On the Design of Filters by Synthesis

Following an introduction to the methods and significance of filter design, this paper deals with definitions and filter synthesis according to prescribed attenuation requirements. It shows how the characteristic function for cases of practical importance can be calculated, particularly for the special cases 1) where this function is a power term in the frequency variable (maximally flat), 2) where it is a Tchebycheff polynomial function, and 3) where the effective attenuation in both the pass band and the stop band is of Tchebycheff character, and, also for 4) the general case of Tchebycheff behavior in the pass band with prescribed attenuation poles. The way, and the separate steps, of the determination of the circuit-element values for the realization of reactance ladder-network filters without mutual inductances are indicated, and the various necessary computation formulas are described. For the first two special cases mentioned above, computation formulas applicable in general are given. New formulas are presented for the transformation of a low-pass into a band-pass configuration with a minimum number of inductances, and into a band-pass with the stray capacitance being taken into consideration. These are illustrated by several numerical examples. Next, the paper deals with the design of filters, taking into consideration additional requirements involving the group delay. Possibilities of solving the problem are described. An actual example, together with the measured results, illustrates the excellent agreement between the theoretical determination and the behavior of the filter constructed accordingly. The Appendix contains a selected section of a table which gives the normalized values of circuit elements for low-pass filters with the effective attenuation in the pass band and the stop band behaving in the Tchebycheff sense, for the degrees
n = 6, 7, 8, 9, 10
, and
11
. Graphs are also appended for easy estimation of the degree needed for a filter design and, thereby, the complexity of the resulting structure.