R'V' Srinivasa Murthy

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• R.V. Srinivasa Murthy
• Assistant Professor
• Dept. of E C
• A.P.S. College of Engineering
• Bangalore

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Resonant Circuits

• Resonance is an important phenomenon which may
  occur in circuits containing both inductors and
  capacitors.
•  
• In a two terminal electrical network containing
  at least one inductor and one capacitor, we
  define resonance as the condition, which exists
  when the input impedance of the network is purely
  resistive. In other words a network is in
  resonance when the voltage and current at the
  network in put terminals are in phase.

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Resonance condition is achieved either by keeping
inductor and capacitor same and varying frequency
or by keeping the frequency same and varying
inductor and capacitor.

• Study of resonance is very useful in the area of
  communication.
• The ability of a radio receiver to select the
  correct frequency transmitted by a broad casting
  station and to eliminate frequencies from other
  stations is based on the principle of resonance.

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The resonance circuits can be classified in to
two categories 1. Series Resonance Circuits2.
Parallel Resonance Circuits

• 1.Series Resonance Circuit
• A series resonance circuit is one in which a coil
  and a capacitance are connected in series across
  an alternating voltage of varying frequency as
  shown in figure.

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The response I of the circuit is dependent on
the impedance of the circuit, Where Z R
j XL - j XCand I V at any value of
frequency Z

• We have XL 2 p f L ? XL varies as f
  
• and XC 1 ? XC varies
  inversely as f
• 2pfC
• In other words, by varying the frequency it is
  possible to reach a point where XL XC .
  In that case Z R and hence circuit will
  be under resonance.

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Hence the series A.C. circuit is to be under
resonance, when inductive reactance of the
circuit is equal to the capacitive reactance.

• The frequency at which the resonance occurs is
  called as resonant frequency ( fr)

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Expression for Resonant Frequency ( fr )At
resonance XL XC i.e. ? r L
1 ? r C ?2 r L C
1 ?2 r 1 L C
f 2 r 1
4p 2LC

• f r 1
  2p

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Salient Features of Resonant circuit

• At resonance XL XC
• At resonance Z R i.e. impedance is minimum
  and hence I V is maximum Z
• The current at resonance (Ir) is in phase with
  the voltage
• The circuit power factor is unity
• Voltage across the capacitor is equal and
  opposite to the voltage across the inductor.

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Frequency response of a series resonance circuit

• For a R-L-C series circuit the current I
  V ______
• R j ( XL- XC )
  
  
• At resonance XL XC and hence the current
  at resonance (Ir) is given by Ir V/R
• At off resonance frequencies since the impedance
  of the circuit increases, the current in the
  circuit will reduce. At frequencies f gt fr , the
  impedance is going to be more inductive.
  Similarly at frequencies f lt fr , the circuit
  impedance is going to be more capacitive.

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Thus the resonance curve will be as shown in
figure.
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Qualify factor (or Q factor)

• Another feature of a resonant circuit is the
• Q rise of voltage across the resonating
  elements.
• If V is the applied voltage across a series
  resonance circuit at resonance, I r V
• R
• The voltage across the inductance L V L
  I r XL
• V L V ? r L
• R
• V L Q V where Q ? r L is the
  Q factor
• R
• From the above equation it is seen that the
  voltage across the inductive coil is Q times the
  applied voltage (V) The response of the series
  resonant circuit is largely dependent on Q of the
  coil.

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Band width concept

• Any circuit response, which is frequency
  dependent, has certain limitations.
• The output response during limited band of
  frequencies only will be in the useful range.
• If the out put power is equal to or more than
  half of the maximum power output, that band of
  frequencies is considered to be the useful band.

