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Parallel Resonance
Lecture 17
In the last lecture, we covered the selectivity properties like cut-off
frequency, bandwidth, etc. for a series resonant circuit. Then we
solved two numerical examples for the series circuit.
In this lecture we study the behavior of a parallel RLC circuit when
the frequency is varied. The parallel resonant circuit to be studies
consists of ideal R, L, and C elements connected in parallel with a
current source. The following items are to be covered,
 Conditions of parallel resonant circuit
 Quality factor of the parallel resonant circuit
 The properties of circuit impedance and
voltage versus frequency
 Selectivity conditions like cut-off frequency
and bandwidth
Parallel Resonant Circuit
The basic format of the series resonant circuit is a series RLC
combination in series with an applied voltage source. The parallel
resonant circuit has the basic configuration of Figure 11, which has R
L and C elements all in parallel with an applied current source.
Fig. 11 Ideal parallel resonant network
For the series circuit, the impedance was a minimum at resonance
producing a significant current that resulted in high VC and VL. For
parallel resonant circuit, the impedance is relatively high at resonance
producing a significant voltage for VC and VL. Note the relationship
VL=IZT and VC=IZT.
The admittance seen by the current source, which is reciprocal of
impedance ZT, is
YT  YR  YL  YC
1
1
= 
 j C
R j L
1
= R1  j(C 
)
L
The circuit is said to be in resonance when the imaginary part of
admittance YT is zero, or,
C 
1
 0  p 
L
or ,
fp 
1
LC
1
2 LC
The expressions for ωp and fP for a parallel resonant circuit with ideal
components are the same as for the series resonant circuit. In the
above, the subscript p stands for parallel circuit.
Then, at resonance, the voltage of the parallel circuit,
V=IR
The parallel LC combination acts as an open circuit, called a tank
circuit.
The Quality Factor Q for a Parallel Resonant Circuit
From the definition of the quality factor, for a parallel RLC circuit,
Reactive power
Real power
V2 / XL
R
= 2

V / R p L
Qp 
Also,
Qp =
V 2 / XC
2
V /R
 p RC
The Selectivity Curves for Parallel Resonant Circuits
The impedance versus frequency curve in Figure 12 shows that the
impedance is maximum at the resonant frequency fp. Note, this is
opposite to the case in series resonant circuit when the impedance is
at its minimum value.
Fig.12 Total impedance (ZT) vs. frequency for the parallel resonant
circuit
Since the current I of the current source is constant for any value of
impedance (ZT) or frequency, the voltage across the parallel circuit
will have the same shape as the total impedance (ZT), as shown in
Figure 13. Figure 14 gives the variation of the impedance angle.
Fig. 13 Variations of voltage of the parallel resonant circuit
Observe that in Figures12 and 13, fP represent the resonant
frequency for the parallel circuit, while f1 and f2 are the cut-off
frequencies.
Fig14. Phase plot for parallel resonant circuit
The phase angle plot shows that at low frequency the circuit is
inductive (lagging power factor), while at high frequency it is
capacitive (leading power factor). At resonant frequency, the
inductive reactance cancels the capacitive reactance, and it is a
resistive circuit with unity power factor.
From figures 12 and 13, at half power or cut-off frequency f1 and f2
the voltage of the parallel resonant circuit is 0.707 times the
maximum value.
The impedance at these frequencies is
0.707 Zmax =0.707 R.
0.707Zmax  0.707R
This means the admittance is
2
R
Equating the admittances of the parallel resonant circuit at the cut-off
frequencies,
2  ( 1 )2  (C  1 )2
R
R
L
and following similar steps as in the series resonant circuit, it can be
shown that,
1
1 2 1
 (
) 
2RC
2RC
LC
1
1 2 1
1  
 (
) 
2RC
2RC
LC
2 
From the relationships of the quality factor Qp for the parallel resonant
circuit,
Qp  p RC
Or,
p
1

RC Qp
The expressions for the cut-off frequency (in radian per second) can
be written as,
2 
p
2Qp
1  
 p (
p
2Qp
1 2
) 1
2Qp
 p (
1 2
) 1
2Qp
Observe that these expressions are exactly the same as for the
series resonant circuit when expressed in terms of ωp and Qp.
The expressions for f1 and f2 can be obtained by dividing ω with 2π.
The expressions for the bandwidth (in radian per second) is,
p
1
2  1 

RC Q p
Similarly,
fp
1
f 2  f1 

2 RC Q p
The definition of selectivity of a circuit is
p
2  1

fp
f 2  f1
Which is the same as Qp for a parallel resonant circuit. Comparative
selectivity curves are shown in Figure 15. It can be seen that the
smaller the bandwidth, the greater is the selectivity. As in series
resonant circuit, the selectivity curve can be considered to be
symmetrical about resonant frequencies for Qp>10.
Fig. 15 Comparative selectivity curves
Example
Given the parallel resonant circuit of Fig.16
composed of ideal
elements:
a) Determine the resonant frequency fp
b) Find the total impedance at resonance
c) Calculate the quality factor and bandwidth of the system
d) Find the voltage VC at resonance
e) Determine current IL and IC
f) Calculate the cut-off frequencies f1 and f2
Fig. 16 Numerical example for parallel resonant circuit
Solution
a)
b)
fp 
1

1
2 LC 2 (1x103 )(1x106)
=5.03 kHz
ZT  R s ZL ZC  R s  10k
c)
Rs
Rs
10 x103
Qp 


X L 2 f p L 2 (5.03x103 )(1x103 )
=316.41
f p 5.03x103
BW 
Qp  316.41  15.9Hz
d)
e)
VC =IZT = (10x10-3)(10x103) =100 V
VC
VL
100
100



 3.16
3

3
X L 2 f p L 2 (5.03x10 )(1x10 ) 31.6
V
100
IC  C 
 3.16
X C 31.6
IL 
f)
f2 
fp
 fp (
1 2
) 1
2Q p
2Qp
5030
1
=
 5030 (
) 2  1  5041.9Hz
2x316.4
2x316.4
fp
1 2
f1  
 fp (
) 1
2Qp
2Q p
=5026.02Hz
Summary
In this lecture, we covered the parallel resonant circuit
with ideal R, L and C elements. The following items
were covered:
 Conditions for an RLC parallel circuit to be in
resonance
 Derived the expressions for quality factor for
the parallel resonant circuit
 Demonstrated the behavior of the circuit
impedance and voltage as a function of
frequency
 Derived the expressions for cut-off
frequency, band width, and selectivity of a
parallel resonant circuit
 Solved a numerical example for a parallel
resonant circuit
SELF-TEST(17)
Consider the parallel RLC resonant circuit with “ideal” elements given
below.
I
IC
IR
IL
2KO
1mH
+
10nF
_
1) The total impedance seen by the source at resonance is
(a) 0
(b) 2KΩ
(c) 2MΩ
(d) infinity
Ans: (b)
2) The resonant frequency approximately is
(a) 10 Hz
500 KHz
(b) 100 KHz
(c) 160 KHz
(d)
Ans: (c)
3) The current through the resistance (IR) at resonance is
(a) 0
(b) 1 mA
(c) 2 mA
(d) 10 mA
Ans: (c)
4) 4) The inductor current IL at resonance is
a) Equal to IR
b) Equal to IC
c) Equal and opposite to IC
d) Equal to I
Ans: (c)