ECE 546 Lecture -13 Scattering Parameters

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ECE 546 Lecture ‐13 Scattering Parameters Spring 2020

Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois [email protected]

ECE 546 –Jose Schutt‐Aine 1 Transfer Function Representation

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

ECE 546 –Jose Schutt‐Aine 2 Y-parameter Representation

I1111122 yV yV

I2211222yV yV

ECE 546 –Jose Schutt‐Aine 3 Y Parameter Calculations

II12 yy11 21 VV11 VV2200

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

ECE 546 –Jose Schutt‐Aine 4 Z Parameters

VzIzI1111122 

VzIzI2211222

ECE 546 –Jose Schutt‐Aine 5 Z-parameter Calculations

VV12 zz11 21 II11 II2200

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

ECE 546 –Jose Schutt‐Aine 6 H Parameters

VhIhV1111122 

I2211222hI hV

ECE 546 –Jose Schutt‐Aine 7 H Parameter Calculations

VI12 hh11 21 II11 VV2200

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

ECE 546 –Jose Schutt‐Aine 8 G Parameters

I1111122 gV g I

VgVgI2211222

ECE 546 –Jose Schutt‐Aine 9 G-Parameter Calculations

IV12 gg11 21 VV11 II2200

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

ECE 546 –Jose Schutt‐Aine 10 TWO‐PORT NETWORK REPRESENTATION

Z Parameters Y Parameters

VZIZI1111122 IYVYV1111122

VZIZI2211222 IYVYV2211222

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

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

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

ECE 546 –Jose Schutt‐Aine 11 Wave Approach

Use a travelling wave approach

VE111 ir E VE222 ir E

EEir11 EEir22 I1 = I2 = Zo Zo - Total voltage and current are made up of sums of forward and backward traveling waves.

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

ECE 546 –Jose Schutt‐Aine 12 Wave Approach

Ei1 Ei2 a1 = a2 = Zo Zo

Er1 Er 2 b1 = b2 = Zo Zo

Zo is the reference impedance of the system

b1 = S11 a1 + S12 a2

b2 = S21 a1 + S22 a2

ECE 546 –Jose Schutt‐Aine 13 Wave Approach

b b1 2 S = | S21 = 11 a2=0 a a1 1 | a2=0

b1 b2 S12 = | a1=0 S22 = a2 a2 | a1=0

To make ai = 0 1) Provide no excitation at port i 2) Match port i to the characteristic impedance of the reference lines.

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

ECE 546 –Jose Schutt‐Aine 14 S‐Parameters of TL

(X)1 2  S = S = 11 22 1 X 22

()X1  2 S = S = 12 21 1 X 22

Zcref Z  R jLG  jC   Zcref Z R  jL Z  Xe  l c GjC 

ECE 546 –Jose Schutt‐Aine 15 S‐Parameters of Lossless TL

(X)1 2  S = S = Zcref Z  LC 11 22 1 X 22   Zcref Z 2 L ()X1  Z  S12 = S 21 = 22  jl c C 1 X  Xe

If Zc = Zref

S11 = S 22 = 0  jl S12 = S 21 = e

ECE 546 –Jose Schutt‐Aine 16 N-Port S Parameters

bSS111121    a bSS   a 221222         bSannnn   b=Sa

If bi = 0, then no reflected wave on port i  port is matched  Vi  ai  Vi : incident voltage wave in port i Zoi V   i : reflected voltage wave in port i Vi bi  Zoi Zoi : impedance in port i

ECE 546 –Jose Schutt‐Aine 17 N-Port S Parameters 1  ia-b  v=Zi va+b Zo (1) (2) (3) Zo Substitute (1) and (2) into (3) 1   Zo a+b Z a-b Zo Defining S such that b = Sa and substituting for b   ZZooU+Sa U-Sa U : unit matrix

SZ ZS -1 1 SZ+UZZZ U ZU+SU-S Zo oo

ECE 546 –Jose Schutt‐Aine 18 N-Port S Parameters If the port reference impedances are different, we define k as

