Chapter 7 Two- network 7.1 definition, examples 7.2 definition, examples 7.3 Hybrid parameters definition, examples 7.4 Transmission parameters definition, examples 7.5 Conversion of the impedance, admittance, chain, and hybrid parameters 7.6 definition, characteristics, examples 7.7 Conversion from impedance, admittance, chain, and hybrid parameters to scattering parameters or vice versa 7.8 Chain scattering parameters

7-1 微波工程講義 7.1 Impedance parameters I1 I2 Basics linear 1. V1 port 1 port 2 V2 network [V ]= Z [I , ][][]I : source, V : response [ ]

V  Z Z I  V = Z I + Z I reference reference 1 = 11 12 1 , 1 11 1 12 2      = + plane 1 plane 2 V2  Z 21 Z 22 I 2  V2 Z 21I1 Z 22 I 2 V = 1 Z11 : open - circuit input impedance at port 1 I1 = I2 0 V = 1 Z12 : open - circuit reverse transimpedance I 2 = I1 0 V = 2 Z 21 : open - circuit forward transimpedance I1 = I2 0 V = 2 Z 22 : open - circuit input impedance at port 2 I 2 = I1 0 7-2 微波工程講義 Lecture 4 How to Determine Z Parameters?

=⋅+⋅ I = 0 VZIZI11111222 ==VV12 ZZ11 21 =⋅+⋅ II11== VZIZI2211222 II2200

= I1 0 ==VV21 ZZ22 12 II22== II1100

I1=0 I2 Z22 Two Port Reciprocity: O.C. V1 V Network 2 = Z12Z 21

1/9/2003 2 Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Discussion I1 I2 V 6I = = Ω = 1 = 2 = 1. Ex.7.1 Z11 Z 22 6 , Z12 6 I 2 = I 2 I1 0 6Ω V 6I 6 6 V1 V2 Z = 2 = 1 = 6,[]Z =   21 1 6 6 I1 I =0 I1   2. Ex.7.2 2 12 0 I1 I2 Z = 12, Z = 3,[]Z =   11 22 2  0 3 12Ω V2 V V V1 = 1 = = 2 = 3Ω Z12 0, Z 21 0 I 2 = I1 = I1 0 I2 0 3. Ex.7.3 18 6 I1 = = []= = + [][] I2 Z11 18, Z 22 9, Z3   Z1 Z 2  6 9 12Ω 3Ω V2 V 6I V 6I V1 = 1 = 2 = = 2 = 1 = 6Ω Z12 6, Z 21 6 I 2 = I 2 I1 = I1 I1 0 I2 0

4. Z12=Z21 → reciprocal circuit Z12=Z21, Z11=Z22 → symmetrical and reciprocal circuit 5. Useful for series circuits analysis. 7-3 微波工程講義 Series Connection of Two-Port Lecture 4 Networks

I1 I2

V1A ZA V2A =+ V1 V2 [Z ] [ZZAB] [ ]

V1B ZB V2B

I1 I2

1/9/2003 10 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Lecture 4 Example: T-Network

I1 ZA ZB I2  V Z ==+1 ZZ I = 0 11 I A C Z V 2  1 I =0 V1 C 2  2  ==V2 ZZ21 C I1 =  I2 0  == = ZZZ12 21 C I1 0   ==+V2 Z22 ZZBC I2 =  I1 0

1/9/2003 5 Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 γ

I1 2 6. Ex. 7.4 γ γ I port1: source V ,port 2 : open Γ = 1 in open γ −2 l − l Zo, 2 → = + = 1 V V1 Vin Vine ,V2 2Vine V V V e−2 l = in − in γ= γ I1 , I 2 0 l γ Z o Zo γ γ V V +V e−2 l γ e l + e − l Z Z = 1 = in in = Z γ = o = Z 11 −2 l o l − − l 22 I1 = Vin Vinγe e e tanh l I2 0 − γ Z o Zo γ V 2V e− l 2 Z Z = 2 = in = Z γ = o = Z 21 −2 l β o l − −γl 12 I1 = Vin Vine e e sinh l I2 0 − Z o Zo β  Zβ Z   o o  Z Z β = = o = = o []=  j tan l j sin l  lossless line Z11 Z 22 , Z12 Z 21 , Z j tan l j sin l  Z o β Z o   j sin l j tan βl 

7-4 微波工程講義 7.2 Admittance parameters Basics I1 I2 1. linear V1 port 1 port 2 V2 []I = Y [][][]V , V : source, I : response [] network I  Y Y V  I = Y V + Y V  1  =  11 12  1 , 1 11 1 12 2 reference reference = + plane 1 plane 2 I 2  Y21 Y22 V2  I 2 Y21V1 Y22V2 I = 1 Y11 : short - circuit input admittance at port 1 V1 = V2 0 I = 1 Y12 : short - circuit reverse transadmittance V2 = V1 0 I = 2 Y21 : short - circuit forward transadmittance V1 = V2 0 I = 2 Y22 : short - circuit input admittance at port 2 V2 = V1 0

7-5 微波工程講義 Lecture 4 Y Parameters − − [VZI]=⋅[ ] [ ] []IZ=⋅ [1 ] V [ ][]YZ= []1

=⋅+⋅ V = 0 I1111122 YVYV 2 ==II12 YY11 21 =⋅+⋅ VV11== I2211222 YVYV VV2200

= V1 0 ==II21 YY22 12 VV22== VV1100

S.C. I1 I2 Y22 Two Port V1=0 Network V2

1/9/2003 3 Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Discussion − 1. Ex.7.5  0.05 0.05 I1 I2 Y = Y = 0.05,[]Y =   11 22 1 − 0.05 0.05  0.05S I − 0.05V V1 V2 = 1 = 2 = − = Y12 0.05 Y21 V2 V =0 V2 2. Ex.7.6 1

