Two Extensions of the Darlington Synthesis Procedure

Abstract

Two methods are presented which make it possible to synthesize a driving-point impedance function by a Darlington-type synthesis without surplus factors. One method relies on the use of nonreciprocal devices (gyrators) in the synthesized network. The additional constraints due to presence of gyrators are discussed. It is shown that the residue condition for physical realizability now becomes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k_{11} k_{22} - k_{12}^2 - \beta^2 &gt; 0</tex> ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta</tex> is the coefficient of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_{12}</tex> term which is of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta \omega_{0}/(s^2 + \omega_{0}^2))</tex> . It is shown also that two real-part conditions now exist. In addition to the usual Brune real-part condition <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re z_{11} Re z_{22} - (Re(z_{12} + z_{21}))^2/4 &gt; 0</tex> for reciprocal elements, there exists another restriction <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Re 1/y_{11} Re 1/z_{22} - (Re(z_{12} - z_{21})/2z_{22})^2 \geq 0</tex> on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</tex> -axis (the equal sign is dropped for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Res &gt; 0</tex> ). The second method relaxes the cascade network requirement, and from a study of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E v Z(s)</tex> allows the synthesis of the network by a Darlington-type synthesis. Examples are given of both new methods.