ECE 546
Lecture -15
X-Parameters
Spring 2014
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
[email protected]
ECE 546 – Jose Schutt-Aine
1
References
[1] J J. Verspecht and D. E. Root, "Polyharmonic Distortion Modeling,"
IEEE Magazine, June 2006, pp. 44-57.
[2] D.E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata,
“Broad-band poly-harmonic distortion (PHD) behavioral models from fast
automated simulations and large-signal vectorial network
measurements,”IEEE Trans. Microwave Theory Tech., vol. 53, no. 11, pp.
3656–3664, Nov. 2005.
[3] D.E. Root, J. Verspecht, J. Horn, J. Wood, and M. Marcu,
“X Parameters”, Cambridge University Press, 2013.
ECE 546 – Jose Schutt-Aine
2
Scattering Parameters
V1  a1  b1
I1 
a1  b1
Zo
For a two-port
b1  S11a1  S12 a2
V2  a2  b2
b2  S21a1  S22 a2
I2

a2  b2
Zo
For a general N-port
N
B  SA
Bi   Sij Aj
Sij 
j 1
“ …most successful behavioral models…”
ECE 546 – Jose Schutt-Aine
Bi
Aj
Ak 0
k j
k 1,..., N
3
X Parameters: Motivation
S Parameters are a very powerful
tool for signal integrity analysis.
Today, X parameters are primarily
used to characterize power
amplifiers and nonlinear devices.
Not yet applied to signal integrity.
ECE 546 – Jose Schutt-Aine
4
X Parameters
Purpose
Characterize nonlinear behavior of devices and
systems
Advantages
- Mathematically robust framework
- Can handle nonlinearities
- Instrument exists (NVNA)
- Blackbox format  vendor IP protection
- Matrix format  easy incorporation in CAD tools
- X Parameters are a superset of S parameters
ECE 546 – Jose Schutt-Aine
5
Challenges in HS Links
High speed Serial channels are pushing the current
limits of simulation. Models/Simulator need to handle
current challenges
– Need to accurately handle very high data rates
– Simulate large number of bits to achieve low BER
– Non-linear blocks with time variant systems
– Model TX/RX equalization
– All types of jitter: (random, deterministic, etc.)
– Crosstalk, loss, dispersion, attenuation, etc…
– Handle and manage vendor specific device settings
– Clock data recovery (CDR) circuits
These cannot be accurately modeled with S parameters
ECE 546 – Jose Schutt-Aine
6
Motivation
Limitation: S Parameters only work for linear
systems. Many networks and systems are nonlinear
• Applications
High-speed links, power amplifiers, mixed-signal circuits
• Existing Methods
Load pull techniques
IBIS models
Models are flawed and incomplete
ECE 546 – Jose Schutt-Aine
7
PHD Modeling
• Polyharmonic distortion (PHD) modeling is a
frequency-domain modeling technique
• PHD model defines X parameters which form a
superset of S parameters
• To construct PHD model, DUT is stimulated by
a set of harmonically related discrete tones
• In stimulus, fundamental tone is dominant and
higher-order harmonics are smaller
ECE 546 – Jose Schutt-Aine
8
PHD Framework
• Signal is represented by a fundamental with
harmonics
• Signals are periodic or narrowband modulated
versions of a fundamental with harmonics
• Harmonic index: 0 for dc contribution, 1 for
fundamental and 2 for second harmonic
• Power level, fundamental frequency can be
varied to generate complete data for DUT
ECE 546 – Jose Schutt-Aine
9
PHD Framework
• Stimulus
A-waves are incident and B-waves are scattered
• Reference System ZC
Default value is 50 ohm
For a given port with voltage V and current I
A
V  ZC I
2
V  ZC I
B
2
ECE 546 – Jose Schutt-Aine
10
PHD Framework
Fpm describes a time-invariant systemdelay
in time domain corresponds to phase shift in
frequency domain
B pm e jm  Fpm  A11e j , A12e j 2 ,..., A21e j , A22e j 2 ,...
For phase normalization, define
P  e j ( A11 )
B pm  Fpm  A11 , A12 P 2 , A13 P 3 ,..., A21P 1 , A22 P 2 ,... P  m
ECE 546 – Jose Schutt-Aine
11
PHD Framework
ECE 546 – Jose Schutt-Aine
12
Cross-Frequency Phase for
Commensurate Tones
•
•
Defined as the phase of each pseudowave when
the fundamental, A1,1, has zero phase.
B2,3 can be related to A2,2 in magnitude and phase.
A1,1
A2,2
B2,3
ECE 546 – Jose Schutt-Aine
13
Time-Invariance Property of
Nonlinear Scattering Function
j
j 2
j 3
Fp,k ( A1,1 e , A1,2 (e ) , A1,3 (e ) ,...)
 Fp,k ( A1,1, A1,2 , A1,3 ,...)(e j )k
• Shifting all of the inputs by the same time means that different harmonic
components are shifted by different phases.
time delay
f0 180º phase shift
2f0 360º phase shift
3f0 540º phase shift
ECE 546 – Jose Schutt-Aine
14
Defining Phase Reference
• Can use time-invariance to separate magnitude and
phase dependence of one incident pseudowave.
Bp,k  Fp,k ( A1,1, A1,2 , A1,3 ,...)
 Fp,k ( A1,1 , A1,2P 2 , A1,3P 3 ,...)P k
Shifting reference to
zero phase of A1,1.
ECE 546 – Jose Schutt-Aine
using
A1,1
j arg( A1,1 )
P
e
A1,1
15
Commensurate Tones
X-Parameter Formalism*
• Define
)
2
3
X p( FB
(
A
,
A
P
,
A
P
,...)
,k
1,1
1,2
1,3
 Fp,k ( A1,1, A1,2 , A1,3 ,...)P  k
)
2
3
k
Bp,k  X p( FB
(
A
,
A
P
,
A
P
,...)
P
,k
1,1
1,2
1,3
• Still difficult to characterize this nonlinear term.
• If only one incident pseudowave, A1,1, is large then the
other smaller inputs can be linearized about the largesignal response of Fp,k to only A1,1.
*D. E. Root, et al., X-Parameters, 2013.
ECE 546 – Jose Schutt-Aine
16
PHD Framework
Define variables
Apm
port
harmonic
Bpm
port
harmonic
• Introduce multivariate complex function Fpm
such that
Bpm  Fpm  A11 , A12 ,..., A21 , A22 ,...
ECE 546 – Jose Schutt-Aine
17
Harmonic Superposition
In many situations, there is only one dominant largesignal input component present. The harmonic frequency
components are relatively small harmonic components
can be superposed
Harmonic superposition principle is key to PHD model
ECE 546 – Jose Schutt-Aine
18
Nonanalytical Mapping*
A nonlinearity described by:
f ( x)   x   x3
Signal is sum of main signal and additional perturbation
term which is assumed to be small
x(t )  xo (t )  x(t )
* see: J. Verspecht and D. E. Root, "Polyharmonic Distortion Modeling," IEEE Magazine, June 2006, pp. 44-57.
ECE 546 – Jose Schutt-Aine
19
Case 1
Consider the signal x(t), given by the sum of a real dc
component and a small tone at frequency f
A is real
xo (t )  A
x(t ) 
 e jt   *e jt
 is a small complex number
2
The linear response in x(t) can be computed by
  y (t )   f  xo (t )  x(t )   f  xo (t ) 
ECE 546 – Jose Schutt-Aine
20
Case 1
  y (t )   f '  xo (t )  x(t )
For case 1, we evaluate the conductance nonlinearity f ’(xo)
at the fixed value xo=A
f '  A     3 A2
After substitution, we get
jt
*  jt



