4
RESONANT CIRCUITS
A communication circuit designer frequently requires means to select (or reject) a
band of frequencies from a wide signal spectrum. Resonant circuits provide such
®ltering. There are well-developed,sophisticated methodologies to meet virtually
any speci®cation. However,a simple circuit suf®ces in many cases. Further,resonant
circuits are an integral part of the frequency-selective ampli®er as well as of the
oscillator designs. These networks are also used for impedance transformation and
matching.
This chapter describes the analysis and design of these simple frequency-selective
circuits,and presents the characteristic behaviors of series and parallel resonant
circuits. Related parameters,such as quality factor,bandwidth,and input impedance,
are introduced that will be used in several subsequent chapters. Transmission lines
with an open or short circuit at their ends are considered next and their relationships
with the resonant circuits are established. Transformer-coupled parallel resonant
circuits are brie¯y discussed because of their signi®cance in the radio frequency
range. The ®nal section summarizes the design procedure for rectangular and
circular cylindrical cavities,and the dielectric resonator.
4.1 SERIES RESONANT CIRCUITS
Consider the series R-L-C circuit shown in Figure 4.1. Since the inductive reactance
is directly proportional to signal frequency,it tries to block the high-frequency
contents of the signal. On the other hand,capacitive reactance is inversely propor-
tional to the frequency. Therefore,it tries to stop its lower frequencies. Note that the
voltage across an ideal inductor leads the current by 90

(i.e.,the phase angle of an
105
Radio-Frequency and Microwave Communication Circuits: Analysis and Design
Devendra K. Misra
Copyright # 2001 John Wiley & Sons,Inc.
ISBNs: 0-471-41253-8 (Hardback); 0-471-22435-9 (Electronic)

ÀI
v
o
tdt  v
o
tv
in
t4:1:1
Taking the Laplace transform of this equation with initial conditions as zero (i.e.,no
energy storage initially),we get
sL
R

1
sRC
 1

V
o
sV
i
s4:1:2
where s is the complex frequency (Laplace variable).
The transfer function of this circuit, Ts,is given by
Ts
V
o
s
V
i

s
1;2
À
R
2L
Æ

R
2L

2
À
1
LC
s
4:1:5
The circuit response will be in¯uenced by the location of these poles. Therefore,
these networks can be characterized as follows.
 If
R
2L
>
1

LC
p
,i.e.,R > 2

L
C

1

LC
p
,i.e.,R < 2

L
C
r
,the two poles of Ts will be complex conjugate
of each other. The circuit is underdamped.
Alternatively,the transfer function may be rearranged as follows:
Ts
sCR
s
2
LC  sRC  1

sCRo
2
o
s
2
 2zo
o
s  o
2
o
4:1:6
where

Àzo
o
Æ jo
o

1 À z
2
p
. As shown in Figure 4.2,the two poles are
complex conjugate of each other. Output transient response will be oscillatory with a
ringing frequency of o
o
1 À z
2
 and an exponentially decaying amplitude. This
circuit is underdamped.
For z  0,the two poles move on the imaginary axis. Transient response will be
oscillatory. It is a critically damped case.
For z  1,the poles are on the negative real axis. Transient response decays
exponentially. In this case,the circuit is overdamped.
SERIES RESONANT CIRCUITS
107
Consider the unit step function shown in Figure 4.3. It is like a direct voltage
source of one volt that is turned on at time t  0. If it represents input voltage v
in
t
then the corresponding output v
o
t can be determined via Laplace transform
technique.

s  zo
o

2
1À z
2
o
2
o
where L
À1
represents inverse Laplace transform operator. Therefore,
v
o
t
2B

1 À z
2
p
e
ÀBo
o
t
sin o
o
t

1 À z
2

LCo À
1
o

or,
V
o
 jo
V
i
 jo
1 
j
RC
o
o
2
o
À
1
o


V
i
 jo
1 
j
o
o

2
R

o
o
L
R
Since o
o
L 
1
o
o
C
,
Q 
o
o
L
R

1
o
o
RC


LC
p
RC

o
 jo
V
i
 jo
 A jo
1
1  jQ
o
o
o
À
o
o
o

4:1:13
The magnitude and phase angle of (4.1.13) are illustrated in Figures 4.5 and 4.6,
respectively. Figure 4.5 shows that the output voltage is equal to the input for a
signal frequency equal to the resonant frequency of the circuit. Further,phase angles
of the two signals in Figure 4.6 are the same at this frequency,irrespective of the
quality factor of the circuit. As signal frequency moves away from this point on
either side,the output voltage decreases. The rate of decrease depends on the quality
factor of the circuit. For higher Q,the magnitude is sharper,indicating a higher
selectivity of the circuit. If signal frequency is below the resonant frequency then
output voltage leads the input. For a signal frequency far below the resonance,output
leads the input almost by 90

. On the other hand,it lags behind the input for higher
frequencies. It converges to À90

V
o
 jo
V
in
 jo
 1 À
1
1  jQ
o
o
o
À
o
o
o


jQ
o
o
o
À
o
o
o

1  jQ
o
o

o
o
o
À
o
o
o

2
Therefore,
Q
o
o
o
À
o
o
o

Æ1
Assuming that o
1
< o
o
< o
2
,
Q
o
1

o
2
À
o
1
o
o
À
o
o
o
1

or,
o
2
À
o
2
o
o
2
Ào
1

o
2
o
o
1

o
 o
1
o
2
4:1:14
112
RESONANT CIRCUITS
and,
o
1
o
o
À
o
o
o
1

À
1
Q
A o
1
À
o
2
o
o
1


o
1
 o
2
p
3 f
o


f
1
 f
2
p


9 Â 11
p
 9:949874 MHz
From (4.1.11) and (4.1.15),
Q 
o
o
L
R

o
o
o

 6:430503 Â 10
À11
F % 64:3pF
The circuit arrangement is shown in Figure 4.7. Its magnitude and phase
characteristics are displayed in Figure 4.8.
Figure 4.7. The ®lter circuit arrangement for Example 4.1.
SERIES RESONANT CIRCUITS
113
Input Impedance
Impedance across the input terminals of a series R-L-C circuit can be determined as
follows.
Z
in
 R  joL 
1
joC
 R  joL 1 À
o
2
o
o
2

