# EM.Tempo Tutorial Lesson 2: Analyzing Scattering From A Sphere

 Tutorial Project: Analyzing Scattering From A Sphere Objective: In this project, you will construct a spherical object, assign different materials to it, simulate plane wave scattering from the object and calculate the radar cross section of the target. Concepts/Features: PEC Material Dielectric Material Sphere Adaptive Mesh Plane Wave Source Radar Cross Section Field Sensor Observable 3D Data Visualization Minimum Version Required: All versions ' Download Link: EMTempo_Lesson2

## What You Will Learn

In this tutorial you will learn to draw spherical objects from the ground up. You will define material groups of different types, will set up a plane wave source to excite your physical structure and define field sensor and radar cross section (RCS) observables for your target.

## Getting Started

For this tutorial lesson, set the following parameters in the new project dialog:

 Name EMTempo_Lesson2 Millimeters GHz 10GHz 1GHz

## Drawing a Spherical Solid Object

Select the Sphere button of the Object Toolbar or select the menu item Object → Solid → Sphere.

 EM.Tempo's Object Toolbar.
The property dialog of sphere.
With the sphere tool selected, click at the origin (0,0,0) of the project workspace to start your drawing, and drag the mouse to draw a sphere of radius R = 15mm. A property dialog pops up on the lower right corner of the screen. You can see the radius of the new sphere changing in this dialog as you drag your mouse. At a center frequency of 10GHz, λ0 = 30mm, therefore, your target has a diameter of one free-space wavelength. Note that since you didn't define any material group on the navigation tree, a default perfect electric conductor (PEC) material group called "PEC_1" is immediately created to hold your new object.
 In EM.Tempo, all the new objects you draw are assumed to be perfect electric conductors (PEC) by default unless you define a material group of a different type.

As soon as you draw the sphere, a default blue domain box is placed around it at an offset of 0.25λ0 away from the largest bounding box of your physical structure. Similar to the previous tutorial lesson, convolutional perfectly matched layer (PML) boundary conditions are assumed at all the six faces of the domain box. This is required for modeling an open-boundary problem.

 The PEC sphere object and the enclosing domain box.

## Defining the Plane Wave Source

You will introduce a plane wave source for this project. Since FDTD is a finite-domain technique, it uses a finite-sized source box to excite the physical structure. To define a plane wave source, right-click on the Plane Waves item of the Sources section in the navigation tree and select Insert New Source… from the contextual menu. The Plane Wave Dialog opens up with a number of default settings.

The default plane wave box is places 0.2λ0 away from the largest bounding box of your entire physical structure. This is usually a good choice, and you will keep it for this project. Your plane wave source has an Amplitude of 1V/m and a zero Phase. You will keep the default TMz polarization. In the “Incident Angle” section of the dialog you need to enter the elevation θ and azimuth φ angles in the standard spherical coordinate system. Accept the default values: θ = 180o and φ = 0o, which represent an X-polarized normally incident plane wave. The above settings mean that the plane wave source will illuminate the spherical target with a modulated Gaussian pulse waveform centered at 10GHz with a bandwidth of 1GHz, where the envelope of the electric field reaches a maximum value of 1V/m. You will see a magenta wireframe box in the project workspace that completely encloses the sphere.

 In EM.Tempo, plane waves are characterized by their unit propagation vector. Therefore, for downward-looking and upward-looking normally incident plane waves along the ∓Z-axis, θ must be 180° and 0°, respectively.
 EM.Tempo's plane wave dialog.

## Defining the Project Observables

To calculate the radar cross section of your structure, you need to define an RCS observable. To do so, right click on the Radar Cross Sections item in the Observables section of the navigation tree, and select Insert New RCS… Accept most of the default values, but set both the Theta and Phi angle increments to 3°. The RCS box appears as a light blue box around your physical structure.

For this project, you will also define two field sensors at XY and YZ planes to observe the electric and magnetic field distributions. To define the field sensors, right-click on the Near Field Sensors item in the “Observables” section of the navigation tree and select Insert New Observable… In the Field Sensor Dialog, enter the point (0, 0, 0) for Coordinates and select Z from the Direction drop-down list for the XY plane. Then define Sensor_2 which is X-directed and centered at (0,0,0) for the YZ plane. Two orthogonal, purple boxes appear around your structure inside the computational domain.

 The Field Sensor dialog.
 The PEC spherical object with the plane wave box, RCS box and field sensor planes.

## Running the FDTD Simulation

Before running the FDTD simulation, check out the FDTD mesh of your physical structure. For this project, you will leave the mesh density at its default value of 20 cells/λeff. You will see that the curvature of the sphere is fairly well captured by EM.Tempo's adaptive mesh generator even at this relatively low mesh density. For this project, you will accept the default convergence criteria, that is a Power Threshold of -30dB and a No. Time Steps equal to 10,000. Run the FDTD simulation and wait until it converges.

