## Finite Arrays

In addition to explicit modeling of finite arrays in Ansys HFSS, there are three other methods based on using unit cells. To learn more about the unit cell, please see part 1 of this blog.

In part 2, I will introduce and compare these 3 methods (1) Finite array defined using the array setting in unit cell, (2) Finite Array using Domain Decomposition Method (FADDM), (3) 3D component arrays (Figure 1). Please note that that the method 3 requires HFSS 2020R1 or newer.

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Figure 1. (a) Unit cell, (b) FADDM, (c) 3D component array.

## Finite Array using Unit Cell

After defining a unit cell (Figure 1a), you may simply define the number of elements, the spacing between them, and the scan angle. The assumption is that there is no mutual coupling, every element has the same radiation pattern and the same excitation. This is a good approximation for large arrays (10×10 or larger). This method may not be accurate enough in some cases, for example for a small number of elements; and when the antenna elements have main beams toward the angles that are close to the plane of the array and toward the other elements, causing a higher level of mutual coupling. However, smaller arrays won’t require as large of a compute resource as a large array.

The advantage of this method is its simulation speed. It requires the minimum memory and time to provide a quick array simulation. To define the array (after running the analysis for unit cell), right-click on Radiation from the Project Manager window. Select Antenna Array Setup, and then Regular Array Setup. In the Antenna Array Setup under Regular Array type, define the location of the first cell, the direction, the distance between the cells and number of cells in each direction.

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Figure 2. Steps to define a finite array, (a) Antenna array setup, (b) & (c) Regular array setup.

## Finite Array using Domain Decomposition Method

General Domain Decomposition (DDM) for a single domain provides a way to reduce memory requirement, however, it does not reduce the meshing time for large explicit arrays. Using Finite Array DDM (FADDM) addresses this shortcoming. The FADDM bypasses the adaptive meshing stage by duplicating the mesh that was generated for a unit cell. While the unit cell is used to create the mesh, the assumption of uniform excitation is no longer present. Each element in FADDM can have different magnitude and phase and is individually modeled, however, the mesh created in the unit cell is used to generate the overall mesh, therefore, no CPU time is spent on generating the mesh. This can be seen in Figure 3, by linking the mesh of the unit cell to the FADDM, the mesh is copied, and no mesh refinement will be needed. You may compare it with the explicit array of the same size (Figure 3(c)) where the entire array has to be meshed and mesh refinement will be necessary for adaptive meshing. This can be a huge simulation time saving when the array size is large.

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Figure 3. (a) Mesh from a unit cell, (b) mesh linked to FADDM, (c) explicit array mesh.

To create FADDM from unit cell, create a new HFSS design. Then copy the unit cell into the new design. In the Project Manager window, right-click on Model and choose Create Array, as shown in Figure 4. This opens the window that allows user to define the number of elements of the array along the lattice directions. By selecting “Active Cells” tab, user can define where active, passive and padding cells are located. This gives the user a means of creating different lattice shapes.

Please note that padding cells defined in the General tab represent the size of vacuum buffer surrounding the array. They are not visible to the users but are included in the FADDM simulation. The same mesh from the unit cell simulation is duplicated to padding cells (Figure 5). It is also possible to add padding cells in the Active Cells tab, and those cells are also invisible, but can be used to create the array lattice of the desired shape (Figure 6).

Figure 4. The steps to create a DDM array, the Padding Cells are used to create the vacuum box and are invisible to the user.

Figure 5. The FADDM needs a padding cell to create a vacuum box around the design. The padding cells are invisible to the user.

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Figure 6. (a) Padding cells can be used to create a lattice, (b) the lattice created does not show the padding cells.

The next step is to link the mesh to the unit cell. First, an analysis setup should be created. Choose Advanced Solution Setup. In the Driven Solution Setup General tab, reduce the number of Maximum Number of Passes to 1, as shown in Figure 7(a), then choose Advanced tab and click on Import Mesh (Figure 7(b)). Click on Setup Link. This is to link the simulation to the mesh of a unit cell. There are two steps needed here. First, choose the file or design that contains the mesh information (Figure 7(c)), second is to map the variables Figure 7(d). The last step to setting up the analysis is selecting Advanced Mesh Operation tab and selecting “Ignore mesh operation in target design” (Figure 7(d)). Now the array is ready and simulation can be run. You notice that adaptive meshing goes to only one pass. If in Setup Link window the option of “Simulate source design as needed” is checked (Figure 7(c)), then if a design variable that affects the geometry is changed, the meshing of the unit cell is repeated as needed. After the simulation is completed the elements magnitude and phases can be changed as a post processing step by “Edit Sources” (right-click on Excitations). The source names provided in the edit sources is slightly different than an explicit array (Figure 8)

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Figure 7. Different windows related to setting up a linked mesh in FADDM.