0.1(0.2 + 0.025) I1 Y = = 0.0692 I2 11 0.1+ 0.2 + 0.025 0.1S 0.2S + 0.2(0.1 0.025) V2 Y = = 0.0769 V1 22 0.1+ 0.2 + 0.025 0.025S I = 1 Y12 : source V2 ,port 1 short V2 = V1 0 I I = Y V = 0.0769V , voltage across 0.1S: V = 2 = 0.615V 2 22 2 2 m 0.1+ 0.025 2 = − = − → = − = I1 0.1Vm 0.0615V2 Y12 0.0615 Y21  0.0692 − 0.0615 []Y =   2 − 0.0615 0.0769 

7-6 微波工程講義 3. Ex.7.7 + = + 0.1(0.2 0.025) = 0.05S Y11 0.05 0.1192 0.1+ 0.2 + 0.025 I1 I2 0.2(0.1+ 0.025) Y = 0.05 + = 0.1269 0.1S 0.2S 22 + + 0.1 0.2 0.025 V2 V1 I 0.025S = 1 Y12 : source V2 ,port 1 short V2 = V1 0 0.05 I = Y V = 0.1269V ,current through 0.05S: I = I = 0.05I 2 22 2 2 n 0.1269 + 0.05 2 2 − = current through 0.2S: I 2 I n 0.0769V2 0.1 current through 0.1S: 0.0769V = 0.0615V 0.1+ 0.025 2 2 = − + = − → = − = I1 (0.0615V2 0.05V2 ) 0.1115V2 Y12 0.1115 Y21  0.1192 − 0.1115 []Y =   = Y + Y [][] 3 − 0.1115 0.1269  1 2 4. Useful for parallel circuits analysis.

7-7 微波工程講義 Parallel Connection of Two-Port Lecture 4 Networks

I1 I1A I2A I2

V1 YA V2 =+ [YY] [ A] [ YB ] I2A I2A

YB

1/9/2003 11 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Lecture 4 Example: Pi-Network

I1 YC I2  I YYY==+1 V = 0 11 A C 2  V1 = V  V2 0 V1 YA YB 2   ==−I2 YY21 C V1 =  V2 0  ==− = YY12 21 YC V1 0   ==+I2 YYY22 BC V2 =  V1 0

1/9/2003 6 Lecture 4. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 γ 1 5. Ex.7.8γ I I2 port1: source V ,port 2 : short Γ = −1 in short Z ,γ −2 l γ o 2 → = − = 1 V V1 Vin Vine ,V2 0 V − V V e 2 l γ 2V = in + in = − in − l I1 , I 2 e γ l Z o Zo Z o −2 l V V γ e γ in + in I Z Z 1 1+ e −2 l 1 Y = 1 = o o = γ = = Y 11 − γ −2 l − −2 l 22 V1 = Vin Vine Z o 1 e Z o tanh l V2 0 γ

2V − − in e lγ β I Z 1 2 1 Y = 2 = o = − γ = − = Y 21 − −2 l l − −γl 12 V1 = Vin Vin e Z o e e Z o sinh l V2 0 β β  1 j   β  1 j jZ tan l Z sin l lossless line Y = Y = ,Y = Y = ,[]Y =  o o  11 22 12 21 j β 1 jZ o tan l Z o sin l    β   Z o sin l jZ o tan l 

7-8 微波工程講義 7.3 Hybrid parameters Basics I1 I2 1. linear V1 port 1 port 2 V2 network V  h h I  V = h I + h V 1 = 11 12 1 , 1 11 1 12 2      = + I 2  h21 h22 V2  I 2 h21I1 h22V2 reference reference plane 1 plane 2 I1 ,V2 : sources, V1, I 2 : responses I = 1 h11 : short - circuit input admittance at port 1 V1 = V2 0 V = 1 h12 : open - circuit reverse voltage V2 = V1 0 I = 2 h21 : short - circuit forward current gain I1 = V2 0 I = 2 h22 : open - circuit input admittance at port 2 V2 = I1 0 7-9 微波工程講義 Discussion 1. Useful for transistor circuits analysis. 2. Ex.7.9 I1 I2 Ω Ω × 12 3 V1 3 6 2 h = = 12 + = 14 1 V 11 + V 6Ω I1 = 3 6 V1 0 I 1 1 h = 2 = = 22 + V2 = 3 6 9 I1 0 V 6I 3 I 6 2 h = 1 = 2 = ,h = 2 = − = − 12 21 + V2 = 9I 2 3 I1 = 6 3 3 I1 0 V2 0

7-10 微波工程講義 7.4 Transmission (ABCD, chain) parameters Basics I1 I2 1. linear V1 port 1 port 2 V2 network V  A B V  V = AV − BI 1 = 2 , 1 2 2    −  = − I1  C D I 2  I1 CV2 DI 2 reference reference plane 1 plane 2 V A = 1 : open - circuit reverse voltage gain V2 = I2 0 V B = 1 : short - circuit reverse transimpedance − I 2 = V2 0 I C = 1 : open - circuit reverse transadmittance V2 = I2 0 I D = 1 : short - circuit reverse current gain − I 2 = V2 0

7-11 微波工程講義 Lecture 4 ABCD Parameters

VAVBI=⋅ +⋅ VVAB  122 12=⋅ =⋅ +⋅   ICVDI122 II12CD 

!!!

I2 1) [ ]− I1 ABCD ? Two Port V1 V Network 2 − 2) Z [ABCD] ?

3) Z [ABCD]− ?