 The default FDTD mesh of the PEC sphere.

## Visualizing the RCS of the PEC Sphere

At the end of the FDTD simulation, the Radar Cross Sections part of the navigation tree is populated with three 3D RCS plots: Theta RCS, Phi RCS and the total RCS. In order to better view the RCS plots, you have to either hide the target object or alternatively “freeze” it. Freezing an object replaces it with a wireframe outline of it, so you can still recognize its location and orientation in the project workspace. To freeze an object or a group of objects, right-click on their name and select Freeze from the contextual menu. To unfreeze a frozen object and get back to its normal display, repeat the same procedure and remove the check mark in front of the “Freeze” item in the contextual menu. Note that the maximum RCS is 0.00957m2 in the forward-scatter direction.

 Frozen objects cannot be selected using the mouse or cannot be highlighted using mouse-over.
 The 3D RCS plot of the PEC spherical target.

Next, open the data manager and plot the data files "RCS_1_RCS_Polar_YZ.ANG" and "RCS_1_RCS_Polar_ZX.ANG". Note that the polar plots of the RCS in the two principal planes are different.

 The 2D polar plot of the RCS of the PEC spherical target in the YZ plane. The 2D polar plot of the RCS of the PEC spherical target in the ZX plane.

## Visualizing the Near Fields of the PEC Sphere

After the end of the FDTD simulation, 14 near-field plots are added under the nodes of each of your two field sensors: “Sensor_1” at XY plane and “Sensor_2” at YZ plane. The figures below show the total electric and magnetic fields at the XY and YZ planes. Note the null fields inside the PEC sphere.

 The total electric field of the PEC sphere on the XY plane. The total magnetic field of the PEC sphere on the XY plane. The total electric field of the PEC sphere on the YZ plane. The total magnetic field of the PEC sphere on the YZ plane.

In this part of the tutorial lesson, you will analyze the scattering characteristics of a dielectric-coated PEC sphere. First, you need to define a new dielectric material group in the same project. Right-click on the Dielectric item under the “Physical Structure” section of the navigation tree and select Insert New Dielectric… The dielectric dialog opens up with the vacuum as the default material type. You can simply enter a name for your material and enter values for its constitutive parameters: relative permittivity εr, relative permeability μr, electric conductivity σ and magnetic conductivity σm. Alternatively, you can click the Material button of this dialog to access EM.Tempo’s Materials List.

 The dielectric material dialog.
 The materials list.
 The dielectric material dialog.

Open the materials list and select "Polycarbonate" from the list. Click the OK button of the dialog to return to the dielectric dialog. You will see that the material properties are filled: εr = 3.2, μr = 1, σ = σm = 0. The last defined material group remains as the active material group. That means that you can now draw new objects under this material group. Draw a new sphere of radius R = 20mm center at (0, 0, 0). The new dielectric sphere in light green color is concentric with the old red PEC sphere, which is completely enclosed inside. If you hover your mouse over the larger sphere, its color becomes translucent and you will be able to see the enclosed smaller sphere.

 The dielectric sphere. Hovering the mouse over the dielectric sphere to view the PEC sphere inside it.

EM.Tempo's FDTD solver is a volumetric solver. This means Maxwell's Equations are solved numerically at every single cell inside the computational domain. Although the two spheres are concentric, the cells inside the interior PEC sphere cannot have dielectric properties. EM.Tempo has a material hierarchy rule that takes care of object overlaps. This feature greatly facilitates the construction of physical structures with different material compositions.

 If two objects share overlap regions, the PEC material always supersedes the dielectric material.

Examine the default FDTD mesh of the dielectric-coated PEC target and run an FDTD analysis using the default convergence criteria.

 The default FDTD mesh of the dielectric-coated PEC sphere.

## Visualizing the RCS and Near Fields of the Coated Sphere

Visualize the 3D RCS of your new target and plots its 2D polar graphs. Note how much the ZX-plane RCS has changed due to the dielectric coating. Also note that the maximum RCS has increased almost five-fold to 0.04775m2 in the forward-scatter direction.

 The 3D RCS plot of the coated spherical target.
 The 2D polar plot of the RCS of the coated spherical target in the YZ plane. The 2D polar plot of the RCS of the coated spherical target in the ZX plane.

Similarly, plot the electric and magnetic field distributions of the new target and compare them to the case of the PEC target without the dielectric coating.

 The total electric field of the coated sphere on the XY plane. The total magnetic field of the coated sphere on the XY plane. The total electric field of the coated sphere on the YZ plane. The total magnetic field of the coated sphere on the YZ plane.