General Network Formulation

V1+ I1+
Z 0 ,1
Port Voltages and Currents
I1 V1− I1−
Vk = Vk+ + Vk− I k = I k+ + I k− V1
port 1
+ +
V I 2 2
I2 V 2
Z 0, 2 + N-port
–
port 2 Network
V2− I 2−
VN
+
–

Characteristic (Port) Impedances

IN
port N
+ − VN+ I N+
V V
k k
Z 0, k = =− + − Z 0, N
I I
k k VN− I N−
1
Note: all current components are defined positive with direction into the positive terminal at each port
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Impedance Matrix
I1 ⎡ V1 ⎤ ⎡ Z11 Z12  Z1N ⎤ ⎡ I1 ⎤
V1 + Port 1 ⎢V ⎥ ⎢ Z
- ⎢ 2 ⎥ = ⎢ 21 Z 22  Z 2 N ⎥⎥ ⎢⎢ I 2 ⎥⎥
I2 ⎢  ⎥ ⎢    ⎥ ⎢  ⎥
V2 + Port 2
- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
N-port ⎣VN ⎦ ⎣ Z N 1 Z N 2  Z NN ⎦ ⎣ I N ⎦
Network
IN [V ] = [Z ][I ]
VN + Port N
- Vi,oc+ Port i
-
Open-Circuit Impedance Parameters
Ij Port j

Vi ,oc N-port
Z ij = Network
2
Ij Port N
I k = 0 for k ≠ j
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Admittance Matrix
I1 ⎡ I1 ⎤ ⎡ Y11 Y12  Y1N ⎤ ⎡ V1 ⎤
V1 + Port 1 ⎢ I ⎥ ⎢ Y ⎥ ⎢ ⎥
- ⎢ 2 ⎥ = ⎢ 21 Y22  Y2 N ⎥ ⎢V2 ⎥
I2 ⎢  ⎥ ⎢    ⎥ ⎢  ⎥
V2 + Port 2
- ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
N-port ⎣ I N ⎦ ⎣YN 1 YN 2  YNN ⎦ ⎣VN ⎦
Network
IN [I ] = [Y ][V ]
VN + Port N
- Ii,sc Port i

Short-Circuit Admittance Parameters +

Vj _ Port j

I i , sc N-port
Yij = Network
Vj Port N
Vk = 0 for k ≠ j 3

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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+ ⎤

⎢ − ⎥ ⎢ ⎢ ⎥
⎢V2 ⎥ = ⎢ S 21 S 22  S 2 N ⎥⎥ ⎢V2+ ⎥ or [V − ] = [S ][V + ]
⎢  ⎥ ⎢    ⎥ ⎢  ⎥
V1+ I1+
⎢ − ⎥ ⎢ ⎥ ⎢ ⎥ Z 0,1
⎢⎣VN ⎥⎦ ⎣ S N 1 SN 2  S NN ⎦ ⎢⎣VN+ ⎦⎥ I1 V1− I1−
V1
+
–

with voltage and current at port n: port 1

V2+ I 2+
I2 V2
Vn = Vn+ + Vn− Z 0, 2 + N-port
–
I n = I n+ + I n! port 2 Network
V2− I 2−
VN
+

+ !
–

IN
= (V ! V n n ) Z0 port N
4 VN+ I N+
Note: S-parameters depend on port impedances Z 0,n = Z 0 Z 0, N
VN− I N−
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Transmission Line Basics
Lossless Transmission Line
V2+
V1+ 0 z
−
Z0,1 1 V V2−Z0,2
+
_
+ + +
_
E1 V1 Z0, ! = "l V2 E2
- -
Port 1 Port 2

V1 = V1+ + V1! V (z) = V0+e! j! z + V0!e+ j! z V2 = V2+ +V2!

