ece580 guest-lecture f2011 - classesclasses.engr.oregonstate.edu/eecs/fall2017/ece580... · prof....

1

Prof. Andreas Weisshaar ― ECE580 Network Theory - Guest Lecture ― Fall Term 2011 1

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



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


General Network Formulation

−

−

+

+

−==k

k

k

kk I

VIVZ ,0

Characteristic (Port) Impedances

−+ += kkk VVV −+ += kkk IIIPort Voltages and Currents

+ –

port 1

port 2

port N

+ – I2

I1

IN

V2

V1

VN

Note: all current components are defined positive with direction into the positive terminal at each port

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z

N-port Network


Impedance Matrix

+ - V1

I1

+ - V2

+ - VN

I2

IN

Port 1

Port 2

Port N

N-port Network

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

NNNNN

N

N

N I

II

ZZZ

ZZZZZZ

V

VV

2

1

21

22221

11211

2

1

[ ] [ ][ ]IZV =

jkIj

ociij

kIV

Z≠=

=for0

,

+ - Vi,oc

Ij

Port i

Port j

Port N

N-port Network

Open-Circuit Impedance Parameters

3


Admittance Matrix

+ - V1

I1

+ - V2

+ - VN

I2

IN

Port 1

Port 2

Port N

N-port Network

jkVj

sciij

kVI

Y≠=

=for0

,

Ii,sc Port i

Port j

Port N

N-port Network

Short-Circuit Admittance Parameters

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

NNNNN

N

N

N V

VV

YYY

YYYYYY

I

II

2

1

21

22221

11211

2

1

+ _ Vj

[ ] [ ][ ]VYI =


+–

port 1

port 2

port N

+–

+–

I2

I1

IN

V2

V1

VN

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z

N-portNetwork

+–

port 1

port 2

port N

+–

+–

I2

I1

IN

V2

V1

VN

+–+–

port 1

port 2

port N

+–

+–

+–

+–

I2

I1

IN

V2

V1

VN

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z

−−22 IV

++22 IV

2,0Z

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z++11 IV

−−11 IV

1,0Z

N-portNetwork

The Scattering Matrix The scattering matrix relates incident and reflected voltage waves at the network ports as (assume ):

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

+

+

−

−

−

NNNNN

N

N

N V

VV

SSS

SSSSSS

V

VV

2

1

21

22221

11211

2

1

−+ += nnn VVVIn = In

+ + In!

= Vn+ !Vn

!( ) Z0

with voltage and current at port n:

][][][ +− = VSVor

Note: S-parameters depend on port impedances Z0,n = Z0

Z0,n = Z0

4


+–

port 1

port 2

port N

+–

+–

I2

I1

IN

V2

V1

VN

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z

N-portNetwork

+–

port 1

port 2

port N

+–

+–

I2

I1

IN

V2

V1

VN

+–+–

port 1

port 2

port N

+–

+–

+–

+–

I2

I1

IN

V2

V1

VN

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z

−−22 IV

++22 IV

2,0Z

−−22 IV

++22 IV

2,0Z

++NN IV

−−NN IV NZ ,0

++NN IV

−−NN IV NZ ,0

++11 IV

−−11 IV

1,0Z++11 IV

−−11 IV

1,0Z

N-portNetwork

The Scattering Matrix The scattering matrix relates incident and reflected voltage waves at the network ports as (assume ):

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

+

+

+

−

−

−

NNNNN

N

N

N V

VV

SSS

SSSSSS

V

VV

2

1

21

22221

11211

2

1

−+ += nnn VVVIn = In

+ + In!

= Vn+ !Vn

!( ) Z0

with voltage and current at port n:

][][][ +− = VSVor

Note: S-parameters depend on port impedances Z0,n = Z0

Z0,n = Z0

Interpretation? Power relationships?


