The finiteelement method (FEM) calculates the screening effect of electromagnetic radiation throughscreens with periodic perforations. Electronic equipment often hasscreening grids, thus design engineers must determine the shieldingeffectiveness of perforated grids in electronic cabinet walls.
FEM analyzes EMF (electromagneticfield) effects in such devices. Emitted EMI (electromagnetic interference)through these lattices disturbs device operation or causes thedeviceto alter the operation of other equipment in the environment. Suchcomplex electronic devices require full and detailed modeling of thegeometric structure.
Note that ventilation screening, introduced to ease thermal heatingeffects, is critical for EMI/ EMC (electromagnetic conformance)performance. Emitted EMI through these lattices disturbs deviceoperation or causes the device to alter the operation of otherequipment in the environment. Such complex electronic devices requirefull and detailed modeling of the geometric structure.
Given EMI/EMC problems, an approximate calculation to predictinterference due to perforated holes is considered satisfactory.Homogenization is the process of replacing grids or screens withelectromagnetically equivalent continuous materials that simplifiessimulation of electronic equipment( Figure1, below ).
|Figure1: A homogeneous sheet replaces a rectangular grid in FEM.|
In this discussion, geometric patterns are replaced not byequivalent material, but by equivalent surface impedance. Thishomogenized characteristic surface impedance is determined via modelingthe unit cell behavior of an infinite periodic grid( Figure 2, below ) Although suchapproximation neglects the grid's edge effects, generated examples showit gives practical accuracy in large grid repetition.
|Figure2: Unit cell of a rectangular grid was cut off for analysis and shows aport above and below the grid.|
Homogenizing a grid
Homogenizing a grid raises two concerns. One is the incident angle'seffect on impedance boundarycondition (Figure 3, below ).The other is the effect of incidentfield polarization .The effect of incident angle is negligiblewhen wavelength is large compared with grid periodicity.
|Figure3. Shown is the equivalent circuit of the surface impedance of the unitcell to vertical polarization.|
Meanwhile, unit cell impedance replacing the grid is different fromx and y polarizations of the electric field, where x and y areorthogonal directions of the lattice. Thus, polarization is vital todefining unit cell impedance if lattice size varies in x and ydirections.
Since polarization of the incident field is unknown in realapplications, homogenized impedance is treated as an anisotropicimpedance boundary condition in FEM. In turn, FEM is modified to copewith problems of modeling anisotropic impedance boundary conditions.
Without limiting the generality of the method, consider atwodimensional rectangular lattice for discussion with a grid to bereplaced by an equivalent homogeneous impedance boundary condition.
To determine the characteristic impedance of this sheet, a unit cellof the grid is cut out and analyzed for x and y polarizations. Applyingpairs of perfect electric conductor(PEC) and perfect magneticconductor (PMC) boundary conditions controls thepolarization ofthe incident wave.
First, derive the condition when the solution of the unit cell isindependent of incident angle. Applying Floquet theory, the propagationconstant in the x direction is:
where ko is the free space wave number, theta ' �'is the incident angle, D is the lattice size in xdirection and m is the mode index. The propagation constant in the z direction (normal to the surfaceof the grid) is:
The field is independent of the incident angle if no propagatingFloquet modes can exist for any theta' �'.The condition for this is:
From conditions 1 and 3, it follows that:
If wavelength exceeds lattice size, the grid will not exhibitresonance with the incident angle. Thus, the field is almostindependent of the incident angle, so normal incidence port excitationscan be used. The next step is to solve the field of the unit cell withtwo ports and boundary conditions, corresponding to verticalpolarization.
At an arbitrary, but low-frequency computation, the impedancematrix of the two-port model can be calculated. Since the equivalentimpedance at reference plane z = 0 is needed, the impedance matrix mustbe de-embedded to this plane.
Entries of the impedance matrix will then contain the equivalentimpedance at the reference plane:
where Z ver is the homogenized, equivalentsurface impedance of the unit cell to vertical polarization. Forsimplicity, vertical and horizontal polarizations used here correspondto y and x directions, respectively.
This procedure is repeated when PEC and PMC boundary condition pairsare exchanged. In this case, the equivalent surface impedance Z hor is calculated. The anisotropic equivalent surface impedance is formedfrom the vertical and horizontal impedance values as:
Knowing the homogenized anisotropic impedance, the screen isreplaced by a sheet representing an anisotropic impedance boundarycondition.
