Analytical Derivation of T-Shaped Structures for Synthesis of Quarter-Wavelength Transmission Lines

T-shaped structures, proposed to synthesize the quarter-wavelength transmission lines, have proved efficient for circuit miniaturization. While the design equations for the symmetric case are published, the existing equations for asymmetric cases, as reported recently, could lead to frequency-shift problems. In this letter, we analytically study the T-shaped structures and present closed-form design equations for synthesis of quarter-wavelength transmission lines. The equations explicitly show that quarter-wavelength transmission lines synthesized with asymmetric T-shaped structures cannot be perfectly matched to the reference impedance. The design of couplers with asymmetric T-shaped structures can be regarded as exploiting reflection cancellations to attain the matched condition at each port, as indicated by the equations. The closed-form equations are clear and concise enough for facilitating the design and application of T-shaped structures.