Abstract:

This paper presents a systematic study on the modeling and stability analysis of fractional-order cascaded RLC networks with time delays. A generalized model of an $n$-stage cascaded RLC network with time delays is developed using the Caputo fractional derivative. The corresponding fractional-order differential equations are derived for both single-stage ($n=1$) and two-stage ($n=2$) configurations. The transcendental characteristic equation of the system is obtained via Laplace transform. By applying the Matignon stability criterion, asymptotic stability conditions are established for systems with and without time delays. It is shown that stability in the delay-free case depends mainly on the fractional order $\alpha$, whereas in the presence of time delays, stability is independent of $\alpha$ and instead governed by the delay parameter $\tau$. Notably, the critical delay threshold $\tau_{\mathrm{max}}$ for system stability is derived analytically. A detailed numerical study (Table I) further elucidates the effects of key parameters, including the resistance $R$, inductance $L$, capacitance $C$, fractional order $\alpha$, and time delay $\tau$ on the stability behavior.

This study provides a theoretical basis and practical design guidelines for tuning parameters to ensure stability in fractional-order circuits with time delays.

Key words: Fractional-order system, Time-delay circuit, Cascaded RLC network