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Chapter 7 Two-Port Network Chapter 7 Two-port network 7.1 Impedance parameters definition, examples 7.2 Admittance parameters 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 Scattering parameters 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 6. Ex. 7.4 I1 I2 port1: source V ,port 2 : open Γ = 1 in open γ −2γl −γl Zo, 2 → = + = 1 V V1 Vin Vine ,V2 2Vine V V V e−2γl = in − in = I1 , I 2 0 l Z o Zo V V +V e−2γl eγl + e −γl Z Z = 1 = in in = Z = o = Z 11 −2γl o γl − −γl γ 22 I1 = Vin Vine e e tanh l I2 0 − Z o Zo V 2V e−γl 2 Z Z = 2 = in = Z = o = Z 21 −2γl o γl − −γl γ 12 I1 = Vin Vine e e sinh l I2 0 − Z o Zo Z Z o o Z Z β β = = o = = o []= j tan l j sin l lossless line Z11 Z 22 , Z12 Z 21 , Z j tan βl j sin βl Z o Z o j sin βl 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 5. Ex.7.8 I1 I2 port1: source V ,port 2 : short Γ = −1 in short Z ,γ −2γl o 2 → = − = 1 V V1 Vin Vine ,V2 0 V V V e−2γl 2V = in + in = − in −γl I1 , I 2 e l Z o Zo Z o V V e −2γl in + in I Z Z 1 1+ e −2γl 1 Y = 1 = o o = = = Y 11 − −2γl − −2γl γ 22 V1 = Vin Vine Z o 1 e Z o tanh l V2 0 2V − in e −γl I Z 1 2 1 Y = 2 = o = − = − = Y 21 − −2γl γl − −γl γ 12 V1 = Vin Vin e Z o e e Z o sinh l V2 0 1 j 1 j jZ tan βl Z sin βl lossless line Y = Y = ,Y = Y = ,[]Y = o o 11 22 β 12 21 β j 1 jZ o tan l Z o sin l β 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 gain 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=⋅ +⋅ VVAB 122 12=⋅ =⋅ +⋅ ICVDI122 II12CD !!! I2 1) [ ]− I1 ABCD ? Two Port V1 V Network 2 − 2) Z [ABCD] ? 3) Z [ABCD]− ? 1/9/2003 7 ELG4105: Microwave Circuits © S.
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