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Resonators

1

Microwave Resonatorsc owave eso ato s

• Series Resonant CircuitjLjRZin

1

ZVZIZIVP

Cjj

2

in2

inin

in

21

21

21

CjLjRI

Z

2

in

1 21

222

RIP

C2

loss 21

2

LIWm2

11141

CICVW Ce 2

22 141

41

2


LW 12• Series Resonant Circuit

em WWjPP 2lossin Z

RCRL

PWQ m

:frequencyresonancenear the

12

i

0

0

loss0

em

IWWjP

IPZ

22

221

loss2in

inLjR

LCLjRZ

Z

11

:frequency resonance near the

2

20

2

2in

in

em

RIPZ

WW

: when occurs Resonance

21

lossin

RQjRLjR 22 0

LC

I

10

21

j1resonatorLossless

:loss with Resonator

00

0in 20 LjZR

Q

QLC

storedenergy average:factor Quality

Q:bandwidth fractionalpower -Half

21resonator Lossless

22

00

em WW

Q

second / lossenergy

Q

RjRQRZ1WBor

2BW 222in

3

lP Q


• Parallel Resonant Circuit1

11

VIZIVP

CjLjR

Z

22

in

1111

11

CjjV

ZVIZIVP

2

ininin

11222

VP

CjLR

V

2

loss1

2

CVW

R

e2

loss

412

LVLIW Lm 2

22 141

414

4


• Parallel Resonant Circuit

em WWjPP 2lossin

RCLR

PWQ m2

00loss

0

em

IWWjP

IPZ

22

221

loss2in

in

CjCjZ

Z

/11

:frequency resonance near the 1

0

in

em

RIPZ

WW

: when occurs Resonance

21

lossin RRCj

CjCjLjR

Z

21 1

00

in

LC

I

10

21

jQCjCj

R:loss with Resonator

/21 212

0

0in 21 CjZR

Q

QLC

storedenergy average:factor Quality

Qj

b d idthf ti lH lf2

1 resonator Lossless 00

em WW

Q

second / lossenergy

QRZ 1WB2

:bandwidth fractionalpower -Half2

2in

5

lPQ2


• Loaded and Unloaded Q High Q

= 0 > 0Low Q

= 0

0 > 0

< 0

Fi 6 3 ( 271) A t i it

Series resonant circuit

L circuitsseriesfor0

Figure 6.3 (p. 271) A resonant circuit connected to an external load, RL.

High Q

LRRQL

Le

circuits parallelfor

circuits seriesfor

High Q

Low Q = 0 < 0

> 0

QQQ

L

eL

1110

= 0 0

Parallel resonant circuit

6

QQQ eL


7

Transmission Line Resonatorsa s ss o e eso ato s

• Short-Circuited /2 Line0

0in tanh

lZZZZ L

0

00in

tt h tanh

tanh

ljlljZlZZ

ZZL

0i

0

0iftan:Observe tanh tan1

tan tanh

ljZZlljljlZ

0

0in

line. TEM small; is ,; tanh1 that Assume

0if tan :Observe

lll

ljZZ

0

0

0

vl

vl

vll

ppp

Figure 6.4 (p. 273) A short-circuited length of lossy transmission line, and the voltage distributions for n = 1 )2( n and )2/( 000

tantan tan

l

8

resonators.


