134 ResonanceMultiplying both sides by [R 2 + (27rfoL)2]//27rf0 C we get
R 2 + (27rf0L)2 = L//C (6.19)
(2rrfoL) 2= (L/C) - R 2
(27rf0) 2 = (1/LC) - (R//L) 2 (dividing both sides by L 2)
2rrf0 = ~/[(1/LC) - (R/L)21
f0 = (1/27r)~/[(1/LC)- (R//L) 2] (6.20)
If the resistance (R) is very much smaller than the inductive reactance (2rrf0L),
which is normally the case, then Equation (6.19) becomes (2rrf0L) 2 = L/C so
that
(27rf0) 2 = 1//LC (dividing both sides by 1//L 2)
2"n'f0 = X/(1/LC)
fo = 1/2rrV'(LC) (6.21)
Compare this with the Equation (6.2) for the resonant frequency of a series
RLC circuit.
Dynamic impedance
Remembering that the equivalent impedance (Zeq) of a parallel combination of
two impedances Z1 and Z2 is given by Zeq = Z1Z2/(Z 1 Jr- Z2) we see that the
general expression for the impedance of the parallel RLC circuit of Fig. 6.16 is
Z = (R + jwL)(-j/wC)/[(R + jwL) + (-j/wC)]
Dividing the numerator and denominator by (-j/wC) we have
Z = (R + jwL)/[(R/-j/wC) + (jwL/-j/wC) + (-j/wC/-j/wC)]
= (R + jwL)/[jwCR - w2LC + 1]
Again, assuming that wL >> R, this reduces to
Z = jwL/[jwCR + (1 - w2LC)] (6.22)At resonance, w^2 =w0^2 = 1 / LCandthen
Z- jwoL/[jwoCR + (1 - 1)] = L/CR
This has the characteristics of a pure resistance and is called the dynamic
impedance (Zd) of the tuned circuit. Thus
Zd - L/CR (6.23)
Just as in the case of the series tuned circuit, in this parallel circuit the ratio
o~oL/R is called the Q-factor of the circuit. In this case, though, as Z~ becomes
larger, so the current becomes smaller and the Q-factor becomes larger.