Introduction to Electric Circuits

124 Resonance

9 The voltage phasor VR = IR is drawn in phase with the current phasor.

The voltage V is the phasor sum of VL, Vc and VR and because VL = -Vc then
V = VR. The phase angle of the circuit (the angle between the applied voltage,
V, and the current, I) is & = 0, so that the power factor is cos & = 1. The
condition XL = Xc occurs when
/
2 rrf L - 1/ 2 rrf C (6.1)

and this can be made to happen by the variation of L or C or f.
The graphs of XL and Xc to a base of frequency are shown in Fig. 6.3 and it
can be seen that:

(1) as f--+ 0, XL --+ 0 and Xc --+ ~

(2) as f--+ oo, XL + ~ and Xc --+ 0

(3) at a particular frequency (f~) XL = Xc in magnitude.

XL

Xcl

Ifo

I J

I I f

Figure 6.3

This frequency is called the resonant frequency of the circuit and may be
calculated from Equation (6.1), putting f = f0. Thus

2rrf0L = 1/2rrf0C

(2rrf0) 2 = 1/LC (multiplying both sides by 2rrf0/L)

2rrf0 - 1/'v/(LC) (taking the square root of both sides)

Finally

f0 = 1/2rrX/(LC) (6.2)

Also, remembering that the angular frequency w = 2rrfwe may write

a~o = 1/~/(LC) (6.3)

Summarizing, when an RLC circuit is in a state of series resonance:

9 the circuit behaves as a pure resistance;

9 the inductive reactance is equal to the capacitive reactance;

9 the applied voltage V = IR;