Duality and resonance in RLC–circuits. Exact formulas for phase, amplitude resonance and bandwidth

The paper is devoted to the study of two-terminal electric circuits consisting of three elements - a resistor, a capacitor, and an inductor. There are eight different possible combinations of such circuits. We apply a standard and nonstandard duality (frequency duality). In this way, all possible circuits are divided into three groups: (2+ 2 +4). The first pair of circuits is extensively covered in most electrical engineering textbooks. The second pair is not interesting from the frequency point of view. We investigate in detail a third group of four circuits, which, by suitable substitution of variables, bring the finding of their frequency characteristics to the study of a single function. For them we find exact formulas for the frequency of phase resonance and amplitude resonance. The values of the minimum (maximum) amplitude at the corresponding resonance frequencies are found. Exact and approximate formulas for the cut-off frequency and bandwidth of the passband-stopband transition are derived.