Chapter 4 Microwave network analysis 4.1 Impedance and equivalent voltages and currents equivalent transmission line model (b, Zo) 4.2 Impedance and admittance matrices not applicable in microwave circuits 4.3 The scattering matrix properties, generalized scattering parameters, VNA measurement 4.4 The transmission (ABCD) matrix cascade network 4.5 Signal flow graph 2-port circuit, TRL calibration 4.6 Discontinuities and modal analysis microstrip discontinuities and compensation

4-1 微波電路講義 4.1 Impedance and equivalent voltages and currents • Equivalent voltages and currents Microwave circuit approach Interest: voltage and current at a set of terminals (ports), power flow through a device, and how to find the response of a network

For a certain mode in the line, the line characteristics are represented by it’s global quantities Zo, b, l. Define: equivalent voltage (wave)  transverse electric field equivalent current (wave)  transverse magnetic field  voltage (wave)/current (wave) = characteristic impedance or wave impedance of the line and voltage  current = power flow of the mode → use transmission line theory to analyze microwave circuit performance at the interested ports

4-2 微波電路講義 • Impedance  characteristic impedance of the medium   

E  wave impedance of the particular mode of wave t Zw   Ht V  characteristic impedance of the line Zo  I  Vz() input impedance at a port of circuit Zz() in Iz()

4-3 微波電路講義 Discussion 1. Transmission line model for the TE10 mode of a rectangular waveguide V (x, z)  E  dl  E dy y   y : x  dependent , non - unique value

x transverse fields (Table 3.2, p.117) transmission line model

V Voo e j b z V  e j b z  j b z  j b z xx Ey ( A e  A e )sin  C1 V sin aaI Ioo e j b z I  e j b z 1 xx H ( A e j b z  A  e j b z )sin  C I sin VVoo x Z a2 a eej b z j b z TE10 ZZoo   Ey k VVoo ZZTE    o Zo  10   H x b IIoo

1  * 1 P E H dxdy PVI oo  * 2  yx 2 4-4 微波電路講義 (derivation of C1 and C2) xx EA eA()sin()sin eC j b V zj zj eVb zj z  e b b yoo aa1  AA VVoo, CC11 1 xx HA eA  eC()sin()sin I eI j b e zj zj b zj z  b b xoo Zaa 2 TE10 AA II   , ooC ZC Z 22TETE1010

 2 11ab ab 2 A 2 PE H   dxdyAV * IC** C   24200 y xo Z o 2C C* Zab 12 TE10 12TE10 V  ACZ C ZZ o 2 TE10 2 =-1 TEo10  ICo AC 11 22 CC  , 12abab

4-5 微波電路講義 2. Ex.4.2

incident TE10 o r Zoa, ba  Zod, bd wave

ZZ  odoa Q: What if the incident wave is ZZ odoa from the other direction? “N” koo k ZZkkoaodo or kk o, ,  ,   bbad

222 222 2 22ffcc baco kkkk dcc b, k ,   avcpc / r Xacmacmband:  2.28624.4721376.56,4.17  kmfGHz  fGHz 1 ccc ac ,,r 1 111 if fGHzkf 102  cm /209, bo ard158,2.54333 mkmm b  ,304

500ZZoaod ,259.6   ,0.316  11 if fGHzkkmkm6201 in  ,147,  br  ocr d 11 126kmkairjocab in  ,54," m TE non exis 10 t" 4-6 微波電路講義 4.2 Impedance and admittance matrices V1+, I1+ reference plane V1¯, I1¯ for port 1 Zo1V1, I1 (plane for port 1  + + V1 0) t1 N-port VN , IN network VN¯, IN¯

port N IN ,VN Zon reference plane  VVVi  tN ii for port N 1  IIIVVi i  i () i  i (plane for Zoi  VN 0 )   VVii Zoi    IIii

11 PVIPVIinc,, iRe{ii *}, in i Re{ i i *} 22 4-7 微波電路講義 • Impedance matrix

VZZZI1111211      N VZZI       22122     N Vresponseii VZIZ    , ,        ij      Isourcejj       Ikk0,0, k  jI  k j      VZZZINNNNNN   12  

