# A Fractional Complex Permittivity Model of Media with Dielectric Relaxation

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## Abstract

**:**

## 1. Introduction

## 2. Fractional Generalized Debye’s Equation

## 3. The Fractional Model

## 4. Numerical Results

- Step 1
- one chooses a frequency range (${\omega}_{min},{\omega}_{max}$) and a test value for $\alpha $;one read the correspondent permittivity experimental values:${\u03f5}^{{}^{\prime}\left(\alpha \right)}\left({\omega}_{min}\right)$, ${\u03f5}^{{}^{\u2033}\left(\alpha \right)}\left({\omega}_{min}\right)$, ${\u03f5}^{{}^{\prime}\left(\alpha \right)}\left({\omega}_{max}\right)$, ${\u03f5}^{{}^{\u2033}\left(\alpha \right)}\left({\omega}_{max}\right)$;one initializes $n=0$ and $m=0$.
- Step 2
- one resolves the system at frequencies ${\omega}_{min}$ and ${\omega}_{max}$.
- Step 3
- If there is a real and positive solution, then if m is not null go to end; otherwise, one puts:$n=n+1$; $m=0$; ${\alpha}_{n}=\alpha $; ${\left({C}_{0}\right)}_{n}={C}_{0}$; ${\left({F}_{0}\right)}_{n}={F}_{0}$; ${\left({C}_{1}\right)}_{n}={C}_{1}$; ${\left({F}_{1}\right)}_{n}={F}_{1}$;it reduces $\alpha $ = $\alpha $ - 0.001 and go back to step 2.
- Step 4
- If n is null, one puts ${\alpha}_{n}=\alpha $ and $\alpha $ = $\alpha $ + 0.001 and go back to step 2; otherwise, one puts $m=m+1$, $n=0$ and goes back to step 2.
- End
- The solution so determined is compared with the predictive permittivity model in [10], at a temperature of 37 °C with reference to the human aorta. This method is equally applicable to biological tissues. From Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 (horizontal axis rad/s), we observe that a predictive fractional model of the complex permittivity is in accordance with experimental data with good approximation. The experimental data are those relating to measure campaign published in [10]. In particular from Figure 9, we see that percentage error relative permittivity and conductivity to experimental data is, respectively, almost always lower and thorough than that of the Ciancio–Kluitenberg model and Cole–Cole extended model [7,8,9,10]. In the frequency range 2.5 × 10${}^{3}$ @ 9.29 × 10${}^{10}$, the maximum relative error to experimental data of permittivity fractional model is −21% at 1.58 × 10${}^{5}$ rad/s; of Ciancio–Kluitenberg’s model is +25% at 1.58 × 10${}^{7}$ rad/s and extended of Cole–Cole’s model is +35% at 1.58 × 10${}^{7}$ rad/s; in the same frequency range, the maximum relative error to experimental data of conductivity fractional model is +15% at 1.95 × 10${}^{10}$ rad/s; of Ciancio–Kluitenberg’s model is +41% at 1.95 × 10${}^{10}$ rad/s and extended of Cole–Cole’s model is −37% at 2.5 × 10${}^{3}$ rad/s. In Figure 10, we show the trend of fractional order with respect to the frequency.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Permittivity and (

**b**) conductivity at frequencies 100 Hz @ 1 KHz. Line dot-dashed fractional model $\alpha =0.983$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 3.**(

**a**) Permittivity and (

**b**) conductivity at frequencies 1 KHz @ 9 KHz. Line dot-dashed fractional model $\alpha =0.985$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 4.**(

**a**) Permittivity and (

**b**) conductivity at frequencies 9 KHz @ 224 KHz. Line dot-dashed fractional model $\alpha =0.990$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 5.**(

**a**) Permittivity and (

**b**) conductivity at frequencies $224\mathrm{KHz}\text{}@\text{}1\mathrm{MHz}$. Line dot-dashed fractional model $\alpha =1$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 6.**(

**a**) Permittivity and (

**b**) conductivity at frequencies 1 MHz @ 10 MHz. Line dot-dashed fractional model $\alpha =0.975$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 7.**(

**a**) Permittivity and (

**b**) conductivity at frequencies 10 MHz @ 100 MHz. Line dot-dashed fractional model $\alpha =0.936$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 8.**(

**a**) Permittivity and (

**b**) conductivity 100 MHz @ 20 GHz. Line dot-dashed fractional model $\alpha =0.983$, line continue ordinary model, line dotted extensive Cole–Cole’s model, points experimental data.

**Figure 9.**Relative percentage error (

**a**) permittivity and (

**b**) conductivity. Line dot-dashed fractional model, line continue Ciancio–Kluitenberg model, line dotted extensive Cole–Cole’s model, points experimental data.

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**MDPI and ACS Style**

Ciancio, A.; Flora, B.F.F.
A Fractional Complex Permittivity Model of Media with Dielectric Relaxation. *Fractal Fract.* **2017**, *1*, 4.
https://doi.org/10.3390/fractalfract1010004

**AMA Style**

Ciancio A, Flora BFF.
A Fractional Complex Permittivity Model of Media with Dielectric Relaxation. *Fractal and Fractional*. 2017; 1(1):4.
https://doi.org/10.3390/fractalfract1010004

**Chicago/Turabian Style**

Ciancio, Armando, and Bruno Felice Filippo Flora.
2017. "A Fractional Complex Permittivity Model of Media with Dielectric Relaxation" *Fractal and Fractional* 1, no. 1: 4.
https://doi.org/10.3390/fractalfract1010004