1/9/2003 7 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Cascade Connection of Two-Port Lecture 4 Networks

I1A I2A I1B I2B

V1A ABCDA V2A V1B ABCDB V2B

=+ [ABCD] [ ABCDA] [ ABCDB ]

1/9/2003 12 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Discussion 1. Useful for cascade circuits analysis. I1 I2 2. Ex.7.10 V I A = 1 = 1, D = 1 = 1 − V2 = I 2 = 1Ω I2 0 V2 0 V1 V2 V I 1 1 B = 1 = 1,C = 1 = 0, −   I 2 = V2 = 0 1 V2 0 I2 0 3. Ex.7.11 1 I2 V I I A = 1 = 1, D = 1 = 1 − V2 = I 2 = I2 0 V2 0 jwS V2 V1 V I  1 0 B = 1 = 0,C = 1 = jw, −   I 2 = V2 =  jw 1 4. Ex.7.12 V2 0 I2 0     +  1Ω 1Ω 1 1 1 0 1 jw 1 I1 I2    =   0 1 jw 1  jw 1

jwS V2 1+ jw 11 1 1+ jw 2 + jw V1    =    jw 10 1  jw 1+ jw 5. AD-BC=1 → reciprocal circuit A=D, AD-BC=1→ symmetrical and reciprocal circuit 7-12 微波工程講義 γ

1 6. Ex.7.13 γ I I2

V1 I1 B = γ , D = ,source : port 1, port 2 : short Γ = -1 − − I 2 = I 2 = γ Z ,γ V2 0 V2 0 o V2 γ γ V1 −2 l − V V e 2V − V = V −V e 2 l ,Vγ = 0, I = in + in , I = − in e l 1 in in 2 1 Z Z 2 γ Z o o o l V V e−2 l in + in −2 l −2 l γ −2 l V −V e − γ γZ Z + = in in = 1 e = = o o = 1 e = B Z o − l Z o sinh l, D γ − l cosh l 2Vin − l 2e 2Vin − l 2e eγ e Z o Z o γ V1 I1 γ A = ,C = ,source : port 1, portγ 2 : open Γ = 1 V2 = V2 = I2 0 I2 0 γ V V e−2 l = + −2 l = − l = in − in = V1 Vin Vine ,V2 γ2Vine , I1 , I 2 0 Z o Zo V V eγ−2 l in − in − V V +V e 2 l γ I Z Z 1 A = 1 = in in = cosh l,C = 1 = o o = sinh l β − l −γl V2 = 2Vine V2 = 2Vine Z o I2 0 I2 0  β  cos l β jZ o sin l lossless line  1   j sinh l cos βl  γ  Z o  7-13 微波工程講義 Lecture 4 ABCD Parameters of TL

I1 I2  +−1 +− VV=+ V, I =( V − V ) V1 l,β V2  11  Z0   =++−βjl − jl β VVe2 Ve 0 l  1 +−β − β  IVeVe=−( jl jl )  2  Z0

VV= cosβ ljIZ+ sinβ l  21 10 cosββljZl sin  []ABCD = 0 = V1 β + β ββ Ij21sin lIl cos jY0 sin l cos l  Z0

1/9/2003 8 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Transformation Between Different Sets of Lecture 4 Parameters • Any set of parameters can be transformed into any other set of parameters R. Ludwig and P. Bretchko, RF Circuit Design: Theory and Applications, Prentice Hall RF Circuit Design: Theory and R. Ludwig and P. Bretchko, 1/9/2003 9 ELG4105: Microwave Circuits © S. Loyka, Winter 2003 Lecture 5 S-Parameters

• Why S (scattering) parameters? • Z, Y and ABCD parameters: O.C. or S.C. terminations – very difficult at microwave frequencies • O.C. & S.C. : standing waves make measurements difficult and can destroy elements • S-parameters: defined in terms of incident/reflected waves • Easy to measure at microwaves: matched terminations

1/16/2003 1 Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 7.5 Conversion of the impedance, admittance, chain, and hybrid parameters see p.267, Table 7.1 7.6 Scattering parameters Basics 1. Z, Y, H, and ABCD parameters require an open or short circuit at port. It is difficult or impossible to determine the parameters of a network at radio and microwave frequencies. 2. a1 b2 b  S S a  b = S a + S a  1  =  11 12  1 , 1 11 1 12 2 linear b  S S a  b = S a + S a port 1 port 2 2 21 22 2 2 21 1 22 2 network a : incident (power) wave at port i i b1 a2 reference bi : reflected (power) wave at port i reference plane 1 plane 2

7-14 微波工程講義 Definition of S-Parameters

− ==bV11 =reflected wave at port 1 S11 + a incident wave at port 1 1 a =0 V1 + = 2 V2 0

− port bV reflected power at port 2 Output S ==22 = 22 a + incident power at port 2 2 a =0 V2 + = 1 V1 0 network − Two-port bV transmitted power at port 2 S ==22 = 21 + incident power at port 1 a1 = V +

a 0 1 = port 2 V2 0 Input − bV transmitted power at port 1 S ==11 = 12 a + incident power at port 2 2 a =0 V2 + = 1 V1 0

1/16/2003 3 Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 3. Measurable parameters for radio frequency and microwave frequency

a1 b2

port 1 linear port 2 network

b1 a2 reference reference plane 1 plane 2 b = 1 S11 : reflection coefficient at port 1 with port 2 terminated by a match a1 = a2 0 b = 2 S 21 : transmission coefficient from port 1 to port 2 with port 2 terminated by a match a1 = a2 0 b = 1 S12 : transmission coefficient from port 2 to port 1 with port 1 terminated by a match a2 = a1 0 b = 2 S 22 : reflection coefficient at port 2 with port 1 terminated by a match a2 = a1 0

7-15 微波工程講義 4. shifting the reference planes

a'1 a1 b2 b'2 linear port 1'port 1 port 2 port 2' network

b'1 b1 a2 a'2

β l1 l2 reference reference reference reference plane 1' planeβ 1 plane 2 plane 2'

