High-Q Resistance-Capacitance Ladder Phase-Shift Networks

Resistance-capacitance ladder lowpass networks, to obtain a phase shift (
\phi
) of 180° between the input and the output, are examined. It is found that there exists a theoretical maximum value of
Q
, defined by
\omega_0/2 |d\phi/d\omega| \omega = \omega_0
where
\omega_0
is the frequency at which the phase shift is 180°. This theoretical limit can be approached in actual practice, but never reached. It is attempted to synthesize such networks having as high a
Q
as possible. In addition, the resulting network possesses the property that the open-circuit input impedance and the short-circuit output admittance are simultaneously maximized. A transformation enables one to get the corresponding highpass structure from the lowpass.