Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

Abstract

AbstractThe LC circuit with a nonlinear capacitor is analyzed using the symplectic conservative perturbation method based on the symplectic matrix. By treating the electric charge and the flux linkage as dual variables, the LC circuit system can be transformed into a Hamiltonian system. Then we solve the nonlinear Hamiltonian matrix equation by the perturbation method. The Hamiltonian matrix can be cast into linear and nonlinear parts. Based on the exact solution of the linear part, the nonlinear part is solved perturbatively by a canonical transformation. Since the coefficient matrix of the obtained equation is still a Hamiltonian matrix, symplectic conservation is guaranteed. The validity of the method is demonstrated, and the stability, precision, and the efficiency of the method are discussed by comparing it with the fourth‐order Runge–Kutta method (RK4), the ordinary perturbation method, the small‐parameter perturbation method based on displacement (SPPD), and the harmonic balance (HB) method. Numerical results show that the proposed method has better stability, precision, and efficiency compared with the other methods and thus has great advantages in long‐term simulations.