Passivity enforcement by modal perturbation
View Patent ↗Residue perturbation is used for enforcing passivity of a linear response model of an electrical component having n>1 ports. According to an exemplary embodiment, a modal perturbation approach can be used, which allows weighted perturbation of the individual modes by the inverse of the corresponding eigenvalues. This provides superior results if the admittance or impedance matrix of the device has a large eigenvalue spread.
1. A method comprising:
computer implemented modeling of linear properties of an electrical component having n >1 ports, whose linear properties are described in a matrix Y relating a voltage applied to the ports to a current through the ports, wherein a dependence of Y on an independent variable s is approximated by a model
Y=F ( s, p 1 , . . . , p K ), [1.1]
with p 1 , . . . , p K being parameters of the model and F being a matrix-valued function describing the dependence of Y on the variable s,
said method comprising a passivation step of ensuring a passivity of said model by perturbing said parameters p 1 , . . . , p K to a perturbed set of parameters p 1 +Δp 1 , . . . , p K +Δp K while ensuring that the perturbed set of parameters fulfills a Boolean valued condition function
C ( p 1 +Δp 1 , . . . , p K +Δp K )=true, [1.2]
wherein said passivation step comprises the step of finding an approximate solution for the equations
[
F
(
s
,
p
1
+
Δ
p
1
,
…
,
p
K
+
Δ
p
K
)
-
F
(
s
,
p
1
,
…
,
p
K
)
]
·
t
i
λ
i
≅
0
,
[
1.3
]
for i=1 . . . n, wherein t i and λ i are the eigenvectors and eigenvalues of matrix Y.
2. The method of claim 1 wherein equation 1.3 is linearized by
[
Δ
p
1
·
∂
∂
p
1
F
(
s
,
p
1
,
…
,
p
K
)
+
…
+
Δ
p
K
·
∂
∂
p
K
F
(
s
,
p
1
,
…
,
p
K
)
]
·
t
i
λ
i
≅
0.
[
2.1
]
3. The method of claim 2 , wherein the condition function C is
eig i ( Re[F ( s, p 1 +Δp 1 , . . . , p K +Δp K )])>0 for all i=1 . . . n
wherein eig i ( ) is an operator returning eigenvalue i of its matrix-valued argument.
4. The method of claim 1 , wherein the condition function C is
eig i( Re[F ( s, p 1 +Δp 1 , . . . , p K +Δp K )])>0 for all i=1 . . . n
wherein eig i ( ) is an operator returning eigenvalue i of its matrix-valued argument.
5. The method of claim 1 , wherein said function F is a function from the group consisting of rational functions, ratio of polynomials, pole-residue functions, state-space models, and combinations thereof.
6. The method of claim 5 wherein equation 1.1 is
Y
=
∑
m
=
1
N
R
m
s
-
a
m
+
D
,
wherein R m with m=1 to N are matrices independent of s, N is the number of poles or resonances taken into account, D is a matrix independent of s, and a m with m=1 to N are the complex angular frequencies of the poles or resonances,
wherein at least one of said matrices R m , D and said poles a m depend on said parameters p 1 , . . . , p K .
7. The method of claim 6 , wherein said matrix Y is one of an impedance matrix, an admittance matrix, and a scattering matrix of said electrical component.
8. The method of claim 6 , wherein said independent variable s is one of a frequency, a time, and a discrete z-domain.
9. The method of claim 6 , wherein each element of said matrices R m and D is one of said parameters p 1 , . . . , p K .
10. The method of claim 9 , wherein said equation 1.3 is
(
∑
m
=
1
n
Δ
R
m
s
-
a
m
+
Δ
D
)
t
i
(
s
)
λ
i
(
s
)
≅
0
for
i
=
1
…
n
.
[
8.1
]
11. The method of claim 6 , wherein each eigenvalue of said matrices R m and D is one of said parameters p 1 , . . . , p K .
12. The method of claim 7 , wherein said equation 1.3 is
(
∑
m
=
1
n
Δ
R
m
s
-
a
m
+
Δ
D
)
t
i
(
s
)
λ
i
(
s
)
≅
0
for
i
=
1
…
n
.
[
8.1
]
13. The method of claim 6 , wherein said equation 1.3 is
(
∑
m
=
1
n
Δ
R
m
s
-
a
m
+
Δ
D
)
t
i
(
s
)
λ
i
(
s
)
≅
0
for
i
=
1
…
n
.
[
8.1
]
14. The method of claim 13 , comprising finding an approximate solution of said equation 1.3 by minimizing a measure of each vector element in said equation 1.3.
15. The method of claim 13 , comprising finding an approximate solution of said equation 1.3 by minimizing the sum of the squares of each vector element in said equation 1.3.
16. The method of claim 1 , comprising finding an approximate solution of said equation 1.3 by minimizing a measure of each vector element in said equation 1.3.
17. The method of claim 1 , comprising finding an approximate solution of said equation 1.3 by minimizing the sum of the squares of each vector element in said equation 1.3.
18. The method of claim 1 , wherein said matrix Y is one of an impedance matrix, an admittance matrix, and a scattering matrix of said electrical component.
19. The method of claim 18 , wherein said independent variable s is one of a frequency, a time, and a discrete z-domain.
20. The method of claim 1 , wherein said independent variable s is one of a frequency, a time, and a discrete z-domain.