ECE 598 JS Lecture 09 Scattering Parametersjsa.ece.illinois.edu/ece598js/Lect_09.pdf · ECE 598‐JS, Spring 2012 Copyright © by Jose E. Schutt‐Aine , All Rights Reserved 21 Dissipated

Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012 11

ECE 598 JS Lecture ‐09Scattering Parameters

Spring 2012

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


Transfer Function Representation

Use a two-terminal representation of system for input and output


Y-parameter Representation

1 11 1 12 2

2 21 1 22 2

I y V y VI y V y V

= += +


Y Parameter Calculations

2 2

1 211 21

1 10 0V V

I Iy yV V

= =

= =

To make V2= 0, place a short at port 2


Z Parameters

1 11 1 12 2

2 21 1 22 2

V z I z IV z I z I

= += +


Z-parameter Calculations

2 2

1 211 21

1 10 0I I

V Vz zI I

= =

= =

To make I2= 0, place an open at port 2


H Parameters

1 11 1 12 2

2 21 1 22 2

V h I h VI h I h V

= += +


H Parameter Calculations

To make V2= 0, place a short at port 2

2 2

1 211 21

1 10 0V V

V Ih hI I

= =

= =


G Parameters

1 11 1 12 2

2 21 1 22 2

I g V g IV g V g I

= += +


G-Parameter Calculations

2 2

1 211 21

1 10 0I I

I Vg gV V

= =

= =

To make I2= 0, place an open at port 2


TWO‐PORT NETWORK REPRESENTATION

- At microwave frequencies, it is more difficult to measure total voltagesand currents.

- Short and open circuits are difficult to achieve at high frequencies.

- Most active devices are not short- or open-circuit stable.

1 11 1 12 2V Z I Z I= +

2 21 1 22 2V Z I Z I= +1 11 1 12 2I Y V Y V= +

2 21 1 22 2I Y V Y V= +

Z Parameters Y Parameters


1 11I = i r

o

E EZ− 2 2

2I = i r

o

E EZ−

- Total voltage and current are made up of sums of forward andbackward traveling waves.

- Traveling waves can be determined from standing-wave ratio.

Use a travelling wave approach

Wave Approach

1 1 1i rV E E= + 2 2 2i rV E E= +


11a = i

o

EZ

22a = i

o

EZ

11b = r

o

EZ

22b = r

o

EZ

Zo is the reference impedance of the system

b1 = S11 a1 + S12 a2

b2 = S21 a1 + S22 a2

Wave Approach


111 a2=0

1

S = | ba

221

1 | a2=0

S = ba

112 a1=0

2

S = | ba

222

2 | a1=0

S = ba

To make ai = 01) Provide no excitation at port i2) Match port i to the characteristic impedance of the reference lines.

CAUTION : ai and bi are the traveling waves in the reference lines.

Wave Approach


2

2 2

1111 22( X )S = S =

XΓ

Γ−−

2

2 2

1112 21( )XS = S =

XΓ

Γ−−

c ref

c ref

Z ZZ Z

Γ−

=+

( )( )R j L G j Cγ ω ω= + +

cR j LZG j C

ωω

+=

+

S‐Parameters of TL

lX e γ−=


2

2 2

1111 22( X )S = S =

XΓ

Γ−−

2

2 2

1112 21( )XS = S =

XΓ

Γ−−

c ref

c ref

Z ZZ Z

Γ−

=+

LCβ ω=

cLZC

=

S‐Parameters of Lossless TL

j lX e β−=

If Zc = Zref

011 22S = S = j l

12 21S = S = e β−


N-Port S Parameters

1 11 12 1

2 21 22 2

n nn n

b S S ab S S a

b S a

⋅ ⋅⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥=

⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⋅ ⋅ ⋅⎣ ⎦ ⎣ ⎦ ⎣ ⎦

b = Sa

ii

oi

VaZ

+

=

If bi = 0, then no reflected wave on port i port is matched

ii

oi

VbZ

−

=

iV +

iV −

oiZ

: incident voltage wave in port i

: reflected voltage wave in port i

: impedance in port i


N-Port S Parameters

( ) ( )1o

o

ZZ

=a + b Z a - b

v = Zi( )1

oZ=i a - b

Substitute (1) and (2) into (3)

