Page 82 - Circuit Analysis II with MATLAB Applications

Chapter 2 Resonance

2.8 A Practical Parallel Resonant Circuit

In our previous discussion, we assumed that the inductors are ideal, but a real inductor has some
resistance. The circuit shown in Figure 2.14 is a practical parallel resonant circuit.

G
L ` C
Y
R

Figure 2.14. A practical parallel resonant circuit
To derive an expression for its resonant frequency, we proceed as follows:

The resonant frequency is independent of the conductance and, for simplicity, it is omitted from
G
the network of Figure 2.14. We will therefore, find an expression for the network of Figure 2.15.

+ I
L I L I C
T `
V C
R

Figure 2.15. Simplified network for derivation of the resonant frequency

For the network of Figure 2.15,

V RjZL –
I = ------------------- = ---------------------------V
L
2
jZL
2
R +
ZL
R +

and
V
I C = --------------------- = jZC V
jZC e
1
where
R
Re I ^` = ---------------------------V
L
2
R + ZL 2
and
– ZL
Im I ^` = ---------------------------V
L
2
R + ZL 2
Also,
Re I ^` = 0
C
and
Im I ^` = ZC V

C
2-16 Circuit Analysis II with MATLAB Applications
Orchard Publications