I1 = I1+ + I1! + ! j! z ! + j! z I 2 = I 2+ + I 2!
I(z) = I e 0 +I e 0

Phase Constant: ! = " LC

6
L V0+ V0!
Characteristic Impedance: Z0 = = + =! !
C I0 I0
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Net Power Flow on Lossless Line

V0−
I(z') ΓL = +
V0
Z 0 = R0 (real)
+
Z 0 γ = jβ V(z') ZL
–
z' 0

V ( z ' ) = V0+ e + jβz ' + ΓL e − jβz '

( )
V0+ + jβz '
I ( z' ) = e ( − ΓL e − jβz ' )
Z0

2
* 2
1
V0+ #
Pave (z) = 2 Re V (z) ( I (z))
{ } = $1! " L %& = const.
2R0
7

9
Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Generalized Scattering Parameters
considerations and definitions
V+
! !0 E
Z0 V− V+ =
I 2
+
+
_ Z0 ZL
E V V − Z L − Z0
- Γ= + =
V Z L + Z0
+ +
ZL !
V = Z0 I V − = −Z 0 I − V= E
Z L + Z0
V = V + +V − I = I+ + I− 1
V+ = 2
(1 + Γ )E
Z0 V−
+
I
1 P + = 12 Re V + ( I + )*
{ }
V = (V + Z 0 I )
2 +
+
_ ZL 2
E V = 12 V + R0
V − = 12 (V − Z 0 I )
8
-
= Pmax
(assuming Z0 is real Z0 = R0)
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

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+
P = 12 Re V ( I ) = 12 V { } R0 a=
R0

− − − * − 2
V−
1 1 b=
P = − Re V ( I ) = V 2 { } 2 R0 R0

a 2 2
R0 b V+ V−
I
+
PL = P + − P − = −
+
_ ZL 2 R0 2 R0
E V
- 1 2 2
= 2
{a −b }
9
(assuming real Z0)
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Scattering Matrix
R0,1 I1
+
_
+
E1 V1 Port 1 ⎡ b1 ⎤ ⎡ S11 S12  S1N ⎤ ⎡ a1 ⎤
- ⎢b ⎥ ⎢ S
N-port ⎢ 2 ⎥ = ⎢ 21 S 22  S 2 N ⎥⎥ ⎢⎢ a2 ⎥⎥
R0,2 I2 Network ⎢  ⎥ ⎢    ⎥ ⎢  ⎥
+ + ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
E2 _ V2 Port 2 ⎣b2 ⎦ ⎣ S N 1 S N 2  S NN ⎦ ⎣a N ⎦
-

R0,N IN bi
Sij =
+ aj
EN +_ VN Port N ak = 0 for k ≠ j
-

Vi − R0,i Vi R0,i
Sij = =10 (i ≠ j )
E j 2 R0, j
( ) E j 2 R0, j
( )
E k = 0 for all k ≠ j E k = 0 for all k ≠ j

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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Scattering Parameters
2
Physical meaning of S ij ( Z0 = R 0 = real)
2 Pi − actual power leaving port i
Sij = = (i ≠ j )
Pmax, j Ek = 0 maximum power from port j Ek = 0
k≠ j k≠ j

R0,1 I1
2
+
Physical meaning of S jj V1 Port 1
-
N-port
bj V j− R0, j Z L , j − R0, j Network
= = R0,j I
j
S jj = +
aj ak = 0 V j R0, j Z L , j + R0, j + Port j
k≠ j Ej _ Vj +
-
ZL,j
2 R0,N IN
PL , j = Pj+ − Pj− = Pmax 1 − S jj + Port N
{ 11
} VN
-
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Relation to Z-Matrix
Impedance matrix: ⎡ Z11  Z1N ⎤
V = ZI ⎢
with Z =   ⎥
⎢ ⎥
⎢⎣ Z N 1  Z NN ⎥⎦
Express V,I in terms of a and b

R10/ 2 (a + b ) = Z R 0−1/ 2 (a − b ) b = (Zn + U)−1 (Zn − U)a

with normalized impedance matrix Z n = R 0−1/ 2 Z R 0−1/ 2
( Z0 = R 0 = real) ⎡ R0,1 0 0 ⎤
⎢ ⎥ −1
and port impedance matrix R10/ 2 = ⎢ 0  0 ⎥ R0−1/ 2 = R10/ 2 ( )
⎢ 0
⎣ 0 R0, N ⎥⎦

−1 12 −1
S = (Zn + U) (Zn − U)= (Zn − U)(Zn + U)
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