e


e
2
  y (t )     3 A  

2


The complex coefficient of term proportional to ejt is
   3 A2 



2


 Linear input-output relationship
ECE 546 – Jose Schutt-Aine
21
Case 2
Now, xo(t) is a periodically time-varying signal:
xo (t )  A cos t 
x(t ) 
 e jt   *e  jt
2
Evaluating the conductance nonlinearity at xo(t) gives
f '( A cos(t ))    3 A2 cos(t )

3 A2  3 A2
  
cos(2t )

2 
2

ECE 546 – Jose Schutt-Aine
22
Case 2
We can evaluate (y(t)) to get:

3 A2  3 A2  e 2 jt  e 2 jt
  y (t )     


2
2
2






  e jt   *e  jt 


2


Now, we have terms proportional to ejt and ej3t and their
complex conjugates. Restrict attention to complex term
proportional to ejt
ECE 546 – Jose Schutt-Aine
23
Case 2
The complex coefficient of term proportional to ejt is
  3 A2 
3 A2 *

 
 
4 
4
2
We observe that the output phasor at frequency w is not just
proportional to the input phasor d at frequency w but has
distinct contributions to both d and d*
 Linearization is not analytic
In Fourier domain, we have:
Yˆ ( )   3 A2 
3 A2 2 jPhase ( )


e


4 
4
Xˆ ( )  2
ECE 546 – Jose Schutt-Aine
24
PHD Derivation
B pm  K pm  A11  P  m
  G pq ,mn  A11  P  m Re( Aqn P  n )
qn
  H pq ,mn  A11  P  m Im( Aqn P  n )
qn
in which
K pm  A11   Fpm  A11 ,0,...,0 
G pq ,mn  A11  
Fpm
 Re  Aqn P  n 
H pq ,mn  A11  
Spectral mapping
is nonanalytic
A11 ,0,...,0
Fpm
 Im  Aqn P  n 
A11 ,0,...,0
ECE 546 – Jose Schutt-Aine
25
PHD Derivation
A P  conj  A P 
Re( A P ) 
Since
n
n
n
qn
qn
qn
2
Im( Aqn P  n ) 
Aqn P  n  conj  Aqn P  n 
2j
we get
B pm  K pm  A11  P  m
  G pq ,mn 
qn
  H pq ,mn 
qn
 Aqn P  n  conj  Aqn P  n  

