Analytical Reflection Coefficient Expressions Utilizing Load-Dependent $X$-Parameters

In this paper, analytical expressions of the input reflection coefficient at the source and the load ports of a two-port active microwave component are proposed. It utilizes the load-dependent X-parameters in the development of these expressions. In contrast to the classical small-signal S-parameters based approach, the cross frequency conversions and harmonic mismatch effects are taken into consideration in this method. In the derivation, the incident and the scattered waves of the amplifier are all represented in a matrix structure. Since the coefficient determinants of the input and output vectors are equals to zero, it is hard to obtain the analytical solutions. Therefore, the order reduction of the matrix under certain principles is introduced to solve this problem. Throughout the derivation process, the matrix transformation and the calculations are performed using only simple complex operations. Therefore, the function analyticity is transmitted from the load-dependent X-parameter mathematical model to the proposed expressions. Finally, the proposed method is experimentally validated on two microwave amplifier circuits.