Data processing: measuring – calibrating – or testing – Measurement system in a specific environment – Electrical signal parameter measurement system
Reexamination Certificate
1999-08-13
2002-06-11
Hilten, John S. (Department: 2863)
Data processing: measuring, calibrating, or testing
Measurement system in a specific environment
Electrical signal parameter measurement system
C702S008000, C702S026000, C702S040000, C702S094000, C324S072000, C324S300000, C324S348000, C324S357000
Reexamination Certificate
active
06405143
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates generally to methods and systems for determining potential fields, and more specifically relates to computer implemented method and system for determining potential fields.
BACKGROUND OF THE INVENTION
With the rapid increase in both component density and operating frequency of integrated circuits, the parasitic coupling capacitance associated with interconnects can pose serious design issues. For example, these issues are prevalent in the design of large-area high-resolution amorphous silicon (“a-Si”) arrays for applications in X-ray imaging or active-matrix liquid-crystal displays. In these two specific areas, the number of parasitic capacitances increase with increasing component density, while the interconnect capacitance due to the crossover of the gate and data lines introduces electronic noise, thus undermining the quality of the image. Further, the capacitive coupling due to geometric overlapping between the gate and source/drain electrodes in the TFT can give rise to charge feed through thus lowering the pixel voltage. To gain insight into the effect of parasitic coupling capacitance on the overall array performance, it is desirable to accurately and efficiently extract the capacitance during the design process. In addition, this can aid in further development of equivalent circuit models for effective SPICE-like simulations for sensitivity analysis and design optimization.
However, there are several computational difficulties in the numerical extraction of capacitance in certain electronic circuits. For example, in a-Si TFTs and imaging arrays at least three computational difficulties arise. One computational difficulty arises from the extreme device geometries, in which the ratio of thin film thickness to other physical dimensions (such as the width and length) can well be of the order of 10
−3
. Another difficulty arises from the floating potential of the glass substrate, because the substrate needs to be included as part of the computational domain. A third problem arises in the treatment of multi-dielectric media, where the electric field is discontinuous across dielectric interfaces, thus requiring the computation of the electric field and therefore necessitating precise computation of the potential field. Overall, the extreme geometry of metal lines and active devices, and the presence of the glass substrate, require a large number N of panels (or mesh elements) to barely resolve the surfaces of interconnects and the interfaces between two dielectric media. This can lead to a very large system of equations.
Other computational difficulties can arise in electrical systems where the determination of capacitance is calculated based on the distribution of charge density on the surface of conductors within the system. Hence, the efficient extraction of capacitance can require the efficient evaluation of the charge density within the system. The charge density problem can be formulated in terms of a system of integral equations involving the electric potential &PHgr; and associated its gradient ∇&PHgr;. However, the overall computational accuracy and performance in solving the system of integral equations lies in the evaluation of the potential &PHgr; and its gradient ∇&PHgr; at all points of interest. Therefore, the efficient evaluation of the capacitance reduces to the efficient evaluation of the potential and the electric field itself, i.e. the gradient of the potential, within the system.
In a more general context, the computation of the potential field is of interest in many research areas ranging from astrophysics, plasma, physics, molecular dynamics, to VLSI and electro-mechanical systems (“MEMS”). In these systems, the potential of a number of particles located throughout a three-dimensional domain and, where each particle has an associated real value (q
i
, i=1 . . . N) is given by the well-known 1/r dependence formula:
Φ
(
X
i
)
=
∑
N
j
=
1
,
j
≠
i
q
j
&LeftDoubleBracketingBar;
X
i
-
X
j
&RightDoubleBracketingBar;
Equation
1
where:
&PHgr; is the potential
N is the number of particles
X
i
, i=1 . . . N is the location of each particle
q
i
, i=1 . . . N is the charge of each particle
∥X
i
−X
j
∥ is the distance r between each particle.
If the evaluation of Equation 1 is performed through direct pairwise particle-to-particle interactions, the computational time is of the order O(N
2
), which in most computers is prohibitively expensive for a large number N of particles. Prior art methods have somewhat reduced this computational time through the use of innovative operations. For example, see J. Barnes and P. Hut, “A hierarchical O(N log N) force-calculation algorithm”,
Nature
, vol 324, pp. 446-449 (“Barnes & Hut”). See also, L. Greengard,
The Rapid Evaluation of Potential Fields in Particle Systems
, MIT Press, 1988 (“Greengard”). See also L. Greengard and V. Rhokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions”, Tech. Rep. YALEU/DCS/RR-1115, Yale University, September 1996 (“Greengard/Rhokhlin”).
Such prior art methods can evaluate the potential in a reduced computational time of only O(N log N) or even O(N). Computational time is reduced by the replacement of a cluster of particles with a single pseudo-particle which is representative of all particles within the cluster. Thus, subsequent steps in the operation need only determine the potential associated with the pseudo-particle. The step involved of creating of the pseudo-particle is usually referred to as a “translation”.
Underlying prior art translations is a multipole expansion (“MP”) operation. In multipole expansions, a tree-like hierarchy of cubes is used in the clustering of the particles and the addition theorem in spherical harmonics is the mathematical background for the translations. The order of the expansion p of the potential in terms of spherical harmonics characterizes the overall performance of the multipole-expansions. The numerical effort in the multipole-expansion method lies in obtaining the expansion coefficients efficiently through a sequence of translations, namely:
1. from source points to multipole centres, defined in the prior art as “Q2M” translations;
2. from multipole centres to multipole centres, defined in the prior art as “M2M” translations;
3. from multipole centres to local centres, defined in the prior art as “M2L” translations;
4. from local centres to local centres, defined in the prior art as “L2L” translations; and,
5. from local centres to the target point for evaluation of the potential at the target, defined in the art as “L2P” translations.
Despite presenting improvements over calculations using Equation 1, certain disadvantages remain with the prior art. For example, the prior art utilizes translations with different structures: one for M2M, another for M2L, and yet another for L2L. Furthermore, Greengard involves a double sum, while Greengard/Rhokhlin involves a single sum with tight coupling between the multipole and local expansion coefficients in the spherical harmonics expansion. Another problem with the prior art is that the accuracy inherently depends on the order of the expansion p in terms of spherical harmonics and on how the M2L translations are performed. See, for example, H. Petersen, E. Smith and D. Soelvason, “Error estimates for the fast multi-pole method. II. The three-dimensional case,”
Proc. R. Soc. Lond. A
, vol. 448, pp. 401-418, 1995. A third problem is that the computation time inherently depends on the order of the expansion and on the trade-off between computational time and memory in choosing the interaction list for M2L translations. The computation time is O(N log N) in Barnes & Hut, O(p
4
) in Greengard and O(p
3
) in Greengard/Rhokhlin. These speeds can be prohibitively high for certain computers and for certain large numbers of particles. A fourth problem is that the prior art can require a large amount of computer memory. The multipole
Nathan Arokia
Pham Hoan Huu
Gowling Lafleur Henderson LLP
Hilten John S.
The University of Waterloo
Vo Hien
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