A11  P  m 


2


 Aqn P  n  conj  Aqn P  n  

A11  P  m 


2j


ECE 546 – Jose Schutt-Aine
26
PHD Model
( FB )
m
(S )
 mn
B pm  X pm
A
P

X
A
P
Aqn
 11 
 pq,mn  11 
qn
(T )
 m n
  X pq
A
P
conj  Aqn 


,mn
11
PHD
Model Equation
qn
X
(S )
p1,m1
 A 
K pm  A11 
11
q, n  1,1 : X
X p(T1,)m1  A11   0
A11
(S )
pq ,mn
q, n  1,1 : X
 A 
(T )
pq ,mn
G pq ,mn  A11   jH pq ,mn  A11 
11
 A 
2
G pq ,mn  A11   jH pq ,mn  A11 
11
ECE 546 – Jose Schutt-Aine
2
27
1-Tone X-Parameter Formalism*
Approximates
Incident Waves
Scattered Waves
)
2
3
Bp,k  X p( FB
(
A
,
A
P
,
A
P
,...)
,k
1,1
1,2
1,3

Nonlinear Mapping
frequency
(FB)
X p,k
( A1,1 ,0,0,...)
A1,1
LargeSignal
frequency
frequency
frequency
Simple Nonlinear Mapping
  X
(S )
p,k ;q ,l

 Aq,l  X
(T )
p,k ;q ,l
 A 
*
q ,l
Nonanalytic Harmonic
Superposition
frequency
frequency
*J. Verspecht, et al., “Linearization…,” 2005.
ECE 546 – Jose Schutt-Aine
28
1-Tone X-Parameter Formalism*
)
k
Bp,k  X p( FB

P

,k
Simple
nonlinear map
q  N , l K

q 1, l 1
( q ,l )(1,1)
X p(S,k);q ,l  Aq ,l  P k l 
Linear harmonic map
function of incident wave
• X-parameters of type FB, S,
and T fully characterize the
nonlinear function.
• Depend on
– frequency
– large signal magnitude, |A1,1|
– DC bias
q  N , l K

q 1, l 1
( q ,l ) (1,1)
X p(T,k);q ,l  Aq* ,l  P k  l
Linear harmonic map
function of conjugate of
incident wave
X
(S )
p ,k ;q ,l
output
output
input
port
harmonic port
P
A1,1
A1,1
input
harmonic
*D. E. Root, et al., X-Parameters, 2013.
ECE 546 – Jose Schutt-Aine
29
Excitation Design
Excitation 2
Excitation 1
Excitation 3
Excitation 4
Each excitation will generate response with fundamental and all harmonics
ECE 546 – Jose Schutt-Aine
30
X-Parameter Data File
TOP: FILE DESCRIPTION
! Created Fri Jul 30 07:44:48 2010
! Version = 2.0
! HB_MaxOrder = 25
! XParamMaxOrder = 12
! NumExtractedPorts = 3
! IDC_1=0 NumPts=1
! IDC_2=0 NumPts=1
! VDC_3=12 NumPts=1
! ZM_2_1=50 NumPts=1
! ZP_2_1=0 NumPts=1
! AN_1_1=100e-03(20.000000dBm) NumPts=1
! fund_1=[100 Hz->1 GHz] NumPts=4
ECE 546 – Jose Schutt-Aine
31
X-Parameter Data File
MIDDLE: FORMAT DESCRIPTION
BEGIN XParamData
% fund_1(real) FV_1(real) FV_2(real) FI_3(real) FB_1_1(complex)
% FB_1_2(complex) FB_1_3(complex) FB_1_4(complex)
% FB_1_7(complex) FB_1_8(complex) FB_1_9(complex)
% FB_1_12(complex) FB_2_1(complex) FB_2_2(complex)
% FB_2_5(complex) FB_2_6(complex) FB_2_7(complex)
% FB_2_10(complex) FB_2_11(complex) FB_2_12(complex)
% T_1_1_1_1(complex) S_1_2_1_1(complex) T_1_2_1_1(complex)
% S_1_4_1_1(complex) T_1_4_1_1(complex) S_1_5_1_1(complex)
% T_1_6_1_1(complex) S_1_7_1_1(complex) T_1_7_1_1(complex)
% S_1_9_1_1(complex) T_1_9_1_1(complex) S_1_10_1_1(complex))
% T_1_11_1_1(complex) S_1_12_1_1(complex) T_1_12_1_1(complex)
% T_2_1_1_1(complex) S_2_2_1_1(complex) T_2_2_1_1(complex)
% S_2_4_1_1(complex) T_2_4_1_1(complex) S_2_5_1_1(complex
% T_2_6_1_1(complex) S_2_7_1_1(complex) T_2_7_1_1(complex)
% S_2_9_1_1(complex) T_2_9_1_1(complex) S_2_10_1_1(complex )
ECE 546 – Jose Schutt-Aine
32
X-Parameter Data File
BOTTOM: DATA LISTING
100
0
-5.8503e-16
-1.25093e-15
-1.38032e-16
-0.0081633
0.000921039
-1.20792e-15
-1.2948e-14
3.39598e-15
2.76565e-14
0.903921
0.0263984
0.316228
-4.19864e-10 -6.37642e-16 -1.6748e-10
-3.79264e-10 -7.91128e-16 -1.51261e-10
-2.09262e-10 0.107122
-5.52212e-08
-2.40901e-08 -0.00739395 -1.21199e-08
-4.82427e-09 -0.00230559 1.07836e-08
-5.09916e-10 -6.95799e-15 -2.56672e-09
3.97284e-10 -7.08201e-15 -2.17127e-09
3.66098e-10 -1.08395e-14 -4.05911e-09
5.60242e-09 2.69755e-14 -6.60802e-10
-5.41159e-09
-4.62314e-16
1.93535e-17
0.0739648
-0.000530768
-0.00288533
-3.25033e-15
-1.43757e-14
1.67366e-14
3.99868e-14
Remarks
 Data is measured or generated from a harmonic
balance simulator
 Data file can be very large
ECE 546 – Jose Schutt-Aine
33
X-Parameter Relationship
bik  Dik  a11  P k 