• Admittance matrix

IYYYV1   11 12 1N   1  IYYV        2   21 2N   2  Iii response IYVY     ,       , ij         Vjj source             Vkk0, k  j V  0, k  j       IYYYVN   N12 N NN   N 

4-8 微波電路講義 Discussion 1. Reciprocal network

t ZZZZ   , ijji t , andZ Y : symmetric matrix YYYY   , ijji (derivation) source port 1 port 2 a V1a, I1a V2a, I2a b V1b, I1b V2b, I2b reciprocity theorem: V1aI1b + V2aI2b =V1bI1a + V2bI2a

V1  Z11 Z12 I1        V2  Z21 Z22I2 

(Z11I1a  Z12I2a )I1b  (Z21I1a  Z22I2a )I2b  (Z11I1b  Z12I2b )I1a  (Z21I1b  Z22I2b )I2a

Z12I2a I1b  Z21I1a I2b  Z12I2b I1a  Z21I1b I2a

(Z12  Z21)(I2a I1b  I2b I1a )  0  Z12  Z21

4-9 微波電路講義 2. T and Π networks

Z1 Z2 Z11-Z12 Z22-Z12

Z3 Z12

()derivation I2=0 Z1 Z2 VZZI      1 11 12 1       I1 V1 V2 VZZI2   21 22   2  Z3

VV12 ZZZZZZ11  1  3, 21   3  12 II111 II2200I =0 Z1 Z2

V2 ZZZ22  2  3 V1 I2 I2 Z3 V2 I1 0

ZZZZZZZZ3  12,, 1  11  12 2  22  12 4-10 微波電路講義 Y3 -Y12

Y1 Y2 Y11+Y12 Y22+Y12

I1 I2 ()derivation Y3 IYYV      1 11 12 1 V1 Y1 Y2 V2=0       IYYV2   21 22   2 

II12 YYYYYY11  1  3, 21    3  12 VV11 I1 I2 VV2200 Y3

I2 YYY22  2  3 V2 V1=0 Y1 Y2 V2 V10

YYYYYYYY1  11  12,, 3   12 2  22  12

4-11 微波電路講義 3. Z- and Y- matrices of a lossless transmission line section

I1 I2 Z1 Z2 ZZjZll bb (cotcsc) V1 Zo, b V2 12 o Z3 ZjZl3  b o csc z ()derivation 0 l  j b z  j b z   j b z  j b z VzVe( )o  VeIzYVe o , ( )  o ( o  Ve o ),B.C. VV1  (0), II 1  (0), VVlI 2  ( ), 2   Il ( ) 1 VVVIYVVVVZI  ,()()         1o o 1 o o o o2 1 o 1 11 VVe j b l  Ve  j b l ( VZIe  )  j b l  ( VZIe  ) j b l  V cos b lj  Z Isinb l .....(1) 2o o22 1 o 1 1 o 1 1 o 1 11 I YVe( j b l  Ve  j b l )  YVZIe [ (  )  j b l  ( VZIe ) j b l ]  jYV sin bb lI cos l .....(2) 2o o o o22 1 o 1 1 o 1 o 1 1

 (2)V1   jZoo I 1 cot b l  jZ I 2 csc b l .....(3)  (3) cotbbll csc (1)V   jZ I cos b l cot b l  jZ I cot b l  jZ I sin b l Z   jZo   2o 1 o 2 o 1 cscbbll cot  =jZoo I12 csc b l  jZ I cot b l

Z1 Z 11 Z 12 jZoo(cot bb l csc l ) Z 2 , Z 3  Z 12 jZ csc b l 4-12 微波電路講義 (1)cossincotcsc...(3)VVl211112b jZ b IlIjYlVjYlV  ooo bb (3)

b(2)sincossin(cotcsc)cosIjY b211112 Vl boooo IljY  bbb VljYlVjYlVl bbjYlVjYlVcsccot oo12 Y3 cotcscbbll YjY  o  bbcsccotll Y1 Y2