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)
13

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 ⎢S ⎥ Z 0, 2
⎣ 21 S 22 ⎦

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
14
from port 1 to port 2: IL = −20 log10 S21 in dB
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Example
Lossless Transmission Line:
V1+ V2+
−
Z0,1V1 V2−Z0,2
+
_
+ + +
_
E1 V1 Z0 θ V2 E2
- -
Port 1 Port 2

If Z0,1 = Z0,2 = Z0, the scattering parameters can be easily obtained by

inspection:
S11 = S22 = 0 S12 = S21 = e − jθ
⎡1 0⎤ ⎡ 0 e − jθ ⎤
[U ] ± [ S ] = ⎢ ⎥ ± ⎢ − jθ ⎥ −1
⎣0 1⎦ ⎣e 0 ⎦ [Z ] = Z0 ([U ] + [S ])([U ] − [S ]) =
−1
−1 ⎡ 1 − e − jθ ⎤ Z 0 ⎡ 1 e − jθ ⎤ ⎡ 1 e − jθ ⎤
= ⎢ ⎥ ⎢ ⎥ =
([U ] − [ S ]) = ⎢ − jθ ⎥ 1 − e − j 2θ ⎣e − jθ 1 ⎦ ⎣e − jθ 1 ⎦
⎣− e 1 ⎦
1 ⎡ 1 e − jθ ⎤ Z 0 ⎡1 + e − j 2θ 2e − jθ ⎤ ⎡ − jZ 0 cot θ − jZ0 sin θ ⎤
= = ⎢ ⎥ = ⎢
⎢ ⎥ 1 − e − j 2θ ⎣ 152e − jθ 1 + e − j 2θ ⎦ ⎣− jZ 0 sin θ − jZ0 cot θ ⎥⎦
1 − e − j 2θ ⎣e − jθ 1 ⎦
17
Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Properties of S-Parameters
Reciprocal networks:
S ij = S ji or [ S ] = [ S ]T Matrix symmetry!

Symmetrical networks:
Electrical Symmetry
S ii = S jj and S ij = 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

⎡ S S12 ⎤ 18
[ S ] = ⎢ 11 ⎥ [ S ]T [ S ]* = ?
⎣S 21 S 22 ⎦
20
Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Properties of S-Parameters
Reciprocal networks:
S ij = S ji or [ S ] = [ S ]T Matrix symmetry!

Symmetrical networks:
Electrical Symmetry
S ii = S jj and S ij = 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

⎡ S S12 ⎤
[ S ] = ⎢ 11 ⎥ 19
[ S ]T [ S ]* = ?
⎣S 21 S 22 ⎦
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Lossless Two-Port Networks
⎡ S S12 ⎤ T⎡ S S 21 ⎤ *
⎡ S11* S12* ⎤
[ S ] = ⎢ 11 ⎥ [ S ] = ⎢ 11 [ S ] = ⎢ * * ⎥
⎣S 21 S 22 ⎦ ⎣ S12 S 22 ⎥⎦ S
⎣ 21 S 22 ⎦

Then
2 2 * ⎤
⎡ S S 21 ⎤ ⎡ S11* S12* ⎤ ⎡ S11 + S 21 S11S12* + S 21S 22
[ S ]T [ S ]* = ⎢ 11 ⎢ * * ⎥
= ⎢ * 2 2 ⎥
⎣ S12 S 22 ⎥⎦ ⎣ S 21 S 22 ⎦ ⎢⎣ S12 S11* + S 22 S 21 S12 + S 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
* * * * S11 + S21 = 0 + e − jθ = 1
S S +S S =0=S S +S S
12 11 22 21 11 12 21 22
*
20 S12 S11* + S22 S21 = e − jθ ⋅ 0 + e − jθ ⋅ 0 = 0

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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Lossless Two-Port Networks

⎡ S S12 ⎤ T⎡ S S 21 ⎤ *
⎡ S11* S12* ⎤
[ S ] = ⎢ 11 ⎥ [ S ] = ⎢ 11 [ S ] = ⎢ * * ⎥
⎣S 21 S 22 ⎦ ⎣ S12 S 22 ⎥⎦ ⎣S 21 S 22 ⎦
Then
2 2 * ⎤
⎡ S S 21 ⎤ ⎡ S11* S12* ⎤ ⎡ S11 + S 21 S11S12* + S 21S 22
[ S ]T [ S ]* = ⎢ 11 ⎢ * * ⎥
= ⎢ * 2 2 ⎥
⎣ S12 S 22 ⎥⎦ ⎣ S 21 S 22 ⎦ ⎢⎣ S12 S11* + S 22 S 21 S12 + S 22 ⎥⎦

