Lecture 28LC and RLC circuits.

Energy in E and B fields

Energy stored in a capacitor

+ + + + + +− − − − − −

E

212

U CV

energy of the electric field

212

U LI

Energy stored in a inductor

B

energy of the magnetic field

RC circuits (review)

+ +− −

C R

Capacitor is initially charged with Q0. What happens when switch is closed?

Current starts at 00 CV QI

R RC

And then decays: 0 tQ

I t e RCRC

t

I0Q

RC

Current through R

t

Q

0Q

Charge in capacitor

t

U20

2

Q

C

Energy in capacitor

RL circuit

+ +− −

C L

Capacitor is initially charged with Q0. What happens when switch is closed?

• Right after it is closed: I = 0. Then current builds up.

• Magnetic field is created in inductor while charge in capacitor is decreasing

• Energy is being transferred form C to L (from electric to magnetic energy)

As C discharges, current should decrease and stop, but inductor keeps it going, so at the end the capacitor is charged again but with the positive charge on the bottom plate.

Then we start all over again… Oscillatory system

Note: No dissipation mechanism (no R) total energy is constant

Q = 0

I = I0

C L B

Q = 0

I = -I0

C L B

Q = -Q0

I = 0

C L+ + + +

E- - - -

Q = Q0

I = 0

C L+ + + +E- - - -

ACT: LC circuit

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 sign of Vb-Va, the voltage across the inductor at time t1?

A. Vb-Va > 0

B. Vb-Va = 0

C. Vb-Va < 0

C L+ + + +- - - -

a

b

Q = 0 VC = 0

But VC = −VL at all

times! So at t =t1, VL = 0

Compare the energy stored in the capacitor and the inductor at time t1:A. UC < UL

B. UC = UL

C. UC > UL

Q = 0 UC = 0

UL is then maximum.

LC circuit: math

C L2

2

2

2

Kirchhoff : 0

0

0

1 0

C LV V

Q dIL

C dtQ d Q

LC dtd Q

QLCdt

This is the SHM equation!!

Solution:

0 cosQ Q t

22

2

10 with

d QQ

dt LC

0 cosQ Q t 0 sinI Q t

0 cosC

QQV t

C C 2

0 cosL

dIV L LQ t

dt

22

20 cos2 2C

QQU t

C C 2 2 2 2

0

1 1sin

2 2LU LI LQ t

2

20 sin2

Qt

C

0

VC VL

UL

UC

t0

2 2 2 20 0 012 2 2

LQ LQ Q

LC C

Energy is continuously transformed: electric magneticCompare to mass and spring (SHM): kinetic potential

Real inductor (with resistance)

Then, our LC circuit becomes an RLC circuit…

C L

R

RLC circuit

C L

R We’ve seen this already:

Simple Harmonic Motion + Damping

R = 0

Q

0

t t

0

Q

R > 0

RLC circuit: math

C L

R

2RL

0 0cos( )tQ Q e t

2

0 2

1

4

RLC L

2

2

Kirchhoff :

0

0

Q dII R L

C dtQ dQ d Q

R LC dt dt

Frequency is a little lower

Indicates how fast is the decay