Find articles by Lisa F. Shatz Craig W. Christensen Find articles by Craig W. Christensen Timothy C. Conceived and designed the experiments: LS CC.

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Find articles by Lisa F. Shatz Craig W. Christensen Find articles by Craig W. Christensen Timothy C. Conceived and designed the experiments: LS CC. Performed the experiments: LS CC. Analyzed the data: LS CC. Wrote the paper: LS CC. Received Oct 21; Accepted Sep This article has been cited by other articles in PMC. Abstract A method of calculating inductances based on first principles is presented, which has the advantage over the more popular simulators in that fundamental formulas are explicitly used so that a deeper understanding of the inductance calculation is obtained with no need for explicit discretization of the inductor.

It also has the advantage over the traditional method of formulas or table lookups in that it can be used for a wider range of configurations. It relies on the use of fast computers with a sophisticated mathematical computing language such as Mathematica to perform the required integration numerically so that the researcher can focus on the physics of the inductance calculation and not on the numerical integration.

Introduction Inductors are used to provide filtering or energy storage within many types of electrical systems.

They are often coiled conductors whose time changing currents induce voltages either in the conductor itself or nearby conductors. They are used extensively in analog circuitry and are an essential component of every electric utility power grid. For most of the twentieth century, calculating inductances was done mainly through the use of formulas or look-up tables [1] , [2] because the inductance calculation which contains an integrable singularity had been intractable.

For the past thirty years or so, simulation tools such as FastHenry [3] , [4] , and Ansys Maxwell [5] , have been used to calculate inductances. Also, for these tools, the user is unlikely to be familiar with their algorithms, so the researcher is less likely to develop an understanding of the fundamentals of inductance calculations; moreover an additional discretization of the inductor may be needed.

Now, with the use of modern high speed computers that use sophisticated mathematical computing languages, the numerical integration is tractable and so inductances can now be computed using first principles, using a multi-functional computational program that may already be available to the user. With this method, the workings of the numerical integration are performed by the computational program, and so there is no need for the user to become an expert in sophisticated numerical integration techniques or in methods for discretizing the geometry.

This method is particularly useful for the design of air core reactors, which protect equipment from potentially damaging power transients. Air reactors, which can consist of tens or even hundreds of interlaced coils, are often custom-designed for a particular operating environment, and inductance simulations of these reactors can be challenging for simulators that were designed with VLSI integrated circuits in mind.

Therefore, we studied a method that uses the computational program Mathematica [6] to determine inductances. The mutual inductance between two conductors has a similar formula except that the field points are in one conductor and the source points in another, 2 Although there are few highly symmetric configurations that allow Eqns. FastHenry, Ansys Maxwell, and Mathematica FastHenry determines inductances and resistances by approximating a conductor as a a series of discretized rectangular filaments, each having a lumped resistance and inductance.

Using mesh analysis, it derives a set of complex linear equations for the filaments whose number may be in the thousands and uses advanced numerical tools such as GEMRES [8] , along with a multipole approach [9] to approximate the integrals of 1 and 2 that are performed for each filament.

The multipole approach uses multipole expansions for 1 and 2 which may be valid for large and can give an accurate approximation of the integral with far fewer computations. The user specifies the discretization by inputting the coordinates and dimensions of the rectangular filaments.

Ansys Maxwell [5] does not solve Eqns. The coefficients of the basis functions are solved for using standard matrix techniques. Since Ansys Maxwell is a large finite element program for general electromagnetic problems, the overhead to use it in terms of cost and training is high. We use Mathematica to determine the inductance by having Mathematica numerically integrate 1 or 2 over the entire volume of the conductor. Mathematica uses a global adaptive strategy, that tries to reach the required precision and accuracy goals of the integral estimate by recursive bisection of the subregion with the largest error estimate into two halves, and computes integral and error estimates for each half.

The user inputs a Mathematica file which describes the integral, and does not need to perform any discretization. Mathematica determines the number of recursive bisections to determine an accurate result. To demonstrate the accuracy of this technique, we use configurations with known inductances to check our results such as that of a ring, a rectangular loop, a solenoid, and a spiral.

We used Mathematica 7 and 9 on a Dell Optiplex 2. We also test our results against a simpler version of air core reactors used in the field, a two-layer, two-coiled spiral, which was built to test this method, since a known solution for this configuration does not exist.

Results Inductance calculations for a ring A conductor in the shape of a solid ring is illustrated in Figure 1. Let us define the coordinates of a point on the conductor as, 3.


Inductance Calculations

Product Details This authoritative compilation of formulas and tables simplifies the design of inductors for electrical engineers. It features a single simple formula for virtually every type of inductor, together with tables from which essential numerical factors may be interpolated. Although compiled in the s, before calculators and computers, this book provides fundamental equations that professionals and practitioners can use to produce algorithms for computer programs and spreadsheets. Starting with a survey of general principles, it explains circuits with straight filaments; parallel elements of equal length; mutual inductance of unequal parallel filaments and filaments inclined at an angle to each other; and inductance of single-layer coils on rectangular winding forms. Additional topics include the mutual inductance of coaxial circular filaments and of coaxial circular coils; self-inductance of circular coils of rectangular cross section; mutual inductance of solenoid and a coaxial circular filament and coaxial single-layer coils; single-layer coils on cylindrical winding forms; and special types of single-layer coil.



Introduction In I was building a tuned loop antenna for one of my homebrew receivers, and needed to find an accurate inductance formula for a flat spiral loop of large diameter. I was unable to find anything suitable over the internet, and started digging through some old books. However, while this book remains a standard reference to this day, it was published in when electronic calculators and computers were generally unavailable. As a result, most of the complex formulae were put into table form in order to make hand calculation easier. I wanted to use a spreadsheet to do the calculations, and so I began to research the literature to find the original formulae from which these tables were derived. Many of the underlying formulae were series expansions of elliptic integral formulae or other similarly intractable formulae. The purpose of this webpage is to discuss some of the methods available to calculate inductance using the power of the computer to its best advantage.

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