Defining S such that b = Sa and substituting for b( ) ( )

o oZ Z=U + S a U - S a

( )( ) 1oZ −=Z U + S U - S ( ) ( )o oZ Z= −-1S Z + U Z U

S Z Z S

(3)(2)(1)( )oZ=v a + b

U : unit matrix


N-Port S Parameters

( )-1i = k a - b

If the port reference impedances are different, we define k as

v = k(a + b)

1

2

o

o

on

Z

Z

Z

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⋅⎢ ⎥⎢ ⎥⎣ ⎦

k

( )= -1k(a + b) Zk a - b

( )( )-1Z = k U + S U - S k( )( )-1 -1S = Zk + k Zk - k

S ZZ S

and and


NormalizationAssume original S parameters as S1 with system k1. Then the representation S2 on system k2 is given by

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦-1-1 -1

2 1 1 1 1 2 2 1 1 1 1 2 2S = k (U + S )(U - S ) k k + k k (U + S )(U - S ) k k - k

Transformation Equation

If Z is symmetric, S is also symmetric


Dissipated Power( )1

2dP = T T * *a U - S S a

The dissipation matrix D is given by:T *D = U - S S

Passivity insures that the system will always be stable provided that it is connected to another passive network

For passivity‐ (1) the determinant of D must be‐ (2) the determinant of the principal minors must be

0≥0≥


Dissipated Power

=T *S S U

When the dissipation matrix is 0, we have a lossless network

The S matrix is unitary.

2 211 21 1S S+ =

2 222 12 1S S+ =

For a lossless two‐port:

If in addition the network is reciprocal, then2

12 21 11 22 12and 1S S S S S= = = −


Lossy and Dispersive Line

( )2

11 22 2 2

11

S S−

= =−

α ρ

ρ α( )2

21 12 2 2

11

S S−

= =−

ρ α

ρ α

le−= γα( )( )

c o

c o

Z ZZ Z

−=

+ω

ρω


Frequency-Domain Formulation*

* J. E. Schutt-Aine and R. Mittra, "Scattering Parameter Transient analysis of transmission lines loaded with nonlinear terminations," IEEE Trans. Microwave Theory Tech., vol. MTT-36, pp. 529-536, March 1988.


( ) ( )1 11 1 12 2( ) ( ) ( )B S A S A= +ω ω ω ω ω

( ) ( )2 21 1 22 2( ) ( ) ( )B S A S A= +ω ω ω ω ω

Frequency-Domain


Time-Domain Formulation



1 11 1 12 2( ) ( )* ( ) ( )* ( )b t s t a t s t a t= +

2 21 1 22 2( ) ( )* ( ) ( )* ( )b t s t a t s t a t= +

1 1 1 1 1( ) ( ) ( ) ( ) ( )a t t b t T t g t= Γ +

2 2 2 2 2( ) ( ) ( ) ( ) ( )a t t b t T t g t= Γ +

( )( )

oi

i o

ZT tZ t Z

=+

( )( )( )

i oi

i o

Z t ZtZ t Z

−Γ =

+


Time-Domain Solutions

[ ]'2 22 1 1 1 1

1

1 ( ) (0) ( ) ( ) ( ) ( )( )

( )t s T t g t t M t

a tt

⎡ ⎤− Γ + Γ⎣ ⎦=Δ

[ ]'1 12 2 2 2 2( ) (0) ( ) ( ) ( ) ( )


t

⎡ ⎤Γ + Γ⎣ ⎦+Δ

[ ]'1 11 2 2 2 2

2

1 ( ) (0) ( ) ( ) ( ) ( )( )


a tt

⎡ ⎤− Γ + Γ⎣ ⎦=Δ

[ ]'2 21 1 1 1 1( ) (0) ( ) ( ) ( ) ( )


t

⎡ ⎤Γ + Γ⎣ ⎦+Δ



' ' ' '1 11 2 22 1 12 2 21( ) 1 ( ) (0) 1 ( ) (0) ( ) (0) ( ) (0)t t s t s t s t s⎡ ⎤ ⎡ ⎤Δ = − Γ − Γ − Γ Γ⎣ ⎦ ⎣ ⎦

