== An EM.Libera Primer ==Â EM.Libera is a 3D free-space structure simulator for modeling metallic structures. It features a full-wave Method of Moments (MoM) engine for analyzing wire structures or wire-frame models of metallic surfaces and solids. Many RF systems utilize wire antennas like dipoles, monopoles, loops, or arrays of wire antennas. The [[3D Method of Moments|3D method of moments ]] (MoM) based on Pocklington's integral equation can accurately model such antennas and their coupling effects. Moreover, in many applications, metallic surface or [[Solid Objects|solid objects]] can be approximately modeled as wire-frame structures. EM.Libera's Wire MoM simulator can provide an adequate numerical solution of wire-frame structures. Examples of this sort are wire antennas in the presence of large reflectors or scatterers. Wire-frame models also make a good approximation of metallic target structures for radar cross section (RCS) analysis.
EM.Libera's Wire MoM simulator is seamlessly interfaced with [[EM.Cube|EM.CUBE]]'s other simulation engines. The solution of a wire-frame structure can be imported to [[EM.Cube]]'s other modules as a set of short dipole sources with proper amplitudes and phases.
Click here to learn more about the theory of [[3D Method of Moments]].
Â
=== Free Space Greenâs Function ===
Â
The Greenâs functions are the analytical solutions of boundary value problems when they are excited by an elementary source. This is usually an infinitesimally small vectorial point source. In order for the Greenâs functions to be computationally useful, they must have analytical closed forms. This can be a mathematical expression or a more complex recursive process. It is no surprise that only very few electromagnetic boundary value problems have closed-form Greenâs functions. The total electric ('''E''') field can be expressed in terms of the electric current in the following way:
Â
:<math> \mathbf{E = E^{inc}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } d \nu' + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') } d \nu' </math>
Â
Â
:<math> \mathbf{H = H^{inc}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } d \nu' + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') } d \nu' </math>
<!--[[File:PMOM1(1).png]]-->
Â
where is the dyadic Greenâs functions for electric fields due to electric current sources and '''E<sup>i</sup>''' is the incident or impressed electric field. The incident or impressed field provides the excitation of the structure. It may come from an incident plane wave or a gap source on a line, etc. The simplest background structure is the unbounded free space, which is represented by the following Greenâs function:
Â
:<math> \mathbf{ \overline{\overline{G}}_{EJ}(r|r') = (\overline{\overline{I}} + \nabla\nabla) } G_{\Lambda} (\mathbf{r|r'}), \quad G_{\Lambda} (\mathbf{r|r'}) = \frac{ e^{-jk_0 \mathbf{|r-r'|}} }{ 4\pi \mathbf{|r-r'|} } </math>
<!--[[File:03_freespace_tn.gif]]-->
Â
where <math>\mathbf{\overline{\overline{I}}}</math> is the unit dyad, <math>\nabla</math> is the gradient operator, '''r''' and '''r'''' are the position vectors of the observation and source points, respectively, and k<sub>0</sub> is the free-space propagation constant. This implies that electromagnetic waves propagate in free space in a spherical form away from the source. Note that the Greenâs function has a singularity at the source, i.e. when '''r''' = '''r''''. This singularity must be removed when solving the integral equations.
Â
=== 3D Integral Equations ===
Â
In the more general formulation of the field integration equations, both electric and magnetic currents are included. In that case, the total electric and magnetic fields are given by the following equations:
Â
:<math> \mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') } </math>
Â
Â
:<math> \mathbf{H = H^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') } </math>
<!--[[File:image001_tn.gif]]-->
Â
The above coupled equations involve four types of dyadic Green's functions that represent the electric and magnetic field radiated by an electric or a magnetic current. The incident or impressed electric and magnetic fields Ei and Hi exist independently of the given structures and are related to each other depending on the type of excitation source.
