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 微波電路講義 6.1 Series and parallel resonant circuits • series RLC resonator

I R L Zin () 3dB BW + Zin () V C 2R - R 1  Zin 0 V 11 Zin ( )   R  j  L   R  j  L (1  2 ) I j C LC Q大 2 2 2     ω>ωo R  j  L(1 oo )  R  j  L Q小 22

Pin P loss2 j  ( W m  W e ) 1 ω=ωo  , o  ω<ωo II22/2 /2 LC

12 1 2 1 2 1 2 1 PIRWILWVCI,,    loss2 m 4 e 4 c 4 2C 6-2 微波電路講義 average energy stored WW quality factor Q ( )    me energy loss/second Ploss 2 2WL2IL / 4  1 QRQ( ) mo     ,  (loss  )   o o o 2 PlossIR/2 R o RC

6-3 微波電路講義 • parallel RLC resonator Zin ( ) I R + R 2 V R L C 3dB BW -

1  0 Zin 2 Z () 1 11 11 11 in Zjino( C ) ()  [  (1  )]  ,  2 R j LR j  L o LC 22P ( P ) j W W ω<ωo in loss m e Q大 II222

2 11V 12 1 22 1 Q小 PWlossm I Le ,, L  V W C V ω>ωo 24RL 44  2 2 =ωo 2We 2CV / 4 R QRC(o ),  o (loss RQ   ) oo 2      PLlosso VR/2  6-4 微波電路講義 Discussion 1. At resonance, Wm=We, Pin=Ploss 2. Near resonant frequency

21RQ o L series resonator:2,(ZRino ) j  LR  jQ   ooRRC RRR parallel resonator: ZQRC,( )   inoo 12jRCL  12 jQ o  3. Half-power fractional bandwidth o

* 22 1 1VV 12 Z 1VI 1 I 1 PRe[ VI** ]= Re[ ]  V Re[in ]  R  Re[ I ]= G in, av2 2ZZY** 2 Z 2 in22 2 2 in in in inZYin in in 2 1V 1 2RQ series resonator:@3dB ,RXPRPRX  , in , av (  3 dB ) 2  in , av (  o )    222R o

2 1I 1 2GQ parallel resonator:@3dB ,GBP  , in , av (  3 dB ) GPGB2 in, av()  o    222G o  1 Q(  ) o  微波電路講義 o 2 B W 6-5 (derivation of series resonator case) 112 ZRj( LRjLRjL)()()    o in j CLC 22222    o   ()2   RjLRjLRjL oooo 

o Rj L2 (derivation of parallel resonator case)

o  1 111 1 1 Zin()[][()]    j  C    j  o   C R j L R j() o   L 1 1 1 1   (1  )   2 o    o(1   /  o )  o  o  o  o

1 111 1 j  Zin()[][]   22  j  o C  j  C    j  C R jo L j  o L R  o L 1 R [ jC 2  ]1  R12 j  RC 6-6 微波電路講義 4. Locus of r= ± x on the Smith chart (derivation) 22 1 112 ririi j     zrjxj  2222 11(1)(1)   ririri  j    2222 rx 1221 riirii 2   222  ri (1)(   2) Locus of g = ± b on the Smith chart (derivation) 22 1 1 r j  i 1   r   i 2  i y g  jb   2 2  j 2 2 1 1 rj i (1  r )  i (1  r )  i 2 2 2 2 gb1r i 2  i r i 2  i 1 2 2 2 2  ri (  1)  ( 2)