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If I r is the maximum current at resonance
thenPower at resonance Pmax I2r R

• But If I I r then corresponding power is
  given by
• v2
• I2 R 1 I2r R 1 ( Max Power)
• 2 2
• Hence the useful range of frequencies will be
  frequencies where current will be equal to or
  more than Ir 0.707 Ir v2

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Consider the frequency response characterstic of
a series resonant circuit as shown in figure
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In the figure it is seen that there are two
frequencies where the out put power is half of
the maximum power. These frequencies are called
as half power points f1 and f2

• A frequency f1 which is below fr where power is
  half of maximum power is called as lower half
  power frequency (or lower cut off frequency).
  
• Similarly frequency f2 which is above fr is
  called upper half power frequency (or upper
  cut-off frequency)
• The band of frequencies between f2 and f1 are
  said to be useful band of frequencies since
  during these frequencies of operation the out put
  power in the circuit is more than half of the
  maximum power. Thus their band of frequencies is
  called as Bandwidth. i.e Band width B W
  f2 - f1

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Selectivity Selectivity is a useful
characteristic of the resonant circuit.
Selectivity is defined as the ratio of band
width to resonant frequency

• Selectivity f2- f1
  frIt can be seen that selectivity is the
  reciprocal of Quality factor. Hence larger the
  value of Q , smaller will be the
  selectivity.The Selectivity of a resonant
  circuit depends on how sharp the out put is
  contained with in limited band of frequencies.
  The circuit is said to be highly selective if the
  resonance curve falls very sharply at off
  resonant frequencies.

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Relation between Resonant frequency and cut-off
frequencies

• Let fr be the resonant frequency of a series
  resonant circuit consisting of R, L and C
  elements .
• From the Characteristic it is seen that at both
  half frequencies f2 and f1 , the output
  current is 0.707 Io which means that the
  magnitude of the impedance is same at these
  points.

0.707 Ir
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At lower cut-off frequency f1Impedance is given
by v R2 (XC1 XL1)2At upper cut-off
frequency f2 Impedance is given by v R2 (XL2
XC2 )2

• But these impedances are equalHence v
  R2 (XC1 XL1)2 v R2 (XL2
  XC2)2 XC1 XL1 XL2 XC2 XC1
  XC2 XL1 XL2 1 1 1
  L ?1 ?2 C
  ? 1 ? 2 ? 1 ? 2
  1 LCWe have f
  r 1 ? ? r 1
  2pvLC
  vLCHence ?r2 _1__ ? ?r2 ?1 ?2
  or fr2 f1 f2 f r v f1 f2
  LC
• i.e. Resonant frequency is the geometrical mean
  of half power frequencies.

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Relation between resonant frequency (f r), Band
width and quality factor (Q)

• At lower cut- off frequency (f 1)
• I V
• v R2 (Xc1 XL1)2
  
• 
  
  
• But I1 Ir /v2 V / v2 R
• Hence V V
• v2 R vR2 (Xc1 XL1)2
•  vR2 (Xc1 XL1)2 v2 R
• i.e. Xc1 XL1 R (1)

At upper cut-off frequency, I2 V
v R2 (XL2 XC2)2 But I2 Ir /v2
V/ v2 R V
V v2 R v R2 (XL2 XC2)2 v R2
(XL2 XC2) 2 v2 R   XL2
XC2 R (2)
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Adding equations (1) and (2),we get Xc1 XL1
Xc2 XL2 2R
Band width f r / Q
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Resonance by varying Inductance Resonance in RLC
series circuit can also be obtained by varying
resonating circuit elements . Let us consider a
circuit where in inductance is varied as shown in
figure.

• At resonance XL Xc , ? L r 1/ ?C
• L r 1/ ?2C , where L r value of inductance
  at resonance.