4:1:16
Figure 4.8 Magnitude (a) and phase (b) plots of A ( jo) for the circuit in Figure 4.7.
114
RESONANT CIRCUITS
At resonance,the inductive reactance cancels out the capacitive reactance.
Therefore,the input impedance reduces to total resistance of the circuit. If signal
frequency changes from the resonant frequency by Ædo,the input impedance can be

o
j2Q

L
 j2 o À o
o
1  j
1
2Q

L 4:1:18
Therefore,a series resonant circuit can be analyzed with R as zero (i.e.,assuming
that the circuit is lossless). The losses can be included subsequently by replacing a
real resonant frequency, o
o
,by the complex frequency,o
o
1  j
1
2Q

.
At resonance,current through the circuit,I
r
,
I
r

V
in

in
4:1:21
Hence,the magnitude of voltage across the inductor is equal to the quality factor
times input voltage while its phase leads 90

. Magnitude of the voltage across the
capacitor is the same as that across the inductor. However,it is 180

out of phase
because it lags behind the input voltage by 90

.
4.2 PARALLEL RESONANT CIRCUITS
Consider an R-L-C circuit in which the three components are connected in parallel,
as shown in Figure 4.9. A subscript p is used to differentiate the circuit elements
from those used in the series circuit of the preceding section. A current source, i
in
t,
PARALLEL RESONANT CIRCUITS
115
is connected across its terminals and i
o
t is current through the resistor R
p
. Voltage
across this circuit is v
o
t. From Kirchhoff's current law,
i
in

p
 sR
p
C
p
 1
!
I
o
s
Hence,
I
o
s
I
in
s

sL
p
R
p
s
2
L
p
C
p
 s
L

p
p
4:2:3
Hence,
z 
1
2o
o
R
p
C
p

1
2R
p

L
p
C
p
s
4:2:4
Figure 4.9 A parallel R-L-C circuit.
116
RESONANT CIRCUITS
The quality factor, Q
p
,and the impedance,Z
p

Z
p

V
o
 jo
I
in
io

I
o
 joR
p
I
in
 jo

joL
p
Ào
2
L
p
C
p
 jo
L
p
R

o
2
o
o
2

4:2:7
Hence,input admittance will be equal to 1=R
p
at the resonance. It will become zero
(that means the impedance will be in®nite) for a lossless circuit. It can be
approximated around the resonance, o
o
Æ do,as follows.
Y
in
%
1
R
p
 j2doC
p

1
R
p
 j
2doQ
o
o

and current through the inductor, I
L
,is
I
L

1
jo
o
L
p
R
p
I
in
ÀjQI
in
4:2:11
Thus,current through the inductor is equal in magnitude but opposite in phase to
that through the capacitor. Further,these currents are larger than the input current by
a factor of Q.
Quality Factor of a Resonant Circuit
If resistance R represents losses in the resonant circuit, Q given by the preceding
formulas is known as the unloaded Q. If the power loss due to external load coupling
PARALLEL RESONANT CIRCUITS
117
is included through an additional resistance R
L
then the external Q
e

,of a resonant circuit includes internal losses as well as the
power extracted by the external load. It is de®ned as follows:
Q
L

o
o
L
R
L
 R
for series resonant circuit
R
L
kR
p
o
o
L
p
for parallel resonant circuit
8
>
>
>
<
>
>
>
:

loaded Q of this circuit.
o
o

1

L
p
C
p
p

1

10
À5
 10
À11
p
 10
8
rad=s
The unloaded Q 
R
p
o
o
L
p


p
kR
L
o
o
L
p

QQ
e
Q  Q
e

50 Â 10
3
10
8
 10
À5
 50 .
4.3 TRANSFORMER-COUPLED CIRCUITS
Transformers are used as a means of coupling as well as of impedance transforming
in electronic circuits. Transformers with tuned circuits in one or both of their sides
are employed in voltage ampli®ers and oscillators operating at radio frequencies.
This section presents an equivalent model and an analytical procedure for the
transformer-coupled circuits.
Consider a load impedance Z
L
that is coupled to the voltage source V
s

1

LC
p
1

L
p
C
p
p
Damping factor, z
R
2

C
L
r
1
2R
p

L
p
C
p
s
Unloaded Q
o
o

R
L
C
R
L
o
o
L
p
 o
o
R
L
C
p
Loaded Q  Q
L
Q Â Q
e
Q  Q
e
Q Â Q
e
Q  Q
e
Input impedance, Z
in
,around resonance R  j
2RQdo
o

 Z
1

nV
2
ÀI
2
=n
 n
2
V
2
ÀI
2
 n
2
Z
2
4:3:5
There are several equivalent circuits available for a transformer. We consider one
of these that is most useful in analyzing the communication circuits. This equivalent
circuit is illustrated in Figure 4.11 below. The following equations for phasor
voltages and currents may be formulated using the notations indicated in Figure
4.11.
V
1
 jo1 À xL
1
I
1

I
2
n

4:3:7
If the circuit shown in Figure 4.11 is equivalent to that shown in Figure 4.10 then
these two equations represent the same voltages as those of (4.3.1) and (4.3.2).
Figure 4.10 A transformer-coupled circuit.
120
RESONANT CIRCUITS