Â
Enforcing the boundary conditions on the integral definitions of the '''E''' and '''H''' fields results in a system of integral equations as follows:
Â
:<math> \mathcal{L}_E(E) = \mathcal{L}_E \left( \mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') } \right) = 0 </math>
Â
Â
:<math> \mathcal{L}_H(H) = \mathcal{L}_H \left( \mathbf{E = E^{i}} + \mathbf{\iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') } + \mathbf{\iiint_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') } \right) = 0 </math>
<!--[[File:image016_tn.gif]]-->
Â
where <math>\mathcal{L}_E(E)</math> is the boundary value operator for the electric field and <math>\mathcal{L}_H(H)</math> is the boundary value operator for the magnetic field. For example, they may require that the tangential components the '''E''' field vanish on perfect electric conductors. Or they may require that the tangential components the '''E''' and '''H''' fields be continuous across an aperture in a perfect ground plane. Given the fact that the dyadic Greenâs functions and the incident or impressed fields are all known, one can solve the above system of integral equations to find the unknown currents '''J''' and '''M'''. Therefore, through these relationships you can easily cast the above integral equations in terms of unknown '''E''' and '''H''' fields.
Â
=== Galerkin Testing ===
Â
The integral equation derived in the previous section can be solved numerically by discretizing the computational domain using a proper meshing scheme. The original functional equation is reduced to a set of discretized linear algebraic equations over elementary cells. The unknown quantities are found by solving this system of linear equations, and many other [[parameters]] can be computed thereafter. This method of numerical solution of integral equations is known as the Method of Moments (MoM). In this method, the unknown electric current is represented by an expansion of basis functions as follows:
Â
:<math> \mathbf{J(r)} = \sum_{n=1}^N {I_n}^{(J)} \mathbf{ {f_n}^{(J)}(r) }</math>
<!--[[File:07_numerical-solutions_tn.gif]]-->
Â
where <math>\mathbf{ {f_n}^{(J)} }</math> are the generalized vector basis functions for the expansion of electric currents, and <math>{I_n}^{(J)}</math> are the unknown complex amplitudes of these basis functions, which have to be determined. Substituting these expansions yields the following discretized integral equation:
Â
:<math> \mathcal{L}_E \left( \mathbf{E^i} +\iiint_V \mathbf{ \overline{\overline{G}}_{EJ}(r|r') } \cdot \sum_{n=1}^N {I_n}^{(J)} \mathbf{ {f_n}^{(J)}(r') } \, d\nu' \right) = 0 </math>
<!--[[File:10_numerical-solution_tn.gif]]-->
Â
In order to solve the above equation, the method of moments uses Galerkin's technique to turn it into a set of linear algebraic equations. This is accomplished by testing the above equations using the basis functions, leading to the following linear system:
Â
:<math>\mathbf{[Z] \cdot [I] = [V]}</math>
<!--[[File:11_numerical-solution_tn.gif]]-->
Â
where
Â
:<math> Z_{ij} = \iiint_{V_i} \mathbf{ {f_i}^{(J)}(r) } \, d\nu \cdot \iiint_{V_j} \mathbf{ \overline{\overline{G}}_{EJ}(r|r') \cdot {f_j}^{(J)}(r') } \, d\nu' </math>
<!--[[File:12_numerical-solution_tn.gif]]-->
Â
and
Â
:<math> V_i = \iiint_{V_i} \mathbf{ {f_i}^{(J)}(r) \cdot E^i(r) } \, d\nu </math>
<!--[[File:13_numerical-solution_tn.gif]]-->
Â
Using a rooftop expansion of the currents on the wires, we can discretize the Pocklington integral equation. In order to convert the discretized integral equation into a system of linear system of algebraic equations, we use Galerkinâs testing process, in which the testing functions are chosen to be identical to the expansion basis functions. However, to avoid the source singularity at r=râ, the expansion functions are placed at the center of the wires, while the test functions are evaluated on the surface of the wires, assuming a finite non-zero radius for all wires. The solution vector [I] is then found as:
Â
:<math>\mathbf{[I] = [Z]^{-1} \cdot [V] } </math>
<!--[[File:24_galerkin_tn.gif]]-->
Â
where [Z]<sup>-1</sup> is the inverse of the impedance matrix and [V] is the excitation vector.
