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Potential Coefficient Equations for Non-Parallel, Non-Orthogonal Cells

IP.com Disclosure Number: IPCOM000062078D
Original Publication Date: 1986-Oct-01
Included in the Prior Art Database: 2005-Mar-09
Document File: 3 page(s) / 28K

Publishing Venue

IBM

Related People

Johnson, TA: AUTHOR

Abstract

The technique presented permits calculations of the potential coefficients necessary for capacitance evaluation between conductors which are non-parallel and non-orthogonal but are inclined at an angle to one another. Reference [1] gives the equation (1) which represents the potential coefficient for any arbitrary cell shape. (Image Omitted) Si and Sj represent the areas of cells i and j, respectively, and Green's function, G, describes the interaction between the cells. For the assumptions of [1], namely, parallel or orthogonal cells, G reduces to 1/R. The case now considered is shown in Fig. 1, where two cells are inclined at an angle to one another. Using the representations shown in Fig. 1, Green's function takes on the following form.

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Potential Coefficient Equations for Non-Parallel, Non-Orthogonal Cells

The technique presented permits calculations of the potential coefficients necessary for capacitance evaluation between conductors which are non-parallel and non-orthogonal but are inclined at an angle to one another. Reference [1] gives the equation (1) which represents the potential coefficient for any arbitrary cell shape.

(Image Omitted)

Si and Sj represent the areas of cells i and j, respectively, and Green's function, G, describes the interaction between the cells. For the assumptions of [1], namely, parallel or orthogonal cells, G reduces to 1/R. The case now considered is shown in Fig. 1, where two cells are inclined at an angle to one another. Using the representations shown in Fig. 1, Green's function takes on the following form.

(Image Omitted)

For rectangular cells, this results in the potential coefficient, Ps, to be

(Image Omitted)

After solving for the first three integrals, Ps reduces to

(Image Omitted)

where If equation (4) is expanded and written explicitly in terms of the remaining integration variable X", the following equation (8) results. where Equation (8) has sixteen separate terms. A general form for the solution for each term is given. FORM 1: FORM 2: Both of these expressions are relatively simple and may be evaluated. Solutions may be found in [2]. FORM 3: The solution to this integral is: where FORM 4: The solution to this integral is: FORM 5: The solution to this integral is: FORM 6: The solution to this integral is: FORM 7: The solution to this integral is: FORM 8: The solution to this integral is: FORM 9: The solution to this integral is: FORM 10: The solution to this integral is: where The solution to this integral is: where FORM 12: The solution to this integral is given as a sum of six parts. They are: where FORM 13: The solution of this integral is best performed after a change of variables. If one uses the following definitions: then the integral may be rewritten as: The solution of this integral may be written as the sum of two parts. where D(u) is an irreducible quartic in u, and N(u) is a polynomial in u. The degree of N(u) is 4. N(u)/D(u) may be written as: where the ri are the roots of D(u). This solution assumes that the four roots are...