W9_parallel_resonance.eps

328 Week 9: Alternating Current Circuits

In a later section we’ll see at least one or two places one canusean seriesLRCcircuit to do
useful things, but first we have to study an even more useful circuit, theparallelLRCcircuit.

The Parallel LRC Circuit

r

I(t)

I (t) I (t)

V sin( t)o ω

L C I (t)R

L C R

Figure 130: A parallelLRCcircuit, with a voltage that has an “internal resistance” that limits its
ability to deliver current. This circuit is ideal for the construction ofa simple AM crystal radio.

The parallel LRC circuit drawn in figure 130 above is actually muchsimplerthan the series as
far as understanding the solution is concerned. In this figure we have added the internal resistance
rof the power supply or antenna, as in the latter case especially the fact that the voltage cannot
supply an infinite amount of power is essential to understanding howthis circuit can be used to
build a crystal radio. Note that we didn’t bother doing this in the caseof the seriesLRCcircuit
because the resistanceRin that case was thetotalresistance from all sources in the single circuit
loop.

It is simple to analyze because thesamevoltage dropV 0 sin(ωt) occurs across allthreecompo-
nents, and so we can just write down the currents through each component using the elementary
single-component rules above:

IR =

V 0

R

sin(ωt) (765)

IL = V^0

χL

sin(ωt−π/2) (766)

IC =

V 0

χCsin(ωt+π/2) (767)

Note well that we use the rules we derived where the current through the inductor isπ/ 2 behind
the voltage (which is thereforeπ/ 2 aheadof the current) and vice versa for the capacitor. To find the
total current provided by the voltage, we simply add these three currents according to Kirchhoff’s
junction rule. Of course, we are adding three trig functions with different relative phases, so we once
again must accomplish this with suitable phasors:

Itot =

V 0

R

sin(ωt) +

V 0

χL

sin(ωt−π/2) +

V 0

χC

sin(ωt+π/2)

= V^0

Z

sin(ωt+φ)

= I 0 sin(ωt+φ) (768)

As before, we can factor out the commonV 0 and look at the resulting triangle addition of the
inverseresistance and reactances to obtain a sum rule for theinverseimpedanceZ: From figure 132