Djordjevic-sarkar Model

[Raguel Charrette]

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

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].

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

In HFSS you can assign conductivity either directly as bulkconductivity, or as a loss tangent. This provides flexibility, but you shouldonly provide the loss once. The solver uses the loss values just as they areentered.

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).

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

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

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]

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.

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).

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).

The Cole Cole Model is not an option in the materialdefinition, however, it is possible to generate the frequency dependentdatasets and use Frequency Dependent option to upload these values. In fact ANSYSHuman Body Models are built based on the data from IFACdatabase and Frequency Dependent option.

Frequency-dependent properties can be plotted in a fewdifferent ways. In View/Edit Material dialog right-click andchoose View Property vs. Frequency. In addition, the dialogs for each ofthe frequency dependent material setup options contain plots displayingfrequency dependence of the properties.

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-Sarkarmodel described above to objects with constant material permittivity greaterthan 1 and dielectric loss tangent greater than 0. Keep in mind, not only isthis feature simple to use, but the Djordjevic-Sarkar model satisfies theKramers-Kronig conditions for causality which is particularly preferred for widebandapplications and where time-domain results will also be needed. Please notethat if the assigned material is already frequency dependent,automatic creation of frequency dependent lossy materials is ignored.

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Understanding the electrical properties of dielectrics is one of the fundamental thing in Electromagnetics and especially in Signal Integrity. In the initial days of my career, I was always pondering why should dielectric permittivity (εr) vary over the frequency. Before we discuss on the material properties further more and understand about Debye model and Sarkar models and their usage, let us understand few critical parameters that define these material parameters. Honestly, I don't know how to start this topic but let us go with the below discussion. How many times have you heard whether your material is causal or your system is causal ?

Causal System: In physical systems, the output cannot react to an input before the input has happened. It is like expecting that you reached a destination before you even drove the car/bike. It is a relationship between cause and effects. You honk in the car and then you should hear the sound. You should not listen to sound before you even honk.

Another instance that I struggled is with the calibration traces. In any 3D simulation tool, for extracting the performance of calibration traces, we mount the SMA connector and run the simulation. However in the reality you may see calibration trace going haywire due to various reasons. In one of our experiment, we wanted to run the below mentioned nominal channel. The simulated channel performance was casual until 80 GHz. However, when we measured the channel performance, we started experiencing issues after 50 GHz. ISI and channel operating margin were significantly up. Notice in the IL performance, how suddenly the curve shoots up causing causality issues.

Check the de-embedding nature of the file. De-embedding is the most common reason for non causal S parameters. What's the big deal if the S-parameters are non causal ? You don't want to have false performance degradation. Another quick fix to make sure is that all dielectric material properties are frequency dependent and causal. In 3D simulations, make sure to adopt to some of the frequency dependent models available in tools and define the dielectric permittivity and loss tangent using those models. This is again an egg and chicken story. You don't simulate, you don't manufacture and when you don't manufacture, you don't encounter these issues. Summary: In a nut shell, make sure your system is causal and passive.

We answered above why we need our system to be causal and now we will discuss what hampers the dielectric material properties. Have you ever asked why should the dielectric permittivity and loss tangent vary over the frequency ? Let me explain the analysis in a gist.

Take a look at Figure 3: Without an applied electric field, the electron clouds around the nuclei are symmetric and at rest. When a field is applied, the EM wave pushes the electrons away from the nuclei producing clouds that are offset and the motion of charges emits secondary waves that interfere with the applied wave to produce a slowing effect on the wave.

In electromagnetism, the electric susceptibility is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field. The greater the electric susceptibility, the greater the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material(and store energy).

He described the problem of atom-field interactions in these terms. Lorentz thought of an atom as a mass ( the nucleus ) connected to another smaller mass ( the electron ) by a spring. The spring would be set into motion by an electric field interacting with the charge of the electron. The field would either repel or attract the electron which would result in either compressing or stretching the spring. Lorentz was not positing the existence of a physical spring connecting the electron to an atom; however, he did postulate that the force binding the two could be described by Hooke's Law: where y is the displacement from equilibrium. If Lorentz's system comes into contact with an electric field, then the electron will simply be displaced from equilibrium. The oscillating electric field of the electromagnetic wave will set the electron into harmonic motion. The effect of the magnetic field can be omitted because it is miniscule compared to the electric field.

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