Analysis of resonance phenomena in a series RLC circuit with supercapacitor

Artykuł dotyczy analizy warunków zajścia zjawiska rezonansu fazowego i amplitudowego dla szeregowego obwodu RLC z superkondensatorem. Zachowanie superkondensatorów różni się od klasycznych kondensatorów, dlatego ich modele matematyczne są bardziej skomplikowane, wykorzystują m.in. rachunek różniczkowo-całkowy ułamkowego rzędu. Do analizy i symulacji wybrano model oparty o bieguny i zera ułamkowego rzędu. Dobierając odpowiednio parametry obwodu rezonansowego R, L , RS i C można uzyskać w obwodzie rezonans fazowy i amplitudowy.The phase and magnitude resonance phenomenon in a series RLC circuit with classic reactive components: the inductance L and the capacitance C is widely known in the circuit theory and thoroughly described. The subject of the paper is an analysis of phase resonance conditions in a series RLC circuit with supercapacitor. Supercapacitors differ from classic dielectric capacitors by the construction, behavior and their applications, described in Section 1. There have been created several kinds of mathematical and circuit models (Section 2). Among them, fractional order models demonstrate the best fit to the frequency characteristics as well as dynamic processes occurring in supercapacitors [3], [4] and [10]. The model based on fractional poles and zeros, defined by formulas (2) and (3), was chosen for analysis. The relations for the circuit impedance (10-13) and resonance frequency (14) were derived. They are illustrated in Figs. 2-11. The formula for resonance frequency is a transcendental equation, therefore it is possible to solve it for the investigated system only numerically. The resonance frequency in the examined series RLC circuit whose model is based on fractional poles and zeros depends on five parameters. The results were compared with those obtained from the analysis conducted in paper [6],where frez depends on only three parameters. A more precise supercapacitor model enables more insightful analysis of phase and magnitude resonance phenomenon