Two Extensions of the Darlington Synthesis Procedure

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
k_{11} k_{22} - k_{12}^2 - \beta^2 > 0
(
\beta
is the coefficient of the
z_{12}
term which is of the form
\beta \omega_{0}/(s^2 + \omega_{0}^2))
. It is shown also that two real-part conditions now exist. In addition to the usual Brune real-part condition
Re z_{11} Re z_{22} - (Re(z_{12} + z_{21}))^2/4 > 0
for reciprocal elements, there exists another restriction
Re 1/y_{11} Re 1/z_{22} - (Re(z_{12} - z_{21})/2z_{22})^2 \geq 0
on the
j
-axis (the equal sign is dropped for
Res > 0
). The second method relaxes the cascade network requirement, and from a study of
E v Z(s)
allows the synthesis of the network by a Darlington-type synthesis. Examples are given of both new methods.