The bilinear transformation in microwaves: a unified approach

Probably the first encounter of a student of microwave circuits with a bilinear transformation takes place when he studies impedance transformation and matching. Later, when he advances in the study of high frequencies, he will find this kind of transformation in issues like synthesis of signal separation structures and n-port reflectometer calibration and measurement. However, for each appearing transformation, there is generally an individual treatment with no connection to the others, and with a minimum interpretation of its geometrical significance, which impedes a thorough appraisal of generalized transformation attributes. In order to give a global perspective of bilinear transformation uses in microwaves, this paper presents a unified notion and includes geometrical exegesis for some particular applications. In this way, the bilinear transformations between parameters, parameter matrices, and matrix eigenvalues related to microwaves are presented in a congruous and connected form. These transformations are single-valued functional relations of a complex variable between immittance (impedance and admittance) and reflection coefficient parameters, reflection coefficient and power parameters, and immittance and scattering parameters.