Resonance curves are perfect circles

TL;DR

This paper introduces a novel geometric approach to understanding electric resonance in RLC circuits, revealing that resonance curves form perfect circles and enabling easy calculation of phase differences and conjugate frequencies.

Contribution

It presents a new geometric method for analyzing resonance phenomena, extending the mathematics to relate currents at different frequencies and simplifying phase and conjugate frequency calculations.

Findings

Resonance curves are perfect circles in the proposed geometric framework.

A simple geometric construction allows phase difference calculation at any frequency.

The method reveals a relationship between conjugate frequencies producing the same current.

Abstract

The ability to approach a physical phenomenon and grasp its major importance is a remarkable quality of understanding. This paper presents a rather elegant and novel way of looking at the resonance phenomenon, which among others shares a common conceptual basis in various fields of physics. For the sake of simplicity, the discussion will be restricted to the case of electric current resonance in series RLC circuits. The mathematics of electric resonance is thus meticulously extended to the extent that it ultimately unravels an invaluable relationship between the current at a certain driving frequency and that at the resonance frequency. Much further it gives rise to an elaborate correlation between any two driving frequencies that give the same current. Arising from the previously mentioned relations, a new technique is devised, a simple geometrical construction, that without running into tedious calculations, allows the computation of the phase difference between the current and the impressed voltage at any given frequency. Not to mention another geometric utility, that is miraculously constructed to correlate any frequency with its corresponding ``conjugate'' one, that allows the same current in the circuit.