# 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:

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:

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

## See Also

- 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
- Glossary
- FDTD References
- FDTD Resources

## References

- Balanis – Advanced Engineering Electromagnetics, 1989