YYYjY11112   o (cotcscbllY b ),csc YjYl b 23 o

YYjYll bb (cotcsc) 4. Reciprocal lossless network 12 o YjYl3 bo csc

Re{Zij}  0 5. Problems to use Z- or Y-matrix in microwave circuits 1) difficult in defining voltage and current for non-TEM lines 2) no equipment available to measure voltage and current in complex value (eg. sampling scope in microwave range, impedance meter <3GHz) 3) difficult to make open and short circuits over broadband 4) active devices not stable as terminated with open or short circuit 4-13 微波電路講義 4.3 The scattering matrix

t1’ t1 + V1’ V1+ port 1 V1’¯ V1¯ N-port tN q1=bl1 network tN’ port N VN+ VN’+ VN¯ VN’¯

qN=blN

 VV11 SSS11 12 1N       VVSS   22 21 2N   V  response VSVS      ,          , ii        ij      Vjjsource         Vkk0, k  j V  0, k  j     VVNN SSSN12 N NN  

4-14 微波電路講義 Discussion 1. Ex 4.4 a 3dB attenuator (Zo=50Ω)

Zin 8.56  41.44  50  ZZin o V1    SV111 V 1  V0( 1  V 1 )  V 1  0     ZZVin o 1 41.44 501 1 V0,0.707 V   V  VV  V V S V  2 2 2 1141.44 18.56 50 1 8.56 21 1 22 1 0 reciprocal SS 2 21 12 S : lossy symmetric SS11 22 1 0 2

 2 1 V1 incident power to port 1:Pinc,1  2 Zo 2 1 22 V1 1VV21 12 1 1 transmitted power from port 2: Ptrans,2  :3dB  att   enuation 2Zo 2 ZZ oo 2 2  input match attenuator design,  RR12 attenuation value 4-15 微波電路講義 2. T-type attenuator design

Zo R1 R1

Eth V1 R2 V2 Zo

Zin

1 ZRZRRZ // RZ inoo 121 1 1 o V RRZ//    Z  2 21o o   2 S21   RZ2  2 o VRRRZRZ11211  //  oo 1

1 2 1 RZZ 1 1 oo 3dB attenuator   1 21 2 2 RZZ22 2 1 2 oo

4-16 微波電路講義 3. Relation of [Z], [Y], and [S] S  Z U1 Z U, Y   Z1 (derivation)

 VVVn n n Let Z1,on   IVVn n n  VZ  IVVZ  VV    ()    Z VVZ   VV      ()()Z U  VZ  U   V   VS  VZ  U  S ()() VZ  U  V     SZ () UZ ()  U 1    

4-17 微波電路講義 4. Reciprocal network S S t, S: symmetric matrix (derivation)   VVVVVIn n  nnn  n () / 2 Let Z1,on   IVVVVIn n  nnn  n () / 2 11 VV  IZ()   U ()  I       22 11 VV  IZ()   U ()  I       22   11 VZ ()()()() U   Z  U   VSZ U Z  U         from 3 SZt (() UZ ) () UZ ()11tt () UZ   US          

4-18 微波電路講義 5. Lossless network (unitary property)

N t * * 1 i  j S S  U , Ski Skj   k1 0 i  j (derivation)

Let Zon  1 lossless (incident power=transmitted power) net averaged input power =0 Pin, i i

11t * ** PV IVRe()  V Re[() ()]VV    t     Im in 22        1 ttt *** t * Re[V VV  VV  VV  V  )] 0 2       VSV     ttt ***  t * V  VV  VV  S  S  V      SSUt  *   N * 6. Lossy network Ski Ski  1 k1

