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

(1)

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.

(2)

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].

Fig. 1. α, β and γ regions of dielectric permittivity

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

(3)

where τ is the relaxation time.

(4)

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

(5)
(6)

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]

(7)

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

(8)

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”.

Fig. 2. Edit Libraries screen shot.

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).

Fig. 3. (Left) View/Edit Material window, (Right) Frequency Dependent Material Setup Option.

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).

Fig. 4. Piecewise Linear Frequency Dependent Material Input window.

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).

Fig. 5. (Left) Project data set selection, (right) defined data set for the material.
Fig. 6. A sample data set.

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).

Fig. 7. This window provides options of choosing which material property is frequency dependent and enter the data set associated with it.

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.

(9)

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]

(10)
(11)

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

(12)
(13)

where

(14)

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

(15)

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.

Fig. 8. The required values to calculate permittivity using Djordjevic-Sarkar model.

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.

(16)
(17)
(18)
(19)
(20)

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

(21)

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).

Fig. 9. The required values for Single Pole Debye model.

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).

Fig. 10. The required values for Multipole Debye model.

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.

Fig. 11. Causal material can be enforced in HFSS Design Settings.

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.

If you would like more information or have any questions about ANSYS products please email info@padtinc.com

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/.
  • ANSYS HFSS Online Help, Nov. 2013, Assigning Materials.
  • 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.
  • ANSYS HFSS Online Help, 2019, Materials Technical Notes.

Useful Links

Piecewise Linear Input

Debye Model Input

Multipole Debye Model Input

Djordjevic-Sarkar

Enter Frequency Dependent Data Points

Modifying Datasets.

“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)

Fig. 1. (a) Select Draw -> Equation Based Surface
Fig. 1. (b) click on the icon that is highlighted

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.

Fig. 2. (a) Equation Based Surface window
Fig. 2. (b) Clikc on the “…” next to X and see the “Edit Equation: window to build the equation for X

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

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

Fig. 3. Elliptical surface equation

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

Fig. 4. Elliptical surface equation

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)
Fig. 5. Planar meandered antenna (a) on xy plane, (b) moved to a height of 5cm
  • 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.
Fig. 6. Thicken the equation based surface to generate the substrate
  • 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.
Fig. 7. Steps for transferring the design to the curved surface (a)

Fig. 7. Steps for transferring the design to the curved surface (b)

Fig. 7. Steps for transferring the design to the curved surface (c)
  • 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.
Fig. 8. Non-planar meandered antenna with non-planar ground

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).
Fig. 9. Use Equation Based Curve to start a new wave port (a) Equation Based Curve definition window (b) wave pot terminal created using equation based curve and sweep along vector
  • 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.
Fig. 10. Sweep along vector to create the new wave port location

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.

Fig. 11. (a) Equation based surface definition using “cos” function, (b), (c), & (d) three different surfaces generated by this equation based surface.

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.

Fig. 12. Effect of curvature on the resonance frequency.

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.

You can also click here to download a copy of this example.