Transmission Line Basics

+ - V1

Port 1 Port 2

+ _ E1

Z0,1

+ _ E2

Z0,2

+1V

+2V

−2V

−1V

+ - V2 Z0, ! = "l

Lossless Transmission Line

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

!e+ j!z

I(z) = I0+e! j!z + I0

!e+ j!z

Z0 =LC=V0

+

I0+= !

V0!

I0!

Characteristic Impedance:

! =" LCPhase Constant:

V2 =V2+ +V2

!

I2 = I2+ + I2

!

V1 =V1+ +V1

!

I1 = I1+ + I1

!

z0

5


Net Power Flow on Lossless Line

Pave (z) = 12 Re V (z) I(z)( )*{ }=

V0+ 2

2R01! "L

2#$

%&= const.

( )''0)'( zj

Lzj eeVzV ββ −++ Γ+=

( )''

0

0)'( zjL

zj eeZVzI ββ −+

+

Γ−=

'z 0

LZβγ jZ =0

+

−

=Γ0

0

VV

L)'(zI

+–)'(zV

'z 0

LZβγ jZ =0

+

−

=Γ0

0

VV

L)'(zI )'(zI

+–)'(zV

+–)'(zVZ0 = R0 (real)


Generalized Scattering Parameters considerations and definitions

I + _ E

0Z+

- V LZ0Z

+V−V

2EV =+

−+ += VVV −+ += III

++ = IZV 0−− −= IZV 0

( )IZVV 021 +=+

( )IZVV 021 −=−

0

0

ZZZZ

VV

L

L

+

−==Γ

+

−

( )E

EZZ

ZVL

L

Γ+=

+=

1210

{ }

max

0

2

21

*21 )(Re

P

RV

IVP

=

=

=

+

+++

+ _ E

0Z+

- V LZ

+V−VI

(assuming Z0 is real Z0 = R0)

!

!! 0

6


Normalized Wave Quantities

+ _ E

0R+

- V LZ

ab

I

{ } 0

2

21*

21 )(Re RVIVP ++++ ==

§  It is useful to express power P without characteristic impedance (port impedance) Z0 = R0 (but P still depends on R0)

0RVa

+

=

{ } 0

2

21*

21 )(Re RVIVP −−−− =−=

0RVb

−

=

{ }2221

0

2

0

2

22

ba

RV

RV

PPPL

−=

−=−=−+

−+

(assuming real Z0)


Scattering Matrix

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

NNNNN

N

N

a

aa

SSS

SSSSSS

b

bb

2

1

21

22221

11211

2

2

1

jkaj

iij

kabS

≠=

=for0

( ) ( ) )(22

allfor0,0

,0

allfor0,0

,0 jiRERV

RERV

SjkEjj

ii

jkEjj

iiij

kk

≠==

≠=≠=

−

+ - V1

I1

Port 1

Port 2

Port N

N-port Network

+ _ E1

R0,1

+ - V2

I2

+ _ E2

R0,2

+ - VN

IN

+ _ EN

R0,N

7



)(port frompowermaximumportleavingpoweractual

00max,

2ji

ji

PPS

jkE

jkEj

iij

kk

≠==≠=

≠=

−

Physical meaning of 2

ijS

Physical meaning of 2

jjS

jjL

jjL

jj

jj

jkaj

jjj RZ

RZRV

RVab

Sk ,0,

,0,

,0

,0

0 +

−===

+

−

≠=

{ }2max, 1 jjjjjL SPPPP −=−= −+

( = real) 00 RZ =

+-V1

I1

Port 1

Port j

Port N

N-portNetwork

R0,1

+-VjIj

+_Ej

R0,j

+-VN

INR0,NZL,j

+-V1

I1

Port 1

Port j

Port N

N-portNetwork

R0,1

+-VjIj

+_Ej

R0,j

+-VN

INR0,NZL,j


Relation to Z-Matrix

IZV =

( ) ( )baRZbaR −=+ − 2/10

2/10

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

NR

R

,0

1,02/1

0

000000

R

( ) ( )aUZUZb −+= −nn

1

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

NNN

N

ZZ

ZZ

1

111

Z

( ) ( ) ( )( ) 11 −− +−=−+= UZUZUZUZS nnnn

2/10

2/10

−−= RZRZn

Impedance matrix:

with

Express V,I in terms of a and b

with normalized impedance matrix

and port impedance matrix ( ) 12/10

2/10

−− = RR

( = real) 00 RZ =

8



§  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)