The usual FEM is extended to include an anisotropic impedanceboundary condition. The functional arising from an electric fielddescription of the vector wave equation is:
where [Y hom ] is an anisotropic tensor . Considering thex and y directions, entries of an anisotropicadmittance tensor become:
For an arbitrary sheet direction, the homogenized admittance tensormust be transformed appropriately.
To validate homogenization, test cases were solved twice using thehomogenized impedance boundary condition, and then withouthomogenization. The entire procedure studies leakage field from acomplex PCB inside a computer cabinet.
The first example is a simple polarization filter that demonstratesthe effectiveness of an anisotropic impedance boundary condition. Asheet with an anisotropic impedance boundary condition is placed in themiddle of a parallel plate waveguide.
|Figure4. Total reflection is provided in horizontally polarized waves (top)and total transmission in vertically polarized waves (bottom).|
The anisotropic admittance values are (the first entry ishorizontal, the second a vertical one):
[Y hom ]= [ 0 0]
This surface should providetotal reflection with horizontally polarized waves, but should beinvisible with vertical polarized waves (Figure 4, above) .
Consider a simple metal cabinet sized 300mm x 300mm x 150mm. Oneside of the cabinet is a square ventilation grid with a lattice, eachside with 25 holes (Figure 5, below ).
|Figure5: A square ventilation grid in a cabinet provided isotropic|
The total grid area is 30mm x 30mm. A Hertzian dipole is placed inthe middle of the cabinet and excited at 3GHz. The far field of thestructure was solved twice and resulted in isotropic impedance. At3GHz, the impedance is j86 ohms. Results of the far field for both thecontrol and the homogenized project tests show agreement (Figure 6, below ).
|Figure6. Far field of both the control and homogenized project shows goodagreement among results.|
The next case investigated was the incident field penetrationthrough a rectangular ventilation grid in a metal cabinet sized 300mm x300mm x 150mm. The ventilation grid on one side of the box has an areaof 37.5mm x 18.75mm with anisotropic holes measuring 29.5mm x 10.75mm.
Given a 3GHz incident wave frequency, the problem was solved twiceby modeling the grid in detail and replacing it with the homogenizedimpedance boundary condition (Figure7, below ).
|Figure7: Incident field impinging onto the ventilation grid resulted inanisotropic impedance because the grid is rectangular.|
Since the grid is rectangular, the impedance is consideredanisotropic, with the vertical and horizontal values at 3GHz placed atj69 ohms and j25 ohms, respectively.
The electric field distribution calculated along the middle axis ofthe cabinet of both the control and the homogenized project Figure 8, below shows goodagreement, except in the grid area where the field pattern of thecontrol project is not homogeneous.
|Figure8: Electric field distribution along the middle axis of the cabinetshowed good agreement for the control and homogenized projects.|
Finally, a real-life computer cabinet with several ventilationgrids, a conducting gasket and other details was examined. Theexcitation is from the real life PCB (Figure9, below ).
Current distribution in the PCB is first computed using a full-waveFEM for solving planar structures. Currents generated near-field distribution that ispassed on to the 3D tool, which handles computer housing.
|Figure9: PCB in a computer cabinet emits radiated field through severalventilation screens.|
Subsequently, fields inside the cabinet and radiated fields throughthe ventilation grids are computed. The resulting radiated field in Figure 9 above does not show the PCBbecause it is handled through the dynamic link with the planar method.
3D-field simulation of the computer housing required an initial meshof 105,000 tetrahedra with all ventilation holes present. Then, only21,000 tetrahedra were needed after impedance boundaries replaced mostventilation holes. Hence, unknowns were reduced by a factor of five.Radiated fields were very similar in both approaches.
|Figure10: A 200MHz clock in the PCB produces varying electric field at 3mdistance from the cabinet.|
Finally, radiated near and far fields are computed and weighed bythe spectrum of the board's clock signal with the maximum electricfield at a 3m distance shown in Figure10, above ,as a function of frequency, which can be compareddirectly with EMC regulations.
Homogenization is a useful tool to replace highly complex periodicstructures in finite-element simulation. Introducing anisotropicimpedance boundary conditions may simulate polarization effects onrectangular grids. The method's accuracy, though not perfect, issatisfactory for practical applications.
Istvan Bardi is Senior R&DEngineer, Martin Vogel is senior member of technical staff, and ZoltanCendes is chairman and CTO at AnsoftCorp .