A ${Z}$-Transform Method for Synthesis of Unequal-Length Multisection Transmission Lines for Multiband Applications

A method based on the $Z$-transform is demonstrated to cascade transmission line sections of unequal lengths and achieve multiband and/or wideband properties. Unlike previous methods based on the $Z$-transform, the proposed method associates the $z^{-1}$ term with the infinitesimal length of the unit segment. Consequently, any electrical length can be expressed in the $z^{-1}$ form. Therefore, while the previous methods are restricted to cascading lines of a commensurate length only, the proposed method can be applied to those of incommensurate lengths. Furthermore, the $z$-polynomials for triple or more bands can be constructed by multiplying a combination of single- A nd dual-band polynomials, which remarkably simplifies the entire procedure to develop multiband and/or wideband circuits with virtually no limitation in terms of the number and/or combination of frequencies. The validity and practicality of the proposed method is verified with experimental results for various impedance transformers for real loads and Wilkinson power dividers; the results show the independence in the number and combination of operating frequencies as well as a dual-band operation with a center frequency ratio as large as 7.75.