6-7 微波電路講義 j 5. Lossless resonator  lossy resonator: oo (1) (derivation) 2Q

lossless series resonator ZjLjL22()ino   j o L oo (1)  Q 2Q j  L R ZjLjLR 2[(1)]22 jL  o ino 2QQ 11 lossless parallel resonator ZjCjC[ino 2][ 2() ]  j oo (1)  2Q j  C Qo RC 1 ZjCjC 2[(1)][ 2 1 o ][11  2] jC ino 2Q QR 6. Unloaded Q, Qu, loaded Q, QL, external Q, Qe 1 1 1 , QQQL e U resonator RL  o L Qu (Qo)  for series RLC circuit  RL Qe   R  L for parallel RLC circuit  o L 6-8 微波電路講義 6.2 Transmission line resonators • Short-circuited /2 line R L C n/2

Zo, ,  ZRjL2  in ZZlj() Z  1 ino  RZlLC ,,o o o 22 L o o (derivation)

ZLo Ztanh l tanhl  j tan  l Zin Z o  Z o tanh(   j  ) l  Z o ZoL Ztanh  l 1  j tan  l tanh  l ZL 0    v       l <<1 lo op  o , tan  l tan( )  , tanh  l l vp2 v p 2 f o 2 o / 2   o  o  o  Z  lj  o  Llo 2     ZZZo ( l j)  R j 2 L , Q ( ) o o  o o in o o Uo R Z  l222  l  l  1jl oo o 6-9 微波電路講義 • Open-circuited /2 line n/2 R L C

Zo, ,  R Z Z  Z  o in 12j  RC in  l  j Zo  1 o RCL,,   l2  Z 2 C oo o (derivation)  1jl Z Ztanh  l Z 1j tan  l tanh  l  ZZZZL o  o   o in o o o  ZoL Ztanh  l tanh(   j  ) l tanh  l  j tan  l ZL  lj o Z Z/  lR Z  o  o ,()Q    RC   o   o   1j 2  RCU o o o  l 2  Z 2  l 2  l  j1  j oo oo l  6-10 微波電路講義 • Short-circuited /4 line (2n+1)/4 R L C

Zo, ,  R Z Zin  Z  o 12j  RC in  lj Zo  1 2 RCL,,   2 o l4  Z  C oo o (derivation)

ZLo Ztanh l tanhl  j tan  l  j cot  l  j cot  l tanh  l  1 Ztanh(in ZZ oooo )   j  l  ZZ  ZZoLtanh1  ljl tan tanh ljl cot  jl cot tanh l ZL 0     v      ll o  op  o  ,cot    tan( )  vp4 v p 4 f o 4 o / 2  2 22  oo 2o  jl1 2ZZR  Z ZQo  RC   ooo   ,()  in oU o o o  1j 24  RCl  4 Z 2  l  j  l  l  j oo 22oo 6-11 微波電路講義 Discussion  1. Transmission line resonator Qw() o Uo 2 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 104.7 air o 2for QU,  o   ,    c   d 2 c 104.7 2.08 Teflon  R s  0.022Np / m air   b 11 R o 1.84  102   2 ln ( ) , s c  a a b 2   5.813  107 Sm / copper 0.022 2.08 0.032Np / m Teflon k  Teflon  or tan   0.03Np / m d 2  104. 7   2380 air  2 0.022 QU (5 GHz )  104.7 2.08  1218 Teflon 2 (0.022 0.03)  6-12 微波電路講義 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.

Zo50  W  5.08 mm ,  eff  1.8  c lo   2.24 cm 2 2 f eff 2f  o eff 151rad / m , o c

Rs c 0.0724Np / m , ZWo

ko r( eff  1) d tan 0.024Np / m 2eff ( r  1)

oo QU (5 GHz )    783 2 2( cd  )

6-13 微波電路講義 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 6-14 微波電路講義 6.5 Dielectric resonators

L Hz(=0)

2L 1 10   r  100, TE01 mod e δ   1 , Qd  λg tan

Ex.6.5  r  95, tan  0.001, a  0.413cm

 f  3.4GHz, Qd  1000

6-15 微波電路講義 6.6 Excitation of resonators • Critical coupling Zin(o) =Zo

R L C

QU Zo

Zin

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

QU ooLL11  QU  Q e , Q L  (QQU ( oe ), o ( )) 2 RRCZZooo C o Q Z coefficient of coupling g Uo (   1 for critical coupling) QRe

