A Direct Proof, in Terms of , of the Realization of as a Quarter-Wave Impedance Transformer

This paper presents necessary and sufficient conditions on an input impedance function, as a function of and , 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, , rather than the frequency variable of Richards, . 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 , are given for a cascade of equal-length transmission-line sections terminated in a resistance.