W9_parallel_resonance.eps

Week 9: Alternating Current Circuits 329

ωt

Vo

Vo

Vo

φ Vo

Z

χL

χC

R

I(t)

Figure 131: A phasor diagram for the parallelLRCcircuit.

χC

φ

Z

χL

1

1

R

1

1

Figure 132: The impedance diagram for the parallelLRCcircuit.

the pythagorean theorem immediately yields an expression for the inverse of the impedance:

1

Z=

√(

1

R^2 + (

1

χC−

1

χL)

2

)

(769)

which we recognize as the phasor equivalent of the familiar rule for reciprocal addition of resistances
in parallel.

We similarly can easily evaluate the phaseφ:

φ = tan−^1

( 1

χC−

1

χL

1

R

)

= tan−^1

(

RC(ω^2 −ω 02 )

ω

)

(770)

where we have factored out aCand 1/ωfrom the first expression and usedω^20 = 1/LC, the resonance
frequency of the circuit.

Resonance for this circuit is theoppositeof the seriesLRCcircuit we first looked at. It still occurs
at the frequencyω=ω 0 =√^1 LCas before, but nowZ^1 islargestat resonance. To understand how
this can be useful, let us think aboutcurrent flowin the circuit both at and away from resonance.

At resonance, the impedance (resistance to current flow) of theLandCtogether is:

1

ZLC

=

√

(

1

χC

−

1

χL

)^2 = 0 (771)

or
ZLC=∞ (772)

No current flowsinto theLandCin combination – they behave like anopen circuitat the resonant
frequency. All the current that flows from the voltage at this frequency therefore flows through the
resistance (or “load”).