4-19 微波電路講義 7. Ex.4.5 0.1500.8545oo  S=00 0.85450.20 S:not symmetric a non-reciprocal network 22 SS11 21 0.745  1  a lossy network

port 1 RLSdB  20log11 16.5

port 2 RLS  20log dB 22 14

ILSdB 20log21 1.4

port 2 terminated with a matched load L  0 

0.15in S11  ,20logRLdB  0.15 16.5

port 2 terminated with a short circuit L   1 

SS12 21L 0.452,in SRL11  6.9 dB 1S22 L

4-20 微波電路講義 8. Shift property

t1’ t1 t2’ + + V1’ V1 port 1 V2+ V2’+ V1’¯ V1¯ port 2 V2¯ V2’¯ 2 q1=bl1 t q2=bl2

'''' jjjjjj22 q112212 q q q q q SeS1111212112122222 SeS eSeS,,, eSeS

eejj q11 q 0000  0000eejj q22 q  n-port network: SS'              jj q q 0 00 0eeNN

9. S-matrix is not effected by the network arrangement.

4-21 微波電路講義 10. Power waves on a lossless transmission line with Zoi Ii  VVZiioi i I  incident (power) wave=:ai  ZZoioi 2  Vgi oi L VVZiioi i I  Z Vi Z reflected (power) wave=:bi  ZZoioi 2 VZ  bi ioj Pin,i VS  V  bS a ,  S    ij  a VZ j a0 ,k j joi  k V,kk 0 j

111 22 2 PV IabPPRe{} * PSP(1) in,,, ii iiiinc222 irefl iinc iiiL , (derivation) + - + - Vi Vi + - Vi -Vi a i - bi a i = ,bi = ,Vi =Vi +Vi = Z oi (a i + bi ),I i = = Z oi Z oi Z oi Z oi

1 * 1 * 1 2 2 * * Im Pin,i = Re{Vi I i }= Re{(a i + bi )(a i - bi ) }= Re{ a i - bi + a i bi - a ibi } 2 2 2 1 2 1 2 2 bi = a i - bi =Pinc,i - Prefl ,i = Pinc,i (1- S ii ),S ii = 2 2 a i a k=0 ,k¹i 2 2 + - Vi Vi 1 + - + 2 - 2 = Re{Pi - Pi }® Pi = = a i ,Pi = = bi 微波電路講義 2 Z oi 4-22 Z oi 11. Two-port device with its S-matrix

b1 S11  : reflection coefficient at port1 with port 2 matched a1 a2 0

b2 S21  : forward transmission coefficient with port 2 matched a1 a2 0

b1 S12  : reverse transmission coefficient with port1 matched a2 a1 0

b2 S22  : reflection coefficient at port 2 with port1 matched a2 a1 0

b   S S   a  b1  a1S11  a2S12 1 11 12 1       b  a S  a S b2   S 21 S 22   a 2  2 1 21 2 22

4-23 微波電路講義 12. Reflection coefficient and S11, S22

Zg

Zo, b out L g in   ZL

Zin Zout ZZZZZZingoutLLo ingoutLgoL    ,, if  then ZZ ZZZZZZingoutLLo

Zo Zo two-port Zo two-port S Zo 11 network network S22

ZZout o ZZino S22  S11  ZZout o ZZino

4-24 微波電路講義 13. RL and IL

b1 RL at port1: - 20log - 20log S11 a1

b2 IL from port 1 to port 2: - 20log - 20log S21 a1 P insertion loss IL(dB) 10log  L1 PL2 a a 1 1 b2 Zg PL1 Zg PL2

b1 two-port Zo Zo network

usually Zg=Zo 4-25 微波電路講義 14. Two-port S-matrix measurement using VNA V  V  V  V  1 1 DUT 2 2

Zo

a1 b1 b2 a2 → S11 → S21 15. Advantages to use S-matrix in microwave circuit 1) matched load available in broadband application 2) measurable quantity in terms of incident, reflected and transmitted waves 3) termination with Zo causes no oscillation 4) convenient in the use of microwave network analysis

4-26 微波電路講義 16. Generalized scattering parameters Ii VZI i ri i Zgi incident power wave aZi , ri : reference impedance 2 Re Zri Vgi ZL VZIi ri* i Vi reflected power wave bi  2 Re Zri

bi V i Z ri** I i Z L Z ri power wave reflection coefficient Sii    Pin,i ai V i Z ri I i Z L Z ri 22 22bi Z L Z ri * power reflection coefficient SZZSii  ,conjugate match L  ri *  ii  0 ai Z L Z ri