' (0) (0)ij ijs s= Δτ

1 11 12( ) ( ) ( )M t H t H t= +

2 21 22( ) ( ) ( )M t H t H t= +

1

1

( ) ( ) ( )t

ij ij jH t s t a−

=

= − Δ∑τ

τ τ τ

' '1 11 1 12 2 1( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t= + +

' '2 21 1 22 2 2( ) (0) ( ) (0) ( ) ( )b t s a t s a t M t= + +


Special Case – Lossless Line

11 22( ) ( ) 0s t s t= = 12 21( ) ( ) ls t s t tv

⎛ ⎞= = −⎜ ⎟⎝ ⎠

δ

1 2( ) lM t a tv

⎛ ⎞= −⎜ ⎟⎝ ⎠

2 1( ) lM t a tv

⎛ ⎞= −⎜ ⎟⎝ ⎠

( )1 1 1 1 2( ) ( ) ( ) la t T t g t t a tv

⎛ ⎞= + Γ −⎜ ⎟⎝ ⎠

( )2 2 2 2 1( ) ( ) ( ) la t T t g t t a tv

⎛ ⎞= + Γ −⎜ ⎟⎝ ⎠

1 2( ) lb t a tv

⎛ ⎞= −⎜ ⎟⎝ ⎠

2 1( ) lb t a tv

⎛ ⎞= −⎜ ⎟⎝ ⎠

Wave Shifting Solution



1 1 1( ) ( ) ( )v t a t b t= +

2 2 2( ) ( ) ( )v t a t b t= +

1 11

( ) ( )( )o o

a t b ti tZ Z

= −

2 22

( ) ( )( )o o

a t b ti tZ Z

= −


SimulationsLine length = 1.27mZo = 73 Ω

v = 0.142 m/ns


0 100 2000

1

2

3

4

5

6

Near End

Time (ns)

Volts

0 100 200-2

0

2

4

6

Far End

Time (ns)

Volts

Simulations


0 10 20 30 40 50-1

0

1

2

3

4

5

lossylossless

Near End

Time (ns)

Volts

0 10 20 30 40 50-2

0

2

4

6

lossylossless

Far End

Time (ns)

Volts

Simulations

Line length = 25 in

L = 539 nH/m

C = 39 pF/m

Ro = 1 kΩ (GHz)1/2

Pulse magnitude = 4V

Pulse width = 20 ns

Rise and fall times = 1ns


N-Line S-Parameters*

B1 = S11 A1 + S12 A2 B2 = S21 A1 + S22 A2

* J. E. Schutt-Aine and R. Mittra, "Transient analysis of coupled lossy transmission lines with nonlinear terminations," IEEE Trans. Circuit Syst., vol. CAS-36, pp. 959-967, July 1989.


Scattering Parameters for N-Line

[ ][ ]-1-111 22S = S = T Γ -ΨΓΨ 1-ΓΨΓΨ T

[ ] [ ]2 -1-121 12 oS = S = E E 1-Γ Ψ 1-ΓΨΓΨ T

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦-1-1 -1 -1 -1 -1 -1

o o o m o o o mΓ = 1+ EE Z H H Z 1- EE Z H H Z

⎡ ⎤⎣ ⎦-1-1 -1 -1 -1

o o o m oT = 1+ EE Z H H Z EE ( )-lΨ = W


Scattering Parameter MatricesEo : Reference system voltage eigenvector matrix

E : Test system voltage eigenvector matrix

Ho : Reference system current eigenvector matrix

H : Test system current eigenvector matrix

Zo : Reference system modal impedance matrix

Zm : Test system modal impedance matrix


Eigen Analysis

* Diagonalize ZY and YZ and find eigenvalues.* Eigenvalues are complex: λi = αi + jβi

W(u) =

eα1u+ jβ1u

eα2u+ jβ2u

•eαn u+ jβnu

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥


Solution

mV = EV

mI = HI

[ ]( ) ( ) ( )x x x− + ΒmV = W A W

[ ]( ) ( ) ( )x x x− + Β-1m mI = Z W A W

-1 -1m mZ = Λ EZH

-1 -1 -1c m mZ = E Z H = E Λ EZ


Solutions

a1(t) = Δ1−1 1 − Γ1 (t)s'11 (0)[ ]−1 T1 (t)g1 (t) + Γ1 (t)M1(t)[ ]

