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EM.Picasso

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== An EM.Picasso Primer ==
 
EM.Picasso® is a versatile planar structure simulator for modeling and design of printed antennas, planar microwave circuits, and layered periodic structures. EM.Picasso's simulation engine is based on a 2.5-D full-wave Method of Moments (MoM) formulation that provides the ultimate modeling accuracy and computational speed for open-boundary multilayer structures. It can handle planar structures with arbitrary numbers of metal layouts, slot traces, vertical interconnects and lumped elements interspersed among different substrate layers. You can use EM.Picasso to model large finite-sized antenna arrays as well as infinite periodic structures such as frequency selective surfaces.
Click here to learn more about the theory of [[Planar Method of Moments]].
 
=== Multilayer Green’s Functions ===
 
The Green’s functions are the solutions of boundary value problems when they are excited by an elementary source. This is usually assumed to be an infinitesimally small vectorial point source. In order for Green’s functions to be computationally useful, they must have analytical closed forms like a mathematical expression, or one should be able to compute them using a recursive process. It turns out that only very few boundary value problems have closed-form Green’s functions. Planar layered structures with laterally infinite extents are one of those few cases, which can be represented by recursive dyadic Green's functions.
 
In general, a structure may support both electric ('''J''') and magnetic ('''M''') currents. The total electric ('''E''') and magnetic ('''H''') fields can be expressed in terms of the electric and magnetic currents in the following way:
 
:<math>E = E^{inc} + \iiint\limits_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') \, dv'</math>
 
:<math>H = H^{inc} + \iiint\limits_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') \, dv'</math>
<!--[[File:PMOM1(1).png]]-->
 
where '''G<sub>EJ</sub>''', '''G<sub>EM</sub>''', '''G<sub>HJ</sub>''', '''GH<sub>M</sub>''' are the dyadic Green’s functions for the electric and magnetic currents due to electric and magnetic current source, respectively, and '''E<sup>i</sup>''' and '''H<sup>i</sup>''' are the incident or impressed electric and magnetic fields, respectively. In these equations, '''r''' is the position vector of the observation point and '''r'''' is the position vector of the source point. V is the volume that contains all the sources and the volume integration is performed with respect to the primed coordinates. The incident or impressed fields provide the excitation of the structure. They may come from an incident plane wave or a gap source on a microstrip line, a short dipole, etc. The complexity of the Green’s functions depends on what is considered as the background structure. If you remove all the unknown currents from the structure, you are left with the background structure.
 
=== Planar Integral Equations ===
 
To derive a system of integral equations, we enforce the boundary conditions on the integral definitions of the '''E''' and '''H''' fields as follows:
 
:<math>L_E(E) = L_E \bigg\{ E^{inc} + \iiint\limits_V \overline{\overline{G}}_{EJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{EM}(r|r') \cdot M(r') \, dv' \bigg\} </math>
 
:<math>L_H(H) = L_H \bigg\{ H^{inc} + \iiint\limits_V \overline{\overline{G}}_{HJ}(r|r') \cdot J(r') \, dv' + \iiint\limits_V \overline{\overline{G}}_{HM}(r|r') \cdot M(r') \, dv' \bigg\} </math>
<!--[[File:PMOM4(2).png]]-->
 
where '''L<sub>E</sub>''' is the boundary value operator for the electric field and '''L<sub>H</sub>''' is the boundary value operator for the magnetic field. For example, '''L<sub>E</sub>''' may require that the tangential components of the '''E'''field vanish on perfect conductors:
 
:<math> \hat{n} \times \hat{n} \times \mathbf{E} = 0, \quad \mathbf{r} \in PEC </math>
<!--[[File:PMOM65.png]]-->
 
Or '''L<sub>E</sub>''' and '''L<sub>H</sub>''' may require that the tangential components of the '''E''' and '''H''' fields be continuous across an aperture in a perfect ground plane:
 
:<math>\begin{cases} \hat{n} \times \hat{n} \times (\mathbf{E}^+ - \mathbf{E}^-) = 0 \\ \hat{n} \times \hat{n} \times (\mathbf{H}^+ - \mathbf{H}^-) = 0 \end{cases} \quad \Rightarrow \quad \mathbf{M}^+(r) = \mathbf{M}^-(r), \quad r \in PMC </math>
<!--[[File:PMOM66(1).png]]-->
 
Given the fact that the dyadic Green’s 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'''.
 