− − − − = j l1 = j l1 =β j l2 = j l2 a1 a'1 e , b'1 β b1e , a 2 a'2 e , b' 2 b2 e

b' − b β − + = 1 = j 2 l1 = 2 = j (l1 l2 ) S '11 S 11 e , S '21 S 21 e a'1 = a1 = a '2 0 β a 2 0 β

b − + b − = 1 = j (l1 l2 ) = 2 = j 2 l2 S '12 β S 12 e , S '22 S 22 e a 2 = a 2 = a1 0 a1 0 β β − − + β  S e j 2 l1 S e j (l1 l2 )  []= 11 12 S '  − + − β  j (l1 l2 ) j 2 l2 S 21 e S 22 e  7-16 微波工程講義 I1 I2 Discussion linear 1.  V + Z I V1 port 1 port 2 V2  V = V + V = i oi i network  i in ,i ref ,i  Vin ,i  1 →  2 I = (V − V ) V − Z I  i Z in ,i ref ,i V = i oi i  oi  ref ,i reference reference 2 plane 1 plane 2 V V + Z I 1 V ≡ in ,i = i oi i = i + Vin,1 Vref,2 a i ( Z oi I i ) 2Z 2 2Z 2 2 Z oi oi oi linear V V − Z I V ≡ ref ,i = i oi i = 1 i − bi ( Z oi I i ) network 2Z oi 2 2Z oi 2 2 Z oi Vref,1 Vin,2  2 1 1 V * Vin ,i 2  P = Re(V I * ) = Re(V in ,i ) = = a  avs ,i in ,i in ,i in ,i i 2 2 Z oi 2Z oi  2  V * V = 1 = 1 ref ,i = ref ,i = 2 Pref ,i Re(V ref ,i I * ref ,i ) Re(V ref ,i ) bi  2 2 Z oi 2Z oi = 2 − 2 power delivered to the port i : Pd ,i a i bi b V b V b V = → = 1 = ref ,1 = 2 = ref ,2 = 1 = ref ,1 if Zo1 Zo2 S11 ,S21 ,S12 a1 a =0 Vin,1 = a1 a =0 Vin,1 = a2 a =0 Vin,2 = 2 Vin,2 0 2 Vin,2 0 1 Vin,1 0 b V = 2 = ref ,2 S22 a2 a =0 Vin,2 = 1 Vin,1 0 7-17 微波工程講義 2. a1 b2

Zs linear s1 ZL V network

b1 a2

Z1, Γ1 Z2, Γ2

b a a a 1− S Γ b = S a + S a → 1 = S + S 2 ,Γ = 1 → 2 = 11 s 1 11 1 12 2 11 12 s Γ a1 a1 b1 a1 S12 s b a a a 1− S Γ b = S a + S a → 2 = S 1 + S ,Γ = 2 → 1 = 22 L 2 21 1 22 2 21 22 L Γ a2 a2 b2 a2 S21 L b a S S Γ Γ = 1 = S + S 2 = S + 12 21 L 1 11 12 11 − Γ a1 a1 1 S22 L b a S S Γ Γ = 2 = S + S 1 = S + 12 21 s 2 22 21 22 − Γ a2 a2 1 S11 s

7-18 微波工程講義 Measurement of S-Parameters

Measurement Setup = ZL Z0

DUT −− bV bV SS==11 =Γ, == 22 11aa++ 1 21 11VV11

2V2 Forward voltage gain: S21 = VG1

1/16/2003 4 Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 = ∠ ° = ∠ ° = ∠ ° = ∠ ° = → 3. Ex.7.14 V1 10 0 , I1 0.1 40 ,V2 12 30 , I 2 0.15 100 , Z o 50 Vin,1,2 ,Vref ,1,2 V + Z I V − Z I V = i oi i ,V = i oi i in,i 2 ref ,i 2 → Vin,1 ,Vin,2 ,Vref ,1 ,Vref ,2 4. Ex.7.15 Z + Z − Z Z = = o o = Z S11 S 22 + + + ZO ZO Z Z o Z o Z 2Z o Z V o V V 1 Z + Z V +V Z S = ref ,2 = 2 = o = in,1 ref ,1 o 21 + Vin,1 Vin,1 Vin,1 Vin,1 Z Z o Vin,2=0 Vin,2=0 Vin,2=0

Vin, 2=0 2Z + 2Z Z 2Z = o o = o = S + + + 12 Z 2Z o Z Z o Z 2Z o

5. S12=S21 → reciprocal circuit S12=S21, S11=S22 → symmetrical and reciprocal circuit

7-19 微波工程講義 1 for j = k 6. Lossless circuit * = ∑ Sij Sik  Ex.7.15 0 for j ≠ k jX Z jX S = S = = ZO ZO 11 22 + + Z 2Z o jX 2Z o 2Z 2Z S = S = o = o 21 12 + + Z 2Z o jX 2Z o 2 2 2 2 X 4Z ∑ S S * = S + S = + o = 1 ij ik 11 22 2 + 2 2 + 2 j=k X 4Z o X 4Z o  jX 2Z 2Z − jX S S * + S S * = o + o  11 12 21 22 jX + 2Z − jX + 2Z jX + 2Z − jX + 2Z ∑ S S * =  o o o o = 0 ij ik 2Z − jX jX 2Z j≠k S S * + S S * = o + o  12 11 22 21 + − + + − +  jX 2Z o jX 2Z o jX 2Z o jX 2Z o

7-20 微波工程講義 7. Ex.7.16 1 − 1 − − Y1 Yo Yo Y1 Y Y = Y + Y , S = S = = = ZO Y ZO 1 o 11 22 1 1 + + + Yo Y1 2Yo Y Y1 Yo V V V V +V 2Y S = ref ,2 = 2 = 1 = in,1 ref ,1 = 1+ S = o = S 21 11 + 12 Vin,1 Vin,1 Vin,1 Vin,1 2Yo Y Vin,2=0 Vin,2=0 Vin, 2=0 Vin,2=0 I I1 2 8. Ex.7.17 Z o − 2 2 Z 2 ZO V V ZO n Z − Z n −1 2 o 1− n 1 2 S = o o = , S = n = 11 n 2 Z + Z n 2 +1 22 Z 1+ n 2 n:1 o o o + Z n 2 o V V V / n 1 V +V 1+ S 2n S = ref ,2 = 2 = 1 = in,1 ref ,1 = 11 = = S 21 2 + 12 Vin,1 Vin,1 Vin,1 n Vin,1 n n 1 Vin,2=0 Vin,2=0 Vin, 2=0 Vin,2=0

7-21 微波工程講義 9. Ex.7.18 − Z o Z o a1 S = S = = 0 b2 11 22 + γ γ Z o Z o Zo, − b1 a2 V V e l = ref ,2 = in,1 = −γl = S 21 e S12 Vin,1 Vin,1 l Vin,2=0 Vin,2=0

