ECE 451
Black Box Modeling
Jose E. Schutt-Aine
Electrical & Computer Engineering
University of Illinois
jose@emlab.uiuc.edu
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
1
Simulation for Digital Design
Nonlinear
Network
1
Nonlinear
Network
3
Linear N-Port
with Measured
S-Parameters
Nonlinear
Network
4
Objective: Perform time‐
domain simulation of
composite network to
determine timing
waveforms, noise response
or eye diagrams
Nonlinear
Network
2
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
2
Macromodel Implementation
Frequency-Domain
Data
IFFT
MOR
Discrete
Convolution
Recursive
Convolution
STAMP
Circuit Simulator
Output
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
3
Black Box Synthesis
Motivations
• Only measurement data is available
• Actual circuit model is too complex
Methods
• Inverse-Transform & Convolution
• IFFT from frequency domain data
• Convolution in time domain
• Macromodel Approach
• Curve fitting
• Recursive convolution
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
4
IFFT/Convolution Approach to Macromodels
Vg1
Vgi
V1 I1
Zg2
Zgi
Blackbox
[S]
:
:
Ii
Vi
In frequency domain
In time domain
B=SA
b(t) = s(t)*a(t)

Convolution: s( t )* a( t )   s( t   )a(  )d

ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
5
Discrete Convolution
When time is discretized the convolution becomes
t
s( t )* a ( t )   s( t   )a(  )
 1
Isolating a(t)
t 1
s( t )* a( t )  s( 0 )a( t )   s( t   )a(  )
 1
Since a() is known for < t, we have:
t 1
H( t )   s( t   )a(  ) : History
 1
ECE 451
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Termination Conditions
Defining s'(0) =s(0)we finally obtain
b(t) = s'(0)a (t) + H (t)
a(t) = (t)b(t) + T(t)g(t)
By combining these equations, the stamp can be derived
ECE 451
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7
Stamp Generation from Convolution
i( t )  Ystamp v( t )  I stamp
Ystamp  Z
1
o
1  s'( 0 ) 1  s'( 0 )
I stamp  2Z
1
1
o
Termination
1  s'( 0 )
1
i
H( t )
Black Box
+
Ig
Yg
v
Ystamp
Istamp
-
(Yg  Ystamp )v(t )  I g  I stamp
ECE 451
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8
Inverse FFT Procedure
1. The sampled data for the blackbox will come with the following set of
parameters: number of points M, start frequency f1, stop frequency f2 (and
frequency step f).
2. We wish to process the data. For this purpose we will define our own set of
parameters which may or may not be the same as those of the original data.
Number of points N, start frequency fstart, stop frequency fstop. From the discussion
above we learned that the start frequency is best set to zero; therefore, all of the
parameters will be set by the choice of two quantities: N and fstop.
3. Perform extrapolation and interpolation of raw data and map the M points from
the raw data into the N points of the processed data
4. Determine the frequency step, fstep=fstop/N. From this, the time step (tstep=1/(2Nfstep)),
the total duration of the simulation (tstop=Ntstep) are determined; the frequency-domain
data points can now be be arranged as follows
a
b
c
d
e
f
5. Set the imaginary parts of the first and last points to zero: Imag{a}=0, and Imag{g}=0
ECE 451
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9
Inverse FFT Procedure
6. Fold the data with respect to its conjugate mirror image in order to insure that
the time-domain response will be pure real; this looks as follows (* indicates
complex conjugate):
g
f*
e*
d*
c*
b*
a
b
c
d
e
f
where g and a are now real quantities. The total number of points is N2=2N
7. Feed those N2 points to IFFT routine to obtain inverse FFT. After the IFFT call, scale
all the points in the returned array by dividing them by (N2tstep). The data now looks like
o
p
q
r
s
t
u
v
w
x
y
z
8. These points should all be real. They are the time-domain impulse response.
9. If everything is fine, proceed by keeping only the first N points of the sequence
(o to t). They represent the impulse response of the data and will be used for the
time-domain convolution
ECE 451
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10
Importance of Low-Frequency
V( f ) 
Calculating inverse Fourier Transform of:
No DC component
(lowest freq: 10 MHz)
2sin  2 ft 
2 ft
With DC component
1.2
0
1
-0.5
0.8
Volts
Volts
0.6
-1
0.4
0.2
-1.5
0
-2
0
20
40
60
80
100
120
-0.2
0
20
40
60
80
100
120
Time (ns)
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
ECE 451
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11
Recursive Convolution
Limitations
• Convolution is slow
• Linear network block can be large
Motivation for Recursive Convolution
• Faster
• Requires transfer function to have the form:
L


a1i
H ( )   A1  

i 1 1  j / c1i 

ECE 451
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12
Time Domain Solution Using Recursive Convolution
Given the frequency-domain relation:
Y ( )  H ( ) X ( )
If the transfer function is written as
L


ai

H ( )   A  

1
/
j



i 1
ci 

Then, the time domain relationship (for step invariant model) is:
L
y (nT )  Ax  (n  K )T    y pi (nT )
i 1
where
y pi (nT )  ai (1  e ciT ) x  (n  K  1)T   e ciT y pi  ( n  1)T 
This is also called recursive convolution
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
13
Macromodel ‐ Approximation
Objective: Approximate frequency-domain transfer function
to take the form:
L


