The simple, passive differentiator in Fig 1 achieves accurate differentiation even at frequencies approaching the limit, f_O=1/(2[pi]RC). The circuit surpasses the performance of an op-amp-based differentiator. The trick is to add a carefully selected inductor, [phi], to eliminate the first- and second-order terms in the equation for the phase angle, f, between the input and output signals. Ideally, of course, to achieve perfect differentiation, this phase angle should be equal to 90° (ie, cot([phi])=0), regardless of the frequency, f.

You force the first two error terms to vanish by first calculating R and C to satisfy requirements for the circuit's input impedance, such that f_O is at least two to three times the maximum signal frequency the circuit will encounter. Then, set f₁=f_O; that is, L=R²C. The phase error thus reduces to cot([phi])=(f/f_O)³, which remains quite small even for frequencies comparable to f_O. (DI #1400) e