 Z   o1   Z  k  o2         Zon  v=k(a+b) and i=k-1  a-b and k(a + b) Zk-1  a - b

ZS SZ S = Zk-1 + k Zk -1 - k Z=k U+S U-S-1 k

ECE 546 –Jose Schutt‐Aine 19 Normalization

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

Transformation Equation

-1-1 -1 S2111122111122 = k (U + S )(U - S ) k k + k k (U + S )(U - S ) k k - k

If Z is symmetric, S is also symmetric

ECE 546 –Jose Schutt‐Aine 20 Dissipated Power 1 P  aU-SSaTT** d 2 The dissipation matrix D is given by:

D=U-ST* 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 0 be  0

ECE 546 –Jose Schutt‐Aine 21 Dissipated Power

When the dissipation matrix is 0, we have a lossless network SST* U The S matrix is unitary. For a lossless two‐port:

22 SS11 21 1 22 SS22 12 1 If in addition the network is reciprocal, then 2 SS12 21and S 11 S 22 1 S 12

ECE 546 –Jose Schutt‐Aine 22 Lossy and Dispersive Line

1 2  1  2  SS SS 11 22 1  22 21 12 1  22 Z  Z   co l    e Zco Z 

ECE 546 –Jose Schutt‐Aine 23 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.

ECE 546 –Jose Schutt‐Aine 24 Frequency-Domain

BSASA1111122()   ()  ()

BSASA2211222()   ()  ()

ECE 546 –Jose Schutt‐Aine 25 Time-Domain Formulation

ECE 546 –Jose Schutt‐Aine 26 Time-Domain Formulation

bt1111122() s ()* t at () s ()* t a () t Z bt() s ()* t at () s ()* t at () o 2211222 Tti () Zio()tZ

at11111() () tbt ()  Ttgt () () Zio()tZ i ()t at22222() () tbt ()  Ttgt () () Zio()tZ

ECE 546 –Jose Schutt‐Aine 27 Time-Domain Solutions

1 ()ts' (0) Ttg () () t  () tM () t at() 2221 1 1 1 1 ()t

()ts' (0) T () tg () t () tM () t  1122 2 2 2 ()t

1 ()ts' (0) T () tg () t  () tM () t at() 1112 2 2 2 2 ()t

()ts' (0) Ttg () () t () tM () t  22111 1 1 ()t

ECE 546 –Jose Schutt‐Aine 28 Time-Domain Solutions

'' bt11111221() s (0) at () s (0) a () t M () t

'' bt22112222() s (0) at () s (0) at () M () t

'''' ()ttstststs  1 1 () 11 (0) 1  2 () 22 (0)  1 () 12 (0)  2 () 21 (0)

M11112()tHtHt () ()

M 22122()tHtHt () ()

' ssij(0) ij (0) t1 Htij()  st ij ( ) a j ( )  1

ECE 546 –Jose Schutt‐Aine 29 Special Case – Lossless Line  l  st() st () 0 st12() st 21 ()   t  11 22  v 

 l   l  Mt12() a t  Mt21() at    v   v 

 l  at11112 Ttgt() () () tat   v  l at22221 Ttgt() () () tat v  l  bt12() a t  Wave Shifting Solution  v   l  bt21() a t   v 

ECE 546 –Jose Schutt‐Aine 30 Time-Domain Solutions

vt111() at () bt ()

vt222() at () bt ()

at11() bt () it1() ZooZ

at22() bt () it2 () ZooZ

ECE 546 –Jose Schutt‐Aine 31 Simulations

Line length = 1.27m

Zo = 73  v = 0.142 m/ns

ECE 546 –Jose Schutt‐Aine 32 Simulations Near End

3 Volts 2

0 0 100 200 Time (ns)

Far End

2 Volts

-2 0 100 200 Time (ns)

ECE 546 –Jose Schutt‐Aine 33 Simulations

Line length = 25 in C = 39 pF/m Pulse magnitude = 4V

1/2 L = 539 nH/m Ro = 1 k (GHz) Pulse width = 20 ns

Rise and fall times = 1ns

Near End Far End 5 6

4 3

2 lossy 2 lossy lossless lossless Volts Volts

1 0

-2 -1 01020304050 0 1020304050 Time (ns) Time (ns)

ECE 546 –Jose Schutt‐Aine 34 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.

