4. Resonance

8/2/2019 4. Resonance

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Resonance In

AC Circuits


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3.1 Introduction

MM

M

h

An example of resonance in the form ofmechanical : oscillation

Potential energy change to kinetic energy thankinetic energy will change back to potentialenergy.

If there is no lost of energy cause by frictionpotential energy is equal to kinetic energy.

mgh = mv It will oscillate for a long time.

Ep=mgh

Ek= mv v


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Resonance in electrical

circuit

C L

i

i

Ep= CV Em= LI

Potential energy stored in capacitor change tomagnetic energy that stored in inductor. Then

magnetic energy change back to potential energystored in capacitor.

If there is no lost of energy by resistor potentialenergy equal to magnetic energy

CV = LI It will oscillate for a long time.


4/30

Characteristic of

resonancecircuit

The frequency response of acircuit is maximum

The voltage Vs andcurrent I are in phase

The impedances is purelyresistive.

Power factor equal to one

Circuit reactance equal zero because capacitive andinductive are equal in magnitude


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At frequency resonance,

(1)

V

I

jXL-jXC

CV = LI

We know

V = I * XL or V = I * XC = I * L = I * 1/C

= I * 2fL = I / 2fC

Substitute into 1

C (2fLI) = LI

f = L

C (2fL)

f = 1

LC2

CV = LI


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Ideal case ( noresistance)

Practical ( energy loss dueto resistance)

i

i

t

t


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Main objective we analysis resonance circuits to find five resonance parameters :

a) Resonance frequency, oAngular frequency when value of current or voltage is maximum

b) Half power frequency, 1 and 2Frequency where current (or voltage) equal Imax/2 (or Vmax/2 ).

c) Quality factor, QRatio of its resonant frequency to its bandwidth

d) Bandwidth, BWDifference between half power frequency


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3.3 Series Resonance

CircuitsR

VR VL

R j XL

- j XCV

By KVL : V = VR + VL + VC

= VR+ jVL jVC

At resonance XL = XC

Hence V = VR+ 0

= VR

= IR* R

Vc

Figure 1


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Series ResonanceCircuits

R

VR VL

j XL

- j XCV Vc

Figure 1

Z = R + j XL - jXC

= R + j (XL XC)

XL = 2fL

XC =1

2fC

where

XL

R

XCf0

f

|Z|

(XL-XC)


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f

f0

|I||Z|

|I| =|V|

|Z|


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Resonance parameter for

series circuita) Resonant frequency,oThe resonance condition is

oL = 1 / oC or o = 1 / LC rad/s

since o = 2fofo LC Hz2 /1 =

b) Half power frequencies

At certain frequencies = 1, 2, the half power frequencies are obtain by setting Z = 2R

R + (L 1/ C) = 2R


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Solving for, we obtain

1 = - R/2L + (R/2L) + 1/LC rad/s

2 = R/2L + (R/2L) + 1/LC rad/s

Or in term of resonant parameter,

1 = o [ - 1/ 2Q + (1/ 2Q) + 1 ] rad/s

2 = o [ 1/ 2Q + (1/ 2Q) + 1 ] rad/s

c) Quality factor, Q

Ratio of its resonant frequency to its bandwidth.


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Q = VLVS

= [ I ] x XL[ I ] x R

= L ; Q = XL

R R

frL 2 =R

Q = VC

V

= [ I ] x XC[ I ] x R

= 1 ; Q = XC

C RR

= 1 frCR2

or


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d) Bandwidth, BW

BW = 2 1

= o [ 1+ (1/ 2Q) + 1/ 2Q ] - o [ 1+ (1/ 2Q) - 1/ 2Q ]

= o [ 1/ 2Q + 1/2Q ]

= o [2/ 2Q]

= o / Q

Q = oL /R = 1/ oCR

thus,

BW = R / L = o / Q

or, BW = oCR


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3.4 Parallel Resonance

Circuits Resonance can be divided into 2:

a) Ideal parallel circuit

b) Practical parallel circuit

At least 3 important information that is needed to analyze to get the resonances

parameter: In resonance frequency, o the imaginary parts of admittance,Y must be equalto zero.

When in lower cut-off frequency, 1 and in higher cut-off frequency, 2 themagnitude of admittance,Y must be equal to 2/R.

i

+

v

-

RC L

Ideal Parallel RLC circuit

o


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ideal RLC parallel circuitR-jXc jXL

Ideal Parallel RLC circuitYT

+

Lc XXj

R

111YT =

)()(11

jBGL

CjR

+=

+=

Whereas G() is the real part called the conductance

and B() is the imaginary parts called the susceptance.


