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. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
50Ω Transmission Line - 50Ω References
1.0
0.8
0.6
0.4 mag(S (2,1)) mag(S (1,1)) 0.2
0.0 0 2 4 6 8 10 freq, G Hz 18 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
9 50Ω Transmission Line - 100Ω References
1.0
0.8 2 Zin = (50 100)Ω 0.6 = 25Ω
0.4 25 −100 S11 = = −0.6 mag(S (2,1)) mag(S (1,1)) 25 +100 0.2
0.0 0 2 4 6 8 10 freq, G Hz 19 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Properties of S-Parameters Reciprocal networks: T Matrix symmetry! Sij = S ji or [S] = [S] Symmetrical networks: Electrical Symmetry Sii = S jj and Sij = S ji and Matrix symmetry! Lossless networks: For a lossless passive network the scattering matrix [S] is unitary:
transpose complex-conjugate [S]T [S]* =[U]
Example: two-port network
⎡S11 S12 ⎤ T * [S] = ⎢ ⎥ [S] [S] = ? ⎣S21 S22 ⎦ 20 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
10 Properties of S-Parameters Reciprocal networks: T Matrix symmetry! Sij = S ji or [S] = [S] Symmetrical networks: Electrical Symmetry Sii = S jj and Sij = S ji and Matrix symmetry! Lossless networks: For a lossless passive network the scattering matrix [S] is unitary:
transpose complex-conjugate [S]T [S]* =[U]
Example: two-port network
⎡S11 S12 ⎤ T * [S] = ⎢ ⎥ [S] [S] = ? ⎣S21 S22 ⎦ 21 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Lossless Two-Port Networks
* * ⎡S S ⎤ ⎡S11 S21 ⎤ ⎡S S ⎤ [S] = 11 12 [S]T = [S]* = 11 12 ⎢S S ⎥ ⎢S S ⎥ ⎢ * * ⎥ ⎣ 21 22 ⎦ ⎣ 12 22 ⎦ ⎣S21 S22 ⎦ Then 2 2 ⎡S S ⎤⎡S * S * ⎤ ⎡ S + S S S * + S S * ⎤ [S]T [S]* 11 21 11 12 11 21 11 12 21 22 = ⎢ ⎥⎢ * * ⎥ = ⎢ * * 2 2 ⎥ S S S S S S S S S S ⎣ 12 22 ⎦⎣ 21 22 ⎦ ⎣⎢ 12 11 + 22 21 12 + 22 ⎦⎥
From unitary condition follows: Example: lossless TL
⎡ 0 e− jθ ⎤ 2 2 2 2 [S] = ⎢ − jθ ⎥ S11 + S21 =1= S12 + S22 ⎣e 0 ⎦ 2 2 2 2 − jθ * * * * S11 + S21 = 0 + e =1 S12S11 + S22S21 = 0 = S11S12 + S21S22 * * − jθ − jθ S12S11 + S22S21 = e ⋅0 + e ⋅0 = 0
22 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
11 Lossless Two-Port Networks
* * ⎡S S ⎤ ⎡S11 S21 ⎤ ⎡S S ⎤ [S] = 11 12 [S]T = [S]* = 11 12 ⎢S S ⎥ ⎢S S ⎥ ⎢ * * ⎥ ⎣ 21 22 ⎦ ⎣ 12 22 ⎦ ⎣S21 S22 ⎦ Then 2 2 ⎡S S ⎤⎡S * S * ⎤ ⎡ S + S S S * + S S * ⎤ [S]T [S]* 11 21 11 12 11 21 11 12 21 22 = ⎢ ⎥⎢ * * ⎥ = ⎢ * * 2 2 ⎥ S S S S S S S S S S ⎣ 12 22 ⎦⎣ 21 22 ⎦ ⎣⎢ 12 11 + 22 21 12 + 22 ⎦⎥