a1i
H ( )   A1  

i 1 1  j / c1i 

Methods
•
•
•
•
•
AWE – Pade
Pade via Lanczos (Krylov methods)
Rational Function
Chebyshev-Rational function
Vector Fitting Method
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Attributes for Macromodel
• Numerical Robustness
– Accuracy
– Passivity
– Causality
• User Flexibility
– Automatic selection of starting poles
– Automatic determination of order
• Compensation Features
– Frequency truncation
– No DC information
– Measurement noise
ECE 451
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
15
Causality - Hilbert Transform Analysis
Enforce causality
h(t )  0, t  0
Every function can be considered as the sum of an even function and an odd function
h(t )  he (t )  ho (t )
1
 h(t )  h(t )
2
1
ho (t )   h(t )  h(t ) 
2
he (t ) 
Even function
Odd function
 he (t ), t  0
ho (t )  
 he (t ), t  0
ho (t )  sgn(t ) he (t )
ECE 451
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16
Causality - Hilbert Transform Analysis
h(t )  he (t )  sgn(t ) he (t )
In frequency domain this becomes
H ( f )  He ( f ) 
1
j f
* He ( f )
H ( f )  H e ( f )  jHˆ e ( f )
Hˆ e ( f ) is the Hilbert transform of H e ( f )

1 1 x( )
xˆ (t )  x(t ) *  
d
 t   t  
ECE 451
Imaginary part of transfer
function is related to the
real part through the
Hilbert transform
Discrete Hilbert Transform
hn
2
,

 
k
n

n
odd
HT (hn )  hˆk  
 2  hn ,
  n even k  n
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
k even
k odd
17
HT for Coupled Lines: 300 KHz – 6 GHz
Actual is red, HT is blue
Actual
Transform
Actual
Transform
Real Part of S11
0.2
0.1
0.1
Imag(S11)
0.2
Real(S11)
Imaginary Part of S11
0.3
0.3
0
0
-0.1
-0.1
-0.2
-0.2
-0.3
-0.3
0
500
1000
1500
Actual
Transform
2000
2500
3000
0
3500
Actual
500
1000
Transform
1500
2000
2500
3000
3500
A
A
Imag(S12)
1
Real(S12)
1
0.5
Real(S12)
Imag(S12)
0.5
0
0
-0.5
-0.5
-1
0
500
1000
1500
2000
2500
3000
3500
-1
0
500
1000
1500
2000
2500
3000
3500
A
A
ECE 451
Observation: Good agreement
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
18
HT for Via: 1 MHz – 20 GHz
Actual
Transform
Actual
Transform
Actual is red, HT is blue
Real(S11) - Via
Imag(S11) - Via
0.3
0.5
0.2
0.1
Actual
Actual
0
-0.1
0
-0.2
-0.3
-0.4
0
100
200
300
400
500
600
700
-0.5
Actual
Transform
Actual
100 Transform
200
0
300
400
500
600
700
A
A
Real(S12) - Via
Imag(S12) - Via
1
1
0.5
0.5
Real(S12)
Imag(S12)
0
0
-0.5
-0.5
-1
-1.5
0
100
200
300
400
A
500
600
700
-1
0
100
200
300
400
500
600
700
Observation: Poor agreement (because frequency range is limited)
ECE 451
A
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19
Approximation function
Measurement Data
Measurement Data
S-parameter
Approximation function
S-parameter
S-parameter
Orders of Approximation
Approximation function
Measurement Data
Frequency
Frequency
Frequency
Low order
LT
Medium order
LT
LT
CT
ECE 451
Higher order
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LT
CT
CT
20
Measurement Data
1.- DISC: Transmission line with discontinuities
Length = 7 inches
2.- COUP: Coupled transmission line2
Line A
u
Line B
v
x
dx
y
w
z
dx = 51/2 inches
Frequency sweep: 300 KHz – 6 GHz
ECE 451
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DISC: Approximation Results
DISC: Approximation order 90
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22
COUP: Approximation Results
COUP: Approximation order 75 – Before Passivity Enforcement
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23
Results After Passivity Check
TL with capacitive discontinuity
Magnitude plot of Y11, measured data
and the 30-th order rational
approximation with passivity check
ECE 451
Phase plot of Y11, measured data and the 30-th
order rational approximation with passivity
check
Copyright © by Jose E. Schutt‐Aine , All Rights Reserved
24
Coupled Lines – Measured on ANA
c
a
Port 1: a – Port 2: d
Data from 300 KHz to 6 GHz
d
b
x
Exact
Convolution
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
B
B
d
y
x
0.1
0.2
0.1
0
0
-0.1
-0.1
-0.2
30
35
40
45
50
-0.2
30
35
A
40
45
50
A
Observation: Good agreement
ECE 451
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25
Microstrip with Steps – Measured on ANA
Microstrip line with discontinuities
Data from 300 KHz to 6 GHz
1
2
Exact
Exact
1.2
1
1
0.8
0.8
0.6
0.6
B
B
1.2
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
30
32
34
36
38
40
30
32
A
34
36
38
40
A
Observation: Good agreement
ECE 451
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26