• Short-Circuited /2 Line 0

jlZjlZZ

2

1 00

00in

LjR

jlZlj

ZZ

2 0

0

ZL

lZR 2nl

1

2

20

0

L

C 3, 2, ,1 ,2

for occurs also Resonance

0

nnl

2 2 0

lRLQ

9


• Example 6.1 Q of Half-Wave Coaxial Line Resonatorsmm,1 resonator, line coaxial 2/Copper a

S/m, 10813.5 :Note . and Find GHz, 5frequency Resonance mm. 4

7copperTeflonair

QQb

a

b.0004.0tan and 08.2Teflon for rSolution

b

air71042 f

2/

11

1084.12

20

s

R

R Teflon08.27.104air7.1042

cf r

11

Np/m 022.011ln2 0

air ;

s

c

Rbaab

Rα

23807.1042

Q

Q

Np/m 030.02tan ,0

Np/m 032.011ln2

0Teflon;air;

Teflon ;

rdd

sc

kαα

baabRα

121803003202

08.27.1042

2380022.022

Teflon

air

Q

Q

10

p, 0Teflon;air ; rdd 03.0032.022Teflon


• Short-Circuited /4 Line

lZZ h

lljljl

ljZlZZlZZZZ

L

L

cottanh1tantanh

tanh tanh tanh

00

00in

412 nl

llllljllljZ

lljljlZ

4/ ;tanh1 that Assume cot tanh

cottanh1 tanh tan1 tantanh

00

412 nl

llll

line. TEM small; is ,

0

0

Z

l

vvvl

ppp

tancotcot

22

0

1

4

00

0

RCQL

ZC

lZR

CjRjl

ZjlljZZ

l

211

222 1

22tan

22cotcot

000in

000

2 4 1 02

0

l

RCQC

L

11

CjRjljl 2122 000in


• Open-Circuited /2 Line0

0in tanh

lZZZZ L

0

00in

t ht1 coth

tanh

lljljZlZZ

ZZL

0i

0

0ifcot:Observe tan tanh

tanh tan1

ljZZljllljZ

0

0in

line. TEM small; is ,; tanh1 that Assume

0if cot :Observe

lll

ljZZ

0

0

0

vl

vl

vll

ppp

000

tantan tan

l

Figure 6.5 (p. 276) A open-circuited length of lossy transmission line, and the voltage distributions for n = 1 )2( n and )2/(

12

resonators.


• Open-Circuited /2 Line 1 00

ZljZZ

1

000in

jljlZZ

21

0

lZR

CjR

2nl

2

00

Z

C

l

1 20

C

L

22

3,2,,1,2/for occurs also Resonance

0

lRCQ

nnl

13

2 2 l


• Example 6.2 A Half-Wave Microstrip ResonatorTeflon is substrate theresonator, line microstrip circuited-open 50 2/

field.) fringing (Ignore .resonantor theof Find GHz, 5frequency Resonancecopper. :Conductors cm. 0.159 inckness with th,)0004.0tan ,08.2(

Qr

S/m. 10813.5 :Note 7copper

Solution

2

801cm5080

1084.12

20

W

Rs

tan1

Np/m 0724.01008.550

1084.13

2

0

kWZRs

c

cm. 24.2222

80.1 cm,508.0 eff

fc

fv

l

W

p

0.151

Np/m 024.012

tan1

eff

eff0

k

r

rd

rad/m 0.151 2 2

222

eff

eff

cf

vf

ff

783

024.00724.020.151

2

Q

14

cvp

Rectangular / Circularecta gu a / C cu aWaveguide Cavities (optional)

Figure 6.8 (p. 283)A li d i l t it d th l t i fi ldA cylindrical resonant cavity, and the electric field distribution for resonant modes with .2or 1

Figure 6.6 (p. 278)A rectangular resonant cavity, and the electric field

distributions for the TE101 and TE102 resonant modes.