ECE 546 –Jose Schutt‐Aine 35 Scattering Parameters for N-Line

-1 -1 S=S=EE1-21 122 o  ΓΨ 1-ΓΨΓΨ T

-1 -1 S=S=T11 22 Γ - ΨΓΨ1-ΓΨΓΨ T -1 -1 -1 -1 -1 -1 -1 Γ =1+EEZHHZooo m 1-EEZHHZ ooo m -1 -1 -1 -1 -1 T = 1+ EEooo Z H H Z m EE o Ψ =W-l

ECE 546 –Jose Schutt‐Aine 36 Scattering Parameter Matrices

Eo : 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

ECE 546 –Jose Schutt‐Aine 37 Eigen Analysis

* Diagonalize ZY and YZ and find eigenvalues. * Eigenvalues are complex: i = i + ji

e 1uj1u     e2uj2u  W(u)        en u jnu 

ECE 546 –Jose Schutt‐Aine 38 Solution

V=EVm

I=HIm

V=WAWm ()x  (xx ) () 

-1 I=ZWAWmm()x  (xx ) () 

-1 -1 Z=mmΛ EZH

-1 -1 -1 Z=EZH=EcmΛ mEZ

ECE 546 –Jose Schutt‐Aine 39 Solutions

1 1 at111111111() 1  () ts ' (0)  Ttgt () ()  () tMt ()

1 11 11112221()'(0)1()'(0) ts   ts  

12122 ()ts ' (0) T () tg () t 2 () tM 2 () t

1 1 at222222222() 1  () ts ' (0)  Ttgt () ()  () tMt ()

1 11 22221111()'(0)1()'(0) ts   ts  

11211()ts ' (0) Ttg () () t 1 () tM 1 () t

ECE 546 –Jose Schutt‐Aine 40 Solutions

-1 '''-1 1(ttstststs ) 1- 1-  1 ( ) 11 (0) 1-  2 ( ) ' 22 (0)  1 ( ) 21 (0)  2 ( ) 12 (0)

-1 -1 ''' 2()ttstststs 1-1-  2 () ' 22 (0) 1-  1 () 11 (0)  2 () 12 (0)  1 () 21 (0)

'' bt11111221() s (0) at () s (0) a () t M () t

'' bt22112222() s (0) at () s (0) at () M () t

ECE 546 –Jose Schutt‐Aine 41 Solutions

1 vtmo111() atbt () () vtEatbt 1 () 11 ()  ()

1 vtmo222() atbt () () vtEatbt 2 () 22 ()  ()

z=0 z=l

V1(z)

V2(z) ...

V3(z) L, C

ECE 546 –Jose Schutt‐Aine 42 Lossless Case – Wave Shifting

st21() st 12 ()  ( t - m )

M12()tat ( - m )

M 21()tat ( - m )

at11112() Ttgt () () () tat ( - m )

at22331() Ttgt () () () tat ( - m )

bt12() a ( t - m )

bt22() at ( - m )

ECE 546 –Jose Schutt‐Aine 43 Solution for Lossless Lines

(t   )   m1   (t m2)  (t m )          (t mn )

a1(t  m1)   a2 (t m2) a (t  )  i m      a n (t mn )

ECE 546 –Jose Schutt‐Aine 44 Why Use S Parameters?

Y-Parameter S-Parameter Reference Test Reference Line Line Test line: Zc,  Line Zo Test line: Zc,  Zo short 2 l (e1 ) S  2 l 11 2 l Y  1 e 1  2e 11 2 l Ze(1 ) c Z  Z co Z : microstrip characteristic impedance   c Z  Z  : complex propagation constant co l : length of microstrip Y can be unstable 11 S11 is always stable

ECE 546 –Jose Schutt‐Aine 45 Choice of Reference

Zcref Z R  jL  Zc  Zcref Z GjC 

Zref is arbitrary What is the best choice for Zref ?