( j ,l )  (1,1)
 Sik , jl  a11  P k l a jl  Tik , jl  a11  P k l a*jl 
P : Phase of a11
Dik : B  type X parameter
Sik , jl : S  type X parameter
Tik , jl : T  type X parameter
ECE 546 – Jose Schutt-Aine
34
X Parameters of CMOS
-3
x 10
-50
-2
T11,11 - Amplitude (dB)
S11,11 - Amplitude (dB)
-1
0.5 GHz
1 GHz
-3
-4
-5
-6
-7
-5
0
5
10
15
|A11| (dBm)
20
25
0.5 GHz
1 GHz
-80
-90
0
5
10
15
|A11| (dBm)
20
25
30
10
15
|A11| (dBm)
20
25
30
-10
T21,11 - Amplitude (dB)
S21,11 - Amplitude (dB)
-70
-100
-5
30
0
-5
0.5 GHz
1 GHz
-10
-15
-20
-5
-60
0
5
10
15
|A11| (dBm)
20
25
30
0.5 GHz
1 GHz
-20
-30
-40
-50
-60
-5
ECE 546 – Jose Schutt-Aine
0
5
35
-35
-40
-40
-50
T12,11 - Amplitude (dB)
S12,11 - Amplitude (dB)
X Parameters of CMOS
-45
-50
-55
0.5 GHz
1 GHz
-60
-65
-5
0
5
10
15
|A11| (dBm)
20
25
0.5 GHz
1 GHz
-80
-90
0
5
10
15
|A11| (dBm)
20
25
30
-20
0.5 GHz
1 GHz
-25
T22,11 - Amplitude (dB)
S22,11 - Amplitude (dB)
-70
-100
-5
30
-20
-30
-35
-40
-45
-50
-5
-60
0
5
10
15
|A11| (dBm)
20
25
30
-40
-60
0.5 GHz
1 GHz
-80
-100
-5
ECE 546 – Jose Schutt-Aine
0
5
10
15
|A11| (dBm)
20
25
30
36
Large-Signal Reflection
Microwave amplifier with fundamental frequency at 9.9 GHz
ECE 546 – Jose Schutt-Aine
37
Compression and AM-PM
Microwave amplifier with fundamental frequency at 9.9 GHz
ECE 546 – Jose Schutt-Aine
38
T22,11
Microwave amplifier with fundamental frequency at 9.9 GHz
ECE 546 – Jose Schutt-Aine
39
Special Terms
• T-Type X Parameter
 Spectral mapping is non-analytical
 Real and imaginary parts in FD are treated differently
 Even and odd parts in TD are treated differently
 T involves non-causal component of signal
• Phase Term P
 P is phase of large-signal excitation (a11)
 Contributions to B waves will depend on P
 In measurements, system must be calibrated for phase
ECE 546 – Jose Schutt-Aine
40
Notation Change
Define
Sik , jl  a11   X ik( S, )jl (| A11 |)
Tik , jl  a11   X ik(T, )jl (| A11 |)
Dik  a11   X ik( FB )  a11 
ECE 546 – Jose Schutt-Aine
41
Handling Phase Term
bik  Dik  a11  P k 

