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Lecture 11
Scattering Parameters
Scattering Matrix – Common Impedance Definition
the S matrix relates the voltage/current waves incident on all ports (incoming waves) to the voltage/current waves leaving the ports (outgoing or scattered waves)
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 2
Scattering Matrix – Common Impedance Definition
1 11 12 1 1212 2
1
N
N NNN N
VV S S SS VV
S SV V
incident
scattered
for now assume common characteristic impedance at all ports
0 0 , 1, ,nZ Z n N
0 ˆ( ) , 1, ,n nnSV Z ds n N
E h z
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 3
(see L10)
Physical Meaning of Scattering Parameters
0 for all k
iij
j V k j
VSV
• to find Sij, launch an incident wave on port jand measure the wave leaving port i
• meanwhile match all ports so that
0 for all kV k j
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 4
Scattering Matrix: Example 1 (3 dB attenuator)
0 50 Z 0 50 Z
,1141.8 58.568.56 50 141.8 58.56inZ
11 0S
no reflection at port 1
22 11 (symmetry)S S
2,2 02
,1 0111 1 0
,1 01 0 L
inV
inV Z Z
Z ZVSV Z Z
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 5
1V
1V
2V
2 0V
Scattering Matrix: Example 1 (3-dB Attenuator) (2)
0 50 Z 0 50 Z
input power2
11
0
1 | |2av
VPZ
output power2
2 21 122
0
21
10
1 | | | | 0.70722 21
a av vV
Z ZV S PP P
2
221
1 0
141.8 || 58.56 50 0.707(8.56 141.8 || 58.56) 58.56V
VSV
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 6
1V 2V
2 0V 1V
Scattering Matrix: Example 2 (Shunt Element)
2 0V
circuit is symmetric
2
2 22 0121 1
12
0 01 1 10
(1 ) 212 2V
VV V Y YS SV V V Y Y Y Y
port 11
1
10 0Z Y Y 0Y
2
21
,1 ,1in inZ Y
port 2 matched
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 7
0
0 0
0
0 0
22 2
22 2
Y YY Y Y Y
Y YY Y Y Y
S
11V
2,2 0 ,2 0
,1 0 0 ,111 0
,1 0 0 ,1
0 011 22
0 0 0
1
( )( ) 2
L L
in inV
in inZ Z Y Y
Z Z Y YSZ Z Y Y
Y Y Y YS SY Y Y Y Y
Scattering Matrix: Example 3 (Series Element)
2,2 0
0 011 0
0 0
11 220
,1 01
,1 0
2
L
VZ Z
in
in
Z ZSZ Z
Z
Z Z ZZ Z
SZ
Z
SZ
circuit is symmetric
2
2 11 0 0 021 121
0 0 01 1 101
1 1 2(1 )( ) ( ) 2V
V V Z Z ZS V SV V Z Z V Z Z Z Z
port 11
1
0Zport 2
2
2
0ZZ port 1
1
1
0Z
2
2
0ZZ
,1inZ
2 0V
port 2 matched
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 8
0
0 0
0
0 0
22 2
22 2
Z ZZ Z Z Z
Z ZZ Z Z Z
S
22' 2V V
Scattering Matrix: Example 4 (Segment of a TL)
2
2,1 0 2 0 0
11 1 ,1 0 20 2,1 0 2 0 0
22
11 222 22
1 where , 1
( 1)1
Lin s
inV Lin sL
L
Z Z e Z ZS Z ZZ Z e Z Z
eS Se
port 11
1
0 ,Z port 2
2
2
0sZ0sZL
port 11
1
0 ,Z
2
2
0sZ0sZ
,1inZ L
see L08/sl.4
2 0V
port 2 matched
• special case of matched line: 0 0 2 11, 0 0sZ Z S
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 9
g
• note that S22 ≠ Γ2
Scattering Matrix: Example 4 (Segment of a TL) (2)
2matched port 2, 0V
port 11
1
0 ,Z
2
2
0sZ0sZ
,1inZ L
21
210
2
VVS V
22 2(1 )iV V
111 11 2 2(1 ) LLiS V VV e e
2 21
111
LLiV e
Ve
S
211 22 2
212 2
22
122 22
2 22
(1 )(1 1 1
(1 ) 1
) ( 1)1L L
L
LL L
L
L
Se e e e
e S
S ee
e
2iV 2V
1V
special case of matched line: 0 0 2 21, 0 LsZ Z S e
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 10
incident wave translated back to 1-1ꞌ
reflected wave translated back to 1-1ꞌ
Relations between Z and S Matrices
assume all ports have the same characteristic impedance Z0
at each (nth) port
0
( ) /n n n
n n n
V V VZI I I
V V VI I I I V V
0
Z
ZV ZVZI V V V
U is the identity matrix compare with V SV
1( ) ( ) S Z U Z U
for a 1-port network
0/ ZZ Z
11 011
11 0
11
in
in
z Z ZSz Z Z
1 ( ) ( )
( ) ( )
Z U V Z U V
V Z U Z U V
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 11
Relations between Z and S Matrices (2)
the relations S ↔ Z
1( ) ( ) S Z U Z U 1( )( ) Z U S U S
the relations S ↔ Y are derived likewise
there is a one-to-one relation between any two sets of network parameters for a given device (see textbook for conversion formulas, Table 4.2 in Pozar 3rd ed.)
