A Direct Proof, in Terms ofcos heta, of the Realization of1 + P(cos^2 heta)as a Quarter-Wave Impedance Transformer

This paper presents necessary and sufficient conditions on an input impedance function, as a function of
\sin \theta
and
\cos \theta
, for it to be realized as a cascade of equal-length transmission-line sections terminated in a resistance. The proofs are given in terms of the variable,
\cos \theta
, rather than the frequency variable of Richards,
p = -j \cos\theta / \sin\theta
. The method is direct, elementary, and applicable to the realization of the input impendance function of a lowpass ladder network of capacitances and inductances terminated in a resistance. Finally, necessary and sufficient conditions on the input reflection coefficient and then on the insertion-loss function, in terms of
\cos\theta
, are given for a cascade of equal-length transmission-line sections terminated in a resistance.