(1) Resonant Circuits11

8/13/2019 (1) Resonant Circuits11

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Resonance ircuits

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Resonance In Electric ircuits

Any passive electric circuit will resonate if it has an inductor

and capacitor

esonance is characterized by the input voltage and currentbeing in phase.

The impedance (or admittance) is completely real when thiscondition exists.

Basically ,there are two types of resonant circuits :(a) series resonance, and

(b) parallel resonance.

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3

Series Resonance

Consider the series RLC circuitshown below.

The input impedance is given by:

The current in the circuit is:

The magnitude of the circuitcurrent is;

1

( )Z R j wL wC= +

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Variation of inductive and capacitive reactance as the frequency fof the source is varied:

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5

When f= 0 , XL = 0 and XC = .

As f increases , the XL increases and the XC decreases till at a

frequency fr the two reactances become equal .

With further increase in f , XL > XC

At fr the net reactance of the circuit = 0

The impedanceof the circuit z= R and the current in the circuit = V/R

fr is known as resonance frequency and the circuit , is said to be in

resonance.

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Therefore resonance occurs when,

It applies to bothto remember.m portant equat ionnThis isseries and parallel resonant c ircuit

66

1wL

wC=

LCf rr

2

1

2==


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The following figure shows the variation in the impedance of the circuit

as the frequency varies from 0 to

At low frequency Xc >XL and the

circuit is capacitive .

As f goes on increasing , the net reactancegoes on decreasing , and theimpedance also goes on decreasing .

At f = fr , the net reactance = 0,

and the circuit impedance is the min. Z = R

.

When f >fr XL >XC and the circuit is

inductive.

As fincr . , Z is incr. too.

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88

The variation in the magnitude is plotted in the following fig. :

Since the current is proportional to

Z, the current incr. with increasing

of f .

At f = fr the current is max.

(Imax).

As f incr. beyond f r I decr.

The voltages across XL and XC are :

VL = I XL , VC = I XC

At fr XL = XC VL =VC

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Series Circuit Current at Resonanc e

The frequency response curve of a series resonance circuit shows that the

.magnitude of the current is a function of frequency

Since the current is proport ional to Z, the current incr. with increasing

of f .

At f = fr the current is max. (Imax).

As f incr. beyond f r I decr.

The voltages across XL and XC are :

VL = I XL , VC = I XC

At fr XL = XC VL =VC

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exampleA series RLC circuit has : R = 80 , L = 100 H and C = 300 pF

Find the resonance frequencyand the current at resonance if E = 10v

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solut iona

1111

LCfr

2

1=

[ ] MHzHz 919.010919.0)1030010100(2

1 6

2

1126 ===

current at resonance E / R = 10 / 80 = 0.125A


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Bandw idth of a Ser ies Reson ance Circu i t

The bandwidth (B) of a seriesRLC circuit is :

BW = f = f2 - f1

Where f1 and f2 are thefrequencies of which the powerdelivered to the circuit is power delivered at resonance .

These known as half powerpoints.

The power delivered atresonance is :

Therefore ,

Thus the currents I1 and I2 ,athalf power point are:

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From the eq. We can write

It is seen that at hpp : (XL XC) / R = 1

At resonance XL XC = 0 ,then when freq. incr. from fr to f2 , XL must

incr. by 0.5 R and XC must decr. by 0.5R to satisfy the eq.

thus :

2 f2 L - 2 fr L = 0.5R or

Similarly when freq. decr. from fr to f1 , XC incr. by 0.5R and XL decr. by

the same value ,thus :

2 fr L - 2 f1 L = 0.5R or

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Bandw idth of a Ser ies Reson ance Circu i t

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Q Facto r

The ratio of the resonance frequency to the BW is known as factor

Q :

from eq. f = R/ 2L and from eq.

Thus Q can be incr. by decr. R or by incr. L/C ratio .

1515

f

fQ r

=

)2

1(

2

LCR

LQ

=

LCfr

21=


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The following fig. shows the current versus frequency graphs for

circuits with different values of Q :

A circuit with high Q has a narrow B

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1xerciseser ies RLC circu i t has L = 50H C = 2000 pF and R

50 . Calcu late Q factor of the c i rcu i t

Find the new value of C required for resonance athe same frequenc y i f the ind uctanc e is do ubled .

Find the new value of Q factor


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2xerciseco nstant vo l tage at frequency of 1 MHz is app l ied to a coi l

n series w ith a variable capacit or .hen the capacitor is set at 500 pF the current in the circui t is

maximum.hen the capacitor is set at 600 pF the current is h alf the maxi.valu e .

ind Resistanc e Indu ctance and Q factor of the coi l .


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3xerciseser ies resonance netwo rk cons ist ing of a resistor of 30 acapaci tor of 2uF and an induc tor of 20mH is con nected across a

sinuso idal sup ply vol tage which h as a con stant output o f 9 vol ts at al lfrequencies.

alculate:he resonant frequency

he cur rent at reso nancehe vol tage across the indu ctor

nd capaci tor at reson ancehe qual i ty facto r

he bandwid th of the ci rcui t .lso s ketch the correspond ing cu rrent waveform for al l f requencies .

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4xerciseser ies circ ui t con sists o f a resistanc e of 4 an inductance of

500mH and a variable capac itance co nn ected acro ss a 100V 50Hzsupply.

alculate:The capacitance require to g ive ser ies resonanc e

The vol tages generated acro ss b oth th e indu cto r and th e capacitor .

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Paral lel Resonanc eThe admittance of the circuit is :

If the source frequency is

adjusted so as to make XL = XC

,then :

Y =1/R and Z = R , I = V/R

This is the condi tion of the

parallel resonance .

The frequency f r at which parallelresonance take place :

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2323

Parallel Resonance

Consider the circuits shown below :

I R L C

V

I

RL

CV

++=

jwLjwC

RVI 11

++=

jwCjwLRIV 1


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a

24

++=

jwLjwC

RVI

11

++=jwC

jwLRIV 1

We notice the above equations are the same provided:

VI

RR 1

CL

If we make the inner-change,

then one equation becomes

the same as the other.

For such case, we say the one

circuit is the dual of the other.

Duality

If we make the inner-change,

then one equation becomes

the same as the other.

For such case, we say the one

circuit is the dual of the other.


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Parallel Resonance

What this means is that for all the equations we have

derived for the parallel resonant circuit, we can use

for the series resonant circuit provided we make

the substitutions:

RbereplacedR

1

LbyreplacedC

CbyreplacedL

What this means is that for all the equations we have

derived for the parallel resonant circuit, we can use

for the series resonant circuit provided we make

the substitutions:

What th is means is that for all the equations we have

derived for the series resonant circuit, we can use

for the parallel resonant circuit provided we make

the substitutions:


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a

Serial Reson ance Paral lel resonance


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When a parallel circuit at resonance ,the following characteristics can be noted :

1. The total current in the circuit is MIN.

2. The current is in phase with the supply voltage and the circuitacts as pure resistive circuit .

3. Admittance of the circuit is MIN. ,therefore the impedance ofthe circuit is MAX.


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:5Exercise

A coil of 20 resistance has an inductance of 0.2 H and is

connected in parallel with a 100 F capacitor.Calculate

- The resonance frequency.

- The quality factor

- The bandwidth of the circuit

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(1) Resonant Circuits11