB

time

(a) = 30 (b) = 45 (c) = 60

Lecture 18, Act 3• At t=0 the current flowing through the circuit is 1/2

of its maximum value.

– Which of the following plots best represents UB, the energy stored in the inductor as a function of time?

3A

(a) (b) (c)

00

UB

time

00

UB

time

00

UB

time

• The key here is to realize that the energy stored in the inductor is proportional to the CURRENT SQUARED.

• Therefore, if the current at t=0 is 1/2 its maximum value, the energy stored in the inductor will be 1/4 of its maximum value!!

LC+ +

- -

I

Q

Lecture 18, Act 3• At t=0 the current flowing through the circuit is 1/2

of its maximum value.

– Which of the following is a possible value for the phase , when the charge on the capacitor is described by: Q(t) = Q0cos(t + )

(a) = 30 (b) = 45 (c) = 60

3B

• We are given a form for the charge on the capacitor as a function of time, but we need to know the current as a function of time.

)sin()( 000 φtωQωdt

dQtI

• At t = 0, the current is given by: φQωI sin)0( 001 1

max 0 02 2 ( )I ω Q

• Therefore, the phase angle must be given by:2

1sin φ 30φ

LC+ +

- -

I

Q

Summary

• Quantitative description

– Frequency of oscillations

– Energy conservation

Text Reference: Chapter 31.1, 31.3, and 31.5

01LC

0

VC

0

VL

• Oscillating voltage and current

• Qualitative description

Next Time...

Reading assignment:Ch. 31.2 through 31.5, 31.7

Examples: 31.1-3, 31.9

•AC power! •AC circuits !

Appendix: LCR DampingFor your interest, we do not derive here, but only illustrate

the following behavior

t

0

Q

0

Q

t

LC+ +

- -

R

40RR

0RR

L

R

2

)'cos(0 teQQ ot

2

2

4

1'

L

R

LCo

In an LRC circuit, depends also on R !