15

101 102


• For comparisonExample 6.3 Copper rectangular waveguide cavity resonator, filled with polyethylene (r = 2.25, tan = 0.0004), with a = 4.755 cm and b = 2.215 cm. f = 5GHz. TE mode: d = 2 20 cm Q = 8403 Q = 2500 Q = 1927TE101 mode: d = 2.20 cm, Qc = 8403, Qd = 2500, Q = 1927TE102 mode: d = 4.40 cm, Qc = 11898, Qd = 2500, Q = 3065

Example 6.4 Copper circular waveguide cavity resonator, filled with Teflon (r = 2.08, tan = 0.0004), with d = 2a. f = 5GHz.TE mode: a = 2 74 cm Q = 29390 Q = 2500 Q = 2300TE011 mode: a = 2.74 cm, Qc = 29390, Qd = 2500, Q = 2300Air-filled→ TE011 mode: a = 3.96 cm, Qc = 42400, Qd = 0, Q = 42400

• In General:Qspherical > Qcylindrical > Qrectangular > Qcoaxial > Qmicrostrip

16

Microwave Resonators (optional)c owave eso ato s (opt o a )

Figure 6.11 (p. 288)Geometry of a cylindrical dielectric resonator.

Figure 6.12 (p. 288)Magnetic wall boundary g ycondition approximation

and distribution of Hzversus z for = 0 of the

1 ,2 tan:mode TE

,10010

01

d

r

QL

first mode of the

cylindrical dielectric resonator.1000GHz8532:cm82550

cm 413.0 ,001.0tan ,95 6.5Ex tan201

r

d

QfLa

Q

17

1000 GHz,853.2 :cm8255.0 dQfL

Excitation of Resonatorsc tat o o eso ato s

Figure 6.13 (p. 291) Coupling to microwave resonators. (a) A microstrip transmission line resonator gap coupled to a microstrip feedline. (b) A rectangular cavity resonator fed by a coaxial probe. (c) A circular cavity resonator aperture coupled to a rectangular waveguide. (d) A dielectric

l d i i f dli

18

resonator coupled to a microstrip feedline.


:coupling,oftCoefficien g• Critical Coupling

circuitsresonant parallelcircuitsresonant series

: coupling, oft Coefficien

0

0unload

ZRRZ

QQg

g

e

coupled critically :1 2.edundercoupl :1 1.

p0

ggQe

Figure 6.14 (p. 292)A series resonant circuit coupled to a feedline

dovercouple :1 3. g

A series resonant circuit coupled to a feedline.

:resonanceat matched resonanceAt

0

ZRRZin

0

00unload

ZL

RLQ

co plingcritical0

0

QQZLQe

19

coupling criticalunload QQe


• A Gap-Coupled Microstrip Resonator2/ 0tan

whenoccurs Resonancebl

2

1

1

0tan

c

c

bjz

bl

1 c

Figure 6.16 (p. 293) Equivalent chart of the g (p ) qgap-coupled microstrip resonator of Figure 6.13a.

ZlZCj

ZZz cot /1 0

lbblj

ZZ

c

tan tan

00

CZb

Figure 6.17 (p. 294) Solutions to (6.78) for the resonant frequencies of the gap-coupled microstrip resonator

20

lbc tan CZbc 0 microstrip resonator.


11 21:frequencycomplex by lossConsider

Qj :coupling criticalFor 0

b

ZR

21

12

2

2 cc

LjR

bj

Qbz

2

22

0

QbZg

Qb

c

c

0

nransmissiocircuit t-open2/:Note

2ZLjR

edundercoupl :12

gQ

b

Rg

c

circuit-open 2/ coupled-But resonatorcircuit parallel :resonator

nransmissiocircuit t-open 2/ :Note

CRLC

dovercouple :12

2

gQ

b

Q

c

c

resonatorcircuit series :resonatoron transmissi

ppRLC

2Q

20

2

:resonanceAt

QbZR

21

22 cQb


dB/cm 01.0nattenuatio ,9.1cm 175.2 , 50 :6.6Ex

eff

0

lZ

GHz 0.52

:eff

approx;0 lcfsol

62822

l

Q

pF0320

05.02

bC

Qb

c

c

1.6%)( GHz 918.4

pF 032.0

0

0

f

ZC c

Figure 6.18 (p. 296) Smith chart plot of input impedance of the gap-coupled microstrip resonator of Example 6.6 versus frequency for various values

22

of the coupling capacitor.

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