LCR circuit

+V0

R

C

1

2

L

V(t)

0t

0tfor

V

0tV

0

I(t)=0 for t<0

0ILC1

dtdI

LR

dtId

VCQ

dtdI

LIR

VVVV

0t

2

2

0

0CLR

0

keI

LC1

LR2

t

20

2

LC12

L2R

L2R

t2

t1

21 ekekI

trial solution

characteristic equation

general solution

LC12

0

L2R

LCR circuit – initial conditions

Initial conditions I(0), ? 0dtdI

No energy stored 00I0LIW 221

L

0

0C

C2

21

C

V0dtdI

L

VV0dtdI

LR0I

00V0CVW

LV

0dtdI 0

+V0

R

C

L

I(t)

LCR circuit 20

2 t2

t1

21 ekekI

overdamped solution

2 20

2,1

LV

kk0dtdI

kk 0kk0I

ekek0tI

021

2121

t2

t1

L2V

k 01

20

2

LC12

0

L2R

tt

CL2

2R

0 ee2

VtI

LCR circuit 20

2

critically damped solution

2 20

21

t21 etkkI

0k 00I 1

t221 ektkk

dtdI

LV

dtdI

k 0

0t2

tetLV

tI 0

LCR circuit 20

2 t2

t1

21 ekekI

2 20

R

2202,1

j

j

2121

tj2

tj1

t

kk 0kk 00I

ekeke0tI RR

0R 1 R 2

t 0

dI Vj k j k

dt L Lj2V

kR

01

tjtjt

22R

CL

0 RR eej2

1e

V0tI

tsineV

0tI Rt

22R

CL

0

t

tL2

R

e

underdamped (oscillatory) solution

LCR circuit 0ILC1

dtdI

LR

dtId2

2

20

2 t2

t1

21 ekekI

overdamped solution

critically damped solution

underdamped (oscillatory) solution

20

2 2,1 β is real

20

2 21 t21 etkkI

20

2 R

2202,1

j

j

LC12

0

L2R

LCR circuit 0ILC1

dtdI

LR

dtId2

2

20

2 t2

t1

21 ekekI

overdamped solution

critically damped solution

underdamped (oscillatory) solution

20

2

20

2

20

2

LC12

0

L2R

tt

CL2

2R

0 ee2

VtI

tetLV

tI 0

tsineV

0tI Rt

22R

CL

0

CP2 September 2003

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This way up

This way up

This way up

LCR circuit 0ILC1

dtdI

LR

dtId2

2

20

2 t2

t1

21 ekekI

overdamped solution

critically damped solution

underdamped (oscillatory) solution

20

2

20

2

20

2

LC12

0

L2R

tt

CL2

2R

0 ee2

VtI

tetLV

tI 0

tsineV

0tI Rt

22R

CL

0