1122 PRe{ V I *}  ( a  b )  P in, i22 i i i i L Ref: K.Kurokawa,"Power waves and the scattering matrix", IEEE-MTT, pp.194-201, March 1965

 traveling wave along a lossless line with real Z0     I Vi V i V i V i a i b i i ai, b i  , V i  V i  V i  Z oi ( a i  b i ), I i   ZZZoi oiZoi oi VVZIVZI a  b i, a  b  I Z  a  i00 i , b  i i i i i i i oi i i Zoi Vi ZL ZZZoi 2200

bi V i Z oi I i Z L Z oi Sii    ,impedance match ZZSL oi  ii  0 ai V i Z oi I i Z L Z oi 4-27 微波電路講義 I Z V i L gi Z VVIi gi, i gi ZZZZL gi L gi Vgi V ZL V 2 i 1 gi Re ZL PVILRe{ i i *} 2 22ZZ L gi Pin,i

ZL 1  Zri VZI ZZZZZZL ri * Re Z ai ri i  VL gi L gi  V ri i gi gi ZZ 2 ReZZri 2 Re ri L gi if ZZL ri * ZL 1  Zri * VZIi ri* i ZZZZL gi L gi bi  V gi 00  S ii  2 ReZZri 2 Re ri

2 1 2 V Re Zri P  P  a  P  gi L in,, i22 i inc i 2 ZZL gi 2 1 Vgi if ZZPL g * L = :maximum power trasfer from source 8 Re ZL 4-28 微波電路講義 a  F() V  Zr  I 

b  F(* V ) Zr   I 

Z r1 0 . 0 0 . 0 0 Z r    0 0 . 0  0 0 0 Z rN

ReZ r1 0 . 0  1 0 . 0 0 F  , 2 0 0 . 0  0 0 0 Re Z rN 1 1 VZIbFZZ       (*)()   rr  ZZ    Fa  

1 1 SFZZZZF  (*)()   rr      

4-29 微波電路講義 4.4 The transmission (ABCD) matrix • Cascade network

I1 I2 I3 A B A B +   +   + V1 _ V2 _ V3 _ C D1 C D2

V1  A B V2  A B A B V3               

I1  C D1 I 2  C D1 C D2 I3  Discussion 1. ABCD matrix of two-port circuits (p.190, Table 4.1) 2. Reciprocal network AD-BC=1 3. S-, Z-, Y-, ABCD-matrix relation of 2-port network (p.192, Table 4.2) 4. Ex. 4.6 ABCD(Z) 4-30 微波電路講義 (derivation) Z (ABCD)

V1  Z11 Z12  I1  V1  A BV2       ,       V2  Z21 Z22 I2  I1  C DI2 

V1 AV2 A Z11    I1 CV2 C I2 0

V1 AV2  BI 2 V2 V2 D Z12    A  B, I1  0  CV2  DI2     I2  I2 I2 I2 C I1 0 I1 0 I1 0 D AD  BC  A( )  B  symmetrical network C C ZZAD1122 V2 V2 1 Z21    reciprocal network I1 CV2  DI2 C I2 0 I2 0 Z12  Z21 V D Z   2  , I  0  CV  DI AD  BC 1 22 1 2 2   AD  BC 1 I2 C I1 0 C C

4-31 微波電路講義 (derivation) S (ABCD)  bSSaVV11112112        AB V111111 VVVV Z I () / 2 o        ,,  bSSaII22122212        CD IVVZVV111111() /() / 2 Z I oo  bV11V112222 Zooo IAVBIZ CVZ DI S11   aVV111122 Z IAVBI o  Zoo CVZ22 DI  aV2200 V2 0 V2 0 A BY CZD I 2  oo  V2  0 A BYoo CZD V2 Zo V2  I2Zo , I2 V2Yo

  V2 V2 b2 V2 V2 2V2 S21      a1 V1  (V1  Zo I1) / 2 AV2  BI 2  ZoCV2  Zo DI2 a2 0 V2 0 I2 V2Yo 2  A  BYo  CZo  D

