**2. Electrical Characteristics of Poultry Eggs**

*2.1. Physical Characteristics of Poultry Eggs*

A complete poultry egg is composed of an eggshell, egg membrane, egg white, yolk, air chamber, etc. as shown in Figure 1a. The main component of the eggshell is calcium carbonate, which accounts for about 11% of the volume of the whole egg. They are hard and play an important role in protecting the egg white and yolk, exchanging gases with the outside world, and providing minerals for embryonic development. Its structure is shown in Figure 1b [24]. The common eggshell includes three layers with slightly different microstructures, and its radial cross section is shown in Figure 1c [25]. The outermost part of the shell is dense, meticulous, and has a certain strength, and thus it is called the cuticle. The middle layer is spongy and densely covered with many small holes, while the innermost layer, called the papillary layer, is pyramidal, and the spaces between the layers can hold air. On the surface of the eggshell lie pores of about 30 microns in diameter. These are called stomata, through which gas exchange and water evaporation occur.

**Figure 1.** Schematic diagram of egg and eggshell structure. (**a**) Structure of the egg. (**b**) Microstructure of the eggshell. (**c**) Radial cross section of the eggshell.

#### *2.2. Model of Electrical Characteristics of Poultry Eggs*

A dielectric in the electric field produces an equivalent bound charge on the atomic scale under the electric field force, and this phenomenon is called dielectric polarization. For an eggshell, when the electric field intensity exceeds a certain value, the bound charge is forced to flow, causing dielectric breakdown and losing its insulation. Therefore, it is very important for the detection of the eggshell cracks to calculate the electrostatic fields of eggs and analyze the current change in the circuit. For this reason, we designed a dynamic detection method for cracks. The microcurrent will be generated at the crack of an eggshell when the egg rotates dynamically in the detection device, which is jointly generated by two models that will be discussed below: one is the electrical breakdown, and the other is capacitance jump. The total current is as follows:

$$I = I\_1 + I\_2 \tag{1}$$

where *I*<sup>1</sup> is the microcurrent generated by electrical breakdown and *I*<sup>2</sup> is the microcurrent generated by the capacitance jump.

### 2.2.1. Model of Capacitance of a Poultry Egg

An electrostatic field with the medium is produced jointly by the bound charge and free charge. In order to represent the electric field, which is under the joint action of both charges, another field vector–electric flux density−→*<sup>D</sup>* , also known as electric displacement, is introduced, which is defined in Table 1, where −→*<sup>E</sup>* is the electric field intensity, −→*<sup>P</sup>* is the electric polarization intensity, and *ε*<sup>0</sup> is the vacuum dielectric constant.

#### **Table 1.** Formula table.


As shown in Figure 2, when there are poultry eggs in the electric field, the properties of the spatial electrostatic field are related to the free charge (*q*0) and the distribution of the dielectric. The macroscopic electrical properties of the dielectric can be replaced by a polarized charge (*q* ), and then the total spatial electrostatic field consists of −→*E*<sup>0</sup> and −→ *E* , as shown in Table 1. Here, −→*E*<sup>0</sup> represents the applied electric field formed by a free charge, and −→ *E* represents the electrolyte polarization electric field formed by a polarized charge. In a linear isotropic dielectric, the electric polarization intensity −→*<sup>P</sup>* is defined as *<sup>ε</sup>*0X*<sup>e</sup>* −→*E* , which can be seen in Table 1, where X*<sup>e</sup>* is the electric polarizability rate. Therefore, we have

$$
\overrightarrow{D} = \varepsilon\_0 (1 + \mathcal{X}\_\varepsilon) \,\overrightarrow{E}' = \varepsilon\_0 \varepsilon\_r \,\overrightarrow{E} \tag{2}
$$

In the above formula, *ε<sup>r</sup>* = (1 + X*e*) stands for relative permittivity, which is a physical parameter characterizing the dielectricity or polarization of dielectric materials, also known as relative permittivity. After the electric displacement vector −→*<sup>D</sup>* is obtained, the Gauss theorem in the medium can be formulated, which is defined in Table 1, where −→*<sup>S</sup>* denotes any closed surface in the medium and q denotes a free charge.

**Figure 2.** Schematic diagram of surface polarization of eggs in electric field.

We can think of the two electrodes and the egg in the middle as one capacitor, as shown in Figure 3a, where the eggshell is an insulator and the egg liquid is approximately a conductor due to a low resistance value. In an equilibrium state, there is no current in the circuit. The egg liquid has a certain conductivity, so the dielectric constant *ε<sup>L</sup>* of the egg liquid is large. If the egg liquid is approximated as a good conductor, according to the position of the upper and lower electrodes and the poor conductivity of the eggshell, the electrical characteristic model under this connection mode can be approximated as the series of two plate capacitors, as is shown in Figure 3b, and then the electric field distribution under the intact eggshell is *U* = *E*1*d*<sup>1</sup> + *E*2*d*2. Therefore, according to the plate capacitance formula, the equivalent capacitance *C*<sup>1</sup> is (*d L*, *d W*, where *L* is the length of the electrode and *W* is the the width of the electrode):

$$\mathbb{C}\_1 = \frac{4\pi\varepsilon\_l\varepsilon\_rLW}{d\_1 + d\_2} \tag{3}$$

where *d*<sup>1</sup> and *d*<sup>2</sup> are the thickness of the upper and lower layers of eggshell, respectively. text

**Figure 3.** Capacitance system diagram. (**a**) Schematic diagram of the capacitor system, composed of the electrode and egg body. (**b**) Schematic diagram of equivalent capacitance of system when the electrode is not at the crack. (**c**) Schematic diagram of equivalent capacitance of the system when electrode is at the crack.

When a crack exists in an eggshell, the electrical characteristics model of the egg change as shown in Figure 3c, and then

$$\mathcal{U}I = \mathcal{U}\_1 + \mathcal{U}\_2 = \frac{\mathcal{Q}}{\mathcal{C}\_1} + \frac{\mathcal{Q}}{\mathcal{C}\_2} \tag{4}$$

$$\mathcal{U}l\_1 = \frac{\mathcal{Q}}{\mathcal{C}\_1} = \frac{\mathcal{U}\mathcal{C}\_1\mathcal{C}\_2}{\mathcal{C}\_1 + \mathcal{C}\_2} = \frac{\mathcal{U}\mathcal{C}\_2}{\mathcal{C}\_1 + \mathcal{C}\_2} = \frac{\mathcal{U}\frac{4\pi\varepsilon\_l\varepsilon\_r\mathcal{S}}{d\_2}}{\frac{4\pi\varepsilon\_l\mathcal{S}}{d\_1} + \frac{4\pi\varepsilon\_l\varepsilon\_r\mathcal{S}}{d\_2}} = \frac{d\_1\varepsilon\_r}{d\_2 + \varepsilon\_r d\_1}\mathcal{U} \tag{5}$$

The electric field at a crack can be defined as

$$E\_1' = \frac{\mathcal{U}\_1}{d\_1} = \frac{\varepsilon\_r}{d\_2 + \varepsilon\_r d\_1} \mathcal{U} \tag{6}$$

When the air breakdown electric field is *E* <sup>1</sup>*<sup>p</sup>* = 30 KV/cm, and *d*<sup>1</sup> = *d*<sup>2</sup> ≈ *d* = 350 μm, then the breakdown voltage *Up* is

$$dI\_p = \frac{d\_2 + \varepsilon\