Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves...

22
Lecture 11 Scattering Parameters

Transcript of Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves...

Page 1: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

Lecture 11

Scattering Parameters

Page 2: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 3: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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)

Page 4: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 5: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 6: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 7: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 8: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 9: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 10: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 11: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 12: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 13: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 14: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 15: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 15

Page 16: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

ElecEng4FJ4 LECTURE 11: SCATTERING PARAMETERS 16

Page 17: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 18: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 19: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 20: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 21: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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

Page 22: Lecture 11 Scattering Parameters - ECE at McMaster S matrix relates the voltage/current waves incident on all ports ... •the S-parameters are the complex reflection and transmission

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