131201 CIRCUIT THEORY

UNIT 3 RESONANT CIRCUITS

RESONANCE CIRCUITS-Series & Parallel

R.Ramaprabha, AP/EEE, SSNCE

Frequency Response

14.1
14.2
14.3
14.4
14.5

Introduction
Transfer Function
Series Resonance
Parallel Resonance
Passive Filters
R.Ramaprabha, AP/EEE, SSNCE

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Introduction
What is FrequencyResponse of a Circuit?

It is the variation in a circuits

behavior with change in signal
frequency and may also be
considered as the variation of the gain
and phase with frequency.
R.Ramaprabha, AP/EEE, SSNCE

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Transfer Function
The transfer function H() of a circuit is the frequencydependent ratio of a phasor output Y() (an element voltage
or current ) to a phasor input X() (source voltage or current).

H( ) =

Y( )
= | H( ) |
X( )

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Transfer Function
Four possible transfer functions:
H( ) = Voltage gain =

Vo ( )
Vi ( )

H( ) =

H( ) = Current gain =

I o ( )
I i ( )

H( ) = Transfer Impedance =

Vo ( )
I i ( )

Y( )
= | H( ) |
X( )

H( ) = Transfer Admittance =

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I o ( )
Vi ( )
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Transfer Function
Example 1
For the RC circuit shown below, obtain the transfer
function Vo/Vs and its frequency response.
Let vs = Vmcost.

R.Ramaprabha, AP/EEE, SSNCE

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Transfer Function
Solution:
The transfer function is
1
V
1
jC
H( ) = o =
=
Vs R + 1/ j C 1 + j RC

The magnitude is

H, ( ) =

1
1 + ( / o ) 2

The phase is = tan

o
1

o = 1/RC

Low Pass Filter

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Transfer Function
Example 2
Obtain the transfer function Vo/Vs of the RL circuit
shown below, assuming vs = Vmcost. Sketch its
frequency response.

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Transfer Function
Solution:
Vo
j L
1
H( ) =
=
=
Vs R + j L 1 + R
j L

The transfer function is

High Pass Filter

The magnitude is

H (, ) =

The phase is = 90 tan 1

= R/L

1+ (

o 2
)

o
R.Ramaprabha,
AP/EEE, SSNCE

9 9

Series Resonance
Resonance is a condition in an RLC circuit in which
the capacitive and inductive reactance are equal
in magnitude, thereby resulting in purely resistive
impedance.
Resonance frequency:
1
rad/s
LC
1
fo =
Hz
2 LC

o =
1
Z = R + j ( L
)
C

R.Ramaprabha, AP/EEE, SSNCE

or

1010

Series Resonance
The features of series resonance:
Z = R + j ( L

1
)
C

The impedance is purely resistive, Z = R;

The supply voltage Vs and the current I are in phase, so
cos = 1;
The magnitude of the transfer function H() = Z() is
minimum;
The inductor voltage and capacitor voltage can be much
more than the source voltage.

R.Ramaprabha, AP/EEE, SSNCE

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Series Resonance
Bandwidth B
The frequency response of the
resonance circuit current is
Vm
I =| I |=
R 2 + ( L 1 / C) 2
The average power absorbed
by the RLC circuit is

P( ) =

The highest power dissipated

occurs at resonance:

Z = R + j ( L

1
)
C

1 2
IR
2

1 Vm2
P(o ) =
2 R

R.Ramaprabha, AP/EEE, SSNCE

1212

Series Resonance
Half-power frequencies 1 and 2 are frequencies at which the
dissipated power is half the maximum value:

1 (Vm / 2 ) 2 Vm2
=
P(1 ) = P(2 ) =
2
R
4R
The half-power frequencies can be obtained by setting Z
equal to 2 R.
1 =

R
R
1
+ ( )2 +
2L
2L
LC

Bandwidth B

2 =

R
R
1
+ ( )2 +
2L
2L
LC

o = 1 2

B = 2 1

R.Ramaprabha, AP/EEE, SSNCE

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Series Resonance
Quality factor,

Q =

The relationship
between the B, Q
and o:

Peak energy stored in the circuit

L
1
= o =
Energy dissipated by the circuit
R
o CR
in one period at resonance

B=

R o
=
= o2 CR
L Q

The quality factor is the ratio of its

resonant frequency to its bandwidth.
If the bandwidth is narrow, the
quality factor of the resonant circuit
must be high.
If the band of frequencies is wide,
the quality factor must be low.
R.Ramaprabha, AP/EEE, SSNCE

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Series Resonance
Example 3
A series-connected circuit has R = 4
and L = 25 mH.
a. Calculate the value of C that will produce a
quality factor of 50.
b. Find 1 and 2, and B.
c. Determine the average power dissipated at
= o, 1, 2. Take Vm= 100V.

R.Ramaprabha, AP/EEE, SSNCE

1515

Parallel Resonance
It occurs when imaginary part of Y is zero

1
1
Y = + j ( C
)
R
L

Resonance frequency:
1
1
rad/s or f o =
Hz
o =
LC
2 LC
R.Ramaprabha, AP/EEE, SSNCE

1616

Parallel Resonance
Summary of series and parallel resonance circuits:
characteristic

Series circuit

Parallel circuit

1
LC

1
LC

o L
1
or
R
o RC

R
or o RC
o L

o
Q
B
1, 2
Q 10, 1, 2

o 1 + (

1 2

) o
2Q
2Q

o 1 + (

B
2

R.Ramaprabha, AP/EEE, SSNCE

1 2 o
)
2Q
2Q

B
2

1717

Parallel Resonance
Example 4

Calculate the resonant frequency of the

circuit in the figure shown below.

Answer: = 19 = 2.179 rad/s

2

R.Ramaprabha, AP/EEE, SSNCE

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In Detail

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SERIES RESONANCE

Resonance is a condition in an RLC circuit in which the

capacitive and inductive reactance are equal in magnitude,

thereby resulting in purely resistive impedance.

The features of series resonance:

The impedance is purely resistive, Z = R;
The supply voltage Vs and the current I are in phase (cos = 1)
The magnitude of the transfer function H() = Z() is minimum;
The inductor voltage and capacitor voltage can be much more
than the source voltage.
R.Ramaprabha, AP/EEE, SSNCE

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Vs ( )
1
1

= R + j L +
= R + j L
I ( )
j C
C

Resonance occurs when circuit is purely resistive

Z ( ) = H ( ) =

Im( Z ) = L

1
1
= 0 o L =
C
o C

o =

o =

1
LC

1
,
LC

Resonance Frequency

fo =

1
2 LC

R.Ramaprabha, AP/EEE, SSNCE

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VR, VL, VC, and I versus frequency for a

series resonant circuit.

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Inductive reactance versus frequency.

Capacitive reactance versus frequency.

Placing the frequency response of the

inductive and capacitive reactance of a
ZT (total impedance) versus frequency for
series R-L-C circuit on the same set of
the series resonant circuit.
axes.
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R.Ramaprabha, AP/EEE, SSNCE

PHASE OF SERIES RESONANCE CIRCUIT

Phase plot for the series resonant circuit.

fo f < f o : Network Capacitive

f > f o : Network Inductive
f = f o : Network Resistive

R.Ramaprabha, AP/EEE, SSNCE

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SERIES RESONANCE

Resonance occurs when the circuit has a complex conjugate pair of

poles.

Resonance allows frequency discrimination in circuits.

Resonance occurs in a circuit that has at least one inductor and one
capacitor.
At Resonance:
1) Impedance is purely resistive.

2) The voltage and current are in phase.

3) The transfer function H()= Z() is Minimum
4) Inductor and capacitor voltages can be much more than (Q times) source
voltage.

Vm
VL = o L = QVm
R
Vm 1
VC =
= QVm
R oC

o L

1
Q=
=
R
oCR

R.Ramaprabha, AP/EEE, SSNCE

25

BANDWIDTH of SERIES RESONANCE

Current versus frequency for the series resonant circuit.

I=I =

Vm
R 2 + ( L 1

)2

Half Power Frequencies

Dissipated power is half of the
maximum value.