− Δ1−1 1 − Γ1(t)s'11 (0)[ ]−1 1 − Γ2 (t)s'22 (0)[ ]−1 ×

Γ1(t)s'21 (0)[ ]T2 (t)g2(t) + Γ2 (t)M2(t)[ ]

a2(t) = Δ2−1 1 − Γ2 (t)s'22 (0)[ ]−1 T2(t)g2 (t) + Γ2 (t)M2 (t)[ ]

− Δ2−1 1 − Γ2 (t)s'22 (0)[ ]−1 1 − Γ1 (t)s'11 (0)[ ]−1 ×

Γ1 (t)s'12 (0)[ ] T1(t)g1 (t) + Γ1 (t)M1(t)[ ]


Solutions

[ ]⎡ ⎤⎣ ⎦-1 -1' ' '

1 1 11 2 22 1 21 2 12Δ (t) = 1- 1-Γ (t)s (0) 1-Γ (t)s' (0) Γ (t)s (0)Γ (t)s (0)

[ ] ⎡ ⎤⎣ ⎦-1-1 ' ' '

2 2 22 1 11 2 12 1 21Δ (t) = 1 - 1 -Γ (t)s' (0) 1 -Γ (t)s (0) Γ (t)s (0)Γ (t)s (0)

' '2 21 1 22 2 2b (t) = s (0)a (t) + s (0)a (t) + M (t)

' '1 11 1 12 2 1b (t) = s (0)a (t) + s (0)a (t) + M (t)


Lossless Case – Wave Shifting

21 12 ms (t) = s (t) = δ(t - τ )

1 2 mM (t) = a (t - τ )

2 1 mM (t) = a (t - τ )

1 1 1 1 2 ma (t) = T (t)g (t) +Γ (t)a (t - τ )

2 2 3 3 1 ma (t) = T (t)g (t) +Γ (t)a (t - τ )

1 2 mb (t) = a (t - τ )

2 2 mb (t) = a (t - τ )


δ(t − τm ) =

δ(t − τm1)δ(t − τm2 )

•δ(t − τmn )

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

ai (t − τm ) =

a1(t − τm1)a2 (t − τm2 )

•an (t − τmn )

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

Solution for Lossless Lines


2111 2(1 )

leY lZ ec

γγ

−+=−−

Zc : microstrip characteristic impedanceγ : complex propagation constantl : length of microstrip

Y11 can be unstable

2111 221

l( e )S le

γ ΓγΓ

−−=

−−

Z Zc oZ Zc o

Γ−

=+

S11 is always stable

Y-Parameter S-Parameter

Test line: Zc, γ

shortZoZo

ReferenceLine

ReferenceLine

Test line: Zc, γ

TestLine

Why Use S Parameters?


c ref

c ref

Z ZZ Z

−Γ =

+ cR j LZG j C

ωω

+=

+

Zref is arbitraryWhat is the best choice for Zref ?

refLZC

=

cLZC

→

11 0S →12j LCd

oS e Xω−→ =

At high frequencies

Thus, if we choose

Choice of Reference


50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

S-Parameter measurements (or simulations) aremade using a 50-ohm system. For a 4-port, the reference impedance is given by:

Zo =

11 1o oS ZZ I ZZ I

−− −⎡ ⎤ ⎡ ⎤= + −⎣ ⎦ ⎣ ⎦

[ ][ ] 1oZ I S I S Z−= + −

Z: Impedance matrix (of blackbox)S: S-parameter matrixZo: Reference impedanceI: Unit matrix

Choice of Reference


328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8

50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

Method: Change reference impedance from uncoupled to coupled system to get new S-parameter representation

Zo =

Zo =

Uncoupled system

Coupled system

as an example…

Reference Transformation


0

0.5

1

1.5

0 2 4 6 8 10

S11 - Linear Magnitude

S11 - 50 Ohm

S11 - Zref

S11

Frequency (GHz)

50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

using

Zo =

as reference…

328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8

using

as reference…

Zo =

Harder toapproximate

Easier to approximate (up to 6 GHz)

