636 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 47, NO. 5, MAY 1999 Short Papers A Generalized Dirichlet Principle for Smoothing Small-Signal Measurements Arthur David Snider and Peter Winson Abstract—We report the theory, implementation, and results of using a Poisson solver to compensate the measured values of drain conductance and transconductance in a nonlinear metal–semiconductor field-effect transistor so as to render them compatible with a large-signal model. The consequent restructuring of the I–V curves is exhibited. Index Terms—FET’s, modeling, nonlinear estimation, smoothing meth- ods. I. GENERAL STATEMENT OF PROBLEM A recurring problem in the modeling of many engineering devices can be described as follows. A certain vector field is, according to theory, supposed to be conservative; it is the gradient of a scalar field . The components of are measured. Due to experimental error or inadequacy of the model, the measured components are nonconservative; curl is small, but not zero. The question is: how does one modify the data so as to construct the scalar ? That is, which genuinely conservative field is closest to the measured ? Specific scenarios for this situation occur in electrostatics (where one needs to construct the electric potential from the electric field ) and in thermodynamics (where the Gibbs free energy is constructed from the volume and the (negative) entropy ) [1]. But perhaps the problem is most crucial in the modeling of nonlinear transistors. Fig. 1 depicts a simple small-signal model of a metal–semi- conductor field-effect transistor (MESFET), corresponding to the large-signal model in Fig. 2. The drain conductance and the transconductance carry the drain current . It is presumed that these conductances are incremental elements of the large-signal drain current , which depends nonlinearly on the gate and drain voltages and as follows: (1) Thus, the vector is a gradient of in “voltage space” and, in theory, one can construct the large-signal current by computing the path-independent integral The zero-curl condition (2) is interpreted as the equality of mixed second partials, and if the values of extracted from -parameter measurements fail to meet this compatibility condition accurately, the integral becomes path-dependent. The question then becomes how the data should be adjusted to achieve consistency with (2), and what values should be assigned to the large-signal drain current . Manuscript received June 25, 1997; revised December 10, 1998. A. D. Snider is with the Department of Electrical Engineering, University of South Florida, Tampa, FL 33620 USA. P. Winson is with the Raytheon Systems Company, Dallas, TX 75265 USA. Publisher Item Identifier S 0018-9480(99)03135-X. Fig. 1. Simplified small-signal FET model. Fig. 2. Simplified large-signal FET model. II. MATHEMATICAL GENERALIZATION In mathematical jargon, we wish to replace a measured vector field with a conservative gradient field so as to minimize the mean- square error For the case , the solution to this variational problem is the classical Dirichlet principle, which states that must be a harmonic function meeting the homogeneous Neumann boundary condition [2]. We shall show that for our situation, satisfies Poisson’s equation (3) where the nonhomogeneity is the divergence of , and the nonho- mogeneous Neumann boundary condition that ’s outward normal derivative must match the outward normal component of (4) Thus, we term our solution a generalized Dirichlet principle. (This problem is discussed in great generality in [3].) The mathematical conditions (3) and (4) can be derived by the following brief argument, modeled after that in [2]. Suppose minimizes Then, for any function , is minimal when . The derivative of this integral with respect to at is by the divergence theorem (Green’s first formula); the final integral is taken over the boundary of the region [4]. We conclude that (5) 0018–9480/99$10.00  1999 IEEE