Variation of the relative permittivity

In this part, the analytical approach is detailed. According to the ratio of resonance method, the electromagnetic simulations, carried out by CST Microwave Studio (CST MWS), one of the 3D CAD software used in our case [Reference Barakat J Moussa15], aim to determine the effective permittivity with respect to the relative permittivity for a given thickness of the substrate.

Based on the permittivity graphs, a setup of a model structure for the trend curves linking the effective permittivity ɛ_reff and the relative permittivity ɛ_r is proposed.

The trend curves of the ɛ_reff with respect to ɛ_r are illustrated in Fig. 4. The variation of the square root of the effective permittivity $\sqrt {\varepsilon _{reff}}$ is a power function with respect to relative permittivity ɛ_r. The equivalent equation has the following form:

(5)$$\sqrt {\varepsilon _{reff}} = a\ast ( {\varepsilon_r} ) ^b. $$

Fig. 4.

Effective permittivity with respect to relative permittivity for different height of substrate.

From the obtained value of the effective permittivity, the objective is to acquire a regression model with a given adjustment function. From Fig. 4, the regression is a power function. In Table 1, we show the different coefficients a _h and b _h from equation (5). In each regression, the coefficient of determination R ², which is a statistical measure of how close the data are to the fitted regression line, is shown. A value of 100% indicates that the model explains all the variability of the response data around its mean value. The value of R ² is close to one which validates our assumption for the regression model.

Table 1.

Power model coefficients for different substrate thickness.

Consequently, the a and b parameters are determined by taking the average of the coefficients a _h and b _h found from each curve in Fig. 4. Finally, the equation of regression of the effective permittivity is:

(6)$$\sqrt {\varepsilon _{reff}} = 1.222\ast ( {\varepsilon_r} ) ^{0.53}$$

The standard deviation for the coefficients a _h and b _h is 3%, which allows us to say that this averaging is acceptable.

Variation of the substrate thickness

After finding the first equation of the effective permittivity with respect to the relative permittivity, the second step is to determine the regression of the effective permittivity with respect to the relative substrate thickness.

Electromagnetic simulations have been conducted using CST MWS: [Reference Barakat J Moussa15]. Based on the ratio of the resonance method, the regression of the effective permittivity with respect to the relative height of the substrate is studied. Thus, the trend curves of $\sqrt {\varepsilon _{reff}}$ with respect to the relative height of the substrate h/d for different relative permittivity ɛ_r have been plotted Fig. 5. The dielectric permittivity could be modelled as quadratic polynomial [Reference Tendeku and Misra16, Reference Ghalichechian and Sertel17]. In addition, the trend curves show a quadratic regression, hence the goal is determining the coefficients of the variation $\sqrt {\varepsilon _{reff}}$ with respect to the relative height h/d. Note that a dipole of a fixed length has been used. In addition, the extraction of the effective permittivity is achieved by using the concept of equation (4)

Fig. 5.

Effective permittivity with respect to relative height for different relative permittivity.

Finally, the form of the quadratic regression will have the following polynomial equation:

(7)$$\sqrt {\varepsilon _{reff}} = a_e( {\;h/d} ) ^2 + b_e( h/d) + c_e. $$

Following the same regression approach, the coefficients a _e, b _e, c _e, are determined from the trend curves relating the effective permittivity and the relative height.

As per Table 2, the majority of the coefficient of determination R ² is close to one, which shows that this quadratic regression is close to the studied data.

Table 2.

Coefficients of the regression for different permittivity.

The next step is to introduce the notion of permittivity into these equations. According to equations (6) and (7), a new form of the quadratic equation is offered by introducing the function of relative permittivity. The coefficients a _e, b _e, c _e from Table 2, are divided by 1.222*(ɛ_r)^0.53 for each value of permittivity to obtain the values of a _es, b _es, c _es in Table 3.

Table 3.

Coefficients of the regression for different permittivity.

Finally, the averaging of the coefficients a _es, b _es, c _es is done to get the final coefficients. Table 4 presents the mean and standard deviation of these coefficients.

Table 4.

Mean and standard deviation for the coefficients.

The final equation of the effective permittivity with respect to the relative permittivity and the relative height which is the dipole length in the air over the substrate thickness is:

(8)$$\sqrt {\varepsilon _{reff}} = 1.222\ast ( \varepsilon _r) ^{0.53}\ast ( {-}0.33( h/d) ^2 + 0.465( h/d) + 0.536).$$

Numerical validation

After the determination of equation (8) linking the permittivity and the relative height, it is convenient to compare the values of the effective permittivity using the analytical equation and the values obtained by electromagnetic simulation.