The half-power frequencies 1 and 2

can be obtained by setting,

Z (1 ) = Z (2 ) = R 2 + ( L 1

Vm

P (1 ) = P(2 ) =
2R

)2 = 2 R

R
1
R
1
R
R
1 =
,

+
+
=
+
+
+
2

2L
2L
2 L LC
2 L LC

R.Ramaprabha, AP/EEE, SSNCE

26

Selectivity

The frequencies corresponding to 0.707 of the maximum current are called the band
frequencies, cutoff frequencies, or half-power frequencies (1, 2).

Half-power frequencies are those frequencies at which the power delivered is onehalf that delivered at resonant frequency.

The range of frequencies between the two are referred to as bandwidth (abbreviated
BW) of the resonant circuit.

Since the resonant circuit is adjusted to select a band of frequencies it is called a
selectivity curve.

The shape of the curve depends on each element of the series R-L-C circuit.

If resistance is made smaller with a fixed inductance and capacitance, the bandwidth
decreases and the selectivity increases.

If the ratio L/C increases with fixed resistance, the bandwidth again decreases with
an increase in selectivity.
R.Ramaprabha, AP/EEE, SSNCE

27

BANDWIDTH OF SERIES
RESONANCE

The width of the response is measured by the BANDWIDTH.

BANDWIDTH is the difference between the half-power
frequencies.

B = 2 1

Resonance frequency can be obtained from the half-power
frequencies.

o = 1 2 , B = 2 1

The SHARPNESS of the resonance is measured by the QUALITY

FACTOR.

QUALITY FACTOR is the ratio of the resonance frequency to the
bandwidth. The higher the Q the smaller is the bandwidth.
o
Q=
B 28
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QUALITY FACTOR OF SERIES RESONANCE

Q=
Q = 2

o
B

Peak Energy Stored

Energy Dissipated in one Period at Resonance
L
1
Q= o =
R
o RC

Q=

o L
R

1
o RC

R o
=
L Q
B
B
1 o , 2 o +
2
2
B=

R.Ramaprabha, AP/EEE, SSNCE

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Effect on Selectivity of R, L, C for Series Resonance

Effect of R on
selectivity

Effect of L and C on
selectivity

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R.Ramaprabha, AP/EEE, SSNCE

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PARALLEL RESONANCE

Resonance is a condition in an RLC circuit in which the capacitive and inductive
reactances are equal in magnitude, resulting in a purely resistive impedance.

Parallel resonance circuit behaves similarly but in opposite fashion compared to
series resonant circuit.

The admitance is minimum at resonance or impedance is maximum.

o =

1
LC

Parallel resonant circuit.

I 1
1
1
1

Y = H ( ) = = + jC +
= + j C
j L R
V R
L

Resonance occurs when admitance is purely resistive

Im(Y ) = L

1
1
= 0 o L =
C
oC

o =

R.Ramaprabha, AP/EEE, SSNCE

1
rad/sec
LC
32

PARALLEL RESONANCE

At Resonance frequency:
1) Admitance is purely resistive.
2) The voltage and current are in phase.
3) The transfer function H()= Y() is Minimum.
4) Inductor and capacitor currents can be much more than the source current.

IL =

Im R
= QI m
o L

I C = o CI m R = QVm

R.Ramaprabha, AP/EEE, SSNCE

33

PARALLEL RESONANCE
Im

V=V =

1
2
1
+ ( C L)
R

Voltage versus frequency for the parallel resonant circuit.

2

The half-power frequencies can be obtained as:

1
1
1
1 =
+
+
2 RC
LC
2 RC
2

1
1
1
+
+
2 = +

2 RC
LC
2 RC

1
o = 1 2 , B = 2 1 =
RC
R.Ramaprabha, AP/EEE, SSNCE

34

Summary of series and parallel resonance circuits:

Characteristic

Series circuit

Parallel circuit

1
LC

1
LC

o L
1
or
R
o RC

B
1, 2
Q 10, 1, 2

R
or o RC
o L

1 + (

o
1 2
)
2Q
2Q

B
2

R.Ramaprabha, AP/EEE, SSNCE

1 + (

1 2
) o
2Q
2Q

B
2

35

R.Ramaprabha, AP/EEE, SSNCE

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