Data processing: measuring – calibrating – or testing – Measurement system – Performance or efficiency evaluation
Reexamination Certificate
2000-09-18
2003-04-29
Assouad, Patrick (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system
Performance or efficiency evaluation
C716S030000, C703S013000
Reexamination Certificate
active
06556954
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to a method and apparatus for determining a malfunction of a technical system that is subject to a malfunction.
2. Description of the Related Art
The numerical simulation of electrical circuits has become of great significance in the development of computer chips in recent years. Simulators have become indispensable due to the high costs of producing a specimen chip and of possible redesign. With simulators it is possible to obtain predictive statements about operating behavior and efficiency of the modeled circuit on a computer. After successful simulation results, a chip is usually burned in silicon.
Using a “Network” approach, a circuit is described by its topological properties, the characteristic equations of circuit elements, and the Kirchhoff rules.
A modified, node analysis known from A. F. Schwarz, Computer-Aided design of microelectronic circuits and systems, vol. 1, Academic Press, London, ISBN 0-12-632431-A, pp. 185-188, 1987 (“Schwarz”) may be utilized for the analysis of a circuit. This leads to a differential-algebraic equation system having the form
C
(
x
(
t
))·
x
(
t
)+
f
(
x
(
t
))+
s
(
t
)=0. (1)
A differential-algebraic equation system is an equation system of the type
F
(
x
(
t
),
x
(
t
)
t
)=0 (2)
with a singular Jacobi matrix F·x of the partial derivations of F according to x(t).
The differential-algebraic equation system (1) is also called a quasi linear-implicit equation system. x(t) indicates a curve of node voltages dependent on a time t, x(t) indicates its derivative according to the time t. f(x(t)) references a first predetermined function that contains conductance values and non-linear elements, C(x(t)) references a predetermined capacitance matrix and s(t) references a second predetermined function that contains independent voltage sources and current sources.
Equation (1) describes only the ideal case for a circuit. In practice, however, noise, i.e., a malfunction of the circuit, cannot be avoided. “Noise” is defined here as an unwanted signal disturbance that is caused, for example, by thermal effects or the discrete structure of the elementary charge. Due to the increasing integration density of integrated circuits, the significance of the predictive analysis of such effects (noise simulation) is increasing.
In the analysis of a circuit taking noise into consideration, Equation (1) can be modeled as:
C
(
x
(
t
))·
x
(
t
)+
f
(
x
(
t
))+
s
(
t
)+
B
(
t, x
(
t
))·&ngr;(&ohgr;,
t
)=0. (3)
where &ngr;(&ohgr;, t), references an m-dimensional vector whose independent components are generalized white noise, where m indicates the number of noise current sources. A matrix B(t, x(t), also referred to as intensity matrix has the dimension n×m. When only thermal noise in linear resistors occurs, then it is constant.
When the circuit is modelled with purely linear elements, then rule (3) becomes a disturbed, linear-implicit differential-algebraic equation system having the form
C·x
(
t
)+
G·x
(
t
)+
s
(
t
)+
B
(
t, x
(
t
))·&ngr;(&ohgr;,
t
)=0. (4)
What is to be understood by the term “index” below is a criterion regarding how far a differential-algebraic equation system “differs” from an explicit, ordinary differential equation system, how many derivation steps are required in order to obtain an explicit, ordinary differential equation system from the differential-algebraic equation system.
Without limitation of the universal validity and given existence of various term definitions of the term “index”, the following definition is employed below for an index of a differential-algebraic equation system:
Let a differential-algebraic equation system of the type
F
(
x
(
t
),
x
(
t
),
t
)=0 (2)
be given. When a lowest natural number i exists, so that the equations
F
(
x
(
t
),
x
(
t
),
t
)=0, (5)
ⅆ
F
(
x
.
(
t
)
,
x
(
t
)
,
t
)
ⅆ
t
=
0
,
(
6
)
⋮
,
ⅆ
(
i
)
F
(
x
.