4-32 微波電路講義 bV VVVVVZ I () / 2 S 11, 222222 o 12  IVVZVVZ() /() / 2 I aV22 222222 oo aV1100 VV  11 V VV 1  DB ,,   1 21AD BC II    ()VZ /22 2 I o 21   CA 22VV 11 ()DVBIZ / 111111 CVZooo AIDVBY V CZo VAV11 2Δ  I1  I2 A BYCZDoo

V1  I1Zo , I1  V1Yo Zo V1 V2

symmetrical b V  V  Z I DV  BI  Z (CV  AI ) S  2  2  2 o 2  1 1 o 1 1 22  S11  S22, A  D a2 V2  V2  Zo I2 DV1  BI1  Zo (CV1  AI1) a1 0 V1 0 reciprocal I1 V1Yo DV1  BYoV1  Zo (CV1  AYoV1)  A  BYo  CZo  D   S12  S21, 1 DV1  BYoV1  Zo (CV1  AYoV1) A  BYo  CZo  D

4-33 微波電路講義 5. Example

t1 Zoc [S] Zom

t2 [S] representation can be obtained from measurement or calculation. coaxial-microstrip transition (a linear circuit) L

Zoc C1 C2 Zom

one possible equivalent circuit

4-34 微波電路講義 4.5 Signal flow graphs • 2-port representation

a1 S21 b2 a1 [S] a2 b1 b2 S11 S22

b1 a2 port 1 port 2 S12

b   S S   a  b1  a1S11  a2S12 1 11 12 1       b  a S  a S b2   S 21 S 22   a 2  2 1 21 2 22

b1 RL at port 1: - 20log - 20log S11 a1

b2 IL from port 1 to port 2: - 20log - 20log S21 a1

4-35 微波電路講義 Discussion Z e e V o ,a  s 1. Source representation s s s Zo  Zs Zo

Zs as } a1 as 1 a1 Vs s b1 s b1 s b1 2. Load representation b2 b2 L ZL L a2 a2

3. Series, parallel, self-loop, splitting rules (p.196, Fig.4.16)

4-36 微波電路講義 4. 2-port circuit representation

as a1 S21 b2 Zs Vs s S11 S22 L s in [S] out LZL

b1 S12 a2

SSΓ 12 21 L as a1 b1 a 1 S 11  a 1 S 21ΓLL S 12(1  S 22 Γ  ...)  a 1 S 11  a 1 1 S22 ΓL

b1 S 12 S 21ΓL s in ΓSin  11  aS11 22 ΓL 1 SS12 21ΓS b b2 a 2 S 22  a 2 S 12ΓSS S 21(1  S 11 Γ  ...)  a 2 S 22  a 2 1 S11ΓS b2

b2 SS12 21ΓS ΓSout  22  out L aS21 11ΓS a2

4-37 微波電路講義 5. TRL (Thru-Reflect-Line) calibration

Find [S]DUT from 2-port measurement using three calibrators DUT [S]

x y

ao b1 a2 b3

e10 S21 e32 e00 e11 S11 S22 e22 e33 e01 S12 e23

bo a1 b2 a3

6 unknowns of [S]x and [S]y to be calibrated to acquire [S]DUT T: Through  3 eqs., R: Reflection  2 eqs., L: Line  3 eqs.  R () and line length (l) can be unknown 4-38 微波電路講義 Calibrators T: Through  3 eqs. x y

R: Reflection  2 eqs. x y

L: Line exp(l)  3 eqs. x y Requirement: connectors and line have same characteristics for 3 calibrators Limitation: operation bandwidth 20°≦βl≦160° 4-39 微波電路講義 (brief derivation) R matrix (wave cascade matrix)

ba1  RR 11   12 2      ab1  RR 21   22 2

R11 RS 1212  S 211 S 11 S 22 S 11 R   R21 RS 2222  S21  1 x xy y  11 1211 12 error matrices  ,R xy  R x21 xy 22 21y 22 0 11  0  Through:  SR TT        1 00  1  00eell Line: SRLL ll    ee00 1/ Reflection:      Thru measurement: RRRRRRmT  x  T  y  x  y   Line measurement: RRRRmL   x  L  y 4-40 微波電路講義 1 ee01 10   MRRRMRRe,   xxLmLmT            00, e11

