Lecture 11 AC CircuitsFrequency Response

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Contents

• Introduction

• Transfer Function

• Series Resonance

• Parallel Resonance

• Passive Filters

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What is Frequency Response of a Circuit?

It is the variation in a circuit’s

behavior with change in signal

frequency and may also be

considered as the variation of the gain

and phase with frequency.

Introduction

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Transfer Function• The transfer function H(ω) of a circuit is the frequency-dependent

ratio of a phasor output Y(ω) (an element voltage or current ) to a

phasor input X(ω) (source voltage or current).

|)(H|

)(X

)(Y )(H

)(V

)(V gain Voltage )(H

i

o

)(I

)(I gain Current )(H

i

o

)(I

)(V ImpedanceTransfer )(H

i

o

)(V

)(I AdmittanceTransfer )(H

i

o

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Transfer Function: Example

Example 1

For the RC circuit shown below, obtain the transfer function Vo/Vs and its frequency response.Let vs = Vmcosωt.

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Solution (Example 1)

Solution:

The transfer function is

,

The magnitude is2)/(1

1)(H

o

The phase iso

1tan

1/RCo

RC j1

1

C j1/ R

Cj

1

V

V)(H

s

o

Low Pass Filter

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Transfer Function: Example

Example 2

Obtain the transfer function Vo/Vs of the RL circuit shown below, assuming vs = Vmcosωt. Sketch its frequency response.

The magnitude is2)(1

1)(H

o

The phase iso

1tan90

R/Lo

L j

R1

1

L jR

Lj

V

V)(H

s

o

High Pass Filter

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Solution (Example 2)

Solution:

The transfer function is

,

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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.

ω Cω Lj R

1Z

Resonance frequency:

HzLC2

1f

rad/sLC

1

o

oro

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Series Resonance

The features of series resonance:

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(ω) = Z(ω) is minimum;

• The inductor voltage and capacitor voltage can be much more than

the source voltage.

ω Cω Lj RZ

1

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Series Resonance

Bandwidth B

The frequency response of the resonance circuit current is

22

m

)C /1L (R

V|I|I

The average power absorbed by the RLC circuit is

RI2

1)(P 2

The highest power dissipated occurs at resonance: R

V

2

1)(P

2

mo

ω Cω Lj RZ

1

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Series Resonance

Half-power frequencies ω1 and ω2 are frequencies at which the dissipated power is half the maximum value:

The half-power frequencies can be obtained by setting Z equal to √ 2 R.

4R

V

R

)2/(V

2

1)(P)(P

2

m

2

m21

LC

1)

2L

R(

2L

R 2

1 LC

1)

2L

R(

2L

R 2

2 21 o

Bandwidth B12 B

R

V

2

1)(P

2

mo

Highest power

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Series Resonance: Quality factor

CR

1

R

L

resonanceat period onein

circuit by the dissipatedEnergy

circuit in the storedenergy Peak Q

o

o

• 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.

The relationship between the B, Q and ωo:

CRQL

RB 2

oo

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Series Resonance

Example 3

A series-connected circuit has R = 4 Ω and L = 25 mH.

a. Calculate the value of C that will produce a quality factor

of 50.

b. Find ω1 and ω2, and B.

c. Determine the average power dissipated at ω = ωo, ω1, ω2.

Take Vm= 100V.

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Parallel Resonance

Resonance frequency:

HzoLC2

1for rad/s

LC

1o

)L

1C ( j

R

1Y

It occurs when imaginary part of Y is zero

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Summary of series and parallel resonance circuits:

LC

1

LC

1

RC

1or

R

L

o

o

ω

ωRCor

L

Ro

o

Q

o

Q

o

2Q )

2Q

1( 1 2 o

o

2Q )

2Q

1( 1 2 o

o

2

Bo

2

Bo

characteristic Series circuit Parallel circuit

ωo

Q

B

ω1, ω2

Q ≥ 10, ω1, ω2

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Parallel Resonance

Example 4

Calculate the resonant frequency of the circuit in the figure shown below.

rad/s2.1792

19Answer:

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Passive Filters

• A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.

• Passive filter consists of only passive element R, L and C.

• There are four types of filters.

Low Pass

High Pass

Band Pass

Band Stop

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Passive Filters

Example 5

For the circuit in the figure below, obtain the transfer function Vo(ω)/Vi(ω).

Identify the type of filter the circuit represents and determine the corner

frequency. Take R1=100W =R2

and L =2mH.

krad/s25Answer: