A. E. Ruehli and P. A. Brennan. Efficient capacitance calculations for threedimensional multiconductor systems. IEEE Trans. on Microwave Theory Tech., MTI-21:76--82, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....produced by raising conductor to one volt while grounding the rest. If the conductors are embedded in a homogeneous dielectric, these potential problems can be solved using an equivalent free space formulation in which the conductordielectric interfaces are replaced by a charge layer of density [22], 23] The charge layer in the free space problem will be the induced charge in the original problem if satisfies the integral equation surfaces (1) where is the known conductor surface potential, is the differential conductor surface area, is the dielectric constant, and is the usual Euclidean ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 76--82, Feb. 1973.

....problems, consider the comparisons in graphs (4) and (6) These graphs compare the self capacitance of a square plate (Fig. 3) and a cube (Fig. 5) calculated using collocation and the Galerkin method. The Galerkin method is somewhat better for coarse discrctizations. This results agrees with [16]. The additional accuracy of the Galerkin method is much more pronounced in problems with multiple dielectrics. Fig. 7 shows two concentric spheres. The inner sphere is a conducting interface of radius I and outer sphere is a dielectric interface of radius 2. The outer sphere is cut to expose the ....

Albert E. Ruehli and Pierce A. Brennan, Efficient Capacitance Calculations for Three- Dimensional Multiconductor Systems, IEEE Transactions on Microwave Theory and Techniques, vol. MTT-21, No. 2, February 1973.

....center point potentials. Using = E b= 15) gives (16) Aq = b as the linear system to solve to for the conductor charge densities. 2. 3 Matrix Solution The standard approach to solving the n x n linear system (16) is to use Gaussian elimination, at a cost of order n 3 operations [7, 5]. For this reason, the equivalent charge formulation approach to capacitance calculation is frequently considered computationally intractable if the number of panels exceeds several hundred. To improve the situation, consider solving (16) using a conjugate residual style iterative method like ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional .multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76-82, February 1973.

....of conductor panel center point potentials. Using D b = 10) gives Aq = b (11) as the linear system to solve to for the conductor charge densities. In the standard approach the n x n linear system (11) is solved using a Gaussian elimina tion algorithm at a cost of order n s operations [8, 6]. Our algorithm uses the multipole accelerated iterative method of the next section which requires only order mn operations. 711 The Multipole Approach The dense linear system of (11) can be solved to compute panel charges from a given set of panel potentials, and the capacitances can be derived ....

A. E. Ruehli and P. A. Brennan. Efficient ca- pacitance calculations for three-dimensional multiconductor systems. IEEE Transactions on Microwave Theory and Techniques, 21(2):76-82, February 1973.

.... and appropriately modified sparsification techniques [12] 13] For problems with arbitrarily shaped dielectric interfaces with moderate permittivity ratios, it is possible to use ECF, though for such problems, the Galerkin boundary element method is far more accurate than centroid collocation [14]. For sufficiently high permittivity ratios, the ECF approach, even when combined with the Galerkin method, can produce arbitrarily inaccurate results. In this paper, we show that any numerical scheme directly discretizing the ECF will fail for high permittivity ratios. The difficulty stems from ....

.... by , and the correction accounting for the finite permittivity (15) As was shown above, the potential results only from the charge on the interface, which is given by the integral equation (13) This is a capacitance problem in homogeneous media and can be solved numerically to high accuracy [4] [14]. The perturbation is set up as a superposition of charges on the conductor surface and the dielectric interface, similar to the definition of the potential in (2) 16) Substituting (16) into (1) one obtains a new set of boundary conditions for the perturbation (17) The second equation holds ....

A. Ruehli and P. Brennan, "Efficient capacitance calculations for threedimensional multiconductor systems," IEEE Trans. Microwave Theory Tech., vol. MTT-2, pp. 76--82, Jan. 1973.