Figure 8. Edit sources gives the ability to change the magnitude and phase of each element.

To compare the run-time and array patterns an example of circular polarized microstrip patch antenna of a 5 ´ 5 element array is shown in Figure 9 and Figure 10. The differences can be seen at angles away from the broadside angle. This shows how the edge effects are ignored in the unit cell approximation. Table 1 shows the comparison of memory and runtime for the three methods.

Table 1. Comparing run time and memory needed for a 5 x 5 array, explicit array vs FADDM.

Figure 9. Comparison of the far-field patterns for LHCP (co-polarization) created using FADDM and unit cell array.

Figure 10. Comparison of the far-field patterns for RHCP (cross-polarization) created using FADDM and unit cell array.

Finally, we compare the co-polarization pattern with an explicit 5 x 5 element array for scan angles of 0 and 30 degrees in  Figure 11 and Figure 12, respectively.

Figure 11. Comparison of far-field LHCP created by explicit array vs. FADDM.

Figure 12. Comparison of scanned far-field LHCP, scan angle of 30 degrees.

## 3D Component Array

In 2020R1 the option of 3D component array was added. This option provides a means of combining different unit cells in one array. The unit cells are defined and imported as 3D components. To create a 3D component unit cell, define each type of the cells in a separate HFSS design, run the analysis, then select all objects in the model. In the Model ribbon, click on Create 3D Component, assign a name (no spaces are allowed in the name), add any information you like to add such as owner, email, company, etc., then click OK. Once all the 3D component cells are created, create a new HFSS design for the 3D component FADDM.

The next step is to create Relative CS for each of the 3D component elements in the HFSS design that will contain the array. For this step you need to plan the array lattice ahead of time, so the components are placed in the proper locations (Figure 13). Overlap is not allowed.

The unit cells should have the followings:

• Identical dimensions of the bounding boxes
• Identical Primary/Secondary (Master/Slave) boundary on the unit cells.

The generation of FADDM is similar to single unit cell array, except that when you select Model->Create Array, the window will be shown as 3D Component Array Properties (Figure 14). After choosing the number of elements and the number of padding cells, the unit cells window (Figure 15) will give you options of choosing one of the 3D component unit cells for each location of the elements in the lattice. The cells can be color coded. In the example shown in Figure 16  there are 3 components, the blank cell, the vertical cell and the horizontal cell. The sources under Edit Sources window are also arranged based on the name of the 3D component cells. At this point there is no option of linking the mesh. Therefore, the number of passes for adaptive mesh should be set to a number that is appropriate for getting a convergence.

Figure 13. 3D component unit cells are arranged to create a 3D component array.

Figure 14. The 3D component array can be created the same way as creating FADDM using a unit cell.

Figure 15. The unit cells are color coded for easier lattice creation.

Figure 16. The result of lattice created using 3D component array.

## Conclusion

Unit cell and Finite Array Domain Decomposition are excellent options for simulating large finite arrays within a reasonable runtime and memory requirements. The 3D component finite array is a nice added feature in 2020R1 that now provides a way to combine unit cells with different geometries in one array.

## Efficient and Accurate Simulation of Antenna Arrays in Ansys HFSS

### Unit-cell in HFSS

HFSS offers different method of creating and simulating a large array. The explicit method, shown in Figure 1(a) might be the first method that comes to our mind. This is where you create the exact CAD of the array and solve it. While this is the most accurate method of simulating an array, it is computationally extensive. This method may be non-feasible for the initial design of a large array. The use of unit cell (Figure 1(b)) and array theory helps us to start with an estimate of the array performance by a few assumptions. Finite Array Domain Decomposition (or FADDM) takes advantage of unit cell simplicity and creates a full model using the meshing information generated in a unit cell. In this blog we will review the creation of unit cell. In the next blog we will explain how a unit cell can be used to simulate a large array and FADDM.

