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Scattering Parameters Scattering Parameters Motivation § Difficult to implement open and short circuit conditions in high frequencies measurements due to parasitic L’s and C’s § Potential stability problems for active devices when measured in non-operating conditions § Difficult to measure V and I at microwave frequencies § Direct measurement of amplitudes/ power and phases of incident and reflected traveling waves 1 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Scattering Parameters Motivation § Difficult to implement open and short circuit conditions in high frequencies measurements due to parasitic L’s and C’s § Potential stability problems for active devices when measured in non-operating conditions § Difficult to measure V and I at microwave frequencies § Direct measurement of amplitudes/ power and phases of incident and reflected traveling waves 2 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 1 General Network Formulation V + I + 1 1 Z Port Voltages and Currents 0,1 I − − + − + − 1 V I V = V +V I = I + I 1 1 k k k k k k V1 port 1 + + V2 I2 I2 V2 Z + N-port 0,2 – port 2 Network − − V2 I2 + VN – I Characteristic (Port) Impedances port N N + − + + VN I N Vk Vk Z0,k = = − + − Z0,N Ik Ik − − VN I N Note: all current components are defined positive with direction into the positive terminal at each port 3 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Impedance Matrix I1 ⎡V1 ⎤ ⎡ Z11 Z12 Z1N ⎤ ⎡ I1 ⎤ + V1 Port 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - V2 Z21 Z22 Z2N I2 ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ I2 + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V2 Port 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - V Z Z Z I N-port ⎣ N ⎦ ⎣ N1 N 2 NN ⎦ ⎣ N ⎦ Network I N [V]= [Z][I] V + Port N N + - V Port i i,oc- Open-Circuit Impedance Parameters Port j Ij N-port Vi,oc Zij = Network I j Port N Ik =0 for k≠ j 4 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 2 Admittance Matrix I1 ⎡ I1 ⎤ ⎡Y11 Y12 Y1N ⎤ ⎡V1 ⎤ + V1 Port 1 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - I2 Y21 Y22 Y2N V2 ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ I2 + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ V2 Port 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ - I Y Y Y V N-port ⎣ N ⎦ ⎣ N1 N 2 NN ⎦ ⎣ N ⎦ Network I N [I]= [Y][V] V + Port N N - Ii,sc Port i Short-Circuit Admittance Parameters _+ Port j Vj N-port Ii,sc Yij = Network V j Port N Vk =0 for k≠ j 5 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 The Scattering Matrix The scattering matrix relates incident and reflected voltage waves at the network ports as (assume Z 0 ,n = Z 0 ): − + ⎡V1 ⎤ ⎡ S11 S12 S1N ⎤ ⎡V1 ⎤ ⎢ − ⎥ ⎢ ⎥ ⎢ + ⎥ V S S S V − + ⎢ 2 ⎥ = ⎢ 21 22 2N ⎥ ⎢ 2 ⎥ or [V ] =[S][V ] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + + V1 I1 ⎢ − ⎥ ⎢ ⎥ ⎢ + ⎥ Z0,1 ⎢VN ⎥ ⎣SN1 SN 2 SNN ⎦ ⎢VN ⎥ ⎣ ⎦ ⎣ ⎦ I1 V − I − V 1 1 1 + with voltage and current at port n: –port 1 + + V2 I2 + − I2 V2 + Vn = Vn +Vn Z N-port 0,2 – + ! port 2 Network − − In = In + In V2 I2 + VN + ! – I = (Vn !Vn ) Z0 port N N + + VN I N Note: S-parameters depend on port impedances Z Z Z0,N 0,n = 0 V − I − N N 6 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 3 The Scattering Matrix The scattering matrix relates incident and reflected voltage waves at the network ports as (assume Z 0 ,n = Z 0 ): − + ⎡V1 ⎤ ⎡ S11 S12 S1N ⎤ ⎡V1 ⎤ ⎢ − ⎥ ⎢ ⎥ ⎢ + ⎥ V S S S V − + ⎢ 2 ⎥ = ⎢ 21 22 2N ⎥ ⎢ 2 ⎥ or [V ] =[S][V ] ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + + V1 I1 ⎢ − ⎥ ⎢ ⎥ ⎢ + ⎥ Z0,1 ⎢VN ⎥ ⎣SN1 SN 2 SNN ⎦ ⎢VN ⎥ ⎣ ⎦ ⎣ ⎦ I1 V − I − Interpretation? V 1 1 1 + with voltage and current at port n: –port 1 + + V2 I2 + − I2 V2 + Vn Power= Vn +Vn relationships?Z N-port 0,2 – + ! port 2 Network − − In = In + In V2 I2 + VN + ! – I = (Vn !Vn ) Z0 port N N + + VN I N Note: S-parameters depend on port impedances Z Z Z0,N 0,n = 0 V − I − N N 7 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Transmission Line Basics Lossless Transmission Line + + V2 V1 0 z V − V − Z0,1 1 2 Z0,2 + + _+ + E1 V Z , ! = "l V _ E - 1 0 2 - 2 Port 1 Port 2 + ! + ! j!z ! + j!z + ! V1 = V1 +V1 V(z) = V0 e +V0 e V2 = V2 +V2 I = I + + I ! + ! 1 1 1 + ! j!z ! + j!z I2 = I2 + I2 I(z) = I0 e + I0 e Phase Constant: ! = " LC L V + V ! Characteristic Impedance: 0 0 Z0 = = + = ! ! C I0 I0 8 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 4 Net Power Flow on Lossless Line V − I(z') 0 ΓL = + V0 + Z0 = R0 (real) Z γ = jβ V(z') Z 0 – L z' 0 + + jβz' − jβz' V (z') = V0 (e + ΓLe ) + V0 + jβz' − jβz' I(z') = (e − ΓLe ) Z0 2 + * V 2 P (z) 1 Re V(z) I(z) 0 #1 % const. ave = 2 { ( ) } = $ ! "L & = 2R0 9 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Generalized Scattering Parameters considerations and definitions V + E ! ! 0 − + Z0 V V = I 2 + − _+ Z Z E 0 V L V Z L − Z0 - Γ = + = V Z L + Z0 ! Z L + + − − V = E V = Z0 I V = −Z0 I Z L + Z0 + − + − V = V +V I I I 1 = + + 1 E V = 2 ( + Γ) − Z V 0 + 1 + + * I + 1 P = 2 Re{V (I ) } V = 2 (V + Z0 I ) + _+ Z 2 E V L 1 + − 1 = 2 V R0 V = 2 (V − Z0 I ) - = Pmax (assuming Z0 is real Z0 = R0) 10 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 5 Normalized Wave Quantities § It is useful to express power P without characteristic impedance (port impedance) Z0 = R0 (but P still depends on R0) + 2 V + 1 + + * 1 + a = P = 2 Re{V (I ) }= 2 V R0 R0 − 2 V − 1 − − * 1 − b = P = − 2 Re{V (I ) }= 2 V R0 R0 a 2 2 R b + − 0 I V V P = P+ − P− = − + L 2R 2R _+ Z 0 0 E V L - 1 2 2 = 2 {a − b } (assuming real Z0) 11 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Scattering Matrix R0,1 I1 + E _+ Port 1 b S S S a 1 V1 ⎡ 1 ⎤ ⎡ 11 12 1N ⎤ ⎡ 1 ⎤ - ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ N-port b S S S a ⎢ 2 ⎥ = ⎢ 21 22 2N ⎥ ⎢ 2 ⎥ R0,2 I2 Network ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ _+ b S S S a E2 V Port 2 ⎣ 2 ⎦ ⎣ N1 N 2 NN ⎦ ⎣ N ⎦ - 2 R bi 0,N IN S = ij a + + j E _ V Port N ak =0 for k≠ j N - N − Vi R0,i Vi R0,i Sij = = (i ≠ j) E j (2 R0, j ) E j (2 R0, j ) Ek =0 for all k≠ j Ek =0 for all k ≠ j 12 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 6 Scattering Parameters 2 Physical meaning of Sij ( Z 0 = R 0 = real) − 2 Pi actual power leavingport i Sij = = (i ≠ j) E =0 Pmax, j Ek =0 maximumpower fromport j k k≠ j k≠ j R I 2 0,1 1 + Physical meaning of S V Port 1 jj - 1 N-port − bj V j R0, j ZL, j − R0, j Network S R0,j I jj = = + = j a j ak =0 V j R0, j ZL, j + R0, j + Port j E +_ V k≠ j j j- ZL,j 2 R + − 0,N IN PL, j = Pj − Pj = Pmax 1− S jj + { } Port N V - N 13 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Relation to Z-Matrix Impedance matrix: ⎡ Z11 Z1N ⎤ with Z ⎢ ⎥ V = Z I = ⎢ ⎥ ⎢Z Z ⎥ Express V,I in terms of a and b ⎣ N1 NN ⎦ 1/ 2 −1/ 2 −1 R0 (a + b) = Z R0 (a − b) b = (Zn + U) (Zn − U)a −1/ 2 −1/ 2 with normalized impedance matrix Zn = R0 Z R0 ( Z 0 = R 0 = real) ⎡ R0,1 0 0 ⎤ ⎢ ⎥ −1 and port impedance matrix R1/ 2 0 0 −1/2 1/2 0 = ⎢ ⎥ R0 = (R0 ) ⎢ 0 0 R ⎥ ⎣ 0,N ⎦ −1 −1 S = (Zn + U) (Zn − U)= (Zn − U)(Zn + U) 14 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 7 Scattering Parameters § Port n is said to be matched when it is terminated with a load having the same impedance as the port impedance Z0,n. § Often, all port impedances are chosen to be equal and Z0,n = 50 Ω. § The values of the scattering (S-) parameters depend on the chosen port impedances. § S-parameters can be algebraically renormalized to different and unequal port impedances. (see later) 15 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 Two-Port Networks Insertion and Return Loss Z 0,1 ⎡S11 S12 ⎤ _+ E ⎢ ⎥ Z0,2 ⎣S21 S22 ⎦ RL IL Return Loss indicates the extend of mismatch in a network in dB port 1: RL = −20 log10 S11 in dB Insertion Loss measure of transmitted fraction of power in dB from port 1 to port 2: IL = −20 log10 S21 in dB 16 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 8 Example Lossless Transmission Line: + + V2 V1 V − V − Z0,1 1 2 Z0,2 + + _+ + E1 V Z V _ E - 1 0 θ 2 - 2 Port 1 Port 2 If Z0,1 = Z0,2 = Z0, the scattering parameters can be easily obtained by inspection: − jθ S11 = S22 = 0 S12 = S21 = e ⎡1 0⎤ ⎡ 0 e− jθ ⎤ [U] [S] ± = ± ⎢ j ⎥ 1 ⎢0 1⎥ e− θ 0 − ⎣ ⎦ ⎣ ⎦ [Z] = Z0 ([U]+[S])([U]−[S]) = 1 j j − jθ − Z ⎡ 1 e− θ ⎤⎡ 1 e− θ ⎤ −1 ⎡ 1 − e ⎤ 0 [U ]−[S] = = − j2θ ⎢ − jθ ⎥⎢ − jθ ⎥ = ( ) ⎢ − jθ ⎥ 1− e e 1 e 1 ⎣− e 1 ⎦ ⎣ ⎦⎣ ⎦ − j2θ − jθ − jθ Z ⎡1+ e 2e ⎤ ⎡ − jZ0 cotθ − jZ0 sinθ ⎤ 1 ⎡ 1 e ⎤ = 0 = = j2 − j2θ ⎢ − jθ − j2θ ⎥ ⎢ ⎥ − θ ⎢ − jθ ⎥ 1− e 2e 1+ e − jZ0 sinθ − jZ0 cotθ 1− e ⎣e 1 ⎦ ⎣ ⎦ ⎣ ⎦ 17 Prof.
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