Two-Port Networks Insertion and Return Loss

+ _ E

1,0Z

2,0Z⎥⎦

⎤⎢⎣

⎡

2221

1211

SSSS

RL IL Return Loss

indicates the extend of mismatch in a network in dB

dBinlog20 1110 SRL −=

Insertion Loss measure of transmitted fraction of power in dB

dBinlog20 2110 SIL −=

port 1:

from port 1 to port 2:

9


Example

+ - V1

Port 1 Port 2

+ _ E1

Z0,1

+ _ E2

Z0,2

+1V

+2V

−2V

−1V

+ - V2 θ0Z

Lossless Transmission Line:

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

02211 == SS θjeSS −== 2112

⎥⎦

⎤⎢⎣

⎡±⎥⎦

⎤⎢⎣

⎡=±

−

−

00

1001

][][θ

θ

j

j

ee

SU

( )

⎥⎦

⎤⎢⎣

⎡

−=

⎥⎦

⎤⎢⎣

⎡

−

−=−

−

−

−

−

−

−−

11

11

11

][][

2

11

θ

θ

θ

θ

θ

j

j

j

j

j

ee

e

ee

SU

( )( ) =−+= −10 ][][][][][ SUSUZZ

⎥⎦

⎤⎢⎣

⎡

+

+

−=

=⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

−=

−−

−−

−

−

−

−

−

−

θθ

θθ

θ

θ

θ

θ

θ

θ

2

2

20

20

1221

1

11

11

1

jj

jj

j

j

j

j

j

j

eeee

eZ

ee

ee

eZ

⎥⎦

⎤⎢⎣

⎡

−−

−−=

θθ

θθ

cotsinsincot

00

00

jZjZjZjZ


50Ω Transmission Line - 50Ω References

2 4 6 80 10

0.2

0.4

0.6

0.8

0.0

1.0

freq, GHz

mag

(S(1,1))

mag

(S(2,1))

10


50Ω Transmission Line - 100Ω References

2 4 6 80 10

0.2

0.4

0.6

0.8

0.0

1.0

freq, GHz

mag

(S(1,1))

mag

(S(2,1)) ( )

Ω=

Ω=

25100502inZ

6.01002510025

11 −=+

−=S


Properties of S-Parameters Reciprocal networks:

Symmetrical networks:

Lossless networks: For a lossless passive network the scattering matrix [S] is unitary:

jiij SS = TSS ][][ =

][][][ * USS T =

Matrix symmetry! or

jjii SS = jiij SS =and

transpose complex-conjugate

Example: two-port network

⎥⎦

⎤⎢⎣

⎡=

2221

1211][SSSS

S ?][][ * =SS T

Electrical Symmetry and Matrix symmetry!

11


Properties of S-Parameters Reciprocal networks:

Symmetrical networks:

Lossless networks: For a lossless passive network the scattering matrix [S] is unitary:

jiij SS = TSS ][][ =

][][][ * USS T =

Matrix symmetry! or

jjii SS = jiij SS =and

transpose complex-conjugate

Example: two-port network

⎥⎦

⎤⎢⎣

⎡=

2221

1211][SSSS

S ?][][ * =SS T

Electrical Symmetry and Matrix symmetry!