....extracting the elements of Vn corresponding to . In most programs, the dense matrix problem in (6) is solved with mine form of Gaussian elimination, and this implies that the calculation grows b a, where again b is the number of current filaments into which the system of conductors is discretized [5]. For complicated packaging structures, b can exceed ten thousand, and solving (6) with Gaussian elimination can take days, even using a high performance scientific work station. 3 Mesh current approach The approach to calculating the frequency depen dent inductance and resistance matrix ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems, " IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76-82, February 1973.

....geometry is summarized by an m#m symmetric matrix C.Thej th column of the capacitance matrix is determined by finding the surface charges on each conductor by raising conductor j to one volt and grounding the rest. The charge on each conductor can be determined by solving the integral equation [2] ##x## Z surfaces ##x 4##0 kx # x # x 2 surfaces# #1# where #x# is the known conductor surface potential, # is the surface charge density, da is the incremental conductor surface area, x, x 2 R 3 ,andkxk is the usual Euclidean length of x given by p x 3 . A standard approach ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," IEEE Trans. on MicrowaveTheoryandTech., vol. 21, pp. 76-82, 1973.

.... for performing 3 D capacitance extraction have focussed on three techniques, the floating random walk method [10] improvements to the finite difference and finite element methods [3, 2] and the so called fast methods based on acceleration of the method of moments or boundary element approach [8, 9, 7]. In this paper we present a new multiscale, or wavelet like, approach to accelerating the boundary element method, and demonstrate the method on several examples. We show that this method has two important features: it can accurately represent the entries of the dense boundary element matrix ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," MTT, vol. 21, pp. 76--82, February 1973.

.... programs avoid forming Z explicitly by reformulating (15) into the sparse tableau form, A 0 h (16) Using direct factorization to solve (16) implies that the calculation grows at least as b 3, where again b is the number of current filaments into which the system of conductors is discretized [7]. For complicated packaging structures, b can exceed ten thousand, and solving (16) with direct factorization will take days, even using a high performance scientific workstation. The Mesh Based Approach The obvious approach to trying to reduce the cost of solving (16) is to apply iterative ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensionM multiconductor systems," IEEE Transactions on Microwave Thcortl and Techniques, vol. 21, pp. 76 82, February 1973.

....i, jth component of the stress tensor. III. ELECTROSTATICS In electrostatic analysis, the conductor potentials are specified and the potential must satisfy the Laplace s equation in the region between the conductors. The charge on each conductor can be determined by solving the integral equation [16] . x,ll.da , x surfaces (16) where , x) is the known conductor surface potential, er is the surface charge density, da is the incremental conductor surface area, x, x R 3, and IIxll is the usual Euclidean length of x given by (8) A standard approach to numerically solving (16) for ....

A. Ruehli and P. A. Brennen, "Efficient capacitance calculations for three-dimensional multiconductor systems;' IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 76-82, Feb. 1973.

....only solve for the surface charges on each conductor produced by raising conductor j to one volt while grounding the rest. These m potential problems can be solved using an equivalent free space formulation in which the conductor dielectric interfaces are replaced by a charge layer of density a [4]. The charge layer in the free space problem will be the induced charge in the original problem if a satisfies the integral equation o (z ) 1 g, x) c 4 eollx x ll da x surfaces. x) where b(x) is the known conductor surface potential, da is the incremental conductor surface area, x, x ....

A. E. Ruehli and P. A. Brennan, "Efficient ca- pacitance calculations for three-dimensional multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 7682, February 1973.

.... extracting the elements of V corresponding to In most programs, the dense matrix problem in (6) is solved with some form of Gaussian elimination, and this implies that the calculation grows b a, where again b is the number of current filaments into which the system of conductors is discretized [5]. For complicated packaging structures, b can exceed ten thousand, and solving (6) with Gaussian elimination can take days, even using a high performance scientific work station. 3 Mesh current approach The approach to calculating the frequency depen dent inductance and resistance matrix ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems, " IEEE Transactions on Microwave Theory. and Techniques, vol. 21, pp. 76-82, February 1973.

