Decomposition Method for Exact Solutions in Coupled Parallel Resonant Circuits

Abstract

The study presents a novel approach for obtaining exact solutions to the equations governing coupled parallel resonant circuits using the decomposition method. Coupled parallel resonant circuits, characterized by their intricate interactions and frequency-dependent behavior, play a crucial role in various applications, including signal processing, communication systems, and electronic filter design. Traditional analytical techniques often struggle to provide closed-form solutions due to the complexity of the coupled equations. This paper addresses this challenge by applying the decomposition method, which simplifies the problem into more manageable sub- problems that can be solved exactly.

The decomposition method involves breaking down the original system of coupled resonant circuit equations into simpler, decoupled sub-systems. This process begins by transforming the coupled differential equations into a form that isolates the individual resonant components. Each of these components is then solved separately, and their solutions are combined to reconstruct the exact solution for the original system. This approach leverages the linearity and additive properties of the resonant circuits to facilitate an exact solution.

The effectiveness of the decomposition method is demonstrated through several examples of coupled parallel resonant circuits. The paper outlines the step-by-step application of the method, including the transformation of the differential equations, the decoupling process, and the final combination of solutions. Detailed solutions are provided for different circuit configurations, showcasing the method's ability to handle varying degrees of coupling and resonance conditions. The results highlight the method's accuracy and computational efficiency, providing a valuable tool for engineers and scientists dealing with complex resonant circuit designs.