Second Derivative General Linear Methods for Transient Analysis: A Robust Approach to Damped RLC Circuits Across Damping Conditions

Transient analysis of an electrical circuit is an important study in determining how electric circuits respond dynamically due to sudden changes, such as switching operations. The parallel RLC circuit has become crucial, both in academics and engineering fields, pertaining to designing various engineering devices: oscillators, filters, and dissipating energy that might be harmful to people or electronics. This paper examines the application of Second Derivative General Linear Methods (SGLMs) for solving the governing second-order ordinary differential equation of a damped parallel RLC circuit. SGLMs with second derivative have given with superior stability and accuracy for non-stiff and stiff systems. This study compared the given methods with other traditional methods under the conditions: as overdamped, critically damped, and underdamped. Results will find wide implications in design and optimization studies of electrical systems by providing a robust framework for accurate, efficient transient analysis. The results show that SGLMs achieved significantly lower absolute errors compared to classical methods. Specifically, the maximum error across all simulations was over 10 times smaller than that of RK4 (Runge-Kutta 4), and the Euler method exhibited even greater deviations. SGLMs remained stable even at larger step sizes (up to h=0.1)where the other methods either became unstable or lost accuracy.