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A generic model for second-order RC sinusoidal oscillators is derived. The model is based on treating an oscillator as a second-order passive network, with an arbitrary unknown structure, terminated at one port by a linear voltage-controlled negative resistor. A modified model which takes into account the fundamentally nonlinear characteristics of the negative resistor is also derived.

This work is aimed at generalizing the design of continuous-time second-order filters to the non-integer-order (fractional-order) domain. In particular, we consider here the case where a filter is constructed using two fractional-order capacitors both of the same order α. A fractional-order capacitor is one whose impedance is Z_c = 1/C(jω)^α, C is the capacitance and α (0 < α ≤ 1) is its order. We generalize the design equations for low-pass, high-pass, band-pass, all-pass and notch filters with stability constraints considered. Several practical active filter design examples are then illustrated supported with numerical and PSpice simulations. Further, we show for the first time experimental results using the fractional capacitive probe described in Ref. 1.

In this work we derive a general non-conservative model for any RC sinusoidal oscillator independent of its particular passive topology or the employed active devices. We consider an arbitrary unknown second-order passive RC network terminated at one port by a negative resistor and proceed to impose oscillation start-up and frequency constraints on a derived state-matrix.