In [[EM.Cube|EM.CUBE]]'s [[Planar Module|Planar module]], magnetic currents are always surface current with units of V/m. Electric currents, however, can be surface currents with units of A/m as in the case of metallic traces like microstrip lines, or they can be volume currents with units of A/m<sup>2</sup> as in the case of perfectly conducting vias. Dielectric inserts are modeled as volume polarization currents that are related to the electric field '''E''' in the following manner:
 
:<math>\mathbf{J}_p(r) = jk_0 Y_0(\varepsilon_r - \varepsilon_b)\mathbf{E}(r)</math>
<!--[[File:PMOM5.png]]-->
 
where k<sub>0</sub> is the free space propagation constant, <math>Y_0 = \tfrac{1}{Z_0} = \tfrac{1}{120\pi}</math> is the free space intrinsic admittance, &epsilon;<sub>r</sub> is the permittivity of the dielectric insert, and &epsilon;<sub>b</sub> is the permittivity of its background layer. In a 2.5-D formulation, it is assumed that the volume currents have only a vertical component along the Z direction, and their circumferential components are negligible.
 
=== Numerical Solution Of Integral Equations ===
 
The planar integral equations derived earlier can be solved numerically by discretizing the unknown currents using a proper meshing scheme. The original functional equations are reduced to 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 and magnetic currents are represented by expansions of basis functions as follows:
 
:<math>J(r) = \sum_{n=1}^N I_n^{(J)} f_n^{(J)} (r)</math>
 
:<math>M(r) = \sum_{k=1}^K V_k^{(M)} f_k^{(M)} (r)</math>
<!--[[File:PMOM2.png]]-->
 
where <math>f_n^{(J)}</math> and <math>f_k^{(M)}</math> are the generalized vector basis functions for the expansion of electric and magnetic currents, respectively, and <math>I_n^{(J)}</math> and <math>V_k^{(M)}</math> are the unknown amplitudes of these basis functions, which have to be determined. Substituting these expansions into the integral equations generates a set of discretized integral equations, which can further be converted to a system of linear algebraic equations. This is accomplished by testing the discretized integral equations using the a set of test functions. In the method of moments, the Galerkin technique is typically used, which chooses the expansion basis functions as test functions. This leads to the following linear system:
 
:<math> \begin{bmatrix} Z^{(EJ)} & T^{(EM)} \\ U^{(HJ)} & Y^{(HM)} \end{bmatrix} \cdot \begin{bmatrix} I^{(J)} \\ V^{(M)} \end{bmatrix} = \begin{bmatrix} V^{(E)} \\ I^{(M)} \end{bmatrix} </math>
<!--[[File:PMOM3.png]]-->
 
where
 
:<math> Z_{ij}^{(EJ)} = \iiint\limits_{V_i} dv f_i^{(J)}(r) \cdot \iiint\limits_{V_j} dv' \overline{\overline{G}}_{EJ}(r|r') \cdot f_i^{(J)}(r')</math>
<!--[[File:PMOM6.png]]-->
 
and
 
:<math> V_i^{(E)} = \iiint\limits_{V_i} dv f_i^{(J)}(r) \cdot E^{inc}(r) </math>
 
 
:<math> I_i^{(H)} = \iiint\limits_{V_i} dv f_i^{(M)}(r) \cdot H^{inc}(r) </math>
<!--[[File:PMOM7.png]]-->
 
Similar expressions can be derived for the T<sup>(EM)</sup>, U<sup>(HJ)</sup> and Y<sup>(HM)</sup>elements of the MoM matrix.
 
=== Discretization Of Electric &amp; Magnetic Currents ===
 
The right choice of the basis functions 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 most popular types of basis functions used in a variety of MoM formulations. The surface currents (whether electric or magnetic) are discretized using 2D rooftop basis functions shown in the figure below:
 
[[File:image055_tn.png]]
 
Figure 1: Rooftop or RWG basis functions built over two rectangular, triangular or mixed cells.
 
The rooftop basis functions are defined over two adjacent cells with a common edge of length. If the two cells are triangular, then the so-called RWG functions are obtained. It is also possible to define rooftop functions over two adjacent rectangular cells or two adjacent rectangular and triangular cells with a common edge. On a rectangular cell, the function is defined as having a (descending or ascending) linear profile in one direction and a constant profile in the other perpendicular direction.
 
The volume polarization currents in 2.5-D MoM have a vertical direction along the Z-axis. These are discretized using prismatic basis functions that have either a rectangular or triangular base with a constant profile along the Z-axis.
 
[[File:image065_tn.png]][[File:image066_tn.png]]
 
Figure 2: Prismatic basis functions built over single triangular and rectangular cells.
== Anatomy Of A Planar Structure ==
You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the '''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tables: '''Available''' sources on the left and '''Associated''' sources on the right. A right arrow ('''--&gt;''') button and a left arrow ('''&lt;--''') button let you move the sources freely between these two tables. You will see in the &quot;Available&quot; table a list of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the &quot;Associated&quot; table on the right. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added port.
{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|transmission lines]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}
[[File:PMOM51(2).png|800px]]
Kazem Sabet
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