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.
The analytical solution to the Infinite Magnetic Line Source problem is from Balanis, Advanced Engineering Electromagnetics, pp.570-574 and is:
|magnetic line source constant current|
|electric permittivity (farads/meter)|
|magnetic permeability (Henrys/meter)|
|distance from source|
|Hankel function of the second kind of order 0.|
The algorithm was run using scilab (an open source numerical computation software package, similar to Matlab).
// // 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
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:
As can be seen, the FDTD simulation matches the analytical solution.
- Fdtd Main Index Page (with example code)
- Maxwell’s Equations
- The Yee Algorithm
- 2D UPML Algorithm and Theory
- The 2D FDTD Wave Algorithm and Theory
- The 2D FDTD Wave Hz Algorithm
- Numerical Methods
- Spherical and Cartesian coordinates
- Vector Algebra
- Fundamental and Derived Units
- FDTD References
- FDTD Resources
- Balanis – Advanced Engineering Electromagnetics, 1989