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# Algorithms for Efficient Inductance Calculation

IP.com Disclosure Number: IPCOM000108120D
Original Publication Date: 1992-Apr-01
Included in the Prior Art Database: 2005-Mar-22
Document File: 5 page(s) / 161K

IBM

## Related People

Huang, CC: AUTHOR

## Abstract

The mutual inductances in a complicated structure can be efficiently evaluated by exploiting the symmetry in the conductor's cross sections, together with the proper use of Gauss Quadrature integration method. Description

This text was extracted from an ASCII text file.
This is the abbreviated version, containing approximately 48% of the total text.

Algorithms for Efficient Inductance Calculation

The mutual inductances in a complicated structure can be
efficiently evaluated by exploiting the symmetry in the conductor's
cross sections, together with the proper use of Gauss Quadrature
integration method.
Description

From (1,2), the mutual inductance L12 between two conductors (1
and 2) can be written as:

(Image Omitted)

where A1 and A2 are the cross sections of these two conductors, and
Lf is the mutual inductance between two "current filaments".  While
closed-form solutions exist in some special cases (Fig. 1), Eq. (1)
needs to be integrated numerically in general. In the following two
algorithms, namely, the Cross-Sectional Symmetry and the Gauss
Quadrature integration method, are exploited to evaluate Eq. (1)
efficiently.
Cross-Sectional Symmetry

Fig. 1 is the cross-section of two rectangular conductors.  The
mutual inductance on the left is the same as that on the right.

As shown in Fig. 1(a), if the conductors are both of symmetric
cross section, and one conductor's axis of symmetry aligns with the
other conductor's axis of symmetry, then a "2-fold" symmetry can be
identified.  It follows that one of the integrals need only be
evaluated over half of that conductor's cross section.  The
implication is that, in case of a 2-fold symmetry, the mutual
inductance is the SAME, whether or not we remove half of a conductor
from the configuration.  Similarly, as shown in Fig. 1(b), a "4-fold"
symmetry can be identified if the two conductors have two axes of
symmetry aligned.  Thus, Eq. (l) can now be re-written as:

(Image Omitted)

where A'1 = A1/2 or A1/4.  In other words, we now only need to carry
out the numerical integration over a much smaller area, if the above
symmetry exists.

To evaluate the mutual inductance between two circular
cylinders, the areas of integral are reduced to much smaller ones
(Fig. 2).

An interesting property can be observed between two circular
conductors.  As shown in Fig. 2(a), a "2-fold" symmetry can be
identified between ANY two circular conductors, if only they are in
the same direction.  Letting a and b be the radii of two conductors,
Eq. (l) now becomes:

(Image Omitted)

In case the two circular conductors are co-axial, an "n-fold"
symmetry is observed (see Fig. 2(b)), and Eq. (1) is further reduced
to

(Image Omitted)

That is, the mutual inductance between two co-axial circular
conductors is the SAME as that between a semi-circular cylinder and a
"weighted" (with p') thin strip.

The concept of cross-sectional symmetry mentioned above can be
applied to many geometries in the packaging application.  As shown in
Fig. 3(a), a solid or mesh p...