....given in Section 6. 2. Formulation. For exposition in this short paper, we consider only the first kind formulation used to compute electrostatic capacitances or forces in three dimensional structures. Electrostatic capacitances are useful figures of merit for designers of electronic packaging [2], and microelectromechanical system designers are interested in electrostatic forces [3] The exterior Dirichlet problem on the associated unbounded, multiply connected domain corresponding to the conductor surfaces in an infinite homogeneous medium is formulated using a single layer charge ....

.... system designers are interested in electrostatic forces [3] The exterior Dirichlet problem on the associated unbounded, multiply connected domain corresponding to the conductor surfaces in an infinite homogeneous medium is formulated using a single layer charge density denoted a [2, 4] (for a second kind formulation of this exterior problem see [5] It then follows that a must satisfy the integral equation (1) x C surfaces. where p(x) is the known conductor surface potential, da is the incremental conductor surface area, x, x C R 3, and IIxll is the Usual Euclidean length ....

[Article contains additional citation context not shown here]

A. E. Ruehl] and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76-82, February 1973.

....dielectric medium. Then to compute the capacitances, Laplace s equation is solved numerically over the charge free region with the conductors providiug boundary conditions. The usual numerical approach is to apply a boundary element technique to the integral form of Laplace s equation[2] as in f, where r is the surface charge density, x, x , da is the incremental surface area, b is the surface potential and is known, and G(x, x ) is the Green s function which in free space is To numerically solve (1) for a, the conductor surfaces are broken into n small panels or tiles. It ....

A. E. Ruehli and P. A. Brennan, "Efficient ca- pacitance calculations for three-dimensional multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76-82, February 1973.

.... with surface charges, represented as a(x) Then in this equivalent free space problem, a(x) is determined by insisting that it produce a po tentiM which matches conductor potentials at conductordielectric interfaces, and satisfies normal electric field conditions at the dielectric interfaces[2, 3]. To numerically compute r, the conductor surfaces and dielectric interfaces are diseretized into n =np . na small panels or tiles, with n r panels on conductor surfaces and na panels on dielectric interfaces. It is then assumed that on each panel i, a charge, qi, is This work was supported by ....

A. Ruehli and P. A. Brennen, "Efficient capacitance calculations for three-dimensional multiconductor sys- tems," IEEE Transactions on Microwave Theory and Techniques, vol. MTT-21, no. 2, pp. 76-82, February 1973.

....1. The charge density in the regular BEM varies rapidly near conductor corners, necessitating more nodes, whereas potential and flux density vary smoothly at the artificial boundaries we have chosen so that fewer nodes suffice; 2. In regular BEM, it is not possible to form the potential matrix [9] as a pre processing step, and it is very costly at run time, therefore macromodeling is not possible; 3. Regular BEM generates dense potential matrices to be inverted, This work was supported by the National Science Foundation under Grant MIP 9216942, and by the Semiconductor Research ....

Ruehli, A.E; Brennan, P.A. "Efficient Capacitance Calculations for Three Dimensional Multiconductor Systems," IEEE Trans. Microwave Theory Tech. MTT-21, pp. 76-82, 1973

.... ##x 0 # 1 4##0 kx # x 0 k da 0 # x 2 surfaces #5# where #x# is the known conductor surface potential, # is the surface charge density, da 0 is the incremental conductor surface area, x, x 0 2 R 3 ,andkxk is the usual Euclidean length of x given by p x 2 1 # x 2 2 # x 2 3 [9]. The most commonly used approach to solving (1) are the finiteelement methods [3] Finite element methods can also be used to solve a partial differential equation form of (5) and is a commonly used approach to solving the coupled electromechanical problem. However, for very complicated ....

