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.
Latin American Journal of Physics Education
Cartwright, K.V. and E. J. Kaminsky, “Finding the minimum input impedance of a second-order twofold-gain Sallen- Key low-pass filter without calculus,” Lat. Am. J. Phys. Educ., Dec. 2014, pp. 4316-1–4316-8.