The MESFET transistor

Communicated by A. Mediavilla (DICOM) and ellaborated by L. Gonzalez-Vega

The design of the MESFET transistor requires the solution of the following nonlinear system of equations for $n\in\{2,3,4\}$:

$$\sum_{j=1}^n a_{ij}\cos(x_j+\beta_{ij})=b_i, \qquad 1\leq i\leq n$$

This system arises when computing the coefficients linked to the derivatives in the Taylor development of the nonlinearities of the capacity appearing in the circuit model for the microwave transistor MESFET. These coefficients are very important when determining the transistor behaviour in the intermodulation distortion and, currently, almost no model takes care of this nonlinearity.

This system is converted to an algebraic one by expanding the cosine function and by introducing the new variables:

$$c_i=\cos(x_i)\quad s_i=\sin(x_i),\qquad 1\leq i\leq n$$

In this way the algebraic problem is reduced to the resolution of the following polynomial system of 2n equations with 2n unknowns:

$$\matrix{ \displaystyle{\sum_{j=1}^n A_{ij}c_j+\sum_{j=1}^n B... ..._j=b_i},& 1\leq i\leq n\cr\cr c_i^2+s_i^2=1,& 1\leq i\leq n\cr}$$

where the A_ij's and the B_ij's are polynomials with integer coefficients in the $\cos(\beta_{ij})$, $\sin(\beta_{ij})$ and a_ij. In this case, only real solutions are required.

The case n=2 is very easily solved and the solution is given by the following:

$$\matrix{ U_4 s_2^4+U_3 s_2^3+U_2 s_2^2+U_1 s_2+ U_0=0\cr \cr... ...s_2)}}\cr \cr s_1=\displaystyle{{V_3(s_2)}\over{V_0(s_2)}}\cr}$$

where the U_k's are polynomials in ${\rm Z\!\!Z}[A_{ij},A_{ij},b_1,b_2]$ and the V_i(s₂)'s are polynomials in ${\rm Z\!\!Z}[A_{ij},B_{ij},b_1,b_2][s_2]$. For example, V₀(s₂) is:

$$\matrix{ 2(-B_{11}A_{21}+A_{11}B_{21}) (A_{11}^2A_{22}B_{22}... ...B_{22}s_2+A_{12}B_{12}A_{21}^2s_2-A_{12}b_1 A_{21}^2)\hfill\cr}$$

Two interesting problems remain to be solved with respect to this problem:

1.

The simplification of the polynomials U_i or V_j(s₂) would provide a better solution: for example in the previous expression of V₀(s₂) the first factor can easily be represented as

$$-B_{11}A_{21}+A_{11}B_{21}=a_{11}a_{21}\sin(\beta_{21}-\beta_{11})$$

2.

The solution of the cases n=3 and n=4 requires the manipulation of very big parametric expressions and one possible way of solving this problem could be the simplification question addressed before.

Finally, we note that the proposer of this problem considered the solution for the case n=2 to be very important for his practical purposes, since he never imagined that such a solution existed.