( j ,l )  (1,1)
 Sik , jl  a11  P k l a jl  Tik , jl  a11  P k l a*jl 
Multiply through by P  k
bik P k  Dik  a11  

( j ,l )  (1,1)
 Sik , jl  a11  P l a jl  Tik , jl  a11  P l a*jl 
P  e j11 where 11 is the phase of a11
we can always express the relationship in terms of modified
power wave variables
bik  Dik  a11  

( j ,l )  (1,1)
 Sik , jl  a11  a jl  Tik , jl  a11  a *jl 
where bik  bik P  k
and
ECE 546 – Jose Schutt-Aine
aik  aik P  k
42
Handling R&I Components
Because of non-analytical nature of spectral mapping, real and imaginary
component interactions must be accounted for separately.
we have
 br   X rr X ri  ar 
b   X


 i   ir X ii  ai 
where
X rr   Sr  Tr  , X ri    Si  Ti 
X ir   Si  Ti  , X ii   Sr  Tr 
ECE 546 – Jose Schutt-Aine
43
Handling Phase Term
Phase term can be accounted for by applying following transformations
 br   X rr
b   X
 i   ir
 cosb
  sin 
b

in which
X ri  ar 

X ii 
a
 i 
 sin b   br'   X rr
 '

cosb   bi   X ir
X ri   cos a
   sin 
X ii 
a
 sin  a   ar' 
 
cos a   ai' 
 br   cosb
 b     sin 
b
 i 
 sin b   br' 
 '

cosb   bi 
 ar   cos a
 a     sin 
a
 i 
 sin  a   ar' 
 '

cos  a   ai 
ECE 546 – Jose Schutt-Aine
44
X Matrix Construction
 Separate real and imaginary components
 Account for real-imaginary interactions
 Account for harmonic-to-harmonic contributions
 Account for harmonic-to-DC contributions
Matrix size is 2mn  2mn
m: number of harmonics
n: number of ports
ECE 546 – Jose Schutt-Aine
45
Matrix Formulation*
size:2mn
 b1 
 a1 
We wish to use:
 b  size:2
a 
2
2

mn
b

a 
 bp 
 ap 
(1)
(1)




a
b
 
pr
pr
 
 bn   (1) 
 an   a (1) 
 pi 
 bpi 
(2)
 a (2)



b
pr
pr
 (2) 
 (2) 
a pi  vector size is 2m
bpi 


ap 
b 
   m: number of harmonics p   

 n: number of ports


  
  
 a(m) 
 b( m) 
(real vectors)
 (prm ) 
 pr

(m) 
a 

*DC term not included
 pi 
 bpi 
b = Xa
ECE 546 – Jose Schutt-Aine
46
 X11
X
X =  21
 

 Xn1
X12
Matrix Formulation*

X 22


Xpq


X1n 
 

 

Xnn 
(11)
 X pqrr
 (11)
 X pqir
(21)
 X pqrr
 (21)
X pqir

Xpq 
 

(real matrix)  
*DC term not included  
 ( m1)
X
 pqir
matrix size is 2mn 2mn
m: number of harmonics
n: number of ports size: 2m
(11)
X pqri
(12)
X pqrr
(12)
X pqri
 
(1m )
X pqrr
(11)
X pqii
(12)
X pqir
(12)
X pqii
 

(21)
X pqri
(22)
X pqrr
(22)
X pqri
 

(21)
X pqii
(22)
X pqir
(22)
X pqii
 




 




 




 

( m1)
X pqii


( mm )
  X pqir
ECE 546 – Jose Schutt-Aine
2m
(1m )

X pqri

 
 

 
 

 
 

( mm ) 
X pqii 
47
X Matrix for 2-Port System*
(2 harmonics)
 X 11(11)
rr
 (11)
 X 11ir
 X 11(21)
rr
 (21)
X 11ir

