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.

1 Analytical Solution
1.1 Scilab Code
2 Results
3 See Also
4 References
5 External Links

Analytical Solution

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

Eqn:6.1

where,

magnetic line source constant current

angular frequency

electric permittivity (farads/meter)

magnetic permeability (Henrys/meter)

distance from source

Hankel 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.

Yee2d.c .c file (FDTD code)

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

2danalyticalsolution

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

References

Balanis – Advanced Engineering Electromagnetics, 1989

External Links

scilab.org

Analytical Solution: Infinite Magnetic Line-source

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