Analytical calculation of the U-shaped circuit

Analytical calculation of elements of the U-shaped circuit is carried out taking into account losses by the method based on the theory of electric circuits. Transformation of the U-shaped circuit to the equivalent regular parallel circuit gives the possibility to express main parameters of the U-shaped circuit on the resonance frequency, depending on the equivalent loaded circuit Q-factor. The feasibility condition for the U-shaped circuit is presented.

A relative error in the calculation of the output capacitance with and without the loss of a U-shaped loop, and the condition of the possibility of neglecting losses in the circuit, depending on its efficiency, are presented.

It is hown that for balanced load the U-shaped circuit must be symmetrical. Symmetrical U-shaped circuit with asymmetric matched load at a certain Q-factor of loaded circuit phase shifter circuit is 90 degrees at the resonance frequency. This property can be used in various radio systems. It is shown the relationship between the U-shaped circuit capacitances and the loaded circuit Q-factor.

Analysis of the symmetric filtering system without losses showed that the its voltage gain module Ku depends on the Q-factor and relative frequency only. In the bandwidth range, the filter has a minimum transfer coefficient at certain frequency, independent of the quality factor. In this case, the bandwidth is equal to twice the frequency observed at a minimum in the Ku in the band of transparency of the filter.

A plot of the minimum of Ku in the transparency band is presented, depending on the loaded circuit Q-factor.