5_MicrowaveResonators

8/3/2019 5_MicrowaveResonators

1/25


2/25

Series Resonant Circuit

2


3/25

3

CjLjRZin

1

ImpedanceInput

CjLjRI

Z

VZIZVIP

inininin

1

21

2

1

2

1*

2

1

2

22

resonatorthetoDeliveredPowerComplex


4/25

4

C

ICV

LI

RIP

c

loss

2

22

2

2

1

4

1

4

1

4

1

21

e

m

W

capacitortheinstoredenergyelectricAverage

W

inductortheinstoredenergymagneticAverage

resistorthebydissipatedPower


5/25

5

22

2

1

22

2

I

WWjP

I

PZ

WWjPP

emlossinin

emlossin

At resonance Wm = We (real),

2

1 2R

I

PZ lossin

Wm = We implies the resonant frequencyLC

10


6/25

6

Quality Factor

loss

em

P

WW

Q

second/lossenergy

storedenergyaverage

RCR

L

P

WQ

loss

m

0

00

12

circuitresonantseriesfor

losslowQHigh


7/257

QBW

1

resonanceatdeliveredthat2

1

iscircuitthetodeliveredpoweraveragereal

2if

bandwidthpowerhalfFractional

22RZin


8/25

Parallel Resonant Circuit

8


9/259

1

11

ImpedanceInput

Cj

LjRZin

CjL

j

RV

VZ

IZVIPin

inin

1

2

1

*

1

2

1

2

1*

2

1resonatorthetoDeliveredPowerComplex

2

22


10/2510

CV

LVLI

R

VP

L

loss

2

e

2

22

m

2

4

1W

capacitorin thestoredenergyelectricAverage

1

4

1

4

1W

inductorin thestoredenergymagneticAverage

2

1

resistorby thedissipatedPower


11/2511

22

2

1

22

2

I

WWjP

I

PZ

WWjPP

emlossinin

emlossin

At resonance Wm = We (real),

2

1 2R

I

PZ lossin

Wm = We implies the resonant frequencyLC

10


12/25

12

Quality Factor

RC

L

R

P

W

Q

loss

m0

0

2

second/lossenergystoredenergyaverage

resonanceatdeliveredthat2

1

iscircuitthetodeliveredpoweraveragereal2

if

bandwidthpowerhalfFractional

22 R

Zin

QBW

1


13/25

Loaded & Unloaded Q In the absence of any loading effects- no external

circuitry- unloaded Q

Usually- external circuit is coupled to the resonantcircuit Q reduced loaded Q (QL)

13

LLL

LL

R/RRwithRLCparallelRRwithRLCseries

RRR


14/25

14

circuitsparallelforQ,External

circuitsseriesforQ,External

0

0

L

RQe

RLQe

L

L

QQQ eL

11

1,Qloaded


15/25

15


16/25

Transmission Line Resonators TL sections various lengths & terminations -

resonators

16

Short-Circuited /2 Line

Lossy TL short circuited at one end

Z0,,,l= /2

ljZlZZin

tanhtanh 00


17/25

17

llj

ljlZZni

tanhtan1

tantanh0

pppvvv

lll

llllTEMAssuming

small)very(,Let

tanh,1lossyTL

0

0


18/25

18

)(,2

00

pvl

0

l

000

tantantan

l

ckt)ResonantRLCseriesofimpedanceinput(2;1

1

0

00

0

00

jLRZl

jlZ

lj

jlZZ

ni

ni

LC

ZLlZR

200

00

1,

2,


19/25

19

The short circuited /2 line resonates for = 0 & its

input impedance at this frequency is Zin = R= Z0lResonance also occurs for l = n/2, n= 1,2,3..

22

0 lR

LQ

l = first resonanceQ decreases as attenuationof the line increases.


20/25

20

Lossy TL short circuited at one end

Z0,,,l= /4

ljZlZZin tanhtanh 00

Short-Circuited /4 Line

ljl

lljZZ

ljdrnrby

llj

ljlZZ

ni

ni

cottanh

cottanh1

cot.&.

tanhtan1

tantanh

0

0


21/25

21

0

0)(,4

Let

l

022

l

000 22tan22cotcot

l

ckt)ResonantRLCparallelofimpedanceinput(21

1;1

22

21

0

0

0

0

0

0

jCR

Zl

jl

Z

jl

lj

ZZ

ni

ni


22/25

22

CL

ZC

l

ZR

20

00

0

1

4

240

lRCQ


23/25

23

Practical resonator

Z0,,,l= /2

ljZlZZin cothcoth 00

Open-Circuited /2 Line

ljl

llj

ZZni

tantanh

tanhtan1

0


24/25

24

0

0)(,2

Let

l

0

l

0

tantan

l

cktResonantRLCparallelofimpedanceinput

0

0

jl

ZZ

ni


25/25

25

CL

ZC

l

ZR

20

00

0

1

2

220

lRCQ

5_MicrowaveResonators

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