## 3 Results
### Quantitative Analysis: Milk Identification  
A few works [17; 28; 29] in the scientific literature present a possible circuit modeling milk, based on a parallel circuit between a capacitance and a series of a capacitance and a resistance. Nevertheless, the fitting operation revealed that the electrode/milk interface cannot be modelled as an ideal capacitor but as a constant phase element (_CPE_), which models a non-ideal behavior of a capacitor [41]. The equivalent electrical circuit of the milk is then reported in Figure 7. The capacitance \(C\) represents the capacitance of the two electrodes separated by the characterized milk, while the resistance \(R\) represents the milk conductance. Figure 6: Real part of the admittance, that is the conductance, of the same milk samples of Figure 4.  
Figure 7: Equivalent circuit of milk. The constant phase element (_CPE_) models the interface electrode/milk; the capacitance \(C\) represents the capacitance of the two electrodes separated by the characterized milk; the resistance \(R\) represents the milk conductance. Working a little on the circuit shown in Figure 7, the series of the _CPE_ admittance (\(Y_{CPE}\)) and of the resistance \(R\) conductance (\(1/R\)) is:  
\[Y_{S}=\frac{Y_{CPE}}{1+RY_{CPE}} \tag{1}\]  
with  
\[Y_{CPE}=Q(j\omega)^{n} \tag{2}\]  
where \(\omega\) is the angular frequency; \(Q\) and \(n\) are the characteristic parameters of the _CPE_. The expression of the admittance of the circuit in Figure 7 is:  
\[Y=\frac{RQ^{2}\omega^{2n}+Q(j\omega)^{n}}{1+R^{2}Q^{2}\omega^{2n}}+j\omega C \tag{3}\]  
and from Equation (3) we can derive the real (\(Re(\gamma)\)) and imaginary part (\(Im(Y)\)) of the admittance, as follows:  
\[Re(Y)=\frac{RQ^{2}\omega^{2n}+Q\omega^{n}\cos\bigl{(}n\frac{\pi}{2}\bigr{)}}{ 1+R^{2}Q^{2}\omega^{2n}} \tag{4}\]  
\[Im(Y)=\frac{Q\omega^{n}\sin\bigl{(}n\frac{\pi}{2}\bigr{)}}{1+R^{2}Q^{2}\omega ^{2n}}+\omega C \tag{5}\]  
Using the Equations (4) and (5), fitting operations were performed on the measured admittances in order to estimate the values of the components that characterize the circuit in Figure 7. In Table 3, the obtained results are summarized, and in Table 4, the maximum errors for the estimation of circuit parameters by fitting operation are reported. For each type of milk, the average values obtained from the measurements on several samples (six for each type) are reported. The measured values deviate less than 5%, 1.5%, 1% and 0.2% from the average values of \(C\), \(R\), \(Q\) and \(n\), respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**MILK SAMPLE** & \(C\) **(nF)** & \(R\) **(\Omega)** & \(Q\) **(\(\times\)10\({}^{-5}\))** & \(n\) \\ \hline WM1 & 1.41 & 37.73 & 7.98 & 0.848 \\ WM2 & 1.36 & 37.57 & 8.12 & 0.838 \\ WM3 & 1.31 & 39.11 & 7.96 & 0.843 \\ WM4 & 1.22 & 37.88 & 8.42 & 0.843 \\ SSM1 & 1.36 & 36.26 & 7.99 & 0.847 \\ SSM2 & 1.51 & 36.16 & 8.09 & 0.838 \\ SSM3 & 1.72 & 37.28 & 8.23 & 0.843 \\ SSM4 & 1.43 & 36.43 & 8.16 & 0.843 \\ LFFM1 & 1.78 & 36.06 & 7.94 & 0.848 \\ FFM2 & 1.84 & 35.83 & 8.01 & 0.838 \\ LFM1 & 1.63 & 39.01 & 8.08 & 0.846 \\ LFM2 & 1.71 & 38.37 & 8.13 & 0.838 \\ LFM3 & 2.03 & 40.53 & 8.15 & 0.843 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of the components of the circuit in Figure 7, obtained by means of fitting operation on the data obtained with the measurements on milk samples. Observing the obtained results, it is clear that the whole milk, for three of the four brands analyzed, has a lower capacity value than the corresponding semi-skimmed milk. This could be explained because fats have a lower dielectric constant than the other components of milk. However, samples 1 (WM1 and SSM1) show no substantial differences in capacity values between the whole milk and semi-skimmed milk. The only noticeable difference between these samples and the others is that these are distributed in plastic bottles, while the others in Tetra Pak.