CHAPTER 4

RESONANCE CIRCUITS

Tunku Muhammad Nizar Bin Tunku MansurPegawai Latihan Vokasional

Pusat Pengajian Kejuruteraan Sistem Elektrik

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Content Series Resonance Parallel Resonance Important Parameters

Resonance Frequency, o

Half-power frequencies, 1 and 2

Bandwidth, Quality Factor, Q

Application

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Introduction

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

Resonance circuits are useful for constructing filters and used in many application.

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Series Resonance Circuit

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At Resonance

At resonance, the impedance consists only resistive component R.

The value of current will be maximum since the total impedance is minimum.

The voltage and current are in phase. Maximum power occurs at resonance

since the power factor is unity.

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Series Resonance

CLTotal jX-jXRZ

R

V

Z

VI m

Total

sm

Total impedance of series RLC Circuit is

At resonance

The impedance now reduce to

CL XX

R ZTotal

)X-j(XRZ CLTotal

The current at resonance

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Resonance Frequency

Resonance frequency is the frequency where the condition of resonance occur.

Also known as center frequency.

Resonance frequency

rad/sLC

1ωo

HzLC2

1

of

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Half-power Frequency

rad/sLC

1

2L

R

2L

Rω

2

2

Half-power frequencies is the frequency when the magnitude of the output voltage or current is decrease by the factor of 1 / 2 from its maximum value.

Also known as cutoff frequencies.

rad/sLC

1

2L

R

2L

Rω

2

1

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Bandwidth,

rad/s)( 12 cc ωωβ

Bandwidth, is define as the difference between the two half power frequencies.The width of the response curve is determine by the bandwidth.

rad/sL

Rβ

10

Current Response Curve

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Voltage Response Curve

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Quality Factor (Q-Factor)

The ratio of resonance frequency to the bandwidth

The “sharpness” of response curve could be measured by the quality factor, Q.

R

LQ oo

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High-Q

It is to be a high-Q circuit when its quality factor is equal or greater than 10.

For a high-Q circuit (Q 10), the half-power frequencies are, for all practical purposes, symmetrical around the resonant frequency and can be approximated as

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o 22

o

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Q-Factor Vs Bandwidth

Higher value of Q, smaller the bandwidth. (Higher the selectivity)

Lower value of Q larger the bandwidth. (Lower the selectivity)

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Maximum Power Dissipated

The average power dissipated by the RLC circuit is

The maximum power dissipated at resonance where

R

V

2

1)P(ω

m2

o

R

VI m

RI2

1)P(ω 2

o

Thus maximum power dissipated is

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Power Dissipated at 1 and 2

At certain frequencies, where ω = ω1 and ω2, the dissipated power is half of maximum power

Hence, ω1 and ω2 are called half-power frequencies.

4R

V

R

)2/(V

2

1)P(ω)P(ω

m22

m21

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Example 14.7If R=2Ω, L=1mH and C=0.4 F, calculate

Resonant frequency, ωo

Half power frequencies, ω1 and ω2

Bandwidth, Amplitude of current at ωo, ω1 and ω2.

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Practice Problem 14.7

A series connected circuit has R=4Ω and L=25mH. Calculate Value of C that will produce a quality

factor of 50. Find 1 , 2 and . Determine average power dissipated

at = o , 1 and 2. Take Vm = 100V

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Parallel Resonance

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Parallel Resonance

The total admittance

ωLωC

1

Resonance occur when

)ωω L1/Cj(R

1YTotal

321Total YYYY

C)(-j/

1

L)(j

1

R

1YTotal

CjωL

j-

R

1YTotal ω

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At Resonance

At resonance, the impedance consists only conductance G.

The value of current will be minimum since the total admittance is minimum.

The voltage and current are in phase.

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Parameters in Parallel Circuit

rad/sLC

1

2RC

1

2RC

1ω

2

1

Parallel resonant circuit has same parameters as the series resonant circuit.

rad/sLC

1ωo

rad/sLC

1

2RC

1

2RC

1ω

2

2

Resonance frequency

Half-power frequencies

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Parameters in Parallel Circuit

RCβ

ωQ o

o

RC

112 ωωβ

Bandwidth

Quality Factor

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Example 14.8If R=8kΩ, L=0.2mH and C=8F, calculate

ωo

Q and ω1 and ω2

Power dissipated at ωo, ω1 and ω2.

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Practice Problem 14.8

A parallel resonant circuit has R=100kΩ, L=25mH and C=5nF. Calculate o

1 and 2 Q

APPLICATION

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PASSIVE FILTERS A filter is a circuit that is designed to

pass signals with desired frequencies and reject or attenuates others

A filter is a Passive Filters if it consists only passive elements which is R, L and C.

Filters that used resonant circuit Bandpass Filter Bandstop Filter

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BANDPASS FILTER

A bandpass filter is designed to pass all frequencies within

ω1 ωo ω2

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BANDPASS FILTER

LC

1

2L

R

2L

Rω

2

1

SERIES RLC CIRCUIT LC

1ωo

2o

CR

L

β

ωQ

L

Rωωβ 12

LC

1

2L

R

2L

Rω

2

2

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BANDPASS FILTER

LC

1

2RC

1

2RC

1ω

2

1

PARALLEL RLC CIRCUIT LC

1ωo

L

CR

β

ωQ

2o

RC

1ωωβ 12

LC

1

2RC

1

2RC

1ω

2

2

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BANDSTOP FILTER

A bandstop or bandreject filter is designed to stop or reject all frequencies within

ω1 ωo ω2

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BANDSTOP FILTER

LC

1

2L

R

2L

Rω

2

1

SERIES RLC CIRCUITLC

1ωo

2o

CR

L

β

ωQ

L

Rωωβ 12

LC

1

2L

R

2L

Rω

2

2

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BANDSTOP FILTER

LC

1

2RC

1

2RC

1ω

2

1

PARALLEL RLC CIRCUIT LC

1ωo

L

CR

β

ωQ

2o

RC

1ωωβ 12

LC

1

2RC

1

2RC

1ω

2

2

EXERCISE