A Brief Description of the Mathematical Problem

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Calculation of the electrical resistance bases on the equation of the spread out resistance:

equation of the spreaded resistance

For calculation of the gradient in the equation above -- the lines of force of the stationary current flux field -- the potential field is required. Such stationary flux fields are described by the Laplacian equation:

Laplacian equation

To solve this partial differential equation for different resistor shape geometries several numerical methods like FEM or BEM can be used. The next picture shows the colored lines of force of a current field within a three times cutted bar resistor (a so called Serpentine-Cut):

gradientfield of a serpentine cut

The electrical resistance can be calculated by a numerical integration if the potential field or current flux field is known at least numerically. (see spread-out resistance equation) To determine a trim characteristic we have to solve the Laplacian equation for every small step of the trim cut path, e.g. with FEM. That means for a lot of small geometrical changes in the domain.
With the current fast computer technique this effort is not such a big deal anymore as it was a couple of years ago. However, this procedure still consumes a couple of minutes. That's still too much to use it within a CAD software or to control a laser trim equipment. But to find a proper resistor layout and trim strategy for high precision trims a simulation based on the exact physical model equations is recommended to confirm the trim strategy for all possible manufacture process circumstances. Furthermore, simulations of that kind can provide additional information, e.g. current density distribution within the domain and some predictions about resistor stability (the so called post-trim drift). The result of a well designed trim strategy is its trim characteristic set, generated during the simulation cycles, which can be used to control the trim process itself.

Some computed flux field plots:

That's an example for a computed power density distribution:


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