10. Ex.7.19 j50 50× (− j25) Z = j50 + = j50 +10 − j20 = 10 + j30 1 − 50 j25 ΖΟ=50 -j25 ΖΟ=50 10 + j30 − 50 S = = 0.74∠117° 11 10 + j30 + 50 (50 + j50) × (− j25) 10 − j30 − 50 Z = = 10 − j30, S = = 0.74∠ −117° 2 50 + j50 − j25 22 10 − j30 + 50 10 − j20 V1 Vref ,2 V 10 + j30 10 − j20 − 40 + j30 S = = 2 = = (1+ ) = 0.67∠ − 90° = S 21 + + 12 Vin,1 Vin,1 Vin,1 10 j30 60 j30 Vin,2=0 Vin,2=0

Vin,2=0

7-22 微波工程講義 Example: T-Network

I1 R RI2 1 2 ==Ω=Ω RR1225 , R 3 100 Z0 =Ω75 V1 R3 V2 S-Parameters ???

S11 , S21 R R =+ + =Ω 1 2 Zin RRR1320() Z 75

SS11=Γ==in 0 22 Zin R3 Z0 + Z RR32() Z 0 1 S ==0 21 + RZ20RRR1320++() Z 2 = SS12 21 1/16/2003 5 Lecture 5. ELG4105: Microwave Circuits © S. Loyka, Winter 2003 The Scattering

EXAMPLE Find the S-parameters of the 3 dB attenuator circuit.

− + ⎡VV11⎤⎡⎤⎡⎤SS11 12 ⎢ − ⎥⎢⎥= ⎢⎥+ ⎣VV22⎦⎣⎦⎣⎦SS21 22

(1) Z (2) Sol Zin in

„ S11 can be found as the reflection coefficient seen at port 1 when port 2 (2) is terminated in a matched loads (Note that Zo= 50Ω and Zin =50Ω)

(1) „ But, Z in = 8.56 + [141 . 8(8.56 + 50)]/ (141 . 8 + 8.56 + 50) = 50Ω ,

so S 11 = 0 .

„ Because of the symmetry of the circuit, S 22 = 0 . The Scattering Matrix

− + ⎡VV11⎤⎡⎤⎡⎤SS11 12 ⎢ − ⎥⎢⎥= ⎢⎥+ ⎣VV22⎦⎣⎦⎣⎦SS21 22

(1) Z (2) Zin in

+ „ S21 can be found by applying an incident wave at port 1, V1 , and - measuring the outcoming wave at port 2, V2 . „ This is equivalent to the transmission coefficient from port 1 to port 2 when port 2 is matched,

- „ From the fact that S11=S22=0, we know that V1 =0 when port 2 is + matched to 50Ω. This also implies that V2 =0. + − „ In this case we then have that V 1 = V 1 andV2 =V2 . The Scattering Matrix

− + ⎡VV11⎤⎡⎤⎡⎤SS11 12 ⎢ − ⎥⎢⎥= ⎢⎥+ ⎣VV22⎦⎣⎦⎣⎦SS21 22

(1) Z (2) Zin in

+ „ So by applying a voltage V1 at port 1 ( V 1 = V1 ) and the voltage cross − the 50 load resistor at port 2 is (VV22= )

where 41.44=141. 8(58.56)/(141. 8+58. 56) .

„ Thus, S12 = S21 = 0.707 . + 2 „ If the input power is V1 / 2Z0 , then the output power is

− 2 + 2 2 + 2 + 2 V2 /2Z0 =| S21V1 | /2Z0 = S21 /2Z0 V1 = V1 /4Z0

which is one-self (-3 dB) of the input power. The Scattering Matrix

EXAMPLE A two-port net work is measured and the following scattering matrix is obtained: ⎡ 0.15∠∠− 0oo 0.85 45 ⎤ []S = ⎢ oo⎥ ⎣0.85∠∠ 45 0.2 0 ⎦ a) determine whether the network is reciprocal or lossless. b) If port two is terminated with a matched load, what will be the return loss at port 1? c) If a short-circuit is placed on port 2, what will be the resulting return loss at port 1? Sol a) Since [S] is not symmetry, the net work is reciprocal. To be lossless, the [S] parameters must satisfy (4.53). Since

2222 |SS11 |+ | 22 |=+= 0.15 0.85 0.745 ≠ 1

Thus, the network is not lossless. The Scattering Matrix

b) When port 2 is terminated with matched load, the reflection coefficient at port 1 is Γ = S11 = 0.15, Thus,

„ When port 2R isLdB terminated= −Γ=−=20lo withg| a short | circuit20lo, weg(0.15) have 16.5 . Thus,

+ − „ The second equation gives V2 = −V2

„ Substituting into the first equation we have

(0.85∠− 45oo )(0.85 ∠ 45 ) =−0.15 =− 0.452 10.2+ The Scattering Matrix

„ So the return loss is RL=−20log | Γ | =− 20log(0.452) = 6.9 dB

NOTE

„ The reflection coefficient looking into port n is not equal to Snn unless all other ports are matched.

„ Similarly, the transmission coefficient from port m to port n is not

equal to Snm, unless all other ports are matched.

