Complex Impedance Transformers Based on Allowed and Forbidden Regions

Three mapping functions are for the first time derived for allowed and forbidden regions of a complex impedance transformer with only one transmission-line section (TL). The divided sections of the allowed and forbidden regions are six for transforming real into complex impedances, and those for both complex termination impedances are 14, indicating that 20 different areas are possible on a Smith chart. Only one TL can be used for both complex termination impedances, but the restrictions are inevitable. To solve them based on the allowed and forbidden regions, three complex impedances transformers are proposed additionally, connecting one TL, one open stub or one short stub to the original TL. For the verification, three complex impedance transformers, the termination impedances of which are located in forbidden regions, are fabricated and measured at a design frequency of 1 GHz. The measured $\left |{ \mathrm {S}_{11} }\right |$ of one transforming (68 + j42.5) into (85 + j17) $\Omega $ is -45.9 dB at 1 GHz, and the bandwidth with the 15-dB return loss is more than 100 %. One complex impedance transformer with (100-j30) and 50 $\Omega $ is designed to have 280 % bandwidth, as well.