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Series and Parallel ResonatorsSeries and Parallel Resonators

Resonators

2

What is a Resonator ?

A resonator is a device or system that exhibits resonance or resonant behavior, i.e., it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others.

3

Series RLC Circuits

Consider the series RLC resonator shown below:

4

Series RLC Circuits

The input impedance Zin is given by

-------(1)

The average complex power delivered to the resonator is

Z R j L jCin

1

P VI Z I I R j L jCin in

1

2

1

2

1

2

12 2*

5

Series RLC Circuits

The average power dissipated by the resistor is

The time-averaged electric energy stored in the capacitor is

Similarly, the time-averaged magnetic energy stored in the inductor is

P I Rloss 1

22

W I Lm 1

42

6

Series RLC Circuits

Input power can be written as

The input impedance can then be expressed as follows:

-----(2)

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Series RLC Circuits

At resonance, the average stored magnetic and electric energies are equal i.e., Wm = We.

So,

and the resonance frequency is defined as

ZP

IRin

loss 2 2/

oLC

1

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Series RLC Circuits

The Quality factor is defined as the product of the angular frequency and the ratio of the average energy stored to energy loss per second

Q is a measure of loss of a resonant circuit. Lower loss implies higher Q and high Q

implies narrower bandwidth.

QW W

Pm e

loss

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Series RLC Circuits

At resonance We = Wm and we have

      ----(3)

When R decreases, Q increases as R dictates the power loss.

QW

P

L

R RCom

loss

o

o

2 1

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Series RLC Circuits

The input impedance can be rewritten in the following form:

Z R j L jC

R j LLC

R j Lino

11

12

2 2

2( )

11

Series RLC Circuits

and

so Zin can be written as

---(4)

2 2 2 o o o( )( )

Z R j L R j L R jRQ

ino

2

22

2

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Series RLC Circuits

Consider the equation

As

---(5)

Z R j LL

Qj Lin

oo 2 2

( )

QL

Ro

Z j Lj Q

j L jQin o

oo 2

22 1

1

2( ) [ ( )]

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Series RLC Circuits

From the EQ.4, when R = 0 for the lossless case, therefore, we can define a complex effective frequency

----(6) so that,

--- (7) to incorporate the loss

Z j Lin 2

o o jQ

' ( ) 11

2

Z j Lin o 2 ( )'

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Series RLC Circuits

From EQ.4 we have Z R jRQ

ino

2

0

2

resonator theof bandwidth fractionalpower -half The

BW

RjRRZ

QBW

BWjRQRRQ

jRZ

in

in

2

1 When

2

0

Series RLC Circuits

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Parallel RLC Circuits

Now let us turn our attention to the parallel RLC resonator:

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Parallel RLC Circuits

The input impedance is equal to

-----(9)

At resonance, and ,

same results as we obtained in series RLC

ZR j L

j Cin

1 11

Z Rin o LC 1

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Parallel RLC Circuits

The quality factor, however, is different

QW

P P

I L

I R

I L

QL

R

V L

V R

R

LRC

om

losso

loss

Lo

R

L

o o

oo

2 2

4

2

2 4

10

2

2

2

2 2

2 2

| |

/

| |

/ ( )

/( )

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Parallel RLC Circuits

Contrary to series RLC, the Q of the parallel RLC increases as R increases.

Similar to series RLC, we can derive an approximate expression of Zin for parallel RLC near resonance .

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Parallel RLC Circuits

Given o

ZR j L

j CR j L

j C j Cino

o

1 1 1 1

1 1

/

ZR

jL j L

j C j Cino o

o

1 11

ZR

jL

LC

j Lj Cin

o

o

o

1 1 2 1

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Parallel RLC Circuits

----(11)

ZR

jL

j Cino

o

12

1

,

ZR

jL LC

j Cin

11

/ ( )

ZR

j Cin

1

21

ZR

j RC

R

j Qino

1 2 1 2 /

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Parallel RLC Circuits

Similar to the series RLC case, the effect of the loss can be incorporated into the lossless result by defining a complex frequency equal to

-----(12) o o jQ

' ( ) 11

2

Parallel RLC Circuits

242

1

1 When

RBWQ

j

RZ

QBW

in

0

1

0

2121)2

1(

0 where,Let

Qj

R

CRj

RCj

RZin

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Loaded and Unloaded Q

Q defined above is a characteristic of the resonant circuit, this will change when the circuit is connected to a load

Resonantcircuit Q

RL

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Loaded and Unloaded Q

if the load is connected with the series RLC, the resistance in the series RLC is given by R’=R+RL, the corresponding

quality factor QL becomes

QL

R

L

R R R

L

R

L

Lo o

L

o

L

o

'

1

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Loaded and Unloaded Q

--- (13) 

On the other hand, if the load is connected with the parallel RLC, we have 1/R’=1/R+1/RL

1 1 1

Q Q QQ

L

RQ

L

RL e

oe

o

L , ,

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Loaded and Unloaded Q

1

1 1 1

1 1 1 1

Q

L

R R R L R L Q QL

o

L o L o e

/ ( / / ) / ( ) / ( )

1

1 1 1

1 1 1 1

Q

L

R R R L R L Q QL

o

L o L o e

/ ( / / ) / ( ) / ( )

1

1 1 1

1 1 1 1

Q

L

R R R L R L Q QL

o

L o L o e

/ ( / / ) / ( ) / ( )

Series and Parallel Resonators

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