X
 X 21(11)rr
 (11)
 X 21ir
 X (21)
rr
 21(21)
 X 21ir
X 11(11)
ri
X 11(12)
rr
X 11(12)
ri
X 12(11)rr
X 12(11)ri
X 12(12)rr
X 11(11)
ii
X 11(12)
ir
X 11(12)
ii
X 12(11)ir
X 12(11)ii
X 12(12)
ir
X 11(21)
ri
X 11(22)
rr
X 11(22)
ri
X 12(21)
rr
X 12(21)
ri
X 12(21)
rr
X 11(21)
ii
X 11(22)
ir
X 11(22)
ii
X 12(21)
ir
X 12(21)
ii
X 12(22)
ir
(11)
X 21
ri
(12)
X 21
rr
(12)
X 21
ri
(11)
X 22
rr
(11)
X 22
ri
(12)
X 22
rr
X 21(11)ii
(12)
X 21
ir
(12)
X 21
ii
X 2(11)
2 ir
X 22(11)ii
(12)
X 22
ir
X 21(21)
ri
X 21(22)
rr
X 21(22)
ri
X 22(21)rr
X 22(21)ri
X 22(22)
rr
X 21(21)
ii
X 21(22)
ir
X 21(22)
ii
X 22(21)ir
X 22(21)ii
X 22(22)
ir
X 12(12)ri 

X 12(12)
ii 

X 12(21)
ri

X 12(22)
ii 
(12) 
X 22
ri
(12) 
X 22
ii 
X 22(22)ri 

(22) 
X 22ii 
(real matrix)
For instance, X(12)21ri is the contribution to the real part
of the 1st harmonic of the wave scattered at port 2 due
to the imaginary part of the 2nd harmonic of the wave
incident port in port 1.
*DC term not included
ECE 546 – Jose Schutt-Aine
48
Polyharmonic Impedance
Linear
Impedance
Polyharmonic
Impedance
- Time invariant
- Linear
- Scalar
- Time invariant
- Linear
- Matrix
- Time variant
- Nonlinear
- Function
[V ( f )]  [ Z ( f )][ I ( f )]
V (t )  Z ( I (t ))
V  ZI
FD & TD
Nonlinear
Impedance
FD only
Model assumes that nonlinear effects are mild and are
captured via harmonic superposition.
ECE 546 – Jose Schutt-Aine
49
Polyharmonic Impedance
4-harmonic system
in frequency domain:
V (1)   Z (11)
 (2)   (21)
V    Z
V (3)   Z (31)
 (4)   (41)
V   Z
Z (12)
Z (13)
Z (22)
Z (23)
Z (32)
Z (33)
Z (42)
Z (43)
Z (14)   I (1) 
 
Z (24)   I (2) 
Z (34)   I (3) 
(44)   (4) 
Z  I 
in time domain:
v(t )  v(1) (t )  v(2) (t )  v(3) (t )  v(4) (t )
i(t )  i (1) (t )  i (2) (t )  i (3) (t )  i (4) (t )
ECE 546 – Jose Schutt-Aine
50
Polyharmonic Impedance
Zo
: Reference impedance matrix
Z
V
: Polyharmonic impedance matrix
I
: Current vector
: Voltage vector
Z = 1 + X 1 - X  Z o
-1
Describes interactions
between harmonic
components of voltage
and current.
V = ZI
ECE 546 – Jose Schutt-Aine
51
Network Formulation
Scattered waves
b = Xa
Termination equations
a = Dv g + Γb
Wave Solution
a = 1 - ΓX Dv g
-1
Voltage Solution
v = 1 + X  a
ECE 546 – Jose Schutt-Aine
52
Steady-State Simulations
cubic term
Time-Domain Response
100
Vin
Vout
80
60
40
X Parameter
Volts
20
0
-20
-40
-60
-80
-100
0
0.5
1
1.5
2
2.5
time(ns)
3
3.5
4
4.5
5
ADS
ECE 546 – Jose Schutt-Aine
53
CMOS Driver/Receiver Channel
Generate X parameters for composite system
Power level: 20 dBm, frequency: 1 GHz
Construct X matrix
Combine with terminations for simulation
ECE 546 – Jose Schutt-Aine
54
CMOS Driver/Receiver - Harmonics
DC+Fundamental
3 Harmonics
8
8
Vin
Vout
6
4
4
2
Volts
Volts
2
0
0
-2
-2
-4
-4
-6
-6
-8
Vin
Vout
6
-8
0
0.5
1
1.5
2
2.5
time(ns)
3
3.5
4
4.5
5
0
0.5
1
1.5
8 Harmonics
3
3.5
4
4.5
5
8
Vin
Vout
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
0
0.5
1
1.5
2
2.5
time(ns)
3
3.5
4
4.5
Vin
Vout
6
Volts
Volts
2.5
time(ns)
12 Harmonics
8
-8
2
5
ECE 546 – Jose Schutt-Aine
0
0.5
1
1.5
2
2.5
time(ns)
3
3.5
4
4.5
5
55
Validation
Time-Domain Response
8
Vin
Vout
6
4
2
Volts
X Parameter
0
-2
-4
-6
-8
0
0.5
1
1.5
2
2.5
time(ns)
3
3.5
4
4.5
5
7
6
5
4
3
ADS
Vout, V
Vin, V
2
1
0
-1
-2
-3
-4
-5
-6
-7
25.0
25.2
25.4
25.6
25.8
26.0
26.2
26.4
26.6
26.8
27.0
27.2
27.4
27.6
27.8
28.0
28.2
28.4
28.6
28.8
29.0
29.2
29.4
29.6
29.8
30.0
time, nsec
ECE 546 – Jose Schutt-Aine
56
Cascading S-Parameter Blocks*
A1(1)
B1(1)
A1(1)
B1
(1)
(1)
[S]
(1)
[T]
B2(1)
A1(2)
A2(1)
B1(2)
B2(1)
A1(2)
A2
(1)
=
=
B1
(2)
(2)
[S]
(2)
[T]
B2(2)
A1(1)
A2(2)
B1(1)
B2(2)
A1(1)
(2)
(1)
A2
[T] = transfer scattering parameters.
B1
(T)
[S]
A2(2)
B2(2)
(T)
[T]
=
(1)
(2)
[T] [T]
A2(2)
B2(2)
Can disregard circuit
behavior at internal node.
*G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed. Prentice-Hall, 1997.
ECE 546 – Jose Schutt-Aine
57
Cascading X-Parameter Blocks*
A1,k(1)
B1,k(1)
(1)
[X]
These equations at
the internal node
must always be
satisfied:
B1,k(1) = A1,k(2),
A1,k(2) = B2,k(1)
for all values of k.
B2,k(1)
A2,k(1)
A1,k(1)
B1,k(1)
=
=
A1,k(2)
B1,k(2)
(T)
[X]
(2)
[X]
B2,k(2)
A2,k(2)
B2,k(2)
A2,k(2)
*D. E. Root, et al., X-Parameters, 2013.
ECE 546 – Jose Schutt-Aine
58
Cascading X Parameters
GOAL: Simulate complete channel by combining Xparameter blocks from different sources into a single
composite X matrix.
Vendor A
Foundry
In-House
Vendor B
In-House
Vendor C
Foundry
X
X-parameters of individual devices can be accurately cascaded within a
harmonic balance simulator environment.
ECE 546 – Jose Schutt-Aine
59
Volterra Series
A linear causal system with memory can be described by the convolution
representation

