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2003/10/7 BMAS20031
SOM-LS: Selective Orthogonal Matrix Least-Squares Method for
Macromodeling MultiportNetworks Characterized by
Sampled Data
Y. Tanji*, T. Watanabe**, H. Asai***
*Kagawa University, Japan
**University of Shizuoka, Japan
***Shizuoka University, Japan
2
1. Introduction2. Selective Orthogonal Matrix Least-
Squares Method3. Macromodeling of Networks
Characterized by Sampled Data4. Examples5. Summary
Contents
3
Which function is dominant ?
Introduction exp
Device Model
x2
sin x
Least-Squares Fitting
4
The least-squares method for system identification selecting the dominant basis functions is presented.This method is based on a learning algorithm of neural network.Macromodeling of networks characterized by sampled data via electromagnetic analysis.The macromodels are described in the format of Verilog-A.
5
1. Introduction2. Selective Orthogonal Matrix Least-
Squares Method3. Macromodeling of Networks
Characterized by Sampled Data4. Examples5. Summary
Contents
6
The device model is made by least-squaresfitting of sampled data
)()(1
xfx i
N
ii∑
=
= KG
Selective Orthogonal Matrix Least-Squares MethodDevice Model:
iabledesign var :function basis:)(matrixconstant :
xxfi
iK
7
The over-determined matrix equation is solved by orthogonal least-squares method.
FPK =
( )[ ] T
Nxxx )(,),(, 21 GGGF K=
=
)()(2
)(1
)2
()2
(2
)2
(1
)1
()1
(2
)1
(1
Nx
Mf
Nxf
Nxf
xM
fxfxf
xM
fxfxf
L
MLMM
L
L
P
Over-determined Equation:
8
The coefficient matrix is rewritten by
The number of basis functions is equal to the number of orthogonal vectorsThe key issue is how to select the column vectors and orthogonalize them.
[ ]Nk ppppppP ,,,,,,, 4321 KK=
[ ]Nkk ppwwwwwP ,,,,,,, 14321 KK −=
[ ]1432111 ,,,,,,, −+−= kkN wwwwwppP KK
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After kth step in the orthogonalization, the residual matrix is defined as
The 2-norm of the residual matrix is proven to be monotonously decreasing function.
kWkkGWFZ −=
kG2-norm
the number of columns
: orthogonal matrix
: intermediate solution
Least Squares Solution
10
Evaluating the 2-norm of the residual matrix, the columns of the matrix P are orthogonalized so that the 2-norm largely decreases at each step.
Selective Orthogonalization
Selective
Normal
2-norm
the number of columns
[ ]Nk pppwwP ,,,,,, 321 KK=
11
1. Introduction2. Selective Orthogonal Matrix Least-
Squares Method3. Macromodeling of Networks
Characterized by Sampled Data4. Examples5. Summary
Contents
12
The SOM-LS method is used for approximating the sampled data with the rational matrix.
Macromodeling of Networks Characterized by Sampled Data
V1
Vi
Vn
I1
Ii
In
EM DeviceV1
Vi
Vn
I1
Ii
In
EM Device
frequency
Frequency Response
13
Sampled Data
Rational Function
1st Level Approximation
Stable Poles
Least-Squares Fitting(scalar approximation)
),,1()()(1)()()(
1
10
NijajajbjbbjY n
ini
mimi
iij
L
L
L
=++++++
=
@ωωωωω
Root Finding
14
Using the SOM-LS method, the dominant poles are extracted and the compact model is obtained.
),,1,0(
)(1
Ni
jpj i
Q
l li
l
K=
=−∑
=
ωω
YKSampled Data
Rational Matrix
2nd Level Approximation
Least-Squares Method(Matrix Approximation)
15
FPK =
=
−−
−−
−−
QpNjpNj
Qpjpj
Qpjpj
ωω
ωω
ωω
1
1
11
2
1
12
11
1
1
11
11
L
MLMM
L
L
P
Orthogonal Least-Squares Method
160 100 200
10–5
100
2–no
rm
degree
Selective
Normal
2-norm of Residual Matrix
17
1. Introduction2. Selective Orthogonal Matrix Least-
Squares Method3. Macromodeling of Networks
Characterized by Sampled Data4. Examples5. Summary
Contents
18
Examples
The marcromodels are described by Verilog-A.
Verilog HDL Verilog-A
Mixed Signal Extension
Verilog-AMS
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)()()( sVsHsV inout ⋅=
2210
10)(sasaa
sbbsH⋅+⋅+
⋅+=
module transfer_func(in, out);
inout in, out;electrical in, out;
analog begin
V(out) <+ laplace_nd(V(in), [b0,b1], [a0, a1, a2]);
end
endmodule
Laplace Transform Description by Verilog-A
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pole-1(real) pole-1(imag) residue-1(real) residue-1(imag)
pole-2(real) pole-2(imag) residue-2(real) residue-2(imag)
:
:
pole-n(real) pole-n(imag) residue-n(real) reside-n(imag)
Verilog-A Model Generation Tool
module model_name(in, out);
inout in, out;electrical in, out;
analog beginV(out) <+ laplace_nd(V(in), [b0,b1], [a0, a1, a2]);
V(out) <+ laplace_nd(V(in), [b0,b1], [a0, a1, a2]);
:
V(out) <+ laplace_nd(V(in), [b0,b1], [a0, a1, a2]);end
endmodule
Verilog-A Model Generation
21
We computed the responses of simple PCB models using Cadence Spectre. The results using the proposed macromodels were compared with the FDTD method on Spectre. The computational speed with the proposed macromodels is two magnitudes faster than the FDTD method on Spectre.
22
Example PCB Model
+5V
V1 V2
1pF
10kΩ
port1
port2
port3
Ground Plane
32mm
24mm
Bottom surface Top surface
(a)
(b)
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Frequency-Domain Response
[dB]
[GHz]0.0
0
10.0
prposed
sampled data
-40
40
24
[V]
[nsec]0.0
5
2.0
0
proposed FDTD Input
Time-Domain Response
25
CPU Time Comparison
4.09 (sec)416.96 (sec)Example3
5.96 (sec)596.19 (sec)Example2
4.27 (sec)550.94 (sec)Example1
Proposed ModelFDTD
26
The selective orthogonal matrix least-squares method is presented.This method is applied to macromodeling of networks characterized by sampled data.The proposed models are described in the format of Verilog-A.Future work: Passivity consideration of the macromodel.
Summary