Lossless Two-Port Networks

⎥⎦

⎤⎢⎣

⎡=

2212

2111][SSSS

S T⎥⎦

⎤⎢⎣

⎡= *

22*21

*12

*11*][

SSSS

S

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡= 2

222

12*2122

*1112

*2221

*1211

221

211

*22

*21

*12

*11

2212

2111*][][SSSSSSSSSSSS

SSSS

SSSS

SS T

⎥⎦

⎤⎢⎣

⎡=

2221

1211][SSSS

S

Then

From unitary condition follows:

222

212

221

211 1 SSSS +==+

*2221

*1211

*2122

*1112 0 SSSSSSSS +==+

Example: lossless TL

⎥⎦

⎤⎢⎣

⎡=

−

−

00

][θ

θ

j

j

ee

S

10222

212

11 =+=+ − θjeSS

000*2122

*1112 =⋅+⋅=+ −− θθ jj eeSSSS

12


Lossless Two-Port Networks

⎥⎦

⎤⎢⎣

⎡=

2212

2111][SSSS

S T⎥⎦

⎤⎢⎣

⎡= *

22*21

*12

*11*][

SSSS

S

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++

++=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡= 2

222

12*2122

*1112

*2221

*1211

221

211

*22

*21

*12

*11

2212

2111*][][SSSSSSSSSSSS

SSSS

SSSS

SS T

⎥⎦

⎤⎢⎣

⎡=

2221

1211][SSSS

S

Then

From unitary condition follows:

222

212

221

211 1 SSSS +==+

*2221

*1211

*2122

*1112 0 SSSSSSSS +==+

≥≤1

for passive (lossy) networks


Applications

+ _ E

SZ

LZ⎥⎦

⎤⎢⎣

⎡

2221

1211

SSSS

inΓ

port 1 refZ ,0port 2

Terminated Port:

one-port network

++− += 2121111 VSVSV++− += 2221212 VSVSV

+++ Γ+Γ= 2221212 VSVSV LL

−+ Γ= 22 VV LLΓ

++

Γ−

Γ= 1

22

212 1

VSSVL

L

22

211211 1 S

SSSL

Lin Γ−

Γ+=Γ

special case: ZL=0 è ΓL = -1

22

211211 1 S

SSSin +−=Γ

All port (reference) impedances are

(Zs = Z0,ref )

