《计算电磁学》第九讲ppt课件.ppt

计算电磁学Part II： 矩量法,Dr. Ping DU (杜平),School of Electronic Science and Applied Physics, Hefei University of Technology,E-mail: ,Chapter 2 Electrostatic Fields (静电场),Dec. 5 , 2011,2,Outline,2.1 Operator Formulation (算子描述), 2.2 Charged Conducting Plate (含电荷的导电平板),3,2.1 Operator Formulation,The static electric intensity E is conveniently found from an electrostatic,potential , which is,(2-1),where denotes the gradient operator.,In a region of constant permittivity and volume charge density ,the electrostatic potential satisfies,(2-2),is the Laplacian operator (拉普拉斯算子).,4,For unique solution, the boundary conditions on are needed.,In other words, the domain of the operator must be specified.,For now, consider fields from charges in unbounded space, in which case,constant as,(2-3),where r is the distance from the coordinate origin(坐标原点), for every of,finite extent.,The differential operator formulation is,(2-4),where,(2-5),5,The domain of L is those functions whose Laplacian exists and have bounded at infinity according to (2-3).,The solution to this problem is,(2-6),where is the distance between the source point,( ) and the field point ( ).,Hence, the inverse operator to L is,(2-7),Note that (2-7) is inverse to (2-5) only for boundary conditions (2-3).,If the boundary conditions are changed, changes.,6,A suitable inner product for electrostatic problems ( constant) is,That (2-8) satisfies the required postulates (1-2), (1-3) and (1-4) is easily verified.,(2-8),where the integration is over all space.,Let us analyze the properties of the operator L.,For this, form the left side of (1-5),(2-9),where,7,Greens identity is,(2-10),where S is the surface bounding the volume V and n is outward direction normal to S.,Let S be a sphere of radius r, so that in the limit the volume V includes all space.,For and satisfying boundary conditions (2-3), and,as .,Hence as . Similarly for .,Since increases only as , the right side of (2-10) vanishes as,. Equation (2-10) then reduces to,(2-11),8,It is evident that the adjoint operator is,(2-12),Since the domain of is that of L, the operator L is self-adjoint (自伴的).,The mathematical concept of self-adjointness in this case is related to the physical concept of reciprocity.,It is evident from (2-5) and (2-7) that L and are real operators.,They are also positive definite because they satisfy (1-6). For L, form,(2-13),and use the vector identity plus the divergence,Theorem (散度定理).,9,The result is,(2-14),where S bounds V.,Again take S a sphere of radius r.,For satisfying (2-3), the last term of (2-14) vanishes as .,Then,(2-15),and, for real and , L is positive definite. In this case, positive,definiteness of L is related to the concept of electrostatic energy (静电能）.,10, 2.2 Charged Conducting Plate (含电荷的导电平板),Consider a square conducting 2a meter on a side and lying on the plane,with center at the origin as shown in Fig. 2-1.,Fig. 2-1. Square conducting plate and subsections,11,Let represent the surface charge density on the plate.,Here, we assume that the thickness is zero.,The electrostatic potential at any point in space is,(2-16),where,The boundary condition is (constant) on the plate.,The integral equation for the problem is,(2-17),12,where , .,The unknown to be determined is the charge density .,A parameter of interest is the capacitance of the plate,(2-18),which is a continuous linear functional of .,Let us first go through a simple subsection and point-matching solution, and later interpret it in terms of more general concepts.,Consider the plate divided into N square subsections, as shown in Fig. 2-1.,Define basis functions,(2-19),13,Thus the charge density can be represented by,(2-20),Substituting (2-20) in (2-17), and satisfying the resultant equation at the mid-point,of each , we obtain the set of equations,(2-21),where,(2-22),14,Note that is the potential at the center of due to a uniform,charge density of unit amplitude over .,A solution to the set (2-21) gives the in terms of which the charge density,is approximated by (2-20).,The corresponding capacitance of the plate, approximating (2-18), is,(2-23),To translate the above results into the language of linear spaces and the method of moments (MoM), let,15,(2-24),(2-25),(2-26),Then is equivalent to (2-17).,A suitable inner product, satisfying (1-2) to (1-4), for which L is self-adjoint, is,(2-27),We choose the functions (2-19) as a subsectional basis.,16,The testing functions are defined as,(2-28),This is the two-dimensional Dirac delta function.,The elements of the l matrix (1-25) are those of (2-22), and the g matrix of,(1-26) is,(2-29),The matrix equation equation (1-24) is identical to the set of equations (2-21).,17,In terms of the inner product (2-27), the capacitance (2-18) can be written,For numerical results, the of (2-22) must be evaluated.,Let denote the side length of each .,The potential at the center of due to unit charge density over its own surface is,(2-31),(2-30),18,The potential at the center of due to unit charge density over unit charge over can be similarly evaluated, but the formula is complicated.,For most purpose, it is sufficiently accurate to treat the charge on as if it were a,point charge, and use,(2-23),19,Homework: Repeat this example. The side length is assume to be equal t o 1.Analyze the size of the subsectional domain V.S. the accuracy.,20,21,22,23,24,25,26,27,28,29,