y (t )   h  x  t    d

where x(t) is the input, y(t) is the output, and h(t) the impulse response of the
system.
A nonlinear system without memory can be described with a Taylor series as:

y (t )   an  x(t ) 
n
n 1
where x(t) is the input and y(t) is the output. The an are Taylor series
coefficients.
ECE 546 – Jose Schutt-Aine
60
Volterra Series
A Volterra series combines the above two representations to describe a
nonlinear system with memory



n
1
y (t )    du1...  dun g n  u1 ,..., un  x(t  ur )
r 1
n 1 n ! 


1
y (t )   du1 g1 (u1 ) x(t  u1 )
1! 




impulse response
1
  du1  du2 g 2 (u1 , u2 ) x(t  u1 )x(t  u2 )
2!  
higher-order impulse responses

1
  du1  du2  du3 g3 (u1 , u2 , u3 ) x(t  u1 )x (t  u2 ) g 2 (u1 , u2 ) x (t  u1 )x (t  u2 )x(t  u3 )
2!  

 ...
where x(t) is the input and y(t) is the output and the gn(u1,…,un) are called the
Volterra kernels
ECE 546 – Jose Schutt-Aine
61
Volterra Series
Application to X parameters:
Take order = 2



1
1
y (t )   du1 g1  u1  x  t  u1    du1  du2 g 2  u1 , u2  x  t  u1  x  t  u2 
1! 
2!  
where the input x(t) is given by
x(t )  exp  j1t   exp  j2t 
This can be written as

1
j t u
j t u
y (t )   du1 g1  u1   e 1 1   e 2  1  
1! 



1
j1 t u1 
j2  t u1 
j1 t u2 
j2  t u2 




du
du
g
u
,
u
e

e
e

e


1
2
2
1
2
 
2! 
ECE 546 – Jose Schutt-Aine
62
Volterra Series
Define
T1  e j1t , T2  e j2t
U11  e  j1u1
U 22  e
y (t )  T1  du1 g1  u1 U11  T2


U 21  e  j2u1
 j2u2
This gives

U12  e  j1u2

 du g u U
1 1
1
21


1
  du1  du2 g 2  u1 , u2  TU
1 11  T2U 21 TU
1 12  T2U 22 
2!  
or
y (t )  I1  I 2  I 3  I 4  I 5  I 6
ECE 546 – Jose Schutt-Aine
63
Volterra Series
in which

I1  T1  du1 g1  u1 U11  I1  T1G1  f1 


I 2  T2  du1 g1  u1 U 21  I 2  T2G1  f 2 

G1(f) is the Fourier transform of g1(u) evaluated at f
ECE 546 – Jose Schutt-Aine
64
Volterra Series


