We see that the results match the measurements to within 2%, which is likely within the accuracy of the measurement considering stray inductance at the ends of the spiral and of the probe leads. The measurements were made using an Agilent LCR meter. The dimensions and the inductances of both the model and the measurements are shown in. To make the integration for the self-inductance more tractable, the numerical integration is performed with the singularities at the limits of the integration range. In this case, of course, we integrate in rectangular coordinates.
For the self-inductance of a side, for a uniformly-distributed current, and for the mutual inductance of the opposite side. The two perpendicular sides do not add mutual inductance because the current densities are perpendicular to each other and therefore. From symmetry, we need only compute the self-inductances of two sides with different lengths, and the two mutual inductances between opposite sides and multiply their sum by two to obtain the self-inductance of the loop. Inductance calculations for a rectangular loop with a rectangular cross-section A conductor in the shape of a rectangular loop (sides of lengths and ) with a rectangular cross-section (height, and width, ) is shown in. However, it only took seconds for both uniform currents (low frequency input) and surface currents (high frequency input), although time was needed to prepare its input files. FastHenry, which uses rectangular segments, defined by the user, produced slightly less accurate results, with the ring discretized into a 100 segments, with cross-sectional areas set to be the same as those of the rings. Timing of the numerical integration was on the order of seconds for the surface current model, and tens of seconds for the uniform current model. Convergence could be obtained by lowering the precision of the result (using the PrecisionGoal option) but it was found that the results were usually not as accurate. In fact, for almost all the inductance calculations described in this paper, Mathematica produced a warning that the global error in the numerical integration had not decreased monotonically yet, as will be shown in this paper, all the results were highly accurate.
The small error that remained is due to the errors near the singularity, where convergence can be difficult. The match between the exact solutions of ) and (13) and the results produced by this method, shown in and, is excellent with the largest error (% deviation from exact solution) being.76% but with most errors less than.01%. Ansys Maxwell does not solve ) and (2) but solves Maxwell's differential equations with boundary conditions and user-specified initial conditions using a finite element method in which the solution is numerically obtained for an arbitrary geometry by breaking it down into tetrahedrons, and using second order quadratic expressions as basis functions to represent the electric or magnetic fields, at the vertices and the midpoints of selected edges. The user specifies the discretization by inputting the coordinates and dimensions of the rectangular filaments. 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. 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, along with a multipole approach to approximate the integrals of (1) and (2) that are performed for each filament.
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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.
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