ABSTRACT
A more practical network, containing a coil of inductance L and resistance R in parallel with a pure capacitance C, is shown in Figure 32.3.
Admittance of coil, YCOIL= 1 R+ jXL
= R− jXL R2+X2L
= R R2+ω2L2 −
jωL
Admittance of capacitor, YC = 1−jXC = j
Xc =jωC
Total circuit admittance, Y = YCOIL + YC
= R R2 +ω2L2 −
jωL
R2 +ω2L2 + jωC (1)
At resonance, the total circuit admittance Y is real (Y =R/(R2+ω2L2)), i.e. the imaginary part is zero. Hence, at resonance:
+ωrC = 0
Therefore ωrL
R2 +ω2r L2 =ωrC and L
Thus ωr L =
C −R
and ω2r = L
CL2 − R
L2 = 1 LC
− R 2
L2 (2)
Hence ωr = √(
1 LC
− R 2
)
and resonant frequency, fr = 1 2π
√( 1 LC
− R 2
) (3)
Note that whenR2/L2 1/(LC) then fr =1/2π√(LC), as for the series R-L-C circuit. Equation (3) is the same as obtained in Chapter 18, page 285; however, the above method may be applied to any parallel network, as demonstrated in Section 32.4 below.
