The Research On Symplectic Method Of LC Circuit In Hamilton System

The symplectic method in Hamilton system is a Hamiltonian algorithm,which has the characteristics of maintaining the basic characteristics of the system.LC circuit is widely applied in communication,electronics and other fields.The analysis of its oscillation rule has important significance.In this paper,the symplectic method is extended to the field of circuit,and the symplectic method is studied to obtain the oscillation rule of LC circuit.Here are the main contributions.(1)LC oscillation circuit is researched by the symplectic method of Hamilton system.The electric quantity q and magnetic linkage ? are treated as the state parameters.By deducing from the Lagrange function with electric quantity q as variable,the LC circuit problem is directed to symplectic system.The dual equation can be solved by segregation variable method and it is converted into the symplectic eigen problem.The oscillation law of LC circuit is obtained through the Hamilton matrix H and the corresponding eigenvalue equation.The solution processes of the First-order,the Second-order and the complex trapezoidal LC oscillating circuit are introduced in detail.The validity and correctness of the method is verified by examples.It is easy to understand and provides a new way for studying linear LC circuits.(2)The symplectic conservative perturbation method based on symplectic matrix is studied for solving nonlinear LC circuit.Starting from the Coulomb volt characteristics of the nonlinear capacitor,electrical charge q of the capacitor and the magnetic linkage ? of the inductance are treated as the state variables and the control equations are written in the form of dual.The Hamilton matrix is written as the sum of linear and nonlinear part.On the basis of the exact solution of the linear part,the problem is transformed into the problem of the multiplication of the symplectic matrix by the regular transformation.After the solution of the corresponding nonlinear part is obtained,the oscillation characteristics of the LC circuit can be obtained.By comparing the conventional perturbation method with the four order Runge Kutta method,it is proved that the symplectic conservative perturbation method based on symplectic matrix is more accurate,stable and efficient,which provides a new effective approach for the research of nonlinear LC oscillator circuit.(3)The perturbation based on displacement method is studied to obtain the oscillation characteristics of LC circuit with nonlinear capacitance.Starting from the Lagrange function of the system,the variable equation is discretized to obtain the time stiffness matrix.The characteristic equation of the circuit is perturbed for the first time,and the original time stiffness matrix is replaced by a perturbed approximate stiffness matrix.On the basis of the relationship between the stiffness matrix and the transfer matrix,the problem is converted into the problem of symplectic transfer matrix,and the oscillation characteristics of the LC circuit can be solved.The comparison between the perturbation based on displacement method and the fourth order Runge Kutta method demonstrates the correctness of the proposed method.It is proved that the proposed method has good precision,efficiency and stability by comparing with the transfer symplectic matrix method.