In a unit cell, the following assumptions are made:

• The pattern of each element is identical.
• The array is uniformly excited in amplitude, but not necessarily in phase.
• Edge affects and mutual coupling are ignored

A unit cell works based on Master/Slave (or Primary/Secondary) boundary around the cell. Master/Slave boundaries are always paired. In a rectangular cell you may use the new Lattice Pair boundary that is introduced in Ansys HFSS 2020R1. These boundaries are means of simulating an infinite array and estimating the performance of a relatively large arrays. The use of unit cell reduces the required RAM and solve time.

Primary/Secondary (Master/Slave) (or P/S) boundaries can be combined with Floquet port, radiation or PML boundary to be used in an infinite array or large array setting, as shown in Figure 3.

To create a unit cell with P/S boundary, first start with a single element with the exact dimensions of the cell. The next step is creating a vacuum or airbox around the cell. For this step, set the padding in the location of P/S boundary to zero. For example, Figure 4 shows a microstrip patch antenna that we intend to create a 2D array based on this model. The array is placed on the XY plane. An air box is created around the unit cell with zero padding in X and Y directions.

You notice that in this example the vacuum box is larger than usual size of quarter wavelength that is usually used in creating a vacuum region around the antenna. We will get to calculation of this size in a bit, for now let’s just assign a value or parameter to it, as it will be determined later. The next step is to define P/S to generate the lattice. In AEDT 2020R1 this boundary is under “Coupled” boundary. There are two methods to create P/S: (1) Lattice Pair, (2) Primary/Secondary boundary.

Lattice Pair

The Lattice Pair works best for square lattices. It automatically assigns the primary and secondary boundaries. To assign a lattice pair boundary select the two sides that are supposed to create infinite periodic cells, right-click->Assign Boundary->Coupled->Lattice Pair, choose a name and enter the scan angles. Note that scan angles can be assigned as parameters. This feature that is introduced in 2020R1 does not require the user to define the UV directions, they are automatically assigned.

### Primary/Secondary

Primary/Secondary boundary is the same as what used to be called Master/Slave boundary. In this case, each Secondary (Slave) boundary should be assigned following a Primary (Master) boundary UV directions. First choose the side of the cell that Primary boundary. Right-click->Assign Boundary->Coupled->Primary. In Primary Boundary window define U vector. Next select the secondary wall, right-click->Assign Boundary->Couple->Secondary, choose the Primary Boundary and define U vector exactly in the same direction as the Primary, add the scan angles (the same as Primary scan angles)

### Floquet Port and Modes Calculator

Floquet port excites and terminates waves propagating down the unit cell. They are similar to waveguide modes. Floquet port is always linked to P/S boundaries. Set of TE and TM modes travel inside the cell. However, keep in mind that the number of modes that are absorbed by the Floquet port are determined by the user. All the other modes are short-circuited back into the model. To assign a Floquet port two major steps should be taken:

Defining Floquet Port

Select the face of the cell that you like to assign the Floquet port. This is determined by the location of P/S boundary. The lattice vectors A and B directions are defined by the direction of lattice (Figure 7).

The number of modes to be included are defined with the help of Modes Calculator. In the Mode Setup tab of the Floquet Port window, choose a high number of modes (e.g. 20) and click on Modes Calculator. The Mode Table Calculator will request your input of Frequency and Scan Angles. After selecting those, a table of modes and their attenuation using dB/length units are created. This is your guide in selecting the height of the unit cell and vaccume box. The attenation multiplied by the height of the unit cell (in the project units, defined in Modeler->Units) should be large enough to make sure the modes are attenuated enough so removing them from the calcuatlion does not cause errors. If the unit cell is too short, then you will see many modes are not attenuated enough. The product of the attenuatin and height of the airbox should be at least 50 dB. After the correct size for the airbox is calcualted and entered, the model with high attenuation can be removed from the Floquet port definition.

The 3D Refinement tab is used to control the inclusion of the modes in the 3D refinement of the mesh. It is recommended not to select them for the antenna arrays.

In our example, Figure 8 shows that the 5th mode has an attenuation of 2.59dB/length. The height of the airbox is around 19.5mm, providing 19.5mm*2.59dB/mm=50.505dB attenuation for the 5th mode. Therefore, only the first 4 modes are kept for the calculations. If the height of the airbox was less than 19.5mm, we would need to increase the height so accordingly for an attenuation of at least 50dB.

A simpler alternative for Floquet port is radiation boundary. It is important to note that the size of the airbox should still be kept around the same size that was calculated for the Floquet port, therefore, higher order modes sufficiently attenuated. In this case the traditional quarter wavelength padding might not be adequate.