Choice of Reference


0

0.5

1

1.5

0 2 4 6 8 10


S11 - 50 Ohm

S11 - Zref

S11

Frequency (GHz)

50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

using

Zo =

as reference…

328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8

using

as reference…

Zo =



Choice of Reference


0

0.05

0.1

0.15

0.2

0.25

0.3

0 2 4 6 8 10


S12 - 50 Ohm

S12 - Zref

S12

Frequency (GHz)

50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

using

Zo =

as reference…

328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8

using

as reference…

Zo =



Choice of Reference


0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10


S31 - 50 Ohm

S31 - ZrefS3

1

Frequency (GHz)

50.0 0.0 0.0 0.00.0 50.0 0.0 0.00.0 0.0 50.0 0.00.0 0.0 0.0 50.0

using

Zo =

as reference…

328.0 69.6 328.9 69.669.6 328.8 69.6 328.9328.9 69.6 328.8 69.669.6 328.9 69.6 328.8

using

as reference…

Zo =


Easier to approximate

Choice of Reference


.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5 3

S11 Magnitude

Zref=ZoZref=80 ohmsZref=100 ohms

Frequency (GHz)

S11

Choice of Reference


.

0.7

0.75

0.8

0.85

0 0.5 1 1.5 2 2.5 3

S21 Magnitude

Zref=ZoZref=80 ohmsZref=100 ohms

Frequency (GHz)

S21

Choice of Reference


Modeling of Discontinuities

1. Tapered Lines

2. Capacitive Discontinuities


General topology of tapered microstrip with dw :width at wide end,dn: width at narrow end, lw: length of wide section, ln : length ofnarrow section, lt: length of tapered section.

Tapered Microstrip


( ) ( )21 1 22( ) ( )* ( ) ( )* ( )j j

j j ju t s t u t s t w t−= +

( 1) ( 1)11 12 1( ) ( )* ( ) ( )* ( )j j

j j jw t s t u t s t w t+ ++= +

* J. E. Schutt-Aine, IEEE Trans. Circuit Syst., vol. CAS-39, pp. 378-385, May 1992.

Tapered Line Analysis Using S Parameters*


-1

0

1

2

3

4

5

0 1 2 3 4 5 6

Vol

ts

Time (ns)

Small EndExcitation at small end

-1

0

1

2

3

4

5

0 1 2 3 4 5 6

Vol

ts

Time (ns)

Wide EndExcitation at small end

-1

0

1

2

3

4

5

0 1 2 3 4 5 6

Vol

ts

Time (ns)

Small EndExcitation at wide end

-1

0

1

2

3

4

5

0 1 2 3 4 5 6

Vol

ts

Time (ns)

Wide EndExcitation at wide end

Tapered Transmission Line


-1

0

1

2

3

4

5

0 1 2 3 4 5Time (ns)

Near End

.0564 mils/in

.1128 mils/in

.2257 mils/in

6 7 8

volts

-1

0

1

2

3

4

5

0 1 2 3 4 5Time (ns)

Far End

.0564 mils/in

.1128 mils/in

.2257 mils/in

6 7 8

volts

Varying tapering rate

Tapered Transmission Line


Zo

Zo

ZoC

-1

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Near End -- C=4 pF

Vol

ts

Time (ns)

-1

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Far end -- C=4 pF

Vol

ts

Time (ns)

Capacitive Load


Zo

Zo

ZoC

-1

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Near end -- C=40 pF

Vol

ts

Time (ns)

-0.5

0

0.5

1

1.5

2

2.5

0 10 20 30 40 50

Far end -- C=40 pF

Vol

ts

Time (ns)

Capacitive Load


Zo

Zstub

ZsL

- Stubs of TL with nonlinear loads- Reduce speed and bandwidth - Limit driving capabilities

Multidrop Buses


Transmission Lines with Capacitive Discontinuities


i r tV V V+ = i r i r t

o o

V V V V VEZ R R Z− −

+ − =

r c c iV T E V= + Γ

Capacitive Discontinuity


( ) ' ( )21 1 22( ) ( )* ( ) ( )* ( )j j

j j ju t s t u t s t w t−= +

' "( ) ( ) ( )j j ju t u t u t= +

' "( ) ( ) ( )j j jw t w t u t= +

Scattering Parameter Analysis


- 1

0

1

2

3

4

0 11.25 22.5 33.75 45

V o l ts

Ti me ( ns)