1 ee  23 32  ,RNRRNRRe yLymTmL           33, e22

ee10 01 reflection measurement at port 1 mx e00  1e11

ee23 32 reflection measurement at port 2 my e33  1e22

Γ mx   S21mT  Γ my   e11 , e 22 , e 10 e 01 , e 23 ee 3210 e 32 e 23 e 01 ,  S12mT  Γ mT 

 e10,, e 01 e23,e 32  Γ,e l

4-41 微波電路講義 (detailed derivation ) m m   x x   x x  el 0 MRRR  11 12 11 12 11 12   x  x L        l m21 m 22   x 21 x 22   x 21 x 22  0 1/ e  m x m x x el xxx e e 11 11 12 21 11  m (111111 )2  (m  m 10 )01  m  0 a   e   l 21 22 11 1200  m x m x x e xxx212121 e 11   21 11 22 21 21  m x m x x/ el xxx  11 12 12 22 12 m(121212 )2  ( m  m )  m  0 b   e  l 21xxx 22 11 1200 m21 x 12 m 22 x 22 x 22 / e 222222 root choice : e10 e 0100 0  11 a  b , e  e 1  a  b y y   n n  el 0 yy RNRR     11 12 11 12 11 12 y    L  y      l  y21 y 22   n 21 n 22  0 1/e yy21 22  y n y n y el yyy ee 11 11 12 21 11 n(111111 )2  ( n  n )  n  0 c   e  23 32  l 12 22 11 21 33  y n y n y e yy1212 ye12 22  21 11 22 21 12 y n y n y/ el yy y  11 12 12 22 12 n(2121 )2  ( n  n )  n  0 de21    l 12yy 22 11 21 y 33  y21 n 12 y 22 n 22 y 22 / e 2222 22 root choice : e23 e 3222 0  33 c  d , e  e 1  c  d

4-42 微波電路講義  ee10 01Γ 1 b  Γ mx Γ mx e00 Γ 1e Γ ea Γ  1111 mx   ee23 32 Γ 1 d  Γ my Γ my e33 Γ  1e2222Γ ecΓ my  e10 e 01 eb 22 1  Γ mT Γ mT ee0011  1e11 ee 2222 a Γ mT

2 b Γ mxmTmTc  Γ my b Γ 1 b Γ eee112210 eb 01 a e ,,() 11,=(e 23 ec 3222- d) e a Γ mxmymTmTd Γ a Γ ea11 Γ  ee S  10 32  21mT  1 ee11 22  e10 eSe 322111 ee 2223mTmT eSe 011211(1),=(1),,, ee 2210 e 01 e 23 e 32  ee23 01 S12mT   1 ee11 22

1 b  Γ mx l m12  Γ...(also for selection), eem1111 ea11  Γ mx a

4-43 微波電路講義 4.6 Discontinuities and modal analysis • equivalent circuit components  E  C,  H  L constant E (V)  parallel connection constant H (I)  serial connection