Perfectly Matched Layer

Although using radiation boundary is much simpler than Floquet port, it is not accurate for large scan angles. It can be a good alternative to Floquet port only if the beam scanning is limited to small angles. Another alternative to Floquet port is to cover the cell by a layer of PML. This is a good compromise and provides very similar results to Floquet port models. However, the P/S boundary need to surround the PML layer as well, which means a few additional steps are required. Here is how you can do it:

1. Reduce the size of the airbox* slightly, so after adding the PML layer, the unit cell height is the same as the one that was generated using the Modes Calculation. (For example, in our model airbox height was 19mm+substrte thickness, the PML height was 3mm, so we reduced the airbox height to 16mm).
2. Choose the top face and add PML boundary.
3. Select each side of the airbox and create an object from that face (Figure 10).
4. Select each side of the PML and create objects from those faces (Figure 10).
5. Select the two faces that are on the same plane from the faces created from airbox and PML and unite them to create a side wall (Figure 10).
6. Then assign P/S boundary to each pair of walls (Figure 10).

*Please note for this method, an auto-size “region” cannot be used, instead draw a box for air/vacuum box. The region does not let you create the faces you need to combine with PML faces.

The advantage of PML termination over Floquet port is that it is simpler and sometimes faster calculation. The advantage over Radiation Boundary termination is that it provides accurate results for large scan angles. For better accuracy the mesh for the PML region can be defined as length based.

Seed the Mesh

To improve the accuracy of the PML model further, an option is to use length-based mesh. To do this select the PML box, from the project tree in Project Manager window right-click on Mesh->Assign Mesh Operation->On Selection->Length Based. Select a length smaller than lambda/10.

Scanning the Angle

In phased array simulation, we are mostly interested in the performance of the unit cell and array at different scan angles. To add the scanning option, the phase of P/S boundary should be defined by project or design parameters. The parameters can be used to run a parametric sweep, like the one shown in Figure 12. In this example the theta angle is scanned from 0 to 60 degrees.

Comparing PML and Floquet Port with Radiation Boundary

To see the accuracy of the radiation boundary vs. PML and Floquet Port, I ran the simulations for scan angles up to 60 degrees for a single element patch antenna. Figure 13 shows that the accuracy of the Radiation boundary drops after around 15 degrees scanning. However, PML and Floquet port show similar performance.

S Parameters

To compare the accuracy, we can also check the S parameters. Figure 14 shows the comparison of active S at port 1 for PML and Floquet port models. Active S parameters were used since the unit cell antenna has two ports. Figure 15 shows how S parameters compare for the model with the radiation boundary and the one with the Floquet port.

### Conclusion

The unit cell definition and options on terminating the cell were discussed here. Stay tuned. In the next blog we discuss how the unit cell is utilized in modeling antenna arrays.

## Defining Antenna Array Excitations with Nested-If Statements in HFSS

HFSS offers various methods to define array excitations. For a large array, you may take advantage of an option “Load from File” to load the magnitude and phase of each port. However, in many situations you may have specific cases of array excitation. For example, changing amplitude tapering or the phase variations that happens due to frequency change. In this blog we will look at using the “Edit Sources” method to change the magnitude and phase of each excitation. There are cases that might not be easily automated using a parametric sweep. If the array is relatively small and there are not many individual cases to examine you may set up the cases using “array parameters” and “nested-if”.

In the following example, I used nested-if statements to parameterize the excitations of the pre-built example “planar_flare_dipole_array”, which can be found by choosing File->Open Examples->HFSS->Antennas (Fig. 1) so you can follow along. The file was then saved as “planar_flare_dipole_array_if”. Then one project was copied to create two examples (Phase Variations, Amplitude Variations).

Fig. 1. Planar_flare_dipole_array with 5 antenna elements (HFSS pre-built example).

