W9_parallel_resonance.eps

Week 9: Alternating Current Circuits 323

with a peak current given by:

I 0 =

V 0

ωL

=

V 0

χL

(733)

We see that the current isπ/ 2 behind in phaseof the voltage drop across the inductor. As before,
in series circuits we will actually use this the other way around, considering the voltage drop across
and inductor in a series circuit as beingπ/ 2 aheadof the (otherwise specified) current through it.
Let’s see how this works.

The Series LRC Circuit

I(t)

0

C

R

L

V sin( t)ω

Figure 126: ALRC(tank) circuit.

In figure 126 above we see aseriesLRCcircuit. To analyze it, we apply Kirchhoff’s voltage/loop
rule to the circuit and get the equation of motion:

V 0 sin(ωt)−LdI

dt

−RI−Q

C

= 0 (734)

or
VL+VR+VC=V 0 sin(ωt) (735)

or
d^2 Q
dt^2

+

R

L

dQ

dt

+

1

LC

Q=

V 0

L

sin(ωt) (736)

There are a number of way to solve this second order, linear,inhomogeneousordinary differential
equation. We will first show a simple one that relies on a “guess”, thenwe will show how if we
use complex exponentials we really don’t have to guess. In fact, if weuse complex exponentials for
everythingassociated with electrical circuits and harmonic oscillation we don’t really need to guess,
but we do need to know more math than a student of introductory physics might initially know.
Wise students will view this as an open invitation tolearn more mathto make the physics easier.

Our goal will be to solve for all voltage drops, the current in the circuit, the power delivered to
each circuit element and the entire circuit as a whole – pretty much everything.

The first thing to note that if we find at least one “particular” solutionQp(t) to the inhomo-
geneous ODE, we can construct a new solution by addinganysolution to thehomogeneousODE
(the undrivenLRCcircuit solved above) and still get a solution. That is, a general solution can be
written:
Q(t) =Qp(t) +Qh(t) (737)

Note that the solutionQh(t) to the homogeneous ODE (equation 718)decays in timeexponentially.
It is atransientcontribution to the overall solution and after many lifetimesτL =R/Lit will
generally be negligible.