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Abstract
The $1\over{\rm f}$ current noise of a laser trimmed resistor is known to be dependent on the shape of the resistor; it is greatest near the tip of a straight cut, and cuts with softer corners give rise to resistors having less noise. This is attributed to current crowding near the corners of these cuts. The geometric factor connected with current flow is explained in the derivation of an expression for the noise as the integral $\int\sb{\rm D}\int\vert$E$\vert\sp4$dxdy, where E = $-\nabla\phi$ is the electric field, $\phi$ is the potential, and D is the face of the resistor. Corners appear as singularities in the integrand. In this situation, it is equivalent to express $\vert$E$\vert$ using the stream function $\psi$, because $\vert\nabla\phi\vert = \vert\nabla\psi\vert$. $\psi$ is the solution of Laplace's equation in the resistor with appropriate boundary conditions.
The present investigation, aided by the formulation of the noise integral given above, addresses the problem of determining the resistor shape which has least noise for a specified resistance. First, the Euler-Lagrange (EL) equation applicable to the problem is derived, but its forboding appearance hinders any effort to find a solution analytically. In addition, the irregular boundary conditions which result from the trimming process generally lead to boundary value problems which are intractable by conventional methods. Consequently, the problem is resolved numerically.
By restricting consideration to polygonal configuration, it is possible to construct a pair of Schwarz-Christoffel (SC) transformations which map each troublesome resistor shape onto a rectangle, where $\psi$ is easily found. The mappings themselves give a representation for $\psi$ and expose the singularities in E. With this information, the most singular part of the noise integral can be expressed in terms of beta functions, and the balance of the integrand can be accurately assessed by rectangular rules after first performing an additional transformation to subdue the effects of the remaining, gentler singularities.
Once the computational challenges have been met, the resistor shape having least noise is identified by incorporating this computation into a steepest descent search. Finally, it is shown that the computer-generated shape, which has surprising simplicity, suggests a non-smooth solution for which the EL equation is not applicable. An independent confirmation of the legitimacy of this solution (in fact, using EL theory) is also presented.