Fromfor passive unitary (lossy) condition networks follows:
2 2 2 2 S11 + S21 ≤=11≥= S12 + S22 * * * * S12S11 + S22S21 = 0 = S11S12 + S21S22
23 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Applications Terminated Port: All port (reference) impedances are
ZS port 1 port 2 Z0,ref
⎡S11 S12 ⎤ _+ E ⎢ ⎥ ZL ⎣S21 S22 ⎦
+ − Γ V2 = ΓLV2 Γin one-port network L − + + Γ S V1 = S11V1 + S12V2 + + + + L 21 + V2 = ΓLS21V1 + ΓLS22V2 V2 = V1 − + + 1− Γ S V2 = S21V1 + S22V2 L 22
special case: ZL=0 è ΓL = -1 Γ S S S L 12 21 Γin = 11 + S12S21 1− ΓL S22 Γin = S11 − 1+ S22 (Zs = Z0,ref ) 24 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
12 Shift in Reference Plane A S12 A S21 A A S11 S22 θ1 = βΔl1 Two-Port θ2 = βΔl2 Network Z0 Z0 B+ − jθ1 A+ B B+ − jθ2 A+ V1 = e V1 A+ B+ [S ] B+ A+ V2 = e V2 V1 V1 V2 V2 A− B− B− A− V V V2 V2 B− jθ1 A− 1 1 B− jθ A− V = e V Δl1 Δl2 2 1 1 A B B A V2 = e V2 shift in shift in reference plane reference plane
⎡e− jθ1 0 ⎤⎡V B− ⎤ ⎡S A S A ⎤ ⎡e jθ1 0 ⎤⎡V B+ ⎤ 1 = 11 12 1 ⎢ − jθ2 ⎥⎢ B− ⎥ ⎢ A A ⎥ ⎢ jθ2 ⎥⎢ B+ ⎥ ⎣ 0 e ⎦⎣V2 ⎦ ⎣S21 S22 ⎦ ⎣ 0 e ⎦⎣V2 ⎦
⎡S B S B ⎤ ⎡e jθ1 0 ⎤⎡S A S A ⎤⎡e jθ1 0 ⎤ 11 12 = 11 12 ⎢ B B ⎥ ⎢ jθ2 ⎥⎢ A A ⎥⎢ jθ2 ⎥ ⎣S21 S22 ⎦ ⎣ 0 e ⎦⎣S21 S22 ⎦⎣ 0 e ⎦ 25 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
S-Matrix Renormalization
−1 −1 S = (Zn + U) (Zn − U)= (Zn − U)(Zn + U)
−1 Zn = (U +S)(U −S)
new −1/2 −1/2 old Zn = R0,new Z R0,new = F Zn F
( Z = R = real) Renormalization matrix 0 0
⎡ Rold Rnew 0 0 ⎤ ⎢ 0,1 0,1 ⎥ F = ⎢ 0 0 ⎥ ⎢ 0 0 Rold Rnew ⎥ ⎣⎢ 0,N 0,N ⎦⎥
26 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
13 Example
Measurement j1 j0.75 Initial ECM j1.5 j0.5 Perturbed & Augmented ECM j2
j2.5
j3 j0.25 S j4 11 j5
S 22
0 S 0.1 0.2 0.4 0.5 0.6 0.8 1 21 2 4 8 16 32 Circuit Model -j5 -j0.25 -j4
-j3 S -j2.5 22 125Ω 155-j0.5Ω -j2 -j1.5 -j0.75 -j1
27 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Comparison of Different Port Normalizations
Z0,1 =125Ω Z0,2 =155Ω Z0,1 = Z0,2 = 50Ω
S11
28 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
14 Voltage Transfer Function from Scattering Parameters I R0 1 I2
+ ⎡S11 S12 ⎤ + _+ V V R E 1 ⎢S S ⎥ 2 0 - ⎣ 21 22 ⎦ -