„ The parameters of a network are properties only of the network itself (assuming the network is linear), and are defined under the condition that all ports are matched. 7.7 Conversion from impedance, admittance, chain, and hybrid parameters to scattering parameters o.r vice versa Discussion 1. See p.288, Table 7.4 2. Derivation ABCD→S V   A B  V  V + Z I V − Z I 1 = 2 , a = i oi i , b = i oi i , Z = Z = Z    −  i i o1 o 2 o  I 1  C D  I 2  2 2Z oi 2 2Z oi = − (1) matched load at port 2, V 2 Z o I 2 V + Z I AV − BI + CZ V − DZ I AV + BV / Z + CZ V + DV = 1 o 1 = 2 2 o 2 o 2 = 2 2 o o 2 2 a1 2 2Z o 2 2Z o 2 2Z o V − Z I AV − BI − CZ V + DZ I AV + BV / Z − CZ V − DV = 1 o 1 = 2 2 o 2 o 2 = 2 2 o o 2 2 b1 2 2Z o 2 2Z o 2 2Z o V − Z I 2V = 2 o 2 = 2 b2 2 2Z o 2 2Z o b AV + BV / Z − CZ V − DV A + B / Z − CZ − D S = 1 = 2 2 o o 2 2 = o o 11 + + + + + + a1 = AV 2 BV 2 / Z o CZ oV 2 DV 2 A B / Z o CZ o D a 2 0 b 2V 2 S = 2 = 2 = 21 + + + + + + a1 = AV 2 BV 2 / Z o CZ oV 2 DV 2 A B / Z o CZ o D a2 0 7-23 微波工程講義 ªV º 1 ªD Bº ª V º V + Z I V − Z I 2 = 1 , a = i oi i , b = i oi i , Z = Z = Z « » ∆ « » «− » i i o1 o 2 o ¬ I 2 ¼ ¬C A¼ ¬ I 1 ¼ 2 2Z oi 2 2Z oi = − (2) matched load at port 1, V1 Z o I 1 V + Z I 1 DV − BI + CZ V − AZ I 1 DV + BV / Z + CZ V + AV a = 2 o 2 = 1 1 o 1 o 1 = 1 1 o o 1 1 2 ∆ ∆ 2 2Z o 2 2Z o 2 2Z o V − Z I 1 DV − BI − CZ V + AZ I 1 DV + BV / Z − CZ V − AV b = 2 o 2 = 1 1 o 1 o 1 = 1 1 o o 1 1 2 ∆ ∆ 2 2Z o 2 2Z o 2 2Z o V − Z I 2V = 1 o 1 = 1 b1 2 2Z o 2 2Z o b DV + BV / Z − CZ V − AV − A + B / Z − CZ + D S = 2 = 1 1 o o 1 1 = o o 22 + + + + + + a 2 = DV 1 BV1 / Z o CZ oV1 AV 1 A B / Z o CZ o D a1 0 b 2∆V 2∆ S = 1 = 1 = 12 + + + + + + a 2 = DV 1 BV1 / Z o CZ oV1 AV 1 A B / Z o CZ o D a1 0 ª + − − ∆ º A B / Z o CZ o D 2 ª º « » S 11 S 12 A + B / Z + CZ + D A + B / Z + CZ + D « » = « o o o o » − A + B / Z − CZ + D ¬S 21 S 22 ¼ « 2 o o » « + + + + + + » ¬ A B / Z o CZ o D A B / Z o CZ o D ¼

7-24 微波工程講義 7.8 Chain scattering (scattering transfer, T-) parameters Basics 1. a  T T b  a = T b + T a 1 = 11 12 2 , 1 11 2 12 2      = + b1  T21 T22 a2  b1 T21b2 T22a2 Discussion 1. See p.289 Table 7.5 for T- and S-parameters conversion b   S S  a   1  =  11 12  1  b2  S 21 S 22 a 2  = (1) matched load at port 2, a 2 0 b = S a , b = S a 1 11 1 2 21 1 Hw #6(due 2 weeks) a a b S a S = 1 = 1 = 1 = 1 = 11 1 = 11 2, 6, 11,21 T11 ,T21 b2 = S 21 a1 S 21 b2 = S 21 a1 S 21 a2 0 a2 0 = (2) matched load at port 1, a1 0 b S + = = − 2 = − 22 T11 b2 T12 a 2 0,T12 T11 a 2 = S 21 a1 0 b b S S − S S S S − S S + 2 = 1 = − = 12 21 11 22 = − 11 22 12 21 T22 T21 ,T22 S 12 T21 S 22 a 2 = a 2 = S 21 S 21 a1 0 a1 0 7-25 微波工程講義 Chapter 9 Signal-flow graphs and applications 9.1 Definitions and manipulation of signal-flow graphs definition, reduction rules 9.2 Signal-flow graph representation of a voltage source 9.3 Signal-flow graph representation of a passive single-port device 9.4 Power gain equations transducer power gain, operating power gain, available power gain

9-1 微波工程講義 9.1 Definitions and manipulation of signal-flow graphs Basics 1. Definitions: signal-flow graph: representation of a linear system node (input and output nodes) : representation of a variable branch: representation of direction and relation between nodes path: a continuous succession of branches traversed in the same direction loop: a path originates and ends at the same node without encountering other nodes more than once along its traverse

9-2 微波工程講義 Discussion 1. Scattering parameters node

1 21 2 a1 b2 a S b

port 1 linear port 2 S11 S22 network b1 a2 b1 a2 S12 reference reference branch plane 1 plane 2 b  S S a  b = S a + S a 1 = 11 12 1 , 1 11 1 12 2      = + b2  S21 S22 a2  b2 S21a1 S22a2 2. Reduction rules (1) Rule 1 5s 2 10s

(2) Rule 2 5s 3 5s+3 9-3 微波工程講義 4 (3) Rule 3 2s 4 1− 2s

4a b = 4a + 2sb → b − 2sb = 4a → b = 1− 2s (4) Rule 4 C2

C2 C4 C1 C4 C1 C4

C3 C3 C4 (5) Rule 5 C2 C1 C2 C4 C4 C1 C C 1 3 C3 C1

9-4 微波工程講義 3. Mason’s gain rule: P ∆ + P ∆ + P ∆ + .... transfer function T(s) = 1 1 2 2 3 3 ∆

Pi : gain of the ith forward path ∆ = 1-∑∑L(1) + L( 2 ) − ∑ L(3) + .... ∆ = (1) + (1) − (1) + 1 1-∑∑L(1) L( 2 ) ∑ L(3) ..... ∆ = (2) + (2) − (2) + 2 1-∑∑L(1) L( 2 ) ∑ L(3) .... ∑ L(1): sum of all first - order loop gains ∑ L( 2 ): sum of all second - order loop gains

