## 3 Results  
### Qualitative Analysis and Milk Classification  
First of all, a qualitative comparison of the results of the measurements was carried out by comparing the plots of the obtained admittances. In Figure 3a,b, we show the imaginary (or susceptance) and real (conductance) part of the admittance of four whole milk samples from the different brands. As is evident, the trend is superimposable for all of the curves. As can be seen from Table 1, the composition of the different types of milk is practically the same, as required by current European legislation. Therefore, it is not surprising that, under the same conditions (temperature, expiration date, measurement electrodes) the admittance trend is of the same type for each milk and the ranges of values only change a little bit. In Figure 3c,d we report the admittance (imaginary and real part) of the samples of semi-skimmed milk, of different brands. Although, in this case the curves also appear to overlap, a deviation from the other three seems evident for one of the brands (brand 3). This deviation is probably due to the fact that the semi-skimmed milk from brand 3 has also undergone a microfiltration process, which slightly altered its electrical characteristics. Anyway, for all of the samples, it can be observed that in the susceptance curves, a peak is present. The low frequency region (lower than the frequency at which the peak occurs) represents the electrical characteristic of the interface between the milk and the electrodes, while the high frequency region provides information about the milk itself. The frequency of the peak represents the relaxation frequency of the electrode polarization [28]. Figure 2: Schematic of the assembly of the electrodes and their positioning inside the sample holder. The overlap of the electrodes is performed in such a way as to minimize parasitic capacitive couplings above the milk level. In Figure 4, the imaginary part of the admittances of different types (whole, semi-skimmed, fat-free, lactose-free) of the milk from the same brand is reported. The blue curve and the red curve refer to the semi-skimmed and whole milk, respectively. For each of the analyzed brands, the maximum of the curve for the semi-skimmed milk is higher (about 4%) than the maximum of the curve for the whole milk. This observation confirms that the imaginary part of the admittance is strictly related to the content of the fats [28]. The higher the fat content, the lower the peak (with the same composition in terms of the remaining nutritional values). The blue curve and the green curve refer to the milk samples with the same content of fat, but the green one is the susceptance of a lactose-free milk. For each of the brands analyzed, the maximum of the curve for the semi-skimmed milk is higher (about 6-8%) than the maximum of the curve for the lactose-free semi-skimmed milk. That is, with the same nutritional values, the lactose-free milk has a susceptance with a lower peak than the milk with lactose. The light blue curve relates to the fat-free and lactose-free milk. Compared to the semi-skimmed milk (blue curve), the absence of fat tends to raise the peak value, but the absence of lactose tends to lower it. As a result, the peak is only slightly higher than in the case of the semi-skimmed milk. Figure 3: (**a**) Imaginary and (**b**) real parts of the measured admittances for whole milk samples from different brands; (**c**) imaginary and (**d**) real parts of the measured admittances for semi-skimmed milk samples from different brands. In Figure 5, a comparison is reported between the fat-free (but with lactose) milk and the semi-skimmed and whole milk from the same brand, but different from the brand in Figure 4.  
In Figure 6, the real part of the admittance is reported, that is the conductance, of the same milk samples of Figure 4. For the conductance, the same considerations made for the susceptance hold, that is that at the higher frequencies (>1 kHz) the value of the conductance increases with a decrease in the fat content.