$-\frac{\delta \text{\hspace{0.05em}}\Omega}{\Omega}\approx \frac{\left({\stackrel{\xaf}{\epsilon}}_{r}-1\right){\epsilon}_{0}{\displaystyle \underset{{V}_{s}}{\int}E\cdot {E}_{0}^{\ast}\text{d}V+\left({\stackrel{\xaf}{\mu}}_{r}-1\right){\mu}_{0}{\displaystyle \underset{{V}_{s}}{\int}H\cdot {H}_{0}^{\ast}}\text{d}V}}{{\displaystyle \underset{{V}_{c}}{\int}\left({D}_{0}\cdot {E}_{0}^{\ast}+{B}_{0}\cdot {H}_{0}^{\ast}\right)\text{d}V}}$ (2)

Two approximations are made in applying Equation (2), based on the assumption that fields in the empty part of the cavity are negligibly changed by the insertion of the sample and that the fields in the sample are uniform over its volume. Both these assumptions can be considered valid if the object is sufficiently small relative to the resonant wavelength. The negative sign in Equation (2) indicates that by introducing the sample the resonant frequency is lowered. Because the permittivity of practical materials is complex, the resonant frequency should also be considered as complex. In Equation (2), the $\text{d}\Omega $ is the complex frequency shift. ${B}_{0}$ , ${H}_{0}$ , ${D}_{0}$ and ${E}_{0}$ are the fields in the unperturbed cavity. $E$ and $H$ are the fields in the interior of the sample. ${\stackrel{\xaf}{\epsilon}}_{r}={{\stackrel{\xaf}{\epsilon}}^{\prime}}_{r}-j{{\stackrel{\xaf}{\epsilon}}^{\u2033}}_{r}$ and ${\stackrel{\xaf}{\mu}}_{r}={{\stackrel{\xaf}{\mu}}^{\prime}}_{r}-j{{\stackrel{\xaf}{\mu}}^{\u2033}}_{r}$ and ${V}_{c}$ and ${V}_{s}$ are the volumes of the cavity and sample respectively. In terms of energy, the numerator of Equation (2) represents the energy stored in the sample and the denominator represents the total energy stored in the cavity. The total energy $W={W}_{e}+{W}_{m}=2{W}_{e}=2{W}_{m}$ . When a dielectric sample is introduced at the position of maximum electric field only the first term in the numerator is significant, since a small change in $\epsilon $ at a point of zero electric field or a small change in $\mu $ at a point of zero magnetic field does not change the resonance frequency. Thus Equation (2) can be reduced to

$-\frac{\delta \text{\hspace{0.05em}}\Omega}{\Omega}\approx \frac{\left({\stackrel{\xaf}{\epsilon}}_{r}-1\right){\displaystyle \underset{{V}_{s}}{\int}E\cdot {E}_{0}^{\ast}{}_{\mathrm{max}}\text{d}V}}{2{\displaystyle \underset{{V}_{c}}{\int}{\left|{E}_{0}\right|}^{2}}\text{d}V}$ (3)

Let ${Q}_{0}$ be the quality factor of the cavity in the unperturbed condition and ${Q}_{s}$ the Q-factor of the cavity loaded with the object. The complex frequency shift is related to measurable quantities by [18]

$\frac{\delta \text{\hspace{0.05em}}\Omega}{\Omega}\approx \frac{\delta \omega}{\omega}+\frac{j}{2}\left[\frac{1}{{Q}_{s}}-\frac{1}{{Q}_{0}}\right]$ (4)

Equating the real and imaginary terms of Equations (3) and (4) we get

$-\frac{{f}_{s}-{f}_{0}}{{f}_{s}}=\frac{\left({{\epsilon}^{\prime}}_{r}-1\right){\displaystyle \underset{{V}_{s}}{\int}E\cdot {E}_{0}^{\ast}{}_{\mathrm{max}}\text{d}V}}{2{\displaystyle \underset{{V}_{c}}{\int}{\left|{E}_{0}\right|}^{2}}\text{d}V}$ (5)

$\frac{1}{2}\left[\frac{1}{{Q}_{s}}-\frac{1}{{Q}_{0}}\right]=\frac{{{\epsilon}^{\u2033}}_{r}{\displaystyle \underset{{V}_{s}}{\int}E\cdot {E}_{0}^{\ast}{}_{\mathrm{max}}\text{d}V}}{2{\displaystyle \underset{{V}_{c}}{\int}{\left|{E}_{0}\right|}^{2}\text{d}V}}$ (6)

We may assume that $E\approx {E}_{0}$ and the value of ${E}_{0}$ in $T{E}_{10p}$ mode as

${E}_{0}={E}_{0}{}_{\mathrm{max}}\mathrm{sin}\left(\frac{m\prod x}{a}\right)\mathrm{sin}\left(\frac{p\prod z}{d}\right)$ where $a$ is the broader dimension of the

waveguide and $d$ is the length of the cavity. Integrating and rearranging the above equations we get

${{\epsilon}^{\prime}}_{r}-1=\frac{{f}_{0}-{f}_{s}}{2{f}_{s}}\left(\frac{{V}_{c}}{{V}_{s}}\right)$ (7)

${{\epsilon}^{\u2033}}_{r}=\frac{{V}_{c}}{4{V}_{s}}\left[\frac{1}{{Q}_{s}}-\frac{1}{{Q}_{0}}\right]$ (8)

If the frequency shift is measured from the resonance frequency ${f}_{t}$ of the cavity loaded with empty capillary rather than that with empty cavity alone the above equations become

${{\epsilon}^{\prime}}_{r}-1=\frac{{f}_{t}-{f}_{s}}{2{f}_{s}}\left(\frac{{V}_{c}}{{V}_{s}}\right)$ (9)

${{\epsilon}^{\u2033}}_{r}=\frac{{V}_{c}}{4{V}_{s}}\left[\frac{1}{{Q}_{s}}-\frac{1}{{Q}_{t}}\right]$ (10)

${Q}_{t}$ is the quality factor of the cavity loaded with empty capillary. ${f}_{s}$ and ${Q}_{s}$ are the resonance frequency and quality factor of the cavity loaded with capillary containing the sample material.