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At lower half pointLet L1 be the value of
inductance. I Io/ v2 and Io V / R
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Resonance by varying capacitance Consider an RLC
series circuit in which capacitance C is varied.
At resonance Xc XL 1 ? L

? Cr
Cr 1 CrCapacitance value
at resonance. ? 2 L

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At lower half point 1 - ? L R
1 R ? L ? C1
? C1   C1 1 Farad
? 2L ?RAt upper half point? L -
1 R 1 ? L - R
? C2 ? C2 C2 1
Farad ? 2L - ?R
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Expressions for maximum capacitor voltage and
maximum inductor voltage
fr
f L max

• From the figure it is clear that voltage across L
  and voltage across C are not maximum at resonant
  frequency (fr). Rather voltages VL and VC are
  equal in magnitude and opposite in phase at fr.
  The voltage Vc is maximum at a frequency fc max
  which is less than fr and the voltage VL is
  maximum at a frequency fL max which is greater
  than fr

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Expression for fcmaxVoltage across capacitor Vc
is given by Vc I Xc I .1/ ? C But I
V/Z. So Vc V

? C vR2 ?L 1/ ?C
2

• To find the frequency at which Vc is maximum we
  have to differentiate Vc with respect to ? and
  equate it to zero VC2
  _________ V2 __________
  ?2 C2 R2 ( ?L 1/ ?C )2
  ________
  V2__________ ?2 R2 C2 ( ?2 L C 1 )2

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d VC2 V2 2 ? R2 C2 2 ( ?2 L C 1 ) (
2 ? L C ) 0d ? ?2 R2 C2
(?2 L C 1)2 2 2?R2C2 2( 2?LC)
(?2LC- 1) 0 2 ?2 L2 C2 2 LC - R2
C2 ?2 1/ (LC - R2 / 2L2 )
?Cmax v 1/LC R2/2L2
rad/sec fCmax 1/2? v
1/LC R2 / 2L2 Hz
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Example 1 A series RLC circuit shown in figure
is connected across an A.C. variable frequency
supply of 200 Volts. Calculate the resonant
frequency and half power frequencies.

• Solution
• fr 1 1_____
  35.6Hz
• 2pvLC 2p v0.5x40x10 6
• Q ?r L 2px35.6 x0.5 1.86
• R 60

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B.W fr 35.6 19 Q
1.86 f2- f1 19
(1)Also fr 2 f1 f2 (35.6) 2
f1 f2 1266 (2)Solving equations
(1) and (2) we get, f1 27.33 Hz f 2
46.33 Hz
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Example 2

A 20 ohm resistor is connected in series with
an inductor, a capacitor and an Ammeter across 25
volts variable frequency a.c supply .When the
frequency is 400 Hz the current is at its
maximum value of 0.5 A and the potential
difference across the capacitor is 150 volts
.Calculatei) The capacitances of the capacitor
ii) The resistance and inductance of the
inductor.
Solution Given Ir 0.5 A, fr
400Hz i)  We have V C I o X C X C V
C 150 300 ohms I o
0.5 1 300O or C 1
3.33 microfarad ?o C
300 ?o
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(ii)    At resonance circuit impedance is pure
resistance given by Z R r V
25 50 O Io 0.5 Where
r Internal resistance of the inductorR 50
R 50 - 20 30 OhmsAt resonance XL XC
300 ohmsL 300 300
0.119 henry ?0 2p x 400
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Example 3A Series RLC circuit consists of R 100
ohms L 0.02H and C 0.02 µF. Calculate
frequency of resonance. A variable frequency
sinusoidal voltage of rms value of 50 volts is
applied to the circuit. Find the frequency at
which voltage across L and C is maximum

• Solution Given R100 ohms
• L 0.02 H , C 0.02 µF
• Resonant frequency
• fo 1 1
  
• 2p vLC 2p v0.02x0.02x10-6
• fo 7.957 KHz

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The frequency at which voltage across C is
maximum is given byfC 1 /2p v 1/ ( LC R2 /
2 L2 )
1 /2p v (1/ 0.02 X 0.02 X 10-6 ) ( 1002 ) /
2( 0.002)2 7.937 KHz (iii) The
frequency at which voltage across L is maximum is
given by fL 1
2p v
LC (R2 C2 / 2) 1 /2p v (0.02 X
0.02 X 10-6) (1002) (0.002 x 10-6)2 /2
7.977 KHz