Simple Nature - Light and Matter

10.5.10 Impedances in series and parallel

How do impedances combine in series and parallel? The beauty
of treating them as complex numbers is that they simply combine
according to the same rules you’ve already learned as resistances.

Series impedance example 35

.A capacitor and an inductor in series with each other are driven

by a sinusoidally oscillating voltage. At what frequency is the cur-

rent maximized?

.Impedances in series, like resistances in series, add. The ca-

pacitor and inductor act as if they were a single circuit element

with an impedance

Z=ZL+ZC

=iωL−

i

ωC

.

The current is then

̃I=

V ̃

iωL−i/ωC

.

We don’t care about the phase of the current, only its amplitude,

which is represented by the absolute value of the complex num-

ber ̃I, and this can be maximized by making|iωL−i/ωC|as small

as possible. But there is some frequency at which this quantity is

zero—

0 =iωL−

i

ωC

1

ωC

=ωL

ω=

1

√

LC

At this frequency, the current is infinite! What is going on phys-

ically? This is an LRC circuit withR= 0. It has a resonance at

this frequency, and because there is no damping, the response

at resonance is infinite. Of course, any real LRC circuit will have

some damping, however small (cf. figure j on page 185).

Resonance with damping example 36

.What is the amplitude of the current in a series LRC circuit?

.Generalizing from example 35, we add a third, real impedance:

| ̃I|=

|V ̃|

|Z|

=

|V ̃|

|R+iωL−i/ωC|

=

|V ̃|

√

R^2 + (ωL− 1 /ωC)^2

This result would have taken pages of algebra without the com-

plex number technique!

Section 10.5 LRC circuits 637