Near End

- 1

0

1

2

3

4

0 11.25 22.5 33.75 45

V o l ts

Ti me ( ns)

Far End

Capacitive Loading


-1

0

1

2

3

4

5

0 1 2 3 4 5Time (ns)

loading period 600 mils300 mils150 mils

6 7 8

volts

-1

0

1

2

3

4

0 1 2 3 4 5Time (ns)

capacitive loading1 pF2 pF4 pF

6 7 8

volts

Computer-simulated near end responses for capacitively loaded transmission line with l = 3.6 in, w = 8 mils, h = 5 mils. Pulse parameters are Vmax = 4 V, tr = tf = 0.5 ns, tw = 4 ns. Left: Varying P with C = 2 pF. Right: Varying C with P = 300 mils.

Capacitive Loading


Linear N-Portwith MeasuredS-Parameters

NonlinearNetwork

1

NonlinearNetwork

2

NonlinearNetwork

3

NonlinearNetwork

4Objective: Perform time‐domain simulation of composite network to determine timing waveforms, noise response or eye diagrams

Blackbox Macromodeling


Output

Frequency-DomainData

IFFT MOR

DiscreteConvolution

RecursiveConvolution

STAMP

Circuit Simulator

Macromodel Implementation


TL Simulation

Copyright © by Jose E. Schutt‐Aine , All Rights ReservedECE 598‐JS, Spring 2012

• Only measurement data is available • Actual circuit model is too complex

Motivations

• Inverse-Transform & Convolution• IFFT from frequency domain data• Convolution in time domain

• Macromodel Approach• Curve fitting• Recursive convolution

Methods

Blackbox Synthesis


Blackbox Synthesis

Terminations are described by a source vector G(ω)and an impedance matrix ZBlackbox is described by its scattering parameter matrix S


Blackbox - Method 1

Scattering Parameters ( ) ( ) ( )B S Aω ω ω=

Terminal conditions ( ) ( ) ( )A B TGω ω ω= Γ +

where11 1

o oU ZZ U ZZ−− −⎡ ⎤ ⎡ ⎤Γ = − + −⎣ ⎦ ⎣ ⎦

and11

oT U ZZ−−⎡ ⎤= +⎣ ⎦

(1)

(2)

U is the unit matrix, Z is the termination impedance matrix and Zo is the reference impedance matrix


Blackbox - Method 1

Combining (1) and (2) [ ] 1( ) ( ) ( )A U S TGω ω ω−= − Γ

and [ ] 1( ) ( ) ( ) ( ) ( ) ( )B S A S U S TGω ω ω ω ω ω−= = − Γ

[ ][ ] 1( ) ( ) ( ) ( ) ( ) ( )V A B U S U S TGω ω ω ω ω ω−= + = + − Γ

[ ] [ ][ ] 11 1( ) ( ) ( ) ( ) ( ) ( )o oI Z A B Z U S U S TGω ω ω ω ω ω−− −= − = − − Γ

{ }( ) ( )v t IFFT V ω=

{ }( ) ( )i t IFFT I ω=


• No Frequency Dependence for TerminationsReactive terminations cannot be simulated

• Only Linear TerminationsTransistors and active nonlinear terminations cannot be described

• StandaloneThis approach cannot be implemented in a simulator

Method 1 - Limitations


B=SA

b(t) = s(t)*a(t)

s( t )* a( t ) s( t )a( )dτ τ τ∞

−∞

= −∫

In frequency domain

In time domain

Convolution:

Blackbox - Method 2


Since a(τ) is known for τ < t, we have:

Isolating a(t)

t 1

1s( t )* a( t ) s( 0 )a( t ) s( t )a( )

τΔτ τ τ Δτ

−

== + −∑

t 1

1H( t ) s( t )a( ) : History

ττ τ Δτ

−

== −∑

t

1s( t )* a ( t ) s( t )a( )