+ 1 V − 1 V Z I V1 = 2 (V1 + Z0 I1 ) 2 = 2 ( 2 − 0 2 )
1+ S11 V1 V1 1− S11 Zin = R0 I1 = = 1− S11 Zin R0 1+ S11 − V2 V2 − R0I2 2V2 V2 S21 = + = = = = (1+ S11 ) V1 V1 + R0I1 ⎛ 1− S ⎞ V1 V 1 11 1⎜ + ⎟ ⎝ 1+ S11 ⎠ V S 2 = 21 V1 1+ S11 29 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Example
Matched 3dB attenuator (Z0 = 50 Ω) (Ref: Pozar pp. 175/6)
Z0 = 50Ω
E _+ 1 Z0 = 50Ω
Zin,1 1 V1 = E1 = 2 E1 Zin,1 = 41.444Ω + 8.56Ω ≈ 50Ω Zin,1 + Z0 ⎛ 41.444 ⎞⎛ 50 ⎞ V2 = V1⎜ ⎟⎜ ⎟ = 0.7077V1 ⎝ 41.444 + 8.56 ⎠⎝ 50 + 8.56 ⎠
S11 = S22 = 0 S12 = S21 = 0.7077 IL = −20 log10 S21 = 3dB
30 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
15 Example using Z-Matrix
Z0 = 50Ω
E _+ 1 Z0 = 50Ω
⎡150.36 141.80⎤ Z = ⎢ ⎥ Ω ⎣141.80 150.36⎦ Z0,1 = Z0,2 = 50Ω ⎡3.0072 2.8360⎤ 0 0.7077 Z R−1/ 2 Z R−1/ 2 ⎡ ⎤ n = 0 0 = ⎢ ⎥ S = ⎢ ⎥ ⎣2.8360 3.0072⎦ ⎣0.7077 0 ⎦
Z0,1 = 50Ω Z0,2 =100Ω ⎡3.0072 2.0054⎤ ⎡0.1670 0.6672 ⎤ Z R−1/ 2 Z R−1/ 2 S = n = 0 0 = ⎢ ⎥ ⎢0.6672 0.3333⎥ ⎣2.0054 1.5036⎦ ⎣ − ⎦ 31 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Z L − Z0,2 100 −50 1 Attenuator terminatedExample in ZL=100Ω andusing Z-Matrix ΓL = = = S-Parameters wrt Z0,1=Z0,2=50Ω ZL + Z0,2 100 + 50 3 Z0 = 50Ω
ΓL S12S21 1 1 1 Γin = S11 + = ΓLS12S21 = = 0.167 E _+ 1− ΓLS22 3 2 2 1 Z0 = 50Ω Z Z Attenuator terminated in ZL=100Ω and L − 0,2 ΓL = = 0 S-Parameters wrt Z0,1=50Ω Z0,2=100Ω Z + Z ⎡150.36 141.80L ⎤ 0,2 S S ΓLZ12= 21⎢ ⎥ Ω Γin = S11 + 141= S11.80= 0.167150.36 1− ΓL S22⎣ ⎦ Z0,1 = Z0,2 = 50Ω ⎡3.0072 2.8360⎤ 0 0.7077 Z R−1/ 2 Z R−1/ 2 ⎡ ⎤ n = 0 0 = ⎢ ⎥ S = ⎢ ⎥ ⎣2.8360 3.0072⎦ ⎣0.7077 0 ⎦
Z0,1 = 50Ω Z0,2 =100Ω ⎡3.0072 2.0054⎤ ⎡0.1670 0.6672 ⎤ Z R−1/ 2 Z R−1/ 2 S = n = 0 0 = ⎢ ⎥ ⎢0.6672 0.3333⎥ ⎣2.0054 1.5036⎦ ⎣ − ⎦ 32 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
16 Conversion between Network Parameters
33 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
Properties of Network Parameters
§ Symmetric Two-Port Network
Z11 = Z22 Y11 =Y22 S11 = S22 A = D
assuming the same port impedances
§ Reciprocal Network
Zij = Z ji Yij = Y ji Sij = S ji AD − BC =1
§ Lossless Network Re{B,C}= 0 Re Z 0 Re Y 0 ST S* I { ij}= { ij}= = Im{A, D}= 0 2 2 e.g. S11 + S21 =1
34 Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
17