13


Shift in Reference Plane

−− = AjB VeV 111θ

11 lΔ= βθ Two-PortNetwork[SB]0Z 0Z

AS11 AS22

AS21

AS12

shift in reference plane


A AB B1lΔ 2lΔ

+AV1+BV2−BV2−AV1

+BV1+AV2

−BV1−AV2

Two-PortNetwork[SB]0Z 0Z

AS11 AS22

AS21

AS12



A AB B1lΔ 2lΔ

+AV1+BV2−BV2−AV1

+BV1+AV2

−BV1−AV2

+−+ = AjB VeV 111θ

−− = AjB VeV 222θ

22 lΔ= βθ

+−+ = AjB VeV 222θ

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+

+

−

−

−

−

B

B

j

j

AA

AA

B

B

j

j

VV

ee

SSSS

VV

ee

2

1

2221

1211

2

12

1

2

1

00

00

θ

θ

θ

θ

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡2

1

2

1

00

00

2221

1211

2221

1211θ

θ

θ

θ

j

j

AA

AA

j

j

BB

BB

ee

SSSS

ee

SSSS


S-Matrix Renormalization

( ) ( ) ( )( ) 11 −− +−=−+= UZUZUZUZS nnnn

( )( ) 1−−+= SUSUZn

FZFRZRZ old2/1new,0

2/1new,0

newnn == −−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=new,0

old,0

new1,0

old1,0

000000

NN RR

RRF

Renormalization matrix ( = real) 00 RZ =

14


0.1 0.2 0.4 0.5 0.6 0.8 1 2 4 8 16 32

j0.25

-j0.25

0

j0.5

-j0.5

0

j0.75

-j0.75

0

j1

-j1

0

j1.5

-j1.5

0

j2

-j2

0

j2.5

-j2.5

0

j3

-j3

0

j4

-j4

0

j5

-j5

0

Measurement Initial ECM Perturbed & Augmented ECM

S11

S21

S22

S22

Example

125Ω

Circuit Model

155Ω


Comparison of Different Port Normalizations

Ω=Ω= 155125 2,01,0 ZZ Ω== 502,01,0 ZZ

S11

15


Voltage Transfer Function from Scattering Parameters

+ _ E

0R

0R⎥⎦

⎤⎢⎣

⎡

2221

1211

SSSS

( )10121

1 IZVV +=+ ( )20221

2 IZVV −=−

11

11

0

111 1

1SS

RV

ZVIin +

−==

11

110 11SSRZin −

+=

+

- V1

+

- V2

( )111

2

11

111

2

101

202

1

221 1

111

2 SVV

SSV

VIRVIRV

VVS +==

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+

=+

−==

+

−

11

21

1

2

1 SS

VV

+=

I1 I2


Example Matched 3dB attenuator (Z0 = 50 Ω) (Ref: Pozar pp. 175/6)

Ω= 500Z+ _ E1

Ω= 500Z

121

01,

1,11 E

ZZZ

EVin

in =+

=

112 7077.056.850

5056.8444.41

444.41 VVV =⎟⎠

⎞⎜⎝

⎛+

⎟⎠

⎞⎜⎝

⎛+

=

02211 == SS

Ω≈Ω+Ω= 5056.8444.411,inZ

7077.02112 == SS dB3log20 2110 =−= SIL

16


Example using Z-Matrix

Ω= 500Z+ _ E1

Ω= 500Z

Ω⎥⎦

⎤⎢⎣

⎡=

36.15080.14180.14136.150

Z

⎥⎦

⎤⎢⎣

⎡== −−

0072.38360.28360.20072.32/1

02/1

0 RZRZn ⎥⎦

⎤⎢⎣

⎡=

07077.07077.00

S

⎥⎦

⎤⎢⎣

⎡== −−

5036.10054.20054.20072.32/1

02/1

0 RZRZn ⎥⎦

⎤⎢⎣

⎡

−=

3333.06672.06672.01670.0

S

Ω== 502,01,0 ZZ

Ω=Ω= 10050 2,01,0 ZZ


Example using Z-Matrix

Ω= 500Z+ _ E1

Ω= 500Z

Ω⎥⎦

⎤⎢⎣

⎡=

36.15080.14180.14136.150

Z

⎥⎦

⎤⎢⎣

⎡== −−

0072.38360.28360.20072.32/1

02/1

0 RZRZn ⎥⎦

⎤⎢⎣

⎡=

07077.07077.00

S

⎥⎦

⎤⎢⎣

⎡== −−

5036.10054.20054.20072.32/1

02/1

0 RZRZn ⎥⎦

⎤⎢⎣

⎡

−=

3333.06672.06672.01670.0

S

Ω== 502,01,0 ZZ

Ω=Ω= 10050 2,01,0 ZZ

Attenuator terminated in ZL=100Ω and S-Parameters wrt Z0,1=Z0,2=50Ω

167.021

21

31

1 211222

211211 ==Γ=

Γ−

Γ+=Γ SS

SSSS LL

Lin

Attenuator terminated in ZL=100Ω and S-Parameters wrt Z0,1=50Ω Z0,2=100Ω

167.01 11

22

211211 ==

Γ−

Γ+=Γ S

SSSSL

Lin

31

5010050100

2,0

2,0 =+

−=

+

−=Γ

ZZZZ

L

LL

02,0

2,0 =+

−=Γ

ZZZZ

L

LL

17


Conversion between Network Parameters


Properties of Network Parameters §  Symmetric Two-Port Network

§  Reciprocal Network

§  Lossless Network

jiij ZZ = jiij YY = jiij SS = 1=− BCAD

2211 ZZ = 2211 YY = DA =2211 SS =

assuming the same port impedances

{ } 0Re =ijZ ISS =*T{ } 0Re =ijY

1221

211 =+ SSe.g.

{ } 0,Re =CB{ } 0,Im =DA

ece580 guest-lecture f2011 - classesclasses.engr.oregonstate.edu/eecs/fall2017/ece580... · prof....

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