ττ τ Δτ

== −∑

When time is discretized the convolution becomes

Discrete Convolution


Defining s'(0) =s(0)Δτ, we finally obtain

a(t) = Γ(t)b(t) + T(t)g(t)

b(t) = s'(0)a (t) + H (t)

By combining these equations, the stamp can be derived

Terminal Conditions


[ ] [ ]1a(t) 1 (t)s'(0) T(t)g(t) (t)H(t)Γ Γ−= − +

b(t) = s'(0)a (t) + H (t)

[ ]o1a( t ) v( t ) Z i( t )2

= +

[ ]o1b( t ) v( t ) Z i( t )2

= −

The solutions for the incident and reflected wave vectors are given by:

Stamp Equation Derivation

The voltage wave vectors can be related to the voltage and current vectors at the terminals


[ ] [ ]o o1 s'( 0 )v( t ) Z i( t ) v( t ) Z i( t ) H( t )2 2

− = + +

[ ]o oZ i( t ) s'(0 )Z i( t ) 2H( t ) 1 s'(0 ) v( t )+ + = −

[ ] [ ]o1 s'(0 ) Z i( t ) 1 s'(0 ) v( t ) 2H( t )+ = − −

[ ] [ ] [ ]1 11 1o oi( t ) Z 1 s'( 0 ) 1 s'( 0 ) v( t ) 2Z 1 s'( 0 ) H( t )− −− −= + − − +

From which we get


or

or

which leads to


[ ] [ ]11stamp oY Z 1 s'( 0 ) 1 s'( 0 )−−= + −

[ ] 11stamp oI 2Z 1 s'( 0 ) H( t )−−= +

stamp stampi( t ) Y v( t ) I= −

i(t) can be written to take the form


in which

and


[ ] [ ]11stamp oY Z 1 s'( 0 ) 1 s'( 0 )−−= + −

[ ] 11stamp oI 2Z 1 s'(0 ) H( t )−−= +

( ) ( )g stamp g stampY Y v t I I+ = +

stamp stampi( t ) Y v( t ) I= −Stamp Equations


No DC Data Point With DC Data Point

If low‐frequency data points are not available, extrapolation must be performed down to DC.

Effects of DC Data


-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Port 1

Port 1 - No DC ExtrapolationPort 1 - With DC Extrapolation

volts

Time (ns)-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16

Port 2

Port 2 - No DC ExtrapolationPort 2 - With DC Extrapolation

Vol

ts

Time (ns)

Effect of Low-Frequency Data


-2

-1.5

-1

-0.5

0

0 20 40 60 80 100 120

No DC component(lowest freq: 10 MHz)

Vol

ts

Time (ns)

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100 120

With DC component

Volts

Time (ns)

Left: IFFT of a sinc pulse sampled from 10 MHz to 10 GHz. Right: IFFT of the same sinc pulse with frequency data ranging from 0-10 GHz. In both cases 1000 points are used

( )2sin 2( )

2ft

V fftπ

π=Calculating inverse Fourier Transform of:

Effect of Low-Frequency Data


=ω ω ωY( ) H( )X ( )

=y( t ) h( t )* x( t )

Frequency-Domain Formulation

= = −∫ τ τ τt

0

y( t ) h( t )* y( t ) h( t )y( )d

== −∑

ττ τ Δτ

t

1h( t )* x( t ) h( t )x( )

−

== −∑

ττ τ Δτ

t 1

1H( t ) h( t )x( ) : History


Convolution

Discrete Convolution

Computing History is computationally expensive Use FD rational approximation and TD recursive convolution

Convolution Limitations


Large Network (>1,000 nodes)

Reduced Order Model

(< 30 poles)

SPICE Y(t) v(t) = i(t) Y(ω) V(ω) = I(ω)

Order Reduction

Y(ω) = Y(ω)~ ~

Recursive Convolution

Y(t) v(t) = i(t) ~

11

1 1

( )1 /

Li

i c i

aY Aj

ωω ω=

⎡ ⎤= +⎢ ⎥+⎣ ⎦

∑

Model Order Reduction


• AWE – Pade• Pade via Lanczos (Krylov methods)• Rational Function• Chebyshev-Rational function• Vector Fitting Method

11

1 1

( )1 /

Li

i c i

aH Aj

ωω ω=

⎡ ⎤= +⎢ ⎥+⎣ ⎦

∑

Objective: Approximate frequency-domain transfer function to take the form:

Methods



Model Order Reduction (MOR)

Question: Why use a rational function approximation?

