Analytical Solution: Infinite Magnetic Line-source

A comparison of the 2D “analytic” solution of an Infinite Magnetic Line-source sine wave (with a carrier frequency of 5 GHz) to the output of the 2D FDTD Yee simulation.

Contents

Analytical Solution

The analytical solution to the Infinite Magnetic Line Source problem is from Balanis, Advanced Engineering Electromagnetics, pp.570-574 and is:

860343133088ed552a404fd68d5ea0c7.png Eqn:6.1

where,

df5e12fd03cdee0c14946806ab633179.pngmagnetic line source constant current
68ce9444e11c8ed7457d9ac9918d32c4.png
ee53ed02e07929d204850b677c1d6929.pngangular frequency
a1f21f33e4cc224c1aa90a478a5339d5.pngelectric permittivity (farads/meter)
2b3a144c31b05e0a773b0b60dc13e991.pngmagnetic permeability (Henrys/meter)
4e5a86924aeba8376b7eaaafbdde58ab.pngdistance from source
666ca07012f4b30d0b419f35349c5042.pngHankel function of the second kind of order 0.

Scilab Code

The algorithm was run using scilab (an open source numerical computation software package, similar to Matlab).

Scilab code:


//
//  Code for infinite magnetic line source
//  usage: scilab
//    getf hello.sci
//    hello
//
function hello

  mu = 1.256637061e-06;
  epsilon = 8.854187818e-12;
  omega = 3.141592654e+10;
  dx = 0.003;
  current = 7.0;

  bta = omega * sqrt( mu * epsilon );
  temp = bta * bta / (4.0 * omega * mu);
  phaseadj = 2.0 * dx * bta;

  fd = mopen( 'foo1.txt' , 'w');
  for istep = 1.0 : 1.0 : 500
    x = istep * dx;
    j = besselj(0,x);
    y = bessely(0,x);
    h = besselh(0,2, bta * x + phaseadj);  // hankel function 2nd kind 0th order
    hz = -current * temp * h;
//    mfprintf(fd, 'i: %f x: %f  j0: %f   y0: %f   h0: %f hz:%fn', istep,x,j,y,h,hz);
    mfprintf(fd, '%d %fn', istep,hz);
  end
  mclose(fd);

endfunction

Results

A 2d FDTD (TE) simulation was run with a grid area of 1000 x 1000.

The source excitation was placed int the center (x=500,y=500) and was an Infinite Magnetic Line-source sine wave with a carrier frequency of 5 GHz.

The simulation was run for 5000 time steps (to allow the simulation to reach steady state) and then a “snapshot” of the values from x = 500 to 1000 was recorded.

The results of the FDTD simulation are compared with the analytical solution:

2danalyticalsolution

As can be seen, the FDTD simulation matches the analytical solution.

See Also

References

  • Balanis – Advanced Engineering Electromagnetics, 1989

External Links

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