Synthesis of Bessel filters with prescribed RLC load

The design is presented of a coupling network between a given RLC load and a resistive generator over a finite frequency band to achieve the Bessel transducer power-gain characteristic of arbitrary order. Youla's theory is used to obtain constraints in terms of the coefficients of the Laurent series expansions of the bounded-real reflection coefficient and two other functions determined by the given load impedance. Formulas are derived for the coefficients of the numerator polynomial of the bounded-real reflection coefficient for networks of order one or two with dc gain not greater than unity and for networks of order three with unit dc gain. Tabulated values are obtained for networks of higher order. Design curves are graphically given which relate the normalized zeros of the all-pass function and the maximum load inductance.