Â
=== Pocklingtonâs Integral Equations for Wire Structures ===
Â
Wire structures are made of linear PEC elements. These may consist of actual physical wires such as a dipole or loop antenna or a wireframe representation of a surface or solid object. In a wire structure, the unknown electric currents are one-dimensional. The integral equation is derived by forcing the tangential component of the electric field to vanish on the surface of the wire. This leads to the following simpler integral equation:
Â
:<math> \mathbf{ \hat{I} \cdot E^i } - jk_0 Z_0 \int_C \left( G_A \mathbf{(r|r')} I(l') \mathbf{ \hat{l} \cdot \hat{l}' } + \frac{1}{{k_0}^2} \frac{\partial G_A}{\partial l} \frac{\partial I}{\partial l'} \right) \, dl' = 0 </math>
<!--[[File:14_pocklingtons_tn.gif]]-->
Â
where G<sub>A</sub> is the free space Greenâs function, I(l) is the unknown linear current in the wire and C is the contour of the wire. and <math>\hat{l}'</math> are the unit vectors along the wire contour. Note that G<sub>A</sub> has a singularity when r = râ, which must be either removed or avoided as will be explained later.
Â
=== Discretization Of Wire Structures ===
Â
The right choice of the basis functions that are used to represent the elementary currents is very important. It will determine the accuracy and computational efficiency of the resulting numerical solution. Rooftop basis functions are one of the more popular types of basis functions used in a variety of MoM formulations. The simplest rooftop function is the one-dimensional triangular functions defined as in the figure below:
Â
[[File:18_meshing_tn.gif]]
Â
This function provides a linear interpolation of the unknown currents or fields in one dimension. Note that the function vanishes at it two ends. This is a desirable feature for basis functions that represent electric currents on metallic wires as the current must vanish at the two ends of a wire. The total current on the wire can be approximated in a linear fashion by a set of one-dimensional rooftop functions as shown in the figure below:
Â
[[File:19_meshing_tn.gif]]
Â
This can be written as
Â
:<math> I(l) = \sum_{n=1}^N a_n f_n(l) \mathbf{\hat{s}_n} </math>
<!--[[File:20_meshing_tn.gif]]-->
Â
where l is the length coordinate along the wire with l=0 at its start point. <math>f_n(l)</math> is the scaled and translated version of the linear basis function <math>f(l)</math> shown in the previous figure. <math>\mathbf{\hat{s}_n}</math> is the unit vector along wire.
== Physical Structure & 3D Mesh Generation ==
Ports are used to order and index gap sources for S parameter calculation. They are defined in the '''Observables''' section of the Navigation Tree. Right click on the '''Port Definition''' item of the Navigation Tree and select '''Insert New Port Definition...''' from the contextual menu. The Port Definition Dialog opens up, showing the total number of existing sources in the workspace. By default, as many ports as the total number of sources are created. You can define any number of ports equal to or less than the total number of sources. This includes both gap sources and active lumped elements (which contain gap sources). In the '''Port Association''' section of this dialog, you can go over each one of the sources and associate them with a desired port. Note that you can associate more than one source with same given port. In this case, you will have a coupled port. All the coupled sources are listed as associated with a single port. However, you cannot associate the same source with more than one port. Finally, you can assign '''Port Impedance''' in Ohms. By default, all port impedances are 50Σ. The table titled '''Port Configuration''' lists all the ports and their associated sources and port impedances.
{{Note|In [[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|EM.CUBE]]]]]]]]]]]]]]]]]] you cannot assign ports to an array object, even if it contains sources on its elements. To calculate the S [[parameters]] of an antenna array, you have to construct it using individual elements, not as an array object.}}
[[File:port-definition.png]]
The radiation patterns of antenna arrays usually have a main beam and several side lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' such as first null level and first null beam width. You can have [[EM.Cube|EM.CUBE]] calculate all such [[parameters]] if you check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. In particular, you can plot such data files at the end of a sweep simulation.
{{Note|Defining an array factor in the radiation pattern dialog simply performs a post-processing calculation. The resulting far field obviously do not take into account any inter-element coupling effects as [[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|[[EM.Cube|EM.CUBE]]]]]]]]]]]]]]]]]] does not construct a real physical array in the project workspace.}}
{{Note|Using an array factor for far field calculation, you cannot assign non-uniform amplitude or phase distribution to the array elements. For this purpose, you have to define an array object.}}