Research On Low-frequency Breakdown Of Electrically Small Objects In Method Of Moments

The electric field integral equation(EFIE)is widely used in the scattering problem,and microwave circuit due to its high precision and efficiency.When deal with low-frequency problems or scarttering problem of electrically small size,the traditional method of moments will result in lowfrequency breakdown.The reason is that as the frequency is getting lower and lower,the contribution of vector potential in the electric field integral equation is much smaller than that of scalar potential.Due to the finite precision of floating point numbers in a computer,the contribution of vector potential may be beyond the numerical error of the contribution of scale potential.At present,there are two main solutions of low-frequency breakdown.The first type is separating the both contributions of vector potential and scalar potential,such as the method of loop-tree basis function decomposition or the method of loop-star basis function decomposition and the method of replacing the electric current density in the scalar potential with the electric charge density by using the current continuity equation,which is called augmented eletric field integral equation.The second type is the generalized eigenvalue method,in which frequency dependence of the contribution of vector potential and the contribution of scalar potential can be adjusted by dealing with the generalized eigenvalue.This thesis describes the principles of electric field integral equations,and introduces the basic theoretical knowledge related to matrix condition numbers and the general solution process of the method of moments.This thesis explain the reason of low-frequency breakdown in terms of the contribution of vector potential and scalar potential.The method of loop-tree basis function decomposition is repeated.The general generation algorithm of loop-tree basis function is derived.To verify the effectiveness of the method of loop-tree basis function decomposition,three different electrical size of metal spheres and two different electrical size of metal cubes is calculated.From the five numerical results,it is observed that even if the contribution of vector potential is not beyond the numerical error of the contribution of scale potential after loop-tree basis function decomposition,the impedance matrix is a seriously ill-conditioned matrix.Finally,this thesis studies the application of the truncated singular value decomposition and the Tikhonov regularization in the iterative solution for matrix equations.Observing the distribution of those singular values of the impedance matrix,it is found that many singular values are near the zero point.Smaller singular values can be truncated by the Truncated singular value decomposition.And the overall singular value distribution of the matrix can be affected by the Tikhonov regularization.In addition,this thesis also studies the difference between the method of generalized-cross-validation and the method of L-curve in how to select the optimal regularization parameters.The regularization parameters selected by those methods are used in truncated singular value decomposition and the Tikhonov regularization in order to compare which method is more appropriate.