(1) ∑ L(1) : sum of all first - order loop gains that do not touch path P1 at any node (1) ∑ L( 2 ) : sum of all second - order loop gains that do not touch path P1 at any node (2) ∑ L(1) : sum of all first - order loop gains that do not touch path P2 at any node 2nd order loop gain : product of 2 first order loops that don't touch at any point 3rd order loop gain : product of 3 first order loops that don't touch at any point

9-5 微波工程講義 4. Ex. 9.5 find the transfer function 6 P2 1 P3 -4 s 11+ s 1 s + 2 113 P1 -3 -5  1 s 3s P = 1×1× ×1× ×3×1 =  1 + + + +  s 1 s 2 (s 1)(s 2) = × × = 3 forward paths  P2 1 6 1 6  1 − 4  P = 1×1× × (−4) ×1 =  3 s +1 s +1  − 3 L =  1 + 2 loops s 1 − 5s L =  2 s + 2 P + P (1− L − L + L L ) + P (1− L ) H(s) = 1 2 1 2 1 2 3 2 − − + 1 L1 L2 L1L2

9-6 微波工程講義 9.2 Signal-flow graph representation of a voltage source

Is 1 Zs bG 1 bs bs as Es Γ Vs Γ s s 1 as bs as

= − + = − + + + Es I s Z s Vs (I s,in I s,ref )Z s (Vs,in Vs,ref ) V −V Z Z = − s,in s,ref + + = − s + + s Z s Vs,in Vs,ref (1 )Vs,in (1 )Vs,ref Zo Zo Zo Z Z − Z → V = o E − o s V s,ref + s + s,in Z o Z s Z o Z s V Z E Z − Z V → s,ref = o s + s o s,in + + 2Z o Zo Z s 2Zo Zo Z s 2Zo → = + Γ bs bG s as

9-7 微波工程講義 9.3 Signal-flow graph representation of a passive single-port device

IL aL 1 aL Γ ΓL VL ZL 1 L bL bL V −V = = + = L,in L,ref VL I L Z L (I L,in I L,ref )Z L Z L Zo = + VL,in VL,ref Z − Z → V = L o V L,ref + L,in Z L Zo V Z − Z V → L,ref = L o L,in + 2Zo Z L Z o 2Z o → = Γ bL L aL

9-8 微波工程講義 Discussion a1 S21 b2 1. Ex.9.6 find Γin

Γ two-port Γin S11 S22 ΓL in ZL network b1 S12 a2 Γ = + Γ + Γ Γ + Γ Γ Γ + (1) in S11 S21 L S12 S21 L S12 S22 L S21 L S12 S22 L S22 L .... S Γ S = S + S Γ S (1+ S Γ + ...) = S + 21 L 12 11 21 L 12 22 L 11 − Γ 1 S22 L (2)Mason's rule  P = S 2 forward paths 1 11 = Γ P2 S21 L S12 = Γ 1 loop L1 S22 L P (1− L ) + P S (1− S Γ ) + S Γ S S Γ S Γ = 1 1 2 = 11 22 L 21 L 12 = S + 21 L 12 in − − Γ 11 − Γ 1 L1 1 S22 L 1 S22 L

9-9 微波工程講義 2. Ex.9.7 find Γ Zs out two-port Γout ZL Vs a1 S21 b2 network

Γs S11 S22 Γout

b1 S12 a2 Γ = + Γ + Γ Γ + Γ Γ Γ + (1) out S22 S21 s S12 S21 s S12 S11 s S21 L S12 S11 s S11 s .... S Γ S = S + S Γ S (1+ S Γ + ...) = S + 21 s 12 22 21 s 12 11 s 11 − Γ 1 S11 s (2)Mason's rule  P = S 2 forward paths 1 22 = Γ P2 S21 s S12 = Γ 1 loop L1 S11 s P (1− L ) + P S (1− S Γ ) + S Γ S S Γ S Γ = 1 1 2 = 22 11 s 21 s 12 = S + 21 s 12 out − − Γ 22 − Γ 1 L1 1 S11 s 1 S11 s

9-10 微波工程講義 3. Ex.9.8 find Pd:power delivered from source, PL:power delivered to the load, Pavs:maximum power available from source source load = + Γ = + Γ = + Γ Γ = + Γ Γ bs aL bs bG as s bG bL s bG aL L s bG bs L s bG 1 1  b  b = G Γs ΓL  s 1− Γ Γ 1 →  L s b − b 1 b b Γ a = s G = ( G − b ) = G L as bL  s Γ Γ − Γ Γ G − Γ Γ  s s 1 L s 1 L s Pd PL 2 2 2 2 2 b b Γ b 2 2 2 P = b − a = G − G L = G (1− Γ ) = b (1− Γ ) = P d s s − Γ Γ − Γ Γ − Γ Γ L s L L 1 L s 1 L s 1 L s = 2 − 2 = 2 − Γ 2 = 2 − Γ 2 PL aL bL aL (1 L ) bs (1 L ) Γ = Γ∗ conjugate match condition : s L

2 2 b 2 b = = G − Γ = G Pavs Pd Γ =Γ∗ 2 (1 s ) 2 s L − Γ − Γ 1 s 1 s

9-11 微波工程講義 4. Ex.9.9 find b3/bs  P = S 2 forward paths 1 31 D = Γ Z P2 S21 L S32 b3 a3  = Γ L1 S11 s a1 b2  = Γ  L2 S22 L Zs  = Γ 3-port L3 S22 D Vs ZL  network  L = S S Γ Γ 8 loop  4 21 12 s L = Γ Γ Γ L5 S31S23S12 s L D b1 a2 L = S S S Γ Γ Γ  6 13 21 32 s L D = Γ Γ ΓD  L7 S23S32 L D  = Γ Γ  L8 S31S13 s D b3 S33 a3 b P ∆ + P ∆ 3 = 1 1 2 2 − + − S31 S23 bs 1 ∑∑∑L(1) L(2) L(3) S32 S31 bG bs a1 b2 ∆ = − Γ ∆ = 1 1 S22 L , 2 1 = + + S21 ∑ L(1) L1 ... L8 Γs 11 22 ΓL S S = + + + + + ∑ L(2) L1L2 L1L3 L2 L3 L3 L4 L1L7 L2 L8 ∑ L(3) = L L L as b1 S12 a2 1 2 3 9-12 微波工程講義 9.4 Power gain equations Basics Zs

