chapter 6 microwave resonators -...

微波電路講義6-1

Chapter 6 Microwave Resonators

6.1 Series and parallel resonant circuits

series and parallel RLC resonators, quality factor Q

6.2 Transmission line resonators

/2 and /4 resonators

6.5 Dielectric resonator

concept

6.6 Excitation of resonators

coefficient of coupling, gap-coupled microstrip resonator,

determine Qu from 2-port measurement


6.1 Series and parallel resonant circuits

• series RLC resonator

2

2 2 2

2 2

2 2

2 2 2 2

2

1 1( ) (1 )

(1 )

2 ( ) 1,

/2 /2

1 1 1 1 1, ,

2 4 4 4

in

o o

in loss m eo

loss m e c

VZ R j L R j L

I j C LC

R j L R j L

P P j W W

LCI I

P I R W I L W V C IC

I R L

C+

V

-

inZ

( )inZ

01

3dB BW

R2

R

ω>ωo

ω<ωo

Q大

Q小

ω=ωo

( )inZ


2

2

average energy storedquality factor ( )

energy loss/second

2 / 42 1( ) , (loss )

/ 2

m e

loss

m oo o o

loss o

W WQ

P

I LW LQ R Q

P R RCI R


R L C

+

V

-

inZ

)(Z in

01

3dB BW

2R

R

• parallel RLC resonator

I

21 1

2

2 2

2

2 2 2

2

2

2

1 1 1 1 1( ) ( ) [ (1 )] ,

2 2 ( )

2

1 1 1 1 1, ,

2 4 4 4

2 / 42( ) , (loss )

/ 2

in o

o

in loss m e

loss m L e

eo o o o

loss o

Z j CR j L R j L LC

P P j W W

I I

VP W I L V W C V

R L

C VW RQ RC R Q

P LV R

ω<ωo

ω>ωo

Q大

Q小

=ωo

( )inZ


Discussion

1. At resonance, Wm=We, Pin=Ploss

2 2*2* *

, 2 2* *

2

3 , 3 ,2

3 , 3

Z1 1 1 1 1 1Re[ ]= Re[ ] Re[ ] Re[ ]=

2 2 2 Z 2 2 2

1 1 2series resonator:@ , , ( ) ( )

2 22

parallel resonator:@ , , ( )

inin av in in

in in in inin in

dB in av dB in av o

o

dB in av dB

V IVV IP VI V R I G

Z Z YZ Y

V RQR X P R P R X

R

G B P

2

,2

1 1 2( )

2 22

1( )

2 W

in av o

o

oo

I GQG P G B

G

QB

3. Half-power fractional bandwidth

2. Near resonant frequency2 1

series resonator: 2 , ( ) oin o

o o

LRQZ R j L R j Q

R RC

parallel resonator: , ( )1 2

1 2in o o

o

o

R R RZ Q RC

j RC Lj Q


(derivation of series resonator case)

2

2 2 2 2 2

1 1( ) ( ) ( )

( ) 2

2

o

o

oin

o o o o

Z R j L R jL R jLj C LC

R jL R jL R jL

R j L

(derivation of parallel resonator case)

1 1

2

1 1

2 2

1

1 1 1 1( ) [ ] [ ( ) ]

( )

1 1 1 1(1 )

(1 / )

1 1 1( ) [ ] [ ]

1[ 2 ]

1 2

o

in o

o

o o o o o o o

in o

o o o

Z j C j CR j L R j L

jZ j C j C j C

R j L j L R L

Rj C

R j RC


4. Locus of r= ± x on the Smith chart

2 2

2 2 2 2

2 2 2 2

2 2 2

1 1 21

1 1 (1 ) (1 )

1 2 2 1 2

( 1) ( 2)

r i r i i

r i r i r i

r i i r i i

r i

jz r jx j

j

r x

(derivation)

Locus of g = ± b on the Smith chart

(derivation)2 2

2 2 2 2

2 2 2 2

2 2 2

1 1 21

1 1 (1 ) (1 )

1 2 2 1 2

( 1) ( 2)

r i r i i

r i r i r i

r i i r i i

r i

jy g jb j

j

g b


5. Lossless resonator lossy resonator: (1 )2

o o

j

Q

6. Unloaded Q, Qu, loaded Q, QL, external Q, Qe

1 1 1,

for series RLC circuit

for parallel RLC circuit

L e U

o

L

e

L

o

Q Q Q

L

RQ

R

L

resonator

Qu (Qo)RL

(derivation)

(1 )2

1 1

(1 )2

1

lossless series resonator 2 2( )

2[ (1 )] 2 22

lossless parallel resonator [ 2 ] [ 2( ) ]

