Finding the Minimum Input Impedance of a Second-Order Twofold-Gain Sallen-Key Low-Pass Filter Without Calculus

We derive an expression for the input complex impedance of a Sallen-Key second-order low-pass filter of twofold gain as a function of the natural frequency ωo and the quality factor . Q From this expression, it is shown that the filter behaves like a Frequency Dependent Negative Resistance (FDNR) element for low frequencies and as a single resistor at high frequencies. Furthermore, the minimum input impedance magnitude is found without using calculus. We discovered that the minimum input impedance magnitude is inversely proportional to Q and can be substantially less than its high-frequency value. Approximations to the minimum input impedance and the frequency at which it occurs are also provided. Additionally, PSpice simulations are presented which verify the theoretical derivations.