(
t
)
,
x
(
t
)
,
t
)
ⅆ
t
(
i
)
=
0
(
7
)
can be transformed into a system of explicit ordinary differential equations, then ‘i’ is referred to as the index of the differential-algebraic equation system. The function F is thereby assumed to be capable of being differentiated an adequate number of times.
A “stochastic differential equation system” is defined by the following differential equation system:
Let a Wiener-Hopf process {W
t
; t·&egr;R
0
+
} be given on a probability space (&OHgr;, A, P) together with a canonic filtration {C
s
; s&egr;[a, b]}. Additionally h and G be: [a, b]×R→R two ((B
[a,b]
×B)−B)-measurable random variables and {tilde over (X)}:&OHgr;→R a (C
a
−B)-measurable function. A stochastic differential equation system is established by the Itô differential
X
s
: &OHgr;→R,
ω
↦
X
~
(
ω
)
+
∫
[
a
,
s
]
h
(
u
,
X
u
(
ω
)
)
ⅆ
λ
(
u
)
+
(
∫
a
s
G
(
u
,
X
u
)
ⅆ
W
u
)
(
ω
)
(
8
)
or, symbolically,
dX
s
=h
(
s, X
s
)
ds+G
(
s, X
s
)
dW
s
, s&egr;[a, b]; X
a
={tilde over (X)}.
(9)
The following method for noise simulation is known from A. Demir et al. Time-domain non-Monte Carlo noise simulation for nonlinear dynamic circuits with arbitrary excitations, IEEE Transactions on Computer-Aided Design of Integrated Circuits and System, Vol. 15, No. 5, pp.493-505, May 1996 (“Demir”)
For the case of a purely additive disturbance, rule (4) can be decoupled into a differential and an algebraic part.
The following applies in the case of a purely additive disturbance:
B
(
t, x
(
t
))≡
B
(
t
), (10)
i.e., the intensity matrix is only dependent on the time t.
(4) thus becomes
C·x
(
t
)+
G·x
(
t
)+
s
(
t
)+
B
(
t
)·&ngr;(&ohgr;,
t
)=0. (11)
Given the assumptions that consistent starting values (x
det
(t
0
), x
det
(t
0
)) are established at a starting time to and given regularity of the matrix brush {&lgr;C+G; &lgr;&egr;C}, an unambiguous solution x
det
for (1) exists in the form
C·x
det
+G·x
det
+s
=0. (12)
A matrix brush {&lgr;C+G; &lgr;&egr;C} is regular when a &lgr;
0
from C exists such that
det(&lgr;
0
C+G
)≠0 (13)
applies.
Consistent starting values (x
det
(t
0
), x
det
(t
0
) can be acquired in that a DC operating point of the system, which is described by (12), is defined, i.e., x
det
=0 is set. NB. J. Leimkuhler et al., Approximation methods for the consistent initialization of differential-algebraic systems of equations, SIAM J. Numer. Anal., Vol. 28, pp. 205-226, 1991 (“Leimkuhler”) discloses a further method for determining consistent starting values (x
det
(t
0
), x
det
(t
0
)).
After a transformation, which is described below, one arrives at rules that are equivalent to rule(11) and have the following form:
y·
[1]
+F
1
·y
[1]
+F
2
·y
[2]
+&sgr;
[1]
+(
{tilde over (B)}
(
t
)·&ngr;(&ohgr;,
t
))
[1]
=0 (14)
and
F
3
·y
[1]
+F
4
·y
[2]
+&sgr;
[2]
+(
{tilde over (B)}
(
t
)·&ngr;(&ohgr;,
t
))
[2]
=0 (15)
with transformed starting conditions
y
[1]
det(t
0
):=(
T
−1
·x
det
(t
0
))
[1]
, (16)
y
[1]
det(t
0
):=(
T
−1
·x
det
(t
0
))
[1]
(17)
and
y
[2]
det(t
0
):=(
T
−1
·x
det
(t
0
))
[2]
, (18)
y
[2]
det(t
0
):=(
T
−1
·x
det
(t
0
))
[2]
(19)
and with
{tilde over (B)}
(
t
):=
S·B
(
t
). (20)
A prescribable
Denk Georg
Schein Oliver
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