1 2
2 1
I 3  T1
du1  du2 g 2  u1 , u2 U11 U12  T1 G2  f1 , f1 

2!  
2


1
1
I 4  T1T2  du1  du2 g 2  u1 , u2 U11 U 22  T1T2G2  f1 , f 2 
2!
2



I5 

1
1
T2T1  du1  du2 g 2  u1 , u2 U 21 U12  T2T1G2  f 2 , f1 
2!
2




1 2
1 2
I 6  T2  du1  du2 g 2  u1 , u2 U 21 U 22  T2 G2  f 2 , f 2 
2!  
2
G2(f1, f2) is the double Fourier transform of g(u,v) evaluated at
(f1, f2)
ECE 546 – Jose Schutt-Aine
65
Volterra Series
So, G1(f) is the Fourier transform of g1(u) evaluated at f and
G2(f1, f2) is the double Fourier transform of g(u,v) evaluated at
(f1, f2)
We can also express y(t) as:
y  t   y1  t   y2  t 
in which
y1  t   T1G1  f1   T2G1  f 2 
and
1 2


y2 t  T1 G2  f1 , f1   T1T2G2  f1 , f 2   T1T2G2  f 2 , f1   T22G2  f 2 , f 2  
2!
ECE 546 – Jose Schutt-Aine
66
Volterra Series
If we take into account the respective amplitudes of the tone,
we have
x(t )  A1 exp  j1t   A2 exp  j2t 
We can make the transformation
T1  AT
1 1
and

y (t ) 
T2  A2T2
j1 t u1 
j2  t u1 


du
g
u
A
e

A
e


2
 11 11



1
j t u
j t u
  du1  du2 g 2  u1 , u2   A1e 1 1   A2 e 2  1  
2!  
  A1e
j1 t u2 
 A2 e
j2  t u2 

y1  t   AT
1 1G1  f1   A2T2G1  f 2 
ATerm  H A
B Term  H B
ECE 546 – Jose Schutt-Aine
67
Volterra
Series

1 2 2


y2 t 
A1 T1 G2  f1 , f1 
2! 
 C Term H C
 AT
1 1 A2T2G2  f1 , f 2   AT
1 1 A2T2G2  f 2 , f1 
D Term  H D

 A22T22G2  f 2 , f 2  

E Term  H E

If we choose f2 = kf1, then T1 f1 and T2  kf1.
In general, 2=k1 so that if HA=A-Term contains terms in f
1
HB=B-Term contains terms in kf1
HC=C-Term contans terms in 2f1
T1  f1 , T2  kf1
HD=D-Term contains terms in 2kf1
HE=E-Term contains terms in (k+1)f1
ECE 546 – Jose Schutt-Aine
68
Volterra Series
- First determine the X parameters of the system
- Next, provide excitation a(t)
a (t )  A1 exp  j1t   A2 exp  j2t 
- Next, calculate b in phasor domain using X parameters
b=Xa
- For each port, the scattered wave will include
contributions from all harmonics
bp  H A  H B  H C  H D  H E
ECE 546 – Jose Schutt-Aine
69
Volterra Series
Finally, a relationship can be obtained to extract Volterra
kernel Fourier transforms
A-Term:
HA
G1  f1  
A1
B-Term:
HB
G1  f 2  
A2
2HC
C-Term: G2  f1 , f1   2
A1
2H D
D-Term: G2  f 2 , f 2   2
A2
HE
E-Term: G2  f1 , f 2  
A1 A2
ECE 546 – Jose Schutt-Aine
70
Volterra Series
TABLE OF VOLTERRA KERNEL TRANSFORMS
Term
Wave
Coefficient
Constant
index
k=1
k=2
k=3
k=3
A-Term
T1
G1(f1)
A1
f1
f1
f1
f1
f1
B-Term
T2
G1(f2)
A2
k f1
f1
2 f1
3 f1
4 f1
C-Term
T12
G2(f1,,f1)
A12/2!
2 f1
2 f1
2 f1
2 f1
2 f1
D-Term
T22
G2(f2,,f2)
A22/2!
2k f1
2 f1
4 f1
6 f1
8 f1
E-Term
T1T2
G2(f1,,f2)
A 1A 2
(k+1) f1
2 f1
3 f1
4 f1
5 f1
ECE 546 – Jose Schutt-Aine
71
Nonlinear Vector Network
Analyzer (NVNA)
NVNA instruments will gradually replace all VNAs
ECE 546 – Jose Schutt-Aine
72
Nonlinear Vector Network
*
Analyzer (NVNA)
*L. Betts, “X-Parameters and NVNA…,” May 9, 2009.
ECE 546 – Jose Schutt-Aine
73