Theory for the determination of conductivity of the materials:

For a dielectric material having non-zero conductivity, the Ampere’s law in phasor form as

$\nabla \times H=\left(\sigma +j\omega \stackrel{\xaf}{\epsilon}\right)E=\left(\sigma +j\omega {\epsilon}^{\u2033}\right)E+j\omega {\epsilon}^{\prime}E$ (11)

where $\epsilon ={\epsilon}^{\prime}-j{\epsilon}^{\u2033}$ is the absolute permittivity of the medium.

The loss tangent

$\mathrm{tan}\delta =\frac{\sigma +\omega {\epsilon}^{\u2033}}{\omega {\epsilon}^{\prime}}$ (12a)

For dielectrics $\sigma =0$ then Equation (12a) becomes

$\mathrm{tan}\delta =\frac{{\epsilon}^{\u2033}}{{\epsilon}^{\prime}}$ (12b)

#Math_53#,is the effective conductivity of the medium.

But

$\mathrm{tan}\delta =\frac{1}{{Q}_{m}}=\left[\frac{1}{{Q}_{s}}-\frac{1}{{Q}_{t}}\right]$ (13)

where is ${Q}_{m}$ is the loaded Q-factor of the cavity with sample alone.

The effective conductivity

${\sigma}_{e}=\frac{\omega {\epsilon}^{\prime}}{{Q}_{m}}=\frac{\omega {{\epsilon}^{\prime}}_{r}{\epsilon}_{0}}{{Q}_{m}}$ (14)

When $\sigma $ is very small, the effective conductivity is reduced to

${\sigma}_{e}=\omega {\epsilon}^{\u2033}=2\text{\pi}f{\epsilon}_{0}{{\epsilon}^{\u2033}}_{r}$ (15)

5. Results and Discussion

The microwave experiment in fruit juices were done using cavity perturbation technique collected from freshly prepared juices as well as packed juices and the results were shown in Figures 2-13. From Figures 2-4 it is observed that freshly prepared juices as well as packed juices exhibit almost similar ranges of dielectric constant even though the measurements were made at different intervals of time. From Figures 5-7 shows the dielectric loss or conductivity of freshly prepared juices. These results are shows that the conductivities are consistent over the extended period of time. From Figures 8-10 shows the dielectric constant of packed

Figure 2. Variation of dielectric constant of pure fruit juice.

Figure 3. Variation of dielectric constant of pure fruit juice after 1 hour.

Figure 4. Variation of dielectric constant of pure fruit juice after 2 hours.

Figure 5. Variation of conductivity of pure fruit juice.

Figure 6. Variation of conductivity of pure fruit juice after 1 hour

Figure 7. Variation of conductivity of pure fruit juice after 2 hours.

Figure 8. Variation of dielectric constant of packed fruit juice.

Figure 9. Variation of dielectric constant of packed fruit juice after 1 hour.

Figure 10. Variation of dielectric constant of packed fruit juice after 2 hours.

Figure 11. Variation of conductivity of packed fruit juice.

Figure 12. Variation of conductivity of packed fruit juice after 1 hour.

Figure 13. Variation of conductivity of packed fruit juice after 2 hours.

fruit juice at different time intervals. These results suggest that the dielectric constant is similar for the different intervals of time. From Figures 11-13 shows the conductivities of packed fruit juice over the extended period of time. These results show that there is a distinct variation of conductivities over the period of time as well as with that of freshly prepared juices. Packed juices exhibit higher conductivity than the freshly prepared juices because of the presence of added preservatives which is necessary to increase its shelf life. So it is very critical that packed fruit juice should be consumed immediately after it is opened. Thus in the S band of microwave (ISM band), freshly prepared fruit juices and packed fruit juice were studied and exhibit distinct variation of conductivity even after certain intervals of time.

6. Conclusion

The microwave characterization has been performed in the freshly prepared juices as well as packed fruit juices using the cavity perturbation technique. The cavity perturbation technique is quick, simple, and accurate and it requires very low volume of sample for measuring the dielectric properties of samples like juices. The limitation of this method is that measurement can lead to more accurate results on liquid samples than on solid samples. From the results, it is observed in the certain dielectric properties, the fresh juices samples and packed juices samples were varying and also varying over a period of time. This measurement method is simple and quick and can be used over a range of juices. These results prove a new method of determining the quality control of juices using microwave principles.

Cite this paper

Lonappan, A. , Afullo, T. and Daniels, W. (2017) Analysis of Certain Fruit Juices Using Microwave Techniques.*Journal of Electromagnetic Analysis and Applications*, **9**, 123-134. doi: 10.4236/jemaa.2017.99011.

Lonappan, A. , Afullo, T. and Daniels, W. (2017) Analysis of Certain Fruit Juices Using Microwave Techniques.

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