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 asω^kand to have a nearly constant angle atkπ/2over an arbitrarily wide frequency range,|k| ≤ 1. Except at the extreme ends, the successive positions of the network poles (along the negative frequency axis) are taken in the ratioρ, in which 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 ratioρ^kfrom a pole. A series string of parallel RL or RC pairs can be used to realize the impedance, according to whetherkis positive or negative. The R/L or 1/RC of successive pairs are in the ratioρ, and the resistors of successive pairs are in the ratioρ^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 anω1/2admittance at a constant 45° angle to within the measurement accuracy of±1 per cent in magnitude and±1° in phase over the frequency range 50 cps to 10,000 cps.