Introduction to Electric Circuits

6.2 Parallel resonance 133

6.2 PARALLEL RESONANCE

Also in Chapter 4 we saw that under certain conditions the parallel RLC circuit

behaves as a pure resistor because the phasors representing the circuit supply

voltage and the total current drawn from the supply are in phase with each

other. This condition is known as parallel resonance and in order to analyse it

more fully the circuit diagram is given again in Fig. 6.16, the relevant phasor

diagram being shown in Fig. 6.17.

v

,t(

Figure 6.16

L ,c

R IL cos ~L

) --.

"1 I V

L ~~ I

I

It. sin Ct_ It.

Figure 6.1 7

The important relationships in the circuit of Fig. 6.16 are:

I = IC + IL phasorially (6.14)

Ic- V//Xc (6.15)

I L -- V//ZL (6.16)

where ZL = R + JXL - V'(R 2 + XL2).

The phasor diagram of Fig. 6.17 is drawn with the supply voltage V as the

reference phasor. The current phasor Ic is drawn leading V by 90~ the current

phasor IL is drawn lagging V by an angle 4~L where the - tan-~ (XL//R) and is the

phase angle of the branch containing the inductance.

From this phasor diagram it is seen that for the circuit supply voltage (V) and

the total current drawn from the supply (I) to be in phase,

Ic = IL sin 4~L (6.17)

where Ic is the current through the capacitor C, IL is the current through the

inductance L, 4~L is the phase angle of the branch containing R and L, and

4u - sin-~ (XI_,//ZL) (6.18)

From Equations (6.15), (6.16) and (6.18) we have, by substitution in Equation

(6.17),

v/xc- VXL/[X/(R ~ + XL ~) • X/(R ~ + X~2)]

Dividing throughout by V and putting Xc - 1/2-rrf0C and XL -- 2"rrf0L (f~ being

the resonant frequency) we have

2-rrfo C - 2-rrfoL/[R 2 + (2"nfoL)21