Lk

k 1 ck

cY( ) H( )X ( ) d X ( )1 j /=

⎡ ⎤= = +⎢ ⎥+⎣ ⎦

∑ω ω ω ωω ω

1

( ) ( ) ( )=

= − + ∑L

pkk

y t dx t T y t

( )( ) ( ) 1 ( )− −= − − + −ω ωck ckT Tpk k pky t a x t T e e y t T

Answer: because the frequency‐domain relation

will lead to a time‐domain recursive convolution:

which is very fast!

where


=

= ++∑ω

ω ω

Lk

k 1 ck

cH( ) d1 j /

Transfer function is approximated as

In order to convert data into rational function form, we need a curve fitting scheme Use Vector Fitting



• 1998 - Original VF formulated by Bjorn Gustavsen and Adam Semlyen*

• 2003 - Time-domain VF (TDVF) by S. Grivet-Talocia.

• 2005 - Orthonormal VF (OVF) by Dirk Deschrijver, Tom Dhaene, et al.

• 2006 - Relaxed VF by Bjorn Gustavsen.

• 2006 - VF re-formulated as Sanathanan-Koerner (SK) iteration by W. Hendrickx, Dirk Deschrijver and Tom Dhaene, et al.

History of Vector Fitting (VF)

* B. Gustavsen and A. Semlyen, “Rational approximation of frequency responses by vector fitting,” IEEE Trans. Power Del., vol. 14, no. 3, pp 1052–1061, Jul. 1999


Measurement Data

Approximation function

Frequency

S-pa

ram

ete r

FrequencyS-

para

met

e r


Measurement Data S-pa

ram

e ter

Frequency

Measurement Data


Low order Medium order Higher order

LT

CT

LT

CT

LT LT

CT

Orders of Approximation


∗ ∗

∗

=

+ ≥ >*

( ) ( ),[ ( ) ( )] , [ ]0 0T T

H s H sz H s H s z s

− ≥ >*( ) ( ) , [ ]0 0TI H s H s s ,for S-parameters.

,for Y or Z-parameters.

= + ⇔ = +[ ] [ ] [ ] ( ) ( ) ( )ω ω ωe o R Ih n h n h n H j H j jH j

•Accurate:- over wide frequency range.

•Stable:- All poles must be in the left-hand side in s-plane or inside in the unit-circle in z-plane.

•Causal:- Hilbert transform needs to be satisfied.

•Passive:- H(s) is analytic

MOR Attributes


• Bandwidth Low-frequency data must be added

• PassivityPassivity enforcement

• High Order of ApproximationOrders > 800 for some serial linksDelay need to be extracted

MOR Problems


Length = 7 inches

1.- DISC: Transmission line with discontinuities

2.- COUP: Coupled transmission line2

dx = 51/2 inches

Frequency sweep: 300 KHz – 6 GHz

dxx y

u

z

v

w

Line A

Line B

Examples


DISC: Approximation order 90

DISC: Approximation Results


-0.2

0

0.2

0.4

0.6

0.8

1

1.2

30 32 34 36 38 40

Exact

B

A

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

30 32 34 36 38 40

Exact

B

A

1 2Microstrip line with discontinuitiesData from 300 KHz to 6 GHz

Observation: Good agreement

DISC: Simulations


COUP: Approximation order 75 – Before Passivity Enforcement

COUP: Approximation Results


-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

30 35 40 45 50

Exact

B

A

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

30 35 40 45 50

Convolution

B

A

Port 1: a – Port 2: dData from 300 KHz to 6 GHz

dxx y

a

b

c

d

Observation: Good agreement

COUP: Simulations

Documents

ECE 598 JS Lecture 09 Scattering Parametersjsa.ece.illinois.edu/ece598js/Lect_09.pdf · ECE 598‐JS, Spring 2012 Copyright © by Jose E. Schutt‐Aine , All Rights Reserved 21 Dissipated