L At high frequencies Z  c C

L Thus, if we choose Z  ref C  jLCd Se12  Xo S11  0

ECE 546 –Jose Schutt‐Aine 46 Choice of Reference S-Parameter measurements (or simulations) are made using a 50-ohm system. For a 4-port, the reference impedance is given by:

50.0 0.0 0.0 0.0 0.0 50.0 0.0 0.0 Z: Impedance matrix (of blackbox) Zo = 0.0 0.0 50.0 0.0 S: S-parameter matrix 0.0 0.0 0.0 50.0 Zo: Reference impedance I: Unit matrix

1 11 SZZIoo ZZI  1 ZISISZ   o

ECE 546 –Jose Schutt‐Aine 47 Reference Transformation Method: Change reference impedance from uncoupled to coupled system to get new S-parameter representation

50.0 0.0 0.0 0.0 0.0 50.0 0.0 0.0 Z = Uncoupled system o 0.0 0.0 50.0 0.0 0.0 0.0 0.0 50.0

328.0 69.6 328.9 69.6 69.6 328.8 69.6 328.9 Z = Coupled system o 328.9 69.6 328.8 69.6 69.6 328.9 69.6 328.8 as an example…

ECE 546 –Jose Schutt‐Aine 48 Choice of Reference

S11 - Linear Magnitude using 1.5 Harder to 50.0 0.0 0.0 0.0 approximate 0.0 50.0 0.0 0.0 Z = 0.0 0.0 50.0 0.0 o 0.0 0.0 0.0 50.0 1 as reference…

using S11

328.0 69.6 328.9 69.6 0.5 S11 - 50 Ohm 69.6 328.8 69.6 328.9 Zo = 328.9 69.6 328.8 69.6 S11 - Zref 69.6 328.9 69.6 328.8 Easier to approximate (up to 6 GHz)

as reference… 0 0246810 Frequency (GHz)

ECE 546 –Jose Schutt‐Aine 49 Choice of Reference

S11 - Linear Magnitude using 1.5

50.0 0.0 0.0 0.0 Harder to 0.0 50.0 0.0 0.0 approximate Zo = 0.0 0.0 50.0 0.0 0.0 0.0 0.0 50.0 as reference… 1 S11 using 0.5 S11 - 50 Ohm 328.0 69.6 328.9 69.6 69.6 328.8 69.6 328.9 S11 - Zref Zo = 328.9 69.6 328.8 69.6 69.6 328.9 69.6 328.8 Easier to approximate (up to 6 GHz) as reference… 0 0246810 Frequency (GHz) ECE 546 –Jose Schutt‐Aine 50 Choice of Reference S12 - Linear Magnitude 0.3 using S12 - 50 Ohm

0.25 50.0 0.0 0.0 0.0 S12 - Zref 0.0 50.0 0.0 0.0 Zo = 0.0 0.0 50.0 0.0 0.0 0.0 0.0 50.0 0.2 Harder to as reference… approximate

0.15 S12

0.1 using 328.0 69.6 328.9 69.6 0.05 69.6 328.8 69.6 328.9 Zo = 328.9 69.6 328.8 69.6 69.6 328.9 69.6 328.8 0 as reference… 0246810 Frequency (GHz) Easier to approximate (up to 6 GHz)

ECE 546 –Jose Schutt‐Aine 51 Choice of Reference S31 - Linear Magnitude using 1.2 328.0 69.6 328.9 69.6 Easier to approximate 69.6 328.8 69.6 328.9 1 Zo = 328.9 69.6 328.8 69.6 69.6 328.9 69.6 328.8

S31 - 50 Ohm as reference… 0.8

S31 - Zref

0.6

using S31

50.0 0.0 0.0 0.0 0.4 0.0 50.0 0.0 0.0 Harder to Zo = 0.0 0.0 50.0 0.0 approximate 0.0 0.0 0.0 50.0 0.2 as reference…