Figure 6 shows the difference between the values obtained from electromagnetic simulation extracted from CST and the analytic values from equation (8). As shown in the graphs, for different values of relative permittivity, the relative error is in the 5–10% range except for ɛ_r = 2 where it is 14% at a relative height h/d = 0.1. The idea is to have a preliminary estimation of the effective permittivity that helps the designer. The designer should start with these values and confirm by electromagnetic simulation of the final design of the antenna for example. In order to validate our design, we have compared our analytic model to the one presented in [Reference Rialet, Sharaiha, Tarot and Delaveaud14], for a dipole operating at 6 GHz printed on a substrate of height h = 350 μm, and with relative permittivity ɛ_r = 6, the effective permittivity in our case is 3.55 with respect to 3.5 from the analytic model of [Reference Rialet, Sharaiha, Tarot and Delaveaud14].

Fig. 6.

Numerical validation of the effective permittivity.

Variation of the relative permittivity

Different expressions, describing the equivalent substrate model of multilayer structure, exist in the literature, among which two could be stated: expression developed by Lee et al.: [Reference Lee, Ho and Dahele18] and Krasweski [Reference Kraszewski19]

The expression developed by Krasweski: [Reference Kraszewski19] is (Fig. 8):

(9)$$\varepsilon _{multi} = [ {\varepsilon_h^{1/2} + v_i( \varepsilon_i^{1/2} -\varepsilon_h^{1/2} ) } ] ^2, \;$$

where v _i is the fractional volume of the layer i and where ɛ_h and ɛ_i are the relative permittivity of the host medium and inclusions. This equation is symmetrical and does not involve the form of structures contrary to many formulas in the literature: [Reference Pavageau20] that is why it is used to determine the equivalent permittivity: [Reference Fu, Vuong, Dussopt and Ndagijimana21].

Fig. 8.

Krasweski model.

The fractional volume of the layer i is calculated by dividing the volume of the layer i over the total volume, where s is the substrate surface

(10)$$v_i = \displaystyle{{{\rm volume\;of\;layer\;}i} \over {{\rm total\;volume}}} = \displaystyle{{s{\rm \ast }h_i} \over {s{\rm \ast }h_i + s{\rm \ast }h_h}} = \displaystyle{{h_i} \over {h_i + h_h}}. $$

Expression of the equivalent permittivity

Subsequently, this expression is used to determine the parameters ɛ_multi and h _eq of the equivalent SOI substrate. This preliminary study could help the designer to start the design. Other parameters could be added when doing the design using the 3D CAD software in order to have the effect of the substrate.

From the information about the substrate, we have used Krasweski model (equations (9) and (10)) to get equivalent parameters of the equivalent substrate presented in Table 5.

Table 5.

Equivalent height and relative permittivity of SOI substrate.

According to Table 5, the equivalent relative permittivity of the SOI substrate is 11.71, with an equivalent height of 357.51 μm.

By applying equation (8), the effective permittivity of the SOI substrate can be determined. Thus, the value of effective permittivity is$\;\sqrt {\varepsilon _{reff}} = 2.516$, (ɛ_reff = 6.33). Bunea et al. [Reference Bunea, Neculoiu and Muller22] have also developed an analytic equation for the effective permittivity using CPW and has considered a high resistive silicon substrate, the analytic result (ɛ_reff = 6.30) is comparable to our obtained values. This study is a step towards the determination of the effective permittivity for dipole antennas integrated on SOI substrate. An estimate of the length of a dipole antenna, operating at 60 GHz and an application of equations (1) and (3) gives us an antenna of length 937 μm. The obtained results were the basis of other design of antenna on high SOI substrate: [Reference Fu, Vuong, Dussopt and Ndagijimana21, Reference Barakat, Delaveaud and Ndagijimana23, Reference Barakat, Delaveaud and Ndagijimana24]. The layers near the antenna are electromagnetically more influential on the parameters of the antenna, but their dimensions are very small compared to the silicon layer, leading to a limited influence on the equivalent relative permittivity.

We have used the same procedure for the design of a 60 GHz fully integrated dipole antenna, incorporating an interdigitated structure, and realized on 0.13 μm SOI process from STMicroelectronics, with the following parameters: substrate dimensions are 1200 *2400 μm², dipole's length = 871 μm (Fig. 9) [Reference Barakat, Delaveaud and Ndagijimana23]. The difference in the size between the initial estimated length (937 μm) of the dipole and the realized one (871 μm) was in the order of 7%. These results have helped to reduce the optimization process when we have designed other antennas [Reference Fu, Vuong, Dussopt and Ndagijimana21,Reference Barakat, Delaveaud and Ndagijimana23,Reference Barakat, Delaveaud and Ndagijimana24].

Fig. 9.

Schematic of the dipole with balun on SOI [Reference Barakat, Delaveaud and Ndagijimana23].