The Design of a Constant-Angle or Power-Law Magnitude Impedance

A passive driving point impedance can be built from RL or RC elements to vary in magnitude nearly as
\omega^k
and to have a nearly constant angle at
k\pi/2
over an arbitrarily wide frequency range,
|k| \leq 1
. Except at the extreme ends, the successive positions of the network poles (along the negative frequency axis) are taken in the ratio
\rho
, in which
\rho
in the range 6 to 25 determines an approximation error in the range 1 per cent to 10 per cent. The zeros of an arbitrarily wide-band network lie between poles, in the ratio
\rho^k
from a pole. A series string of parallel RL or RC pairs can be used to realize the impedance, according to whether
k
is positive or negative. The R/L or 1/RC of successive pairs are in the ratio
\rho
, and the resistors of successive pairs are in the ratio
\rho^k
. One pair "at" each band edge, the "corrector" or "compensation" impedance, is specified by different ratios, so as to account for band-edge effects. An experimental admittance constructed with five capacitors and five resistors approximated an
\omega^{1/2}
admittance at a constant 45° angle to within the measurement accuracy of
\pm
1 per cent in magnitude and
\pm
1° in phase over the frequency range 50 cps to 10,000 cps.