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Chapter 14Resonance Circuits

Huseyin BilgekulEeng 224 Circuit Theory II

Department of Electrical and Electronic Engineering Eastern Mediterranean University

Chapter Objectives: Understand the Concept of Transfer Functions. Be Familiar with the Decibel Scale. Learn how to make Bode Magnitude and Phase plots.

Learn about series and parallel resonant RLC circuits. Know Different Types of Passive and Active Filters and their

Characteristics. Understand the use of scaling in circuit analysis. Be Able to use PSpice to obtain frequency response. Apply what is learnt to radio receiver and touch-tone

telephone.

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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(ω) = Z(ω) is minimum;• The inductor voltage and capacitor voltage can be much more

than the source voltage.

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SERIES RESONANCE

( ) 1 1( ) ( )

( )

occurs when circuit is

1 Resonance Frequenc

Resonance purely resistive

1 1Im( ) 0

y

s

o

o

o

VZ H R j L R j L

I j C

Z LC

LC

C

LC

1 1,

2o of

LC LC

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SERIES RESONANCE 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 series R-L-C circuit on the same set of axes.

ZT (total impedance) versus frequency for the series resonant circuit.

SERIES RESONANCE

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PHASE OF SERIES RESONANCE CIRCUIT

Phase plot for the series resonant circuit.

fo

: Network Capacitive

: Network Inductive

: Network Resistive

o

o

o

f f

f f

f f

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SERIES RESONANCE

At Resonance:

1) Impedance is purely resistive.2) The voltage and current are in phase.3) The transfer function H()= Z() is Minimum4) Inductor and capacitor voltages can be much more than (Q times) source voltage.

1

1

mL m

m oC m

o

o

o

L QR

QR C

VV V

V LV V Q

R CR

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.

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BANDWIDTH of SERIES RESONANCE

• The half-power frequencies 1 and 2 can be obtained by setting,

2 21( )

mVI

R L C

I

2 21 2

2

1 2

2 2

1 2

1( ) ( ) ( ) 2

2 ( )

1 1,

2 2 2

(2

2

)

m

Z

R R R R

L L LC

Z R L RC

V

P PR

L L LC

Current versus frequency for the series resonant circuit.

Half Power FrequenciesDissipated power is half of the maximum value.

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The frequencies corresponding to 0.707 of the maximum current are called the band frequencies, cutoff frequencies, or half-power frequencies (ƒ1, ƒ2).

Half-power frequencies are those frequencies at which the power delivered is one-half 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.

Selectivity

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BANDWIDTH OF SERIES RESONANCE

2 1B

1 2 2 1,o B

oQB

The width of the response is measured by the BANDWIDTH.

BANDWIDTH is the difference between the half-power frequencies.

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

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.

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QUALITY FACTOR OF SERIES RESONANCE

1 2

1

,2 2

o

o

o

o o

LQ

R RC

RB

L Q

B B

oQB

Peak Energy Stored2

Energy Dissipated in one Period at Resonance

1o

o

Q

LQ

R RC

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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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PARALLEL RESONANCE

I 1 1 1 1( )

V

Resonance occurs when admitance is purely resistive

1 1Im(

1rad/s) 0 eco

oo

Y H j C j CR j L R L

Y L LC LCC

1o

LC

Parallel resonant circuit.

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.

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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.

mL m C o m m

o

I RI QI I CI R QV

L

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PARALLEL RESONANCE

The half-power frequencies can be obtained as:

22

V1 1( )

mIV

C LR

2

1

2

2

1 1 1

2 2

1 1 1

2 2

RC RC LC

RC RC LC

1 2 2 1

1,o B

RC

Voltage versus frequency for the parallel resonant circuit.

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Characteristic Series circuit Parallel circuit

ωo

Q

B

ω1, ω2

Q ≥ 10, ω1, ω2

Summary of series and parallel resonance Summary of series and parallel resonance circuits:circuits:

1

LC

1

LC

L 1 or

R RCo

o

ω

ω

R or RC

L oo

Qo

Qo

21 1 ( )

2Q 2Qo

o

21 1 ( )

2Q 2Qo

o

B

2o B

2o

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Dual-channel 15-Dual-channel 15-band “Constant Q” band “Constant Q” GGraphic raphic EEqualizerqualizer

The equalizer changes the contribution of the different frequency components of the music signal according to the listeners’s wish.

Gains given to different frequency components of the music signal.

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