# Phase Variation for Selected Frequencies

In this example, I assumed there were three different frequencies that each had a set of coefficients for the phase shift. Therefore, three array parameters were created. Each array parameter has 5 elements, because the array has 5 excitations:

A1: [0, 0, 0, 0, 0]

A2: [0, 1, 2, 3, 4]

A3: [0, 2, 4, 6, 8]

Then 5 coefficients were created using a nested_if statement. “Freq” is one of built-in HFSS variables that refers to frequency. The simulation was setup for a discrete sweep of 3 frequencies (1.8, 1.9 and 2.0 GHz) (Fig. 2). The coefficients were defined as (Fig. 3):

E1: if(Freq==1.8GHz,A1[0],if(Freq==1.9GHz,A2[0],if(Freq==2.0GHz,A3[0],0)))

E2: if(Freq==1.8GHz,A1[1],if(Freq==1.9GHz,A2[1],if(Freq==2.0GHz,A3[1],0)))

E3: if(Freq==1.8GHz,A1[2],if(Freq==1.9GHz,A2[2],if(Freq==2.0GHz,A3[2],0)))

E4: if(Freq==1.8GHz,A1[3],if(Freq==1.9GHz,A2[3],if(Freq==2.0GHz,A3[3],0)))

E5: if(Freq==1.8GHz,A1[4],if(Freq==1.9GHz,A2[4],if(Freq==2.0GHz,A3[4],0)))

Please note that the last case is the default, so if frequency is none of the three frequencies that were given in the nested-if, the default phase coefficient is chosen (“0” in this case).

Fig. 2. Analysis Setup.

Fig. 3. Parameters definition for phase varaitioin case.

By selecting the menu item HFSS ->Fields->Edit Sources, I defined E1-E5 as coefficients for the phase shift. Note that phase_shift is a variable defined to control the phase, and E1-E5 are meant to be coefficients (Fig. 4):

Fig. 4. Edit sources using the defined variables.

The radiation pattern can now be plotted at each frequency for the phase shifts that were defined (A1 for 1.8 GHz, A2 for 1.9 GHz and A3 for 2.0 GHz) (Figs 5-6):

Fig. 5. Settings for radiation pattern plots.

Fig. 6. Radiation patten for phi=90 degrees and different frequencies, the variation of phase shifts shows how the main beam has shifted for each frequency.

# Amplitude Variation for Selected Cases

In the second example I created three cases that were controlled using the variable “CN”. CN is simply the case number with no units.

The variable definition was similar to the first case. I defined 3 array parameters and 5 coefficients. This time the coefficients were used for the Magnitude. The variable in the nested-if was CN. That means 3 cases and a default case were created. The default coefficient here was chosen as “1” (Figs. 7-8).

A1: [1, 1.5, 2, 1.5, 1]

A2: [1, 1, 1, 1, 1]

A3: [2, 1, 0, 1, 2]

E1: if(CN==1,A1[0],if(CN==2,A2[0],if(CN==3,A3[0],1)))*1W

E2: if(CN==1,A1[1],if(CN==2,A2[1],if(CN==3,A3[1],1)))*1W

E3: if(CN==1,A1[2],if(CN==2,A2[2],if(CN==3,A3[2],1)))*1W

E4: if(CN==1,A1[3],if(CN==2,A2[3],if(CN==3,A3[3],1)))*1W

E5: if(CN==1,A1[4],if(CN==2,A2[4],if(CN==3,A3[4],1)))*1W

Fig. 7. Parameters definition for amplitude varaitioin case.

Fig. 8. Exciation setting for amplitude variation case.

Notice that CN in the parametric definition has the value of “1”. To create the solution for all three cases I used a parametric sweep definition by selecting the menu item Optimetrics->Add->Parametric. In the Add/Edit Sweep I chose the variable “CN”, Start: 1, Stop:3, Step:1. Also, in the Options tab I chose to “Save Fields and Mesh” and “Copy geometrically equivalent meshes”, and “Solve with copied meshes only”. This selection helps not to redo the adaptive meshing as the geometry is not changed (Fig. 9). In plotting the patterns I could now choose the parameter CN and the results of plotting for CN=1, 2, and 3 is shown in Fig. 10. You can see how the tapering of amplitude has affected the side lobe level.

Fig. 9. Parameters definition for amplitude varaitioin case.

Fig. 10. Radiation patten for phi=90 degrees and different cases of amplitude tapering, the variation of amplitude tapering has caused chagne in the beamwidth and side lobe levels.

# Drawback

The drawback of this method is that array parameters are not post-processing variables. This means changing them will create the need to re-run the simulations. Therefore, it is needed that all the possible cases to be defined before running the simulation.

## Frequency Dependent Material Definition in ANSYS HFSS

Electromagnetic models, especially those covering a frequency bandwidth, require a frequency dependent definition of dielectric materials. Material definitions in ANSYS Electronics Desktop can include frequency dependent curves for use in tools such as HFSS and Q3D. However, there are 5 different models to choose from, so you may be asking: What’s the difference?