Vs [ S ] ZL

Γs Γin Γout ΓL Pavs Pavn Pin PL P ≡ L Γ operating power gain GP (S, L ) Pin P ≡ avn Γ available power gain G A (S, S ) Pavs P ≡ L Γ Γ transducer power gain GT (S, S , L ) Pavs = = Pavs Pin Γ =Γ* , Pavn PL Γ =Γ* in S L out

9-13 微波工程講義 source load Discussion bG bs a1 S21 b2 aL 1. b r b = b + Γ Γ b → b = G = a s G s in s s − Γ Γ 1 ΓL 1 s in Γs S11 S22 − Γ 2 2 2 2 2 (1 in ) 2 P = a − b = a (1− Γ ) = b as b1 a2 bL in 1 1 1 in − Γ Γ 2 G 1 s in S a S b Pavs Pin Pavn PL r a = b = 21 1 = 21 G L 2 − Γ − Γ − Γ Γ 1 S 22 L (1 S 22 L )(1 s in ) S 2 (1− Γ 2 ) P = a 2 − b 2 = a 2 (1− Γ 2 ) = 21 L b 2 L L L L L − Γ 2 − Γ Γ 2 G 1 S 22 L 1 s in 2 − Γ 2 − Γ 2 − Γ 2 2 S 21 (1 out ) 2 2 1 S11 s (1 out ) = ∗ = r − Γ Γ = Pavn PL Γ =Γ bG , 1 s in ∗ L out ∗ 2 2 Γ =Γ ∗ 2 − Γ − Γ Γ L out − Γ 1 S 22 out 1 s in 1 S 22 out S 2 = 21 b 2 − Γ 2 − Γ 2 G 1 S11 s (1 out ) = = 1 2 Pavs Pin Γ =Γ∗ 2 bG s in − Γ 1 s

9-14 微波工程講義 2. 2 2 2 (1− Γ ) S (1− Γ ) P = in b 2 , P = 21 L b 2 in − Γ Γ 2 G L − Γ 2 − Γ Γ 2 G 1 s in 1 S 22 L 1 s in S 2 = = 21 2 = = 1 2 Pavn PL Γ =Γ∗ 2 2 bG , Pavs Pin Γ =Γ∗ 2 bG L out − Γ − Γ s in − Γ 1 S11 s (1 out ) 1 s 2 2 P S (1− Γ ) G = L = 21 L P P − Γ 2 − Γ 2 in 1 S 22 L (1 in ) 2 2 P S (1− Γ ) G = avn = 21 s A P − Γ 2 − Γ 2 avs 1 S11 s (1 out ) 2 2 2 2 2 2 P S (1− Γ )(1− Γ ) S (1− Γ )(1− Γ ) G = L = 21 L s = 21 L s T P − Γ 2 − Γ Γ 2 − Γ 2 − Γ Γ 2 avs 1 S 22 L 1 s in 1 S11 s 1 L out

9-15 微波工程講義 5. Unilateral transducer power gain GTU

Input Output Zo [ S ] matching Go matching Zo circuit Gs circuit GL

Γs Γin Γout ΓL

= → Γ = S12 0 in S11 1− Γ 2 1− Γ 2 G = s S 2 L = G G G TU − Γ 2 21 − Γ 2 S o L 1 S11 s 1 S 22 L

∗ ∗ 1 2 1 S = Γ , S = Γ → G = S 11 s 22 L TU max − 2 21 − 2 1 S11 1 S 22

9-16 微波工程講義 6. A 800MHz amplifier (Zo=50Ω) with S11=0.45∠150°, S12=0.01∠- 10°, S21=2.0∠10°, S22=0.4∠-150°, Zs=20Ω, ZL=30Ω→GT, GP, GA Z − Z Z − Z Γ = s o = −0.429,Γ = L o = −0.25 s − L − Z s Z o Z L Z o S S Γ S S Γ Γ = S + 21 12 L = 0.455∠150.32°,Γ = S + 21 12 s = 0.408∠150.87° in 11 − Γ out 22 − Γ 1 S 22 L 1 S11 s 2 2 P S (1− Γ ) G ≡ L = 21 L = 5.937 = 7.7dB P P − Γ 2 − Γ 2 in (1 in )1 S 22 L 2 2 P S (1− Γ ) G ≡ avn = 21 s = 5.855 = 7.7dB A P − Γ 2 − Γ 2 avs (1 out )1 S11 s 2 2 2 P S (1− Γ )(1− Γ ) G ≡ L = 21 s L = 5.487 = 7.4dB T P − Γ Γ 2 − Γ 2 avs 1 s in 1 S 22 L

9-17 微波工程講義 7. A 2GHz amplifier (Zo=50Ω) with S11=0.97∠-43°, S12=0, S21=3.39∠140°, S22=0.63∠-32°, Γs =0.97∠43°, ΓL =0.63∠32°, → GT, GP, GA = → Γ = Γ = S12 0 in S11 , out S 22 Γ = ∗ Γ = ∗ s S11 , L S 22 2 2 2 2 P S (1− Γ ) P S (1 − Γ ) G ≡ L = 21 L ,G ≡ avn = 21 s P P − Γ 2 − Γ 2 A P − Γ 2 − Γ 2 in (1 in )1 S 22 L avs (1 out )1 S11 s 2 2 2 P S (1− Γ )(1 − Γ ) G ≡ L = 21 s L T P − Γ Γ 2 − Γ 2 avs 1 s in 1 S 22 L

1 2 1 G = G = G = G = S = 322.42 = 25dB T A P TU max − 2 21 − 2 1 S11 1 S 22

Hw #7 (due 2 weeks) 5, 10, 14

9-18 微波工程講義