A General Class of Maximally-Flat Amplitude Response Ladders

It is shown that the elliptical Tchebycheff pole array defines a general class of maximally-flat amplitude functions when the number of poles approaches infinity. The infinite-order pole array can be realized as an infinite cascade of identical two-terminal ladders. The amplitude characteristic of the driving-point impedance of the two-terminal ladder is flat up to the nominal cutoff frequency,
\omega_0
. Beyond
\omega_0
, the exact shape of the amplitude characteristic is determined by the eccentricity of the original pole array. The two-terminal ladder is a low-pass
R, L
, and
C
structure that is infinitely long. Two special cases are considered: 1) when the pole array becomes linear, the ladder is a constant-
k
type; 2) when the pole array becomes circular, the shunt conductances and series resistances of the ladder rapidly taper toward zero while its
C
and
L
components taper toward a constant-
k
type.