Lumped-Distributed Resonators Providing Multiple Transmission Zeros in Bandpass Filters With Simple and Mixed Couplings
A. V. Zakharov, Sergii Litvintsev
Abstract
The article proposes three lumped-distributed resonators, each of which introduces two transmission zeros (TZ) at real frequencies into the transfer function of inline bandpass filter (BPF). The use of such resonators in mixed-coupled BPFs without cross-coupling increases the number of TZs from ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N + 1$</tex-math> </inline-formula> ) to (3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N + 1$</tex-math> </inline-formula> ). It improves the filter performance. Lumped-distributed resonators represent a stepped-impedance or uniform transmission line segment to which one capacitance or inductance is connected in cascade. These resonators have two antiresonant frequencies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega _{p1}$</tex-math> </inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega _{p2}$</tex-math> </inline-formula> , which are placed next to the main resonant frequency <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega _{0}$</tex-math> </inline-formula> , which leads to two TZs. Antiresonant frequencies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega _{p1}$</tex-math> </inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\omega _{p2}$</tex-math> </inline-formula> are the poles of the input admittance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Y(j\omega )$</tex-math> </inline-formula> , their position is controlled by the parameters of the resonators. Lumped-distributed resonators can form equidistant TZ pairs, including those of higher degree, allowing the implementation of BPFs with quasielliptic frequency responses and improved selectivity. It has been established that reducing Q-factors of inductor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{L}$</tex-math> </inline-formula> and capacitor <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{C}$</tex-math> </inline-formula> do not increase the insertion loss in the passband of BPF. A design method for BPF with mixed couplings between adjacent resonators is established. It is also suitable for filters that use the proposed resonators. As result, the TZ number of such BPFs is increased from ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$</tex-math> </inline-formula> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$+$</tex-math> </inline-formula> 1) to (3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N + 1$</tex-math> </inline-formula> ).