A small dual-frequency transformer in two sections

One of the most useful transmission-line constructs is the quarter-wave transformer that is used to impedance match a line at a single frequency f₀. The feasibility of an electrically small transformer with two sections and capable of achieving ideal impedance matching at two arbitrary frequencies is demonstrated analytically. To achieve this, the exact solution to the resulting transcendental transmission-line equations for two sections is obtained with no restrictions. The parameters of the transformer are presented in explicit closed form, and are exact. The results of this study are useful for a number of practical design problems, including dual-band antennas and RF circuits in general. In particular, feasibility of ideal operation at the important first harmonic frequency 2f₀ is demonstrated.