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 (*p _{GDR}*) 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, ε_{s }is the permittivity at ωτ>>1, and *j ^{2}*=-1. The magnitude of the dispersion is ∆ε = ε

_{s}-ε

_{∞}.

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 α_{n}, 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, *j ^{2}*=-1, ω

_{A}is the lower angular frequency (below this frequency permittivity approaches its DC value), ω

_{B}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 ε_{∞ }tan δ_{1}.

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.

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.