Finding the Minimum Input Impedance of a Second-Order Unity-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 unity gain as a function of the natural frequency , o  quality factor Q and the ratio of the resistors of the filter. From this expression, it is shown that the filter behaves like a single capacitor 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 highfrequency 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.