0 0246810

Frequency (GHz)

ECE 546 –Jose Schutt‐Aine 52 Choice of Reference

Zref=Zo Zref=80 ohms Zref=100 ohms . S11 Magnitude 0.3

0.25

0.2

0.15 S11

0.1

0.05

0 00.511.522.53 Frequency (GHz)

ECE 546 –Jose Schutt‐Aine 53 Choice of Reference

Zref=Zo Zref=80 ohms S21 Magnitude Zref=100 ohms 0.85

0.8 S21

0.75

0.7 00.511.522.53 Frequency (GHz)

ECE 546 –Jose Schutt‐Aine 54 Modeling of Discontinuities

1. Tapered Lines

2. Capacitive Discontinuities

ECE 546 –Jose Schutt‐Aine 55 Tapered Microstrip

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

ECE 546 –Jose Schutt‐Aine 56 Tapered Line Analysis Using S Parameters*

()jj () utjj() s21 ()* t u 1 () t s 22 ()* t wt j ()

(1)jj (1) wtstutstwtjjj()11 ()* () 12 ()* 1 ()

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

ECE 546 –Jose Schutt‐Aine 57 Tapered Transmission Line

Small End Wide End Excitation at small end Excitation at small end 5 5

4 4

3 3

2 2 s lt o

Volts 1 V 1

0 0

-1 -1 0123456 0123456 Time (ns) Time (ns)

Small End Wide End Excitation at wide end Excitation at wide end 5 5

4 4

3 3

2 2

1 1 Volts Volts 0 0

-1 -1 0123456 0123456 Time (ns) Time (ns)

ECE 546 –Jose Schutt‐Aine 58 Tapered Transmission Line

Varying tapering rate

Near End Far End 5 5 .0564 mils/in .0564 mils/in 4 4 .1128 mils/in .1128 mils/in 3 .2257 mils/in 3 .2257 mils/in

2 2

1 volts 1 volts 0 0

-1 -1 012345678 012345678 Time (ns) Time (ns)

ECE 546 –Jose Schutt‐Aine 59 Capacitive Load

Zo C Zo

Near End -- C=4 pF Far end -- C=4 pF 2.5 2.5

2 2

1.5 1.5

1 1 Volts Volts 0.5 0.5

0 0

-0.5 -0.5

-1 -1 01020304050 01020304050 Time (ns) Time (ns)

ECE 546 –Jose Schutt‐Aine 60 Capacitive Load

Zo C Zo

Near end -- C=40 pF Far end -- C=40 pF 2.5 2.5

2 2 1.5 1.5 1 1 Volts Volts 0.5 0.5 0

-0.5 0

-1 -0.5 01020304050 01020304050 Time (ns) Time (ns)

ECE 546 –Jose Schutt‐Aine 61 Multidrop Buses

ZsL

Zstub Zo

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

ECE 546 –Jose Schutt‐Aine 62 Transmission Lines with Capacitive Discontinuities

ECE 546 –Jose Schutt‐Aine 63 Capacitive Discontinuity

VVir E VV ir V t VVir V t   ZooRRZ

VTEVrc ci

ECE 546 –Jose Schutt‐Aine 64 Scattering Parameter Analysis

()jj ' () utjj() s21 ()* t u 1 () t s 22 ()* t wt j ()

'" utjjj() ut () ut ()

'" wtjjj() wt () ut ()

ECE 546 –Jose Schutt‐Aine 65 Capacitive Loading

Near End Far End 4 4

3 3

2 2

1 V o s l t V o s l t 1

0 0

-1 -1 0 11.25 22.5 33.75 45 Time ( ns) 0 11.25 22.5 33.75 45 Time ( ns)

ECE 546 –Jose Schutt‐Aine 66 Capacitive Loading

loading period 600 mils capacitive loading 5 1 pF 300 mils 4 2 pF 4 150 mils 4 pF 3 3 2 2

1 1 volts volts 0 0

-1 012345678 -1 Time (ns) 012345678 Time (ns)

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.

ECE 546 –Jose Schutt‐Aine 67