A. Ruehli and P. A. Brennen, "Efficient capacitance calculations for three-dimensional multiconductor systems," IEEE Trans.Microwave Theory Tech., vol. MTT-21, pp. 76-82, Feb. 1973.

.... for performing 3 D capacitance extraction have focussed on three techniques, the floating random walk method [10] improvements to the finite difference and finite element methods [3, 2] and the so called fast methods based on acceleration of the method of moments or boundary element approach [8, 9, 7]. In this paper we present a new multiscale, or wavelet like, approach to accelerating the boundary element method, and demonstrate the method on several examples. We show that this method has two important features: it can accurately represent the N 2 entries of the dense boundary element ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," MTT, vol. 21, pp. 76--82, February 1973.

....Introduction In deep submicron VLSI, complex 3 dimensional interconnect structures pose a difficult challenge for parasitic capacitance extraction. Many extraction approaches exist, including 1 D, 2 D and 2 1 2 D analytic models [11, 2, 4, 12, 13, 5, 6, 1, 18] as well as 2 D and 3 D field solvers [10, 15, 14, 9, 7, 8]. These techniques span a wide range of cost accuracy regimes. ffl 1 D analysis uses (per unit length) total capacitance, equivalent to (per unit area) area capacitance and (per unit length) edge capacitance when some wire width is assumed. ffl 2 D analysis uses (per unit area) area capacitance ....

....and (per unit length) lateral and fringing capacitances, where geometries on one or more neighboring layers are explicitly modeled but then lumped additively into above and below corrections to the coupling calculation. ffl 3 D analysis uses numerical techniques (finite element as in [10] and Ansoft s products, or finite difference as in [15, 14] and TMA Raphael) to solve Laplace s equation for potential distribution, then applies Gauss s theorem to yield charge distribution, then applies Q = C]V to determine self and mutual capacitances of all conductors. Boundary element ....

A. E. Ruehli and P. A. Brennan, "Efficient Capacitance Calculations for Three-Dimensional Multiconductor Systems," IEEE Trans. on Microwave Theory Tech., vol. MTI-21, pp. 76-82, 1973.

.... 6 6 4 Z GammaA t A 0 3 7 7 5 2 6 6 4 I b Phi n 3 7 7 5 = 2 6 6 4 0 I s 3 7 7 5 : 16) Using direct factorization to solve (16) implies that the calculation grows at least as b 3 , where again b is the number of current filaments into which the system of conductors is discretized [7]. For complicated packaging structures, b can exceed ten thousand, and solving (16) with direct factorization will take days, even using a high performance scientific workstation. 3 The Mesh Based Approach The obvious approach to trying to reduce the cost of solving (16) is to apply iterative ....

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three-dimensional multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76--82, February 1973.

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A. E. Ruehli and P. A. Brennan. Efficient capacitance calculations for threedimensional multiconductor systems. IEEE Trans. on Microwave Theory Tech., MTI-21:76--82, 1973.

No context found.

A. E. Ruehli and P. A. Brennan. Efficient capacitance calculations for threedimensional multiconductor systems. IEEE Trans. on Microwave Theory Tech., MTI-21:76--82, 1973.

No context found.

A.E. Ruehli and P.A. Brennan, "Efficient Capacitance Calculations for Three-Dimensional Multiconductor Systems," IEEE Trans. Microwave Theory and Techniques, Vol. 21, Feb. 1973, pp. 76--82.

No context found.

A. E. Ruehli and P. A. Brennan, "Efficient capacitance calculations for three- dimensional multiconductor systems," IEEE Transactions on Microwave Theory and Techniques, vol. 21, pp. 76 82, February 1973.

No context found.

Albert E. Ruehli and Pierce A. Brennan: "Efficient Capacitance Calculation for Three-Dimensional Multiconductor Systems", IEEE Transactions on Microwave Theory and Techniques, Feb 1973, Vol. MTT-21, No.2, pp. 76

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