In this blog, I will cover each of the options in detail. At the end, I will also show how to activate the automatic setting for applying a frequency dependent model that satisfies the Kramers-Kronig conditions for causality and requires a single frequency definition.

# Background

Recalling that the dielectric properties of material are coming from the material’s polarization

where D is the electric flux density, E is the electric field intensity, and P is the polarization vector. The material polarization can be written as the convolution of a general dielectric response (pGDR) and the electric field intensity.

The dielectric polarization spectrum is characterized by three dispersion relaxation regions α, β, and γ for low (Hz), medium (KHz to MHz) and high frequencies (GHz and above). For example, in the case of human tissue, tissue permittivity increases and effective conductivity decreases with the increase in frequency [1].

Each of these regions can be modeled with a relaxation time constant

where τ is the relaxation time.

The well-known Debye expression can be found by use of spectral representation of complex permittivity (ε(ω)) and it is given as:

where ε is the permittivity at frequencies where ωτ>>1, εs is the permittivity at ωτ>>1, and j2=-1. The magnitude of the dispersion is ∆ε = εs.

The multiple pole Debye dispersion equation has also been used to characterize dispersive dielectric properties [2]

In particular, the complexity of the structure and composition of biological materials may cause that each dispersion region be broadened by multiple combinations. In that case a distribution parameter is introduced and the Debye model is modified to what is known as Cole-Cole model

where αn, the distribution parameter, is a measure of broadening of the dispersion.

Gabriel et. al [3] measured a number of human tissues in the range of 10 Hz – 100 GHz at the body temperature (37℃). This data is freely available to the public by IFAC [4].

# Frequency Dependent Material Definition in HFSS and Q3D

In HFSS you can assign conductivity either directly as bulk conductivity, or as a loss tangent. This provides flexibility, but you should only provide the loss once. The solver uses the loss values just as they are entered.

To define a user-defined material choose Tools->Edit Libraries->Materials (Fig. 2). In Edit Libraries window either find your material from the library or choose “Add Material”.

To add frequency dependence information, choose “Set Frequency Dependency” from the “View/Edit Material” window, this will open “Frequency Dependent Material Setup Option” that provides five different ways of defining materials properties (Fig. 3).

Before choosing a method of defining the material please note [5]:

• The Piecewise Linear and Frequency Dependent Data Points models apply to both the electric and magnetic properties of the material. However, they do not guarantee that the material satisfies causality conditions, and so they should only be used for frequency-domain applications.
• The Debye, Multipole Debye and Djordjevic-Sarkar models apply only to the electrical properties of dielectric materials. These models satisfy the Kramers-Kronig conditions for causality, and so are preferred for applications (such as TDR or Full-Wave SPICE) where time-domain results are needed. They also include an automatic Djordjevic-Sarkar model to ensure causal solutions when solving frequency sweeps for simple constant material properties.
• HFSS and Q3D can interpolate the property’s values at the desired frequencies during solution generation.

Piecewise Linear

This option is the simplest way to define frequency dependence. It divides the frequency band into three regions. Therefore, two frequencies are needed as input. Lower Frequency and Upper Frequency, and for each frequency Relative Permittivity, Relative Permeability, Dielectric Loss Tangent, and Magnetic Loss Tangent are entered as the input. Between these corner frequencies, both HFSS and Q3D linearly interpolate the material properties; above and below the corner frequencies, HFSS and Q3D extrapolate the property values as constants (Fig. 4).

Once these values are entered, 4 different data sets are created (\$ds_epsr1, \$ds_mur1, \$ds_tande1, \$ds_tandm1). These data sets now can be edited. To do so choose Project ->Data sets, and choose the data set you like to edit and click Edit (Fig. 5). This data set can be modified with additional points if desired (Fig. 6).

Frequency Dependent

Frequency Dependent material definition is similar to Piecewise Linear method, with one difference. After selecting this option, Enter Frequency Dependent Data Point opens that gives the user the option to use which material property is defined as a dataset, and for each one of them a dataset should be defined. The datasets can be defined ahead of time or on-the-fly. Any number of data points may be entered. There is also the option of importing or editing frequency dependent data sets for each material property (Fig. 7).

