On the Form of the Phase Response Curve for a Single Real Pole or Zero

The mathematical description of the phase response of a single real pole or zero on a logarithmic frequency basis is shown not to be "the familiar arctangent curve," but of the form y=arctan ϵx or, by shift of ordinate, Y=arctan tanh x/2. The difference between the arctangent and these functions on a normalized basis is found to be in excess of 15 percent. The best straight-line approximation for phase shift is determined as intersecting the 0° and 90° values at 0.0970 and 10.315 times the corner frequency, so that the often used "one-tenth and ten-times" straight-line approximation is confirmed as being extremely good for practical purposes.