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 12
S-matrix of Reciprocal Networks
1 1
1 1
( ) ( ) ( ) ( ) ( )( ) ( ) ( )
T TT T
S Z U Z U Z U Z UZ U Z U Z U Z U S
T S S
• the scattering matrix is symmetric for reciprocal networks
1( ) ( ) S Z U Z U
• since U = UT (always) and Z = ZT (for reciprocal networks)
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 13
S-matrix of Loss-free Networks• the average power delivered to the network is zero (whatever goes
in, comes out)
*
0
* *
0 real imaginary real
*
0
1 1Re( ) Re ( )( )2 2
1 Re[ ]2
1 ( ) 02
T T Tav
T T T Tav
T Tav
PZ
PZ
PZ
V I V V V V
V V V V V V V V
V V V V
*
0 0
incident power scattered power
1 12 2
T TZ Z
V V V V
(power conservation)
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 14
S-matrix of Loss-free Networks (2)
*TT V VV V ( ) ( )T TT T
U
SV SV S SVV VV1 or ( )T T S S U S S
• the S matrix of a loss-free network is unitary
1 for all ,
N
ki ijkjk
S S i j
1, 0, ij
i ji j
• the product of any column of S with the conjugate of that column is unity; with any other column it is zero
• for a reciprocal loss-free network, the same is true for every row of S1or SS U S S
2 211 21
2 222 12
11 12 21 22
12 11 22 21
2-port network:| | | | 1, 1| | | | 1, 2
0, 1, 20, 2, 1
S S i jS S i j
S S S S i jS S S S i j
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Reference Planes and their Positions along Ports
S-parameters relate incident and scattered voltage/current waves
the position of the port plane impacts the phases of these waves (and their amplitudes, if the ports are lossy transmission lines)
1 1
1 1
1
1
1,0
1,0
L
LVV
VV
ee
1 0z 1 1z L
N Nz L 0Nz
,0
,0
N N
N N
N
NL
N
LNV
VVV
ee
1 1
1 1
1 1,0
1,1 0
L
LVV V
ee
V
1z
Nz
0 , S S
1,0V
1,0V
,0NV
,0NV
original position
shifted position
original position
shifted position
N
,0
,0
N N
N N
N
N
L
LN
N
VVV e
V e
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Reference Planes and their Positions along Ports (2)
1 1
2 2
0
0
' where
0 N N
L
L
L
ee
e
V P V P
1 1
2 2
0
0
' where
0 N N
L
L
L
ee
e
V P V P
1( ) P P
0 0 0 V S Vsubstitute in
10 0' ' ' ( ) ' P V S P V V P S P V
0' '
S
P VP SV
0 S P S P
2 j j
i i j j
Ljj jj
L Lij ij
S S eS S e
loss-free ports, 1, ,k kj k N
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 17
N0NZ, NNV a
,N NV b
Generalized Scattering Parameters
the assumption of identical port impedances (Zk = Z0, k = 1,…,N) is dropped – to accommodate networks terminated with different TLs
the wave quantities at the ports must be now independent of the port impedances
root-power waves (see L10)• incident waves
ˆ( )k
k kkSa ds E h z
• scattered waves
ˆ( )k
k k kSb ds E h z
k k kV Z a
k k kV Z b
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 18
Generalized Scattering Parameters – Relation to Power
average power delivered to a port (port k)
, 0.5Re( )k av k kP V I
( )k k k k kk
k k kkk kk
k k
V V V Z a bV V a bI I I
Z Z
express port voltages and currents in terms of the root-power waves
2 2,
2 2,
0.5Re | | | | ( )0.5 | | 0.5 | |
k av k k k kk k
k av k k
P a b a b a bP a b
power of incident wave power of scattered wave
imaginary
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 19
Generalized and Common-impedance S-parameters
Gb S a G
0 for all k
iij
j a k j
bSa
matched port requirement• in terms of voltage waves
G
0 for all k
ij
V
j
j i k j
i ZVVS
Z
• definition of generalized S matrix
G
0 for all k
jij ij
i V k j
ZS S
Z
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 20
Summary
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 21
• the S-parameters are the complex reflection and transmission coefficients of a network under conditions of impedance match
• the diagonal elements of the S-matrix are the reflection coefficients
• the off-diagonal elements are the transmission coefficients
• the S-parameter magnitudes give the square root of the output-to-input power ratio
• the S-parameter phases give the change in the signal phase as it travels from the input to the output port of the device
• the S-matrix of a reciprocal (linear) device is symmetric
• the S-matrix of a loss-free device is unitary
• the generalized S-matrix is necessary when the device ports have different characteristic impedances
Summary (2)
ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 22
• the root-power waves can be used to compute the power delivered to the device through a given (k-th) port
• changing the reference plane at a port by a length L results in adding a phase factor of exp(−γL) to the respective S-parameter (Lis positive/negative if the reference plane moves away from / toward the device)
2 2, 0.5 | | | |k av k kP a b