Djordjevic-Sarkar

This model was developed initially for FR-4, commonly used in printed circuit boards and packages [6]. In fact, it uses an infinite distribution of poles to model the frequency response, and in particular the nearly constant loss tangent, of these materials.

where ε is the permittivity at very high frequency,  is the conductivity at low (DC) frequency,  j2=-1, ωA is the lower angular frequency (below this frequency permittivity approaches its DC value), ωB is the upper angular frequency (above this frequency permittivity quickly approaches its high-frequency permittivity). The magnitude of the dispersion is ∆ε = εs-ε∞.

Both HFSS and Q3D allow the user to enter the relative permittivity and loss tangent at a single measurement frequency. The relative permittivity and conductivity at DC may optionally be entered. Writing permittivity in the form of complex permittivity [7]

Therefore, at the measurement frequency one can separate real and imaginary parts

where

Therefore, the parameters of Djordjevic-Sarkar can be extracted, if the DC conductivity is known

If DC conductivity is not known, then a heuristic approximation is De = 10 εtan δ1.

The window shown in Fig. 8 is to enter the measurement values.

Debye Model

As explained in the background section single pole Debye model is a good approximation of lossy dispersive dielectric materials within a limited range of frequency. In some materials, up to about a 10 GHz limit, ion and dipole polarization dominate and a single pole Debye model is adequate.

The Debye parameters can be calculated from the two measurements [7]

Both HFSS and Q3D allow you to specify upper and lower measurement frequencies, and the loss tangent and relative permittivity values at these frequencies. You may optionally enter the permittivity at high frequency, the DC conductivity, and a constant relative permeability (Fig. 9).

Multipole Debye Model

For Multipole Debye Model multiple frequency measurements are required. The input window provides entry points for the data of relative permittivity and loss tangent versus frequency. Based on this data the software dynamically generates frequency dependent expressions for relative permittivity and loss tangent through the Multipole Debye Model. The input dialog plots these expressions together with your input data through the linear interpolations (Fig. 10).

## Cole Cole Material Model

The Cole Cole Model is not an option in the material definition, however, it is possible to generate the frequency dependent datasets and use Frequency Dependent option to upload these values. In fact ANSYS Human Body Models are built based on the data from IFAC database and Frequency Dependent option.

## Visualization

Frequency-dependent properties can be plotted in a few different ways. In View/Edit Material dialog right-click and choose View Property vs. Frequency. In addition, the dialogs for each of the frequency dependent material setup options contain plots displaying frequency dependence of the properties.

You can also double-click the material property name to view the plot.

## Automatically use causal materials

As mentioned at the beginning, there is a simple automatic method for applying a frequency dependent model in HFSS. Select the menu item HFSS->Design Setting, and check the box next to Automatically use casual materials under Lossy Dielectrics tab.

This option will automatically apply the Djordjevic-Sarkar model described above to objects with constant material permittivity greater than 1 and dielectric loss tangent greater than 0. Keep in mind, not only is this feature simple to use, but the Djordjevic-Sarkar model satisfies the Kramers-Kronig conditions for causality which is particularly preferred for wideband applications and where time-domain results will also be needed. Please note that if the assigned material is already frequency dependent, automatic creation of frequency dependent lossy materials is ignored.

# References

• D.T. Price, MEMS and electrical impedance spectroscopy (EIS) for non-invasive measurement of cells, in MEMS for Biomedical Applications, 2012, https://www.sciencedirect.com/topics/materials-science/electrical-impedance
• W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,” IEEE Trans. Biomed. Eng, vol. 32, pp. 60-64, 1985.
• S. Gabriel, R. W. Lau, and C. Gabriel. “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues.” Physics in Medicine & Biology, vol. 41, no. 11, pp. 2271, 1996.
• Dielectric Properties of Body Tissues in the Frequency Range 10 Hz – 100 GHz, http://niremf.ifac.cnr.it/tissprop/.
• A. R. Djordjevic, R. D. Biljic, V. D. Likar-Smiljani, and T. K. Sarkar, “Wideband frequency-domain characterization of FR-4 and time-domain causality,” IEEE Trans. on Electromagnetic Compatibility, vol. 43, no. 4, p. 662-667, Nov. 2001.

Piecewise Linear Input

Debye Model Input

Multipole Debye Model Input

Djordjevic-Sarkar

Enter Frequency Dependent Data Points

## “Equation Based Surface” for Conformal and Non-Planar Antenna Design

ANYSY HFSS provides many options for creating non-planar and conformal shapes. In MCAD you may use shapes such as cylinders or spheres, and with some steps, you can design you antennas on various surfaces. In some applications, it is necessary to study the effect of curvatures and shapes on the antenna performance. For example for wearable antennas it is important to study the effect of bending, crumpling and air-gap between antenna and human body.