From
for passive condition
unitary(lossy) follows:
networks

2 2 2 2
S11 + S21 ≤=11≥= S12 + S22
* *
S12 S11* + S 22 S21 = 0 = S11S12* + S21S 22
21

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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011
 Applications
Terminated Port: All port (reference)
impedances are
ZS port 1 port 2 Z 0,ref
⎡ S11 S12 ⎤
+
_
E ⎢ ⎥ ZL
⎣S 21 S 22 ⎦
ΓL V2+ = ΓLV2−
one-port network
Γin
V1− = S11V1+ + S12V2+ ΓL S 21
V2+ = ΓL S 21V1+ + ΓL S 22V2+ V2+ = V1+
V2− = S 21V1+ + S22V2+ 1 − ΓL S 22

special case: ZL=0 è ΓL = -1

ΓL S12 S 21
Γin = S11 + S12 S 21
1 − ΓL S 22 22 Γin = S11 −
1 + S 22
(Z s = Z 0,ref )
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Shift in Reference Plane

S12A
S 21A
S11A S 22A
θ1 = βΔl1 Two-Port θ 2 = βΔl2
Z0 Network Z0
V1B + = e − jθ1V1A+ A+
VB+ [SB] V B+
V2A+ V2B + = e − jθ 2V2A+
V1 1 2
B− B−
V1A− V
1 V 2
V2A−
V1B− = e jθ1V1A− Δl1 Δl2 V2B − = e jθ 2V2A−
A B B A
shift in shift in
reference plane reference plane

⎡e − jθ1 0 ⎤ ⎡V1B − ⎤ ⎡ S11A S12A ⎤ ⎡e jθ1 0 ⎤ ⎡V1B + ⎤

⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ 0 e − jθ 2 ⎦ ⎣V2B − ⎦ ⎣S 21A S 22A ⎦ ⎣ 0 e jθ 2 ⎦ ⎣V2B + ⎦

⎡ S11B S12B ⎤ ⎡e jθ1 0 ⎤ ⎡ S11A S12A ⎤ ⎡e jθ1 0 ⎤

⎢ B B ⎥
= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣S 21 S 22 ⎦ ⎣ 0 e jθ 2 ⎦ ⎣23S 21A S 22A ⎦ ⎣ 0 e jθ 2 ⎦
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
Z n = (U + S )(U − S)
−1/ 2 −1/ 2 old
Znew
n = R 0, new Z R 0, new = F Z n F

( Z0 = R 0 = real)
Renormalization matrix

⎡ R old R new 0 0 ⎤
⎢ 0,1 0,1 ⎥
F = ⎢ 0  0 ⎥
⎢ 0 24 0 R0old new ⎥
,N R0, N ⎥
⎣⎢ ⎦
26
Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011

Voltage Transfer Function from

Scattering Parameters
R0 I1 I2
+ ⎡ S11 S12 ⎤ +
+
_ R0
E V1 ⎢S ⎥ V2
- ⎣ 21 S 22 ⎦ -
1 1
V1+ = 2
(V1 + Z 0 I1 ) V2− = 2
(V2 − Z 0 I 2 )
1 + S11 V1 V1 1 − S11
Z in = R0 =I1 =
1 − S11 Z in R0 1 + S11
−
V V − R0 I 2 2V2 V
S 21 = 2+ = 2 = =  = 2 (1 + S11 )
V1 V1 + R0 I1 ⎛ 1 − S11 ⎞ V1
V1 ⎜⎜1 + ⎟⎟
⎝ 1 + S11 ⎠
V2 S
= 21
V1 1 + S2511
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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
Z ij = Z ji Yij = Y ji S ij = S ji AD − BC = 1

§ Lossless Network

T *
Re{B, C}= 0
Re{Z ij }= 0 Re{Yij } = 0 S S =I Im{A, D} = 0
28 2 2
e.g. S11 + S21 = 1
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Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011