### RCS of a sphere as function of frequency

**Files:**

**Introduction**

This document presents a RCS simulation of a sphere with CAPITOLE-RCS. The model used is a simple sphere with a radius of 0.2 m. The Monostatic RCS calculation will be computed in one direction and for a frequency sweep from 10 MHz to 8 GHz.

*Figure 1 – Sphere mesh model*

**1 – Geometry model**

Use the sphere model described in « Getting Started with CAPITOLE-RCS » example.

The file is sphere200mm.step

**2 – Simulation strategy**

In this case, the frequency range is very large, from 10 MHz to 8 GHz.

Because CAPITOLE-RCS is based on a frequency domain method, it is not advised to use a single mesh for the whole frequency sweep. Instead, you should create several CAPITOLE-RCS projects for a sub-band of frequency range and using a mesh model suitable for this frequency range. Keep in mind that the mesh size should always λ/10 or less at the higher frequency.

File name | Frequency Range | # of triangles | Mesh size | Solver | Calculation Time |

sphere100.h5 | 10-100 MHz | 1380 | λ/1026 – λ/102 | Full | 3s |

sphere1000.h5 | 100-1000 MHz | 1380 | λ/102 – λ/10 | Full | 29s |

Sphere2000.h5 | 1000-2000 MHz | 5426 | λ/20 – λ/10 | MLACA | 1min42s |

Sphere3000.h5 | 2000-3000 MHz | 12194 | λ/15 – λ/10 | MLACA | 4min22s |

Sphere4000.h5 | 3000-4000 MHz | 21664 | λ/13 – λ/10 | MLACA | 7min6s |

Sphere5000.h5 | 4000-5000 MHz | 33858 | λ/13 – λ/10 | MLACA | 11Min26s |

Sphere8000.h5 | 5000-8000 MHz | 136 264 | λ/16 – λ/10 | MLACA | 1h15min18s |

The MLACA solver is the most efficient, because it uses a compression of impedance matrix. But at low frequency, the electric mesh size become smaller and it is impossible to use MLACA solver when mesh size is under λ/100, instead it is possible to use the Full solver.

All simulations were executed on a DELL server PowerEdge R920 with 4 CPUs Intel Xeon E7-8857 v2 3.0 GHz 12 cores, 512Go of memory

All projects file can be opened with POSTPRO3D at the same time. And a single graph can display all results.

*Figure 2 – Sphere RCS as function of frequency
*

There are 3 regions of the RCS of a sphere. It can be observed on this graph.

The Rayleigh region, when λ>10 r

For a 0.2m sphere, it is until 150 MHz

The creeping wave effect occurs when λ=2πr

It corresponds to a frequency of 239 MHz where there is a maximum RCS.

The optical region starts when 2πr/λ>10

For a 0.2m sphere it corresponds to 2.39 GHz

Above this frequency, the RCS of the sphere is independant to the frequency and equal to :

σ= πr^2

For a 0.2m sphere, σ = -9 dBm²

And between, Rayleigh region and optical region, this is the Mie region.

*Figure 3 – Rayleigh, Mie and Optical Region*

Ref 1 : https://www.radartutorial.eu/01.basics/Rayleigh-%20versus%20Mie-Scattering.en.html