Discussion 1. Microstrip discontinuities Cg

Cp Cp Coc

open-end gap

4-44 微波電路講義 L L

C

step L2 L1

C L3

T-junction

L L

C bend 4-45 微波電路講義 2. Microstrip discontinuity compensation w a r

r>3w a=1.8w

swept bend mitered bends

mitered step mitered T-junction

4-46 微波電路講義 Solved Problems Prob. 4.11 Find [S] relative to Z0 Z port port port port Z 1 2 1 2 é ù 1/ Z -1/ Z Z + 2Z o éY ù = ê ú,DY = (Y o +Y11)(Y o +Y 22 ) +Y12Y 21 = ZZ 2 ë û 2 ZZ Z  Z,()()2  Z ZZZZZZoooo 1122 12 21 -1/ Z 1/ Z ZZ o    ë û ZZ é ù (Y o -Y11)(Y o +Y 22 ) +Y12Y 21 -2Y12Y o ()()2Z112212 Zooo 2112 Z  Z Z ZZ Z ê ú ZZ éSù = ê DY DY ú S   ë û 2()()Z21112212 ZZooo 21 Z Z Z Z Z  ê -2Y 21Y o (Y o +Y11)(Y o -Y 22 ) +Y12Y 21 ú  ê ú  ZZ DY DY ë û 2 ZZZoo2  ZZo 2  é 2 ù é ù   1/ Z o 2 / ZZ o Z 2Z o ZZZZZZoo22 ê ú ê ú  ,1SS11 21 2   ê DY DY ú ê Z + 2Z o Z + 2Z o ú 2ZZZoo 2ZZ o = = ,1- S 11 = S 21 ê 2 ú ê ú  2 / ZZ o 1/ Z o 2Z o Z  ZZ   ZZZZoo22 ê ú ê ú

ë DY DY û êë Z + 2Z o Z + 2Z o úû

Prob. 4.12 Find S21 of [SA] in cascade of [SB]

SS21AB 21 S21A S21B S21  1 SS22AB 11 S11A S22A S11B S22B S12A S12B 4-47 微波電路講義 000 Prob. 4.14 0.178 900.6 450.4 450 00 0.6 45000.345  0.4 45000.545 00 00 00.3450.5450   2 2 (1) S12  S42  0.8  1 lossy (2)[S] symmetrical  reciprocal

(3)return loss at port1  20log S11  20log 0.1  20dB

(4)insertion loss between port 2 and port 4  20log S24  20log 0.4  8dB phase delay between port 2 and port 4  450 (5)reflection at port 1 as port 3 is connected to a short circuit 0 0 0 ba11 0.178 90 0.6  45 0.4  45 0    00   b2 0.6 45 0 0 0.3   45 0      00 bb33 0.4 45 0 0 0.5   45    00   b4 0 0.3  45 0.5   45 0 0  0 ba310.4 45 0 0 0 0 0 b10.178  90 a 1  0.4  45 b 3  0.178  90 a 1  0.16  90 a 1  0.018  90 a1

b1 Sj11 0.018 微波電路講義 a1 4-48 Prob. 4.19 given [Sij] of a two-port network normalized to Zo, find its generalized [S’ij] in terms of Zo1 and Zo2

 VVZIi i oi i incident (power) wave:ai  = ZZoi2 oi  VVZIi i oi i reflected (power) wave:bi  = ZZoi2 oi

 b VZi oj V  S  V    b  S  a , S i      ij a VZ j a0 ,k j j oi  k Vk 0 ,k j

'''ZZoo21' SSSSSSS11  11,,, 12  12 21  21 22  S22 ZZoo12

4-49 微波電路講義 Prob. 4.28 find P2/P1 and P3/P1 a1 S12 b2 Γ2 P2  0 S12 0  Γ2 b1 a2   S12 S23 S12 0 S23 P1   b3 S23  0 S23 0  Γ3 P3 Γ3

2 a3 SS12 212 SS12 2 23 b1 aa 11  b 2  a222 1in ,, b 3 a 1 111 2  3SSS  232  3 23   2  3 23 2 2 2 22 22 2 P b a   bSS  (1 )(1 )(1 ) 2 2 2  2 212 212 2 2 2 2 222 2 22 2 P1 a b a (1   ) SS S 1      1 1 1 in 2 2 12 2 2 3 23 12 2 1 2  3S 23 (1)  2 2 1 2  3S 23 22 2 2 2 2 2 2 2 2 2 2 P ba bS(1  SS )(1 )(1 S  )   3  333 3 12223 3 12223 2 2 2 2 222 2 22 2 P1 a ba(1   ) SS S 1      1 1 1 in 2 2 12 22 3 23 12 2 1 2  3S 23 (1)  2 2 1 2  3S 23 ADS examples: Ch4_prj

4-50 微波電路講義