Realizations of biquadratic impedances as five-element bridge networks containing one inductor and one capacitor

This paper is concerned with the realization problem of biquadratic impedances as a five-element bridge network with one inductor and one capacitor. First, a group of networks that can handle all the cases are listed, which are classified as three quartets. Then, together with the principle of duality, the realizability conditions of two quartets among them are derived. In the process of the derivation, the realizability conditions based on the existence of positive roots that satisfy certain conditions are converted into the conditions only in terms of the coefficients of the function. Finally, utilizing a canonical form of biquadratic functions, combining the conditions yields a necessary and sufficient condition for any biquadratic impedance to be realizable as a five-element bridge network with one inductor and one capacitor.