6-16 微波電路講義 • Gap-coupled /2 open-circuited microstrip resonator C /2 /2

Zo Zo, 

Zin

ZZlinoc b 11 tan zjbZinco C  ()0, 1 ZZjooc Cjlbl  tantan 111

 solve the resonant frequency 1

zin  resonance or open? tan l o tan0l 1l  l o  2 2π v p v l p l  v p

bco()   Z  C

6-17 微波電路講義 Discussion 1. Q Q  U L 1 g Zo  series resonator Q  R  1 under coupling QQLU g U   R gQQ1 critical coupling / 2LU Qe  parallel resonator   Zo  1 over coupling QQLU

R L C

1 r  Q Zo g U

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

6-18 微波電路講義 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  j()   1 lossless   b2 Zin  1 c zin ()   Z j()   Rj L 2 ω o   lossy 1  22 2QU bbZZ cc 100 /2 open-circuited microstrip resonator (parallel resonator)  gap-coupling  series resonator dz dz (derivation) zz ( ) (  ) ( )in  ( ) in in in 1 1dd 1 11 tanl  b dz dzd l 1  b2 l j l j z  jc, in  in  j c   in btan ld  dld   bv2 bv 2  b 2 c1 1 c p c p1 c 11 vv o 1 pp j lz    in (  ) 2 (  1 ) : lossless 2 2 2 fb1 1 1 c

j j j j()  1 lossy:11   (1),zin (  ) 2 [  1 (1  )]  2  2 2Q11 bc 2 Q b c 2 Q U b c 6-19 微波電路講義 4. Series resonator @w1   1 b   c :small C, large gap  2QU 2  Zo Z o2 Q U b c   Zin(1 )  R 2 , g    1  b c  22QU b c R  Q U    1 bc  : large C, small gap  2QU

 5. Ex. 6.6 /2 open-circuited microstrip1 resonator, Zo=50, l=2.175cm, eff =1.9, =0.01dB/cm, calculate C for critical coupling and f1.

g c uncoupled line: l n   fo  5 GHz 2 2 f eff

g1 oc2   b QUc    628, b   0.05  C   0 . 032 pF 2 g 2  2l  2 Q U 1 Z o

tanl  bco  0  f1  4.918 GHz  f  5 GHz Q QQU 314   628 LU2 6-20 微波電路講義 6.Determin QU from 2-port measurement S 21 R L C 3dB f 3dB Zo Z o

Z Zo Z o

fo R 2Zo RR RZRZ22  oo @fo : Z  , S .....prob 4.11 RRZ2 o R  RZRZ22oo

oLLLQZ  o  o U2 o QUe= , Q = =  g   RRZQRL2 o e

2Zoog S21 ( f ) S21() fo    g  R2 Zoo 1  g 1  S21 ( f )

1 1 1 1 fo   (1 g ),measure QLUL   Q  Q (1  g ) QL Q e Q U Q U f3 sB 6-21 微波電路講義 Solved problems: Prob. 6.22 A parallel resonator, calculate Co for critical coupling and fr. R=1000Ω,L=1.26nH, o Ω Zo Co R L C C=0.804pF, Z =50

Yin

1 21QRu 3 39 YinUo( )10   jjQ 50.5 10  ,   25.3, =  =31.4  10  R RLoo  o LC  1 11 a yin50 Y in 0.05 jja 2.530.054.36  2 22   o 1 ja 1  a 11  aa 11 zz jja 1   1   1  inin j C ZC Z o oo o zin a 4.36   2   2.530.086    1a  20 oo

fr  f o 0.086 f o  4.57 GHz 1 4.36 Co  0.16 pF rCZ o o yin 微波電路講義 ADS examples: Ch6_prj 6-22