The Esaki Diode as a Network Element

Abstract

The following basic theorem is proved: Theorem: Necessary and sufficient conditions for the physical realizability of a short-circuit admittance matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> port obtained by imbedding an Esaki (tunnel) diode in a reciprocal lossless network are: 1) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">- Y(-p)</tex> is a positive-real matrix with either a pole or a zero at infinity. 2) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[Ev Y]</tex> is of rank one or less. 3) (A_1^{kk} A_2 - p^2 B_1^{kk}B_2)^{1/2} is a polynomial which is even or odd as the degree of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> is even or odd. 4) The sum of the poles <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(P)</tex> does not exceed the reciprocal time constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(l/T)</tex> of the diode. 5) The ratio of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/T</tex> does not exceed the ratio of the determinants of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y^q</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_{sc}^q</tex> at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p = \infty</tex> for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> , or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PT \leq \frac {\det\cdot Y^q} {\det \cdot Y_{sc}^q} \arrowvert_{p = \infty}, q = 1,2,\cdots,n;</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Ev</tex> implies the "even part of," each element of matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</tex> is defined as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_{ik}= \frac {N_{ik} {D} = \frac {p B_1^{ik} - A_1^{ik}} {A_2 - p B_2}</tex> with polynomials <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A_2, B_2, A_1^{ik}, B_1^{ik}</tex> being even or odd as the degree of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> is even or odd. The martix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y_{sc}</tex> is the admittance matrix obtained by short circuiting the diode. The superscript q represents all possible combinations of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> accessible ports (the rest are short circuited).