2[ (1 )] [ 22

oo o

o o

in o

j LQ

Q Ro

in o

in o

j

Q

in o

Z j L j L

LjZ j L j L R j L

Q Q

Z j C j C

jZ j C j C

Q

1 11] [ 2 ]

oQ RCoC j CQ R


6.2 Transmission line resonators

• Short-circuited /2 line

Zo, ,

n/2

( )in o

o

Z Z l j

R L C

2

2

1, ,

2o

in

oo

o

Z R jL

ZR Z l L C

L

0

<<1

(derivation)

tanh tanh tan tanh( )

tanh 1 tan tanh

, tan tan( ) , tanh2 2 2 / 2

(

1

L

L oin o o o

o L Z

lpo o o

p p o o o o o

oin o o

o

Z Z l l j lZ Z Z j l Z

Z Z l j l l

vl l l l

v v f

l j

Z Z Z

j l

2) 2 , ( )

2 2 2

oo

o o o oU o

o o

Z

L ll j R j L Q

R Z l l l


• Open-circuited /2 line

Zo, ,

n/2

o

oin

jl

ZZ

R L C

2

1 2

1, ,

2o

in

o

o o

RZ

j RC

ZR C L

l Z C

(derivation)

1tanh 1 tan tanh

tanh tanh( ) tanh tan

/, ( )

1 2 2 2 21

L

L o o oin o o o

o L Z

o

o o o oU o o o

o o

o o

j lZ Z l Z j l l

Z Z Z ZZ Z l j l l j l

l j

Z Z l ZRQ RC

j RC l Z ll j j

l


• Short-circuited /4 line

Zo, ,

(2n+1)/4

2

oin

o

ZZ

l j

R L C

2

1 2

1, ,

4o

in

o

o o

RZ

j RC

ZR C L

l Z C

0

(derivation)

tanh tanh tan cot cot tanh 1 tanh( )

tanh 1 tan tanh cot cot tanh

,cot tan( )4 4 4 / 2 2 2 2

L

L oin o o o o

o L Z

po o o

p p o o o o

Z Z l l j l j l j l lZ Z Z j l Z Z

Z Z l j l l j l j l l

vl l

v v f

2

12

, ( )1 2 4 4 2

2 2

o

o o o oin o U o o o

o o

o o

j lZ ZR

Z Z Q RCj RC l Z l

j l l j


( )2

oU oQ w

Discussion

1. Transmission line resonator

2

7

104.7 air2, ,

2 104.7 2.08 Teflon

0.022 / air1 1 1.84 10

2 ln ( ) , 2

5.813 10 / copper0.022 2.08 0.032 / Teflon

Te

o roU o c d

s

os

c

fQ

c

RNp m

b R

a a bS m

Np m

flon tan 0.03 /2

104 7 2380 air

2 0.022(5 )

104.7 2.081218 Teflon

2 (0.022 0.03)

o r

d

U

kNp m

.

Q GHz

2. Ex. 6.1 /2 coaxial line resonator, b=4mm, a=1mm, f=5GHz,

Teflon r=2.08, tan=0.0004, calculate QUair and QUTeflon


3. Ex. 6.2 /2 open-circuited microstrip resonator, Zo=50,

h=1.59mm, Teflon substrate r=2.08, tan=0.0004, f=5GHz,

calculate resonator length and QU.

50 5.08 , 1.8

2.242 2

2151 / ,

0.0724 / ,

( 1)tan 0.024 /

2 ( 1)

(5 ) 7832 2( )

o eff

o

eff

o eff

o

sc

o

o r eff

d

eff r

o oU

c d

Z W mm

cl cm

f

frad m

c

RNp m

Z W

kNp m

Q GHz


4. For comparison,

Ex. 6.3 rectangular waveguide cavity resonator, r=2.25(polyethylene),

tan=0.0004, f=5GHz.

TE101 mode d=2.2 cm, Qc=8403, Qd=2500, QU=1927@5GHz

TE102 mode d=4.4 cm, Qc=11898, Qd=2500, QU=2065@10GHz

Ex. 6.4 Teflon-filled cylindrical cavity, r=2.08, tan=0.0004,f=5GHz.

a=2.74cm, d=2a=5.48cm, TE011 mode Qc=29390, Qd=2500,QU=2300

@5GHz

air-filled cylindrical cavity, TE011 mode a=3.96cm, d=2a=7.91cm,

Qc=42400@10GHz

5. In general for QU

Qspherical > Qcylindrical > Qrectangular > Qcoaxial > Qmicrostrip

6. Application of resonator: frequency selective component (o, QU)

eg., frequency meter, oscillator, filter, matching circuits


6.5 Dielectric resonators

1000 ,4.3

413.0 ,001.0tan ,95 Ex.6.5

tan

1 1

2mod ,10010

01

d

r

d

g

r

QGHzf

cma

Q ,λ

Le δTE

Hz(=0)L


6.6 Excitation of resonators

• Critical coupling Zin(o) =Zo

Zo

R L C

inZ

A series RLC resonator is given to match with the feedline, i.e., at resonance.

