Iterative method for characterizing the frequency dependence of the linear properties of an electrical component
View Patent ↗A method and device characterize linear properties of an electrical component having n>1 ports. The linear properties of the component are described in a matrix relating a voltage applied to the ports to a current through the ports. A frequency dependence of the matrix is approximated to preserve eigenvalues of the matrix by a pole-residual model. The method includes: (a) obtaining a set of values of the matrix at discrete frequencies, and obtaining eigenvalues and eigenvectors for each value; (b) fitting a set of vector equations to the eigenvalues and eigenvectors with a first set of pole frequencies; and (c) calculating a second set of pole frequencies by a vector fitting process for all modes of an element of the matrix. Steps (b) and (c) are repeated using the second set of pole frequencies in step (c) in a subsequent step (b) until a stop condition is met.
1. A method for characterizing linear properties of an electrical component having n>1 ports, which linear properties are described in a matrix Y relating a voltage applied to the ports to a current through the ports, wherein a frequency dependence of Y is approximated by
Y
(
s
)
≅
Y
rat
(
s
)
=
∑
m
R
m
s
-
a
m
+
D
+
sE
wherein the subscript “rat” denotes a rational function, m runs from 1 to a number N of poles, R m with m=1 to N are frequency independent matrices, D and E are frequency independent matrices or zero, s denotes a complex angular frequency iω, and a m with m=1 to N are complex angular frequencies of the poles,
said method comprising:
a) obtaining a set of values Y(s) of Y at discrete frequencies s, and, for each value Y(s), obtaining eigenvalues λ i and eigenvectors t i with i=1 to n;
b) fitting, in a processor of a computer processing device, a set of n vector equations
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
to the eigenvalues λ i and eigenvectors t i , wherein {tilde over (r)} m is a frequency-independent scalar with m=1 to N, by using a first set of pole frequencies am and by varying {tilde over (r)} m , R m and, if assumed to be non-zero, D and E;
c) calculating, in the processor of the computer processing device, a second set of pole frequencies from:
{ a m }=eig( A−b·c T )
wherein A is a diagonal matrix holding the first set of pole frequencies am as diagonal elements, b is a vector of ones, c is a vector holding {tilde over (r)} m for m=1 to N as obtained by step b), and eig is an operator returning a vector holding eigenvalues of a matrix;
d) repeating steps b) and c) by using the second set of pole frequencies am of step c) as the first set of pole frequencies in a subsequent step b) until a stop condition is met; and
e) outputting a model approximating the linear properties of the electrical component in which the applied voltage is related to output currents in the electrical component.
2. The method of claim 1 , comprising:
f) setting σ(s)=1 in the equation of step b) and calculating R m .
3. The method of claim 2 , wherein the electrical component is connected to an external device, with linear properties of the external device being described by a matrix Y ext , wherein the equations of step b) include:
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
Y
ext
+
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
.
4. The method of claim 3 , wherein the matrix Y describes the relation between a voltage v applied to the ports of the electrical component and currents i which are to run through the electrical component by
i×Yv.
5. The method of claim 4 , comprising:
utilizing a matrix-valued function F({b i }, s) that depends linearly on a set of parameters {b i } in the equations of step b).
6. The method of claim 1 , wherein the electrical component is connected to an external device and wherein linear properties of the external device are described by a matrix Y ext , wherein the equations of step b) include:
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
Y
ext
+
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
.
7. The method of claim 1 , wherein the matrix Y describes a relation between a voltage v to be applied to the ports of the electrical component and currents i which are to run through the electrical component by:
i=Yv.
8. The method of claim 1 , comprising:
utilizing a matrix-valued function F({b i }, s) that depends linearly on a set of parameters {b i } in the equations of step b).
9. A device for characterizing linear properties of an electrical component having n>1 ports, which linear properties are described in a matrix Y relating a voltage applied to the ports to a current through the ports, wherein a frequency dependence of Y is approximated by:
Y
(
s
)
≅
Y
rat
(
s
)
=
∑
m
R
m
s
-
a
m
+
D
+
sE
wherein the subscript “rat” denotes a rational function, m runs from 1 to a number N of poles, R m with m=1 to N are frequency independent matrices, D and E are frequency independent matrices or zero, s denotes a complex angular frequency iω and a m with m=1 to N are complex angular frequencies of the poles,
said device comprising:
a computer having a processor configured to execute a computer program recorded on a non-transitory computer-readable recording medium, the program causing the processor to execute operations comprising:
a) obtaining a set of values Y(s) of Y at discrete frequencies s, and, for each value Y(s), obtaining eigenvalues λ i and eigenvectors t i with i=1 to n;
b) fitting a set of n vector equations:
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
to the eigenvalues λ i and eigenvectors t i , wherein {tilde over (r)} m is a frequency-independent scalar with m=1 to N, by using a first set of pole frequencies am and by varying {tilde over (r)} m , R m and, if assumed to be non-zero, D and E;
c) calculating a second set of pole frequencies from:
{ a m }=eig( A−b·c T )
wherein A is a diagonal matrix holding the first set of pole frequencies am as diagonal elements, b is a vector of ones, c is a vector holding {tilde over (r)} m for m=1 to N as obtained by step b), and eig is an operator returning a vector holding eigenvalues of a matrix;
d) repeating operations b) and c) by using the second set of pole frequencies am of step c) as the first set of pole frequencies in a subsequent step b) until a stop condition is met; and
e) outputting a model approximating the linear properties of the electrical component in which the applied voltage is related to output currents in the electrical component.
10. The device of claim 9 , wherein a design of the electrical component and/or a desired operational frequency of the electrical component is based on the stop condition.
11. The device of claim 10 , wherein the electrical component is designed for connection to an external device, with linear properties of the external device being described by a matrix Y ext , wherein the equations of operation b) include:
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
Y
ext
+
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
.
12. The device of claim 11 , wherein the matrix Y describes the relation between a voltage v to be applied to the ports of the electrical component and currents i which are to run through the electrical component by:
i=Yv.
13. The device of claim 12 , wherein the equations of operation b) include a matrix-valued function F({b i }, s) that depends linearly on a set of parameters {b i }.
14. The device of claim 9 , wherein the electrical component is designed for connection to an external device, with linear properties of the external device being described by a matrix Y ext , wherein the equations of operation b) include:
λ
i
λ
i
(
∑
m
=
1
N
r
~
m
s
-
a
m
+
1
)
︷
σ
(
s
)
t
i
≅
1
λ
i
(
(
Y
ext
+
∑
m
=
1
N
R
m
s
-
a
m
+
D
+
sE
)
·
t
i
)
.
15. The device of claim 9 , wherein the matrix Y describes a relation between a voltage v to be applied to the ports of the electrical component and currents i which are to run through the electrical component by:
i=Yv.
16. The device of claim 9 , wherein the equations of operation b) includes a matrix-valued function F({b i }, s) that depends linearly on a set of parameters {b i }.