## Equation Based Surface

One of the tools that HFSS offers and can be used to do parametric sweep or optimization, is “Draw equation based surface”. This can be accessed under “Draw” “Equation Based Surface” or by using “Draw” tab and choosing it from the banner (Fig. 1)

Once this is selected the Equation Based Surface window that opens gives you options to enter the equation with the two variables (_u, _v_) to define a surface. Each point of the surface can be a function of (_u,_v). The range of (_u, _v) will also be determined in this window. The types of functions that are available can be seen in “Edit Equation” window, by clicking on “…” next to X, Y or Z (Fig. 2). Alternatively, the equation can be typed inside this window. Project or Design Variables can also be used or introduced here.

For example an elliptical cylinder along y axis can be represented by:

This equation can be entered as shown in Fig. 3.

Variation of this equation can be obtained by changing variables R1, R2, L and beta. Two examples are shown in Fig. 4.

## Application of Equation Based Surface in Conformal and Non-Planar Antennas

To make use of this function to transfer a planar design to a non-planar design of interest, the following steps can be taken:

• Start with a planar design. Keep in mind that changing the surface shape can change the characteristics of the antenna. It is a good idea to use a parameterized model, to be able to change and optimize the dimensions after transferring the design on a non-planar surface. As an example we started with a planar meandered line antenna that works around 700MHz, as shown in Fig. 5. The model is excited by a wave port. Since the cylindrical surface will be built around y-axis, the model is transferred to a height to allow the substrate surface to be made (Fig 5. b)
• Next, using equation based surface, create the desired shape and with the same length as the planar substrate. Make sure that the original deisgn is at a higher location. Select the non-planar surface. Use Modeler->Surface->Thicken Sheet … and thicken the surface with the substrate thickenss. Alternatively, by choosing “Draw” tab, one can expand the Sheet dropdown menu and choose Thicken Sheet. Now select the sheet, change the material to the substrate material.
• At this point you are ready to transfer the antenna design to the curved surface. Select both traces of the antenna and the curved substrate (as shown in Fig. 7). Then use Modeler->Surface->Project Sheet…, this will transfer the traces to the curved surface. Please note that the original substrate is still remaining. You need not delete it.
• Next step is to generate the ground plane and move the wave port. In our example design we have a partial ground plane. For ground plane surface we use the same method to generate an equation based surface. Please keep in mind that the Z coordinate of this surface should be the same as substrate minus the thickness of the substrate. (If you thickened the substrate surface to both sides, this should be the height of substrate minus half of the substrate thickness). Once this sheet is generate assign a Perfect E or Finite Conductivity Boundary (by selecting the surface, right click and Assign Boundary). Delete the old planar ground plane.

## Wave Port Placement using Equation Based Curve

A new wave port can be defined by the following steps:

• Delete the old port.
• Use Draw->Equation Based Curve. Mimicking the equation used for ground plane (Fig. 9).
• Select the line from the Model tree, select Draw->Sweep->Along Vector. Draw a vector in the direction of port height. Then by selecting the SweepAlongVector from Model tree and double clicking, the window allows you to set the correct size of port height and vector start point (Fig. 10).
• Assign wave port to this new surface.

Similar method can be used to generate (sin)^n or (cos)^n surfaces. Some examples are shown in Fig. 11. Fig. 11 (a) shows how the surface was defined.

## Effect of Curvature on Antenna Matching

Bending a substrate can change the transmission line and antenna impedance. By using equation based port the change in transmission line impedance effect is removed. However, the overall radiation surface is also changed that will have effects on S11. The results of S11 for the planar design, cylindrical design (Fig. 8), cos (Fig. 11 b), and cos^3 (Fig. 11 c) designs are shown in Fig. 12. If it is of interest to include the change in the transmission line impedance, the port should be kept in a rectangular shape.

Equation based curves and surfaces can take a bit of time to get used to but with a little practice these methods can really open the door to some sophisticated geometry. It is also interesting to see how much the geometry can impact a simple antenna design, especially with today’s growing popularity in flex circuitry. Be sure to check out this related webinar  that touches on the impact of packaging antennas as well. If you would like more information on how these tools may be able to help you and your design, please let us know at info@padtinc.com.