2

coefficient of coupling ( 1 for critical coupling)

o

UU e L

U o

e

R Z

Q Q Q , Q

Q Zg

Q R

UQ

1 1( ( ) , ( ) )o o

U o e o

o o o o

L LQ Q

R RC Z Z C


• Gap-coupled /2 open-circuited microstrip resonator

Zo

inZ

Zo,

C /2/2

1

1 1 1

1

tan1 1( ) 0,

tan tan

solve the resonant frequency

in o cin c o

o o c

Z Z l bz j b Z C

Z Z j C j l b l

ltan

pv

ll

( )c ob Z C

p

1

v

lπ2

v

l

p

2

resonance or open?

tan 0

oin

o

z

l


Discussion

1.

series resonator

parallel resonator

o

U

e

o

Z

Q Rg

RQ

Z

g>1 g=1 g<1 for series resonator

Zo

R L C

1

1 under coupling

1 critical coupling / 2

1 over coupling

UL

L U

L U

L U

QQ

g

Q Q

g Q Q

Q Q

inZ

UQ1

rg


2. Types of excitation for microwave resonators (p.291, Fig.6.13)

E coupling C, H coupling L

3. Gap-coupled /2 open-circuited microstrip resonator

/2 open-circuited microstrip resonator (parallel resonator)

gap-coupling series resonator

1

2

1

1

2 2

1 0 0

( ) lossless

( )( ) 2

lossy2

cinin

o

U c c

j

bZz

j R j L ωZ

Q b b Z Z

1 1

111 1

1 1 1

2

2 2 2

1

112

1 1 1

1 1

(derivation) ( ) ( ) ( ) ( )

tan 1,

tan

( ) ( ) : lossless2 2 2

lossy: (1

in inin in

c in in cin

c c p c p c

p poin

c

dz dzz z

d d

l b dz dz bd l l j l jz j j

b l d d l d b v b v b

v v jl z

f b

112 2 2

1 1

( )), ( ) [ (1 )]

2 2 2in

c c U c

jj j jz

Q b Q b Q b


5. Ex. 6.6 /2 open-circuited microstrip resonator, Zo=50, l=2.175cm,

eff =1.9, =0.01dB/cm, calculate C for critical coupling and f1.

1

1

1

uncoupled line: 52 2

2 628, 0.05 0 032

2 2 2 2

tan 0 4.918 5

314 6282

g

o

eff

go c

U c

g U o

c o

UL U

cl n f GHz

f

bQ b C . pF

l Q Z

l b f GHz f GHz

QQ Q

:small C, large gap

: large C, small gap

2

1 2

1 2

2( ) , 1

2 2

1 2

c

U

o o U cin c

U c U

c

U

bQ

Z Z Q bZ R g b

Q b R Q

bQ

4. Series resonator @w1

1


6.Determin QU from 2-port measurement

fo

3dB3dBf

21S

2121

21

3

2

2 2@ : , .....prob 4.11

2

2 2

2= , = =

2

2 ( )( )

2 1 1 ( )

1 1 1 1(1 ),measure (1 )

o

o o

o

o

o o

o o o U oU e

L o e

o oo

o o

oL U L

L e U U sB

ZR

R Z R ZR Rf Z S

R R Z R

R Z R Z

L L L Q ZQ Q g

R R Z Q R

Z S fgS f g

R Z g S f

fg Q Q Q g

Q Q Q Q f

Zo Z o

Zo Zo

Z

R L C


Solved problems: Prob. 6.22 A parallel resonator, calculate Co for critical

coupling and fr.

Zo R L CCo

R=1000Ω,L=1.26nH,

C=0.804pF, Zo=50Ω

Yin

3 3 9

2 2 2

2

1 2 1( ) 10 50.5 10 , 25.3, = =31.4 10

LC

1 1 150 0.05 2.53 0.05 4.36

1 1 1 1

1 11 1 1

4.362.53 0.086

1 20

uin U o

o o o

in in

o

in in

o o o o

o o

Q RY j j Q

R R L

ay Y j j a

ja a a a

z z j jaj C Z C Z

a

a

0.086 4.57

14.36 0.16

r o o

o

r o o

f f f GHz

C pFC Z

zin

yin

ADS examples: Ch6_prj

chapter 6 microwave resonators -...

Documents