W9_parallel_resonance.eps

306 Week 9: Alternating Current Circuits

+Q

C

S

L

Figure 112: UndrivenLCcircuit

If we substitute this relation in for theI’s and divide byL, we get the following second order,

linear, homogeneous ordinary differential equation:

d^2 Q

dt^2

+

Q

LC

= 0 (643)

We recognize this as the differential equation for aharmonic oscillator! To solve it, we

“guess”^85 :

Q(t) =Q 0 eαt (644)

and substitute this into the ODE to get the characteristic:

α^2 +

1

LC

= 0 (645)

We solve for:

α=±i

√

1

LC

=±iω 0 (646)

and get:

Q(t) =Q0+e+iω^0 t+Q 0 −e−iω^0 t (647)

or (taking the real part and using the initial conditions):

Q(t) =Q 0 cos(ω 0 t) (648)

• Non-driven LRC circuit:In the figure above, the capacitorCon the left is initially charged

+Q

C

S

L

R

Figure 113: UndrivenLRCcircuit

up to chargeQ 0. At timet= 0 the switch is closed and current begins to flow. If we apply

Kirchhoff’s voltage/loop rule to the circuit, we get:

Q

C

−L

dI

dt

−IR= 0 (649)

where

I=−

dQ

dt

(650)

(^85) Not really.