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a) Resonant frequency,o

Angular resonance frequency is when B()=0.

b) Lower cut-off angular frequency, 1

Produced when the imaginary parts = (-1/R)

sradLC

LC

/1

;01

210

==

=

+

+=

=

LCRCRC

RLC

1

2

1

2

1

11

2

1


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c) Higher cut-off angular frequency, 2.

Produced when the imaginary parts = (1/R)

d) Quality Factor, Q

e) Bandwidth, BW

sradLCRCRC

RLC

/1

2

1

2

1

11

2

1

+

+=

+=

RCQ

L

CR

L

RQ

0

0

=

==

QRCBW 012

1 ===


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Duality Concept

R

VR VL

j XL

- j XCV Vc

Figure 1

Series circuit Parallel circuit

i

+

v

-

RC L


Z = Z1 + Z2 + Z3

Z = R + j XL - jXC

Y = Y1 + Y2 + Y3

Y =

+

Lc XXj

R

111

+=L

Cj

R

11

YZ = R + j (L )C

1


20/30

Duality Concept

R

VR VL

j XL

- j XCV Vc

Figure 1


i

+

v

-

RC L


+=

LCj

R

11YZ = R + j (L )

C

1

R R

1

L C

C L


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Duality Concept

R

VR VL

j XL

- j XCV Vc

Figure 1


i

+

v

-

RC L


1 = - R/2L + ((R/2L) + 1/LC) rad/s2 = R/2L + ((R/2L) + 1/LC) rad/s

1 = - 1/2RC + ((1/2RC) + 1/LC) rad/s2 = 1/2RC + ((1/2RC) + 1/LC) rad/s


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practical RLC parallel circuit

Practical Parallel RLC circuit

i

+

V

-

R1

C

L

I1 IC

I1

IC

Z1 = R1 + jXL = |Z1|/

|I1|cos

|I1

|sin

I


23/30



Practical Parallel RLC circuit

i

+

V

-

R1

C

L

I1 IC

I1

IC

Resonance occur when |I1|sin = IC

|I1|cos

|I1

|sin

I


24/30

Resonance occur when |I1|sin = IC

|I1|sin = IC

|V||Z1|

x XL

|Z1||V|=XC

XL

|Z1|2 = XC

1

2frL

R2

+ (2frL)2

= 2frC

R2 + (2frL)2L

=

C

(2frL)2 =LC

- R2

=

21

2

1

L

R

LCr

f


25/30

Q factor

XL

R

2frL

=

=

Q = current magnification IC

=

|I1|sin

I = |I1|cos

tan

=

R

I1

IC

|I1|cos

|I1

|sin


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i

+

V

-

R1

C

L

Second approach to analyze this circuit is by changing the seriesRL to parallel RL circuit.

The purpose of this transformation is to make it much more easierto get the resonance parameter.


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RL in series

Rl

L

Rl jXp

RL in parallel

pp

T

l

ll

l

llll

T

llT

jXRY

X

XRj

R

jXRjXRY

jXRZ

11

1112222

+=

++

+=

+=

+=


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By matching equation ZT and YT above, we can

get:

Or

By defining the quality factor,

l

lp

l

llp

R

LRR

RXRR

22

22

)(+=

+= and

and

l

lp

l

llp

X

LRX

XXRX

22

22

)(+=

+

=

ll

l

R

L

R

XQ

=

Rp

and Xp

can be write as:

2

2

2

2

2

1

)1(

l

lp

ll

l

lllp

Q

QLL

QRR

XRRR

+=

+=+=


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Resonance parameter

a) Angular resonance frequency, o

b) Lower cut-off angular frequency, 1


L

CR

LCX

R

LC

l

l

l

2

02

2

111

==

sradCLCRCR

RLC

ppp

/1

2

1

2

1

11

2

1

+

+=

=


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c) Higher cut-off angular frequency, 2.

d) Quality Factor, Q

e) Bandwidth, BW


sradCLCRCR

RLC

ppp

/1

2

1

2

1

11

2

2

+

+=

+=

RCQ

L

CR

L

RQ

p

p

p

=

==

QCRBW

p

012

1 ===

Date post:	05-Apr-2018
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4. Resonance

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