广工电磁学实验报告

广东工业大学考试试卷 ( A )

课程名称: Experimental reports for “Theory of electromagnetic field and microwave technology”试卷满分 100 分

考试时间: 2012 年月日 (第周星期 )

Please answer the questions in English.

一．Aninfinite long slot, with rectangle cross-section, consists of three grounded conductive plate and one coverplate with 100 voltage. Please give the potential distributions inside slot (following the below instructions ).(必修题)

1)please demonstrate the ideas and procedures on how to deduce this problem;

2)please list the detailed program codes.

3)Please draw figures about potential distributions.

Ans:

1)please demonstrate the ideas and procedures on how to deduce this problem;

1.1 Finite difference method (FDM)

Finite difference method (Finite Difference Method) is based on the principle of differential A numerical method , which the electromagnetic field within the problem into discrete continuous system problems , which uses the numerical solution of discrete points to approximate a continuous real solution domain Therefore , it is an approximate calculation.

Using the finite difference method [ 4,5 ] for solving electromagnetic field problems , the first field will be split into many seeking grid and node, and instead of using derivative difference quotient , then , will be in the Field of partial differential equations [ 3 ] into the potential of each node is unknown or magnetic potential difference equations, by solving the equations can be obtained upon request each discrete node potential or magnetic potential of the numerical solution .

D field boundary conditions : As described in the electromagnetic field PD

E function of the spatial coordinates , so that only a specific set of boundary conditions to obtain a unique solution . Most of electromagnetic problems involving three types of boundary conditions :Diliheli boundary condition , Newman -type boundary conditions and mixed boundary conditions . Diliheli type boundary conditions and Neumann boundary condition commonly referred to as the first boundary condition and the second boundary condition , mixed boundary conditions for the first boundary condition and a second linear combination of boundary conditions .

1.1.1 Laplace equation finite difference form

Two-dimensional Laplace equation using finite difference method can approximate calculation, first solving the region is divided into grids, and then solving the continuous field distribution within the grid nodes with discrete numerical solution instead. Finer the grid points, the higher the precision. Shown in Figure 1, the field in two dimensions, the area is divided into a number square grid, each grid side length of h, the intersection of two sets of parallel lines is called grid nodes, the potential of grid nodes located is, its potential and down about four nodes, respectively,,,. In the case of sufficiently small h, can be the starting point for the Taylor series expansion:

23

23

23,1,1126i j i j h y h h y y ?????+?=++++????????

2

3

23

23,1,1126i j i j h y h h y y

?????-?=-+-+????????

23

23

231,,1126i j i j h x h h x x ?????-?=-+-+????????

2

3

23

231,,1126i j i j h x h h x x

?????+?=++++????????

Sum the above four formula we can get

,1i j ?++,1i j ?++1,i j ?-+1,i j ?+=,4i j ?+22222

h x y ???? ?++??? ?

????

?? Since the field at any point satisfy Poisson equation:

()22

2

22

,F x y x y

?

?

?=

+=?????,the F(x,y) in the formula is the field source

For non-farm sources, F(x,y)= 0, then the two-dimensional Laplace equation finite difference form is:

()

,,1,11,1,14i j i j i j i j i j ?????+-+-=+++

1.2 Finite Element Method (FEM)

FEM (Finite Element Method) is a variational principle and a split interpolation based numerical method . It from the partial differential equation boundary value problem to find an energy functional [ 4 ] , making it the first boundary conditions are met under the premise of extreme value , converted to an equivalent and Partial Differential Equations conditions variational problem , while the cross-sectional area will solve the electromagnetic field into a finite number of smaller units , each unit in the interior , it is assumed that the solution of any element node function and the

function value , and are in accordance with the changes in the coordinates some kind

of law of change, so you can construct a suitable unit interpolation function, substituted into the functional integral formula, the multiple functions into discrete functional , and each independent variable partial derivative , whichever is the extreme , you can get a set of multiple linear or nonlinear algebraic equations , after the first boundary condition amendments, solving this equation , we obtain the desired approximate solution. Finite element method [ 1,4 ] not only to adapt to the complex geometry and boundary conditions, but can be used successfully in a variety of media and non-uniform continuum problem . For difficult to find the energy functional boundary value problems of partial differential equations , but also by the weighted residual method to derive the corresponding algebraic equations, using the finite element method and thus a wide range .

1.3Finite Difference Method in MATLAB

By simple iterative method, using MATLAB programming method to achieve numerical calculation. Simple iterative method is to use a previous iteration the potential of grid points as the initial value of the next iteration. Its iterative formula is:

1

1,,1,11,1,4n n n n n i j

i j i j i j i j ?????+??

=+++ ?+--+??

2)please list the detailed program codes.

clear;clc %设置网格节点的数目 hx=input('hx='); hy=input('hy=');

v=ones(hy,hx);%生成节点二维数组 %上下两行的边界条件值 v(hy,:)=ones(1,hx)*100; v(1,:)=zeros(1,hx); %左右两列的边界条件值 v(:,1)=0;v(:,hx)=0; v1=v;maxt=1;t=0;%初始化数据

while (maxt>1e-6); maxt=0; for i=2:hy-1; for j=2:hx-1

v1(i,j)=(v(i,j+1)+v(i+1,j)+v(i-1,j)+v(i,j-1))/4;%拉普拉斯方程差分式

t=abs(v1(i,j)-v(i,j));

if(t>maxt) maxt=t;

end,end,end

v=v1;

end;

subplot(1,2,1);mesh(v1);%绘制三维曲面图

axis([0,17,0,11,0,100])%设置坐标轴的范围

title('\fontsize{14}\fontname{黑体}简单迭代法三维曲面图') subplot(1,2,2),contour(v1,15)%绘制等电位线图

hold on

x=1:hx;y=1:hy;

[xx,yy]=meshgrid(x,y);%生成网格

[dx,dy]=gradient(v1,0.6,0.6);%计算梯度值

quiver(xx,yy,dx,dy,0.8,'r')%根据梯度值绘制箭头

axis([-1.5,hx+2.5,-2,13])

plot([1,1,hx,hx,1],[1,hy,hy,1,1],'k')%绘制矩形导体的截面title('\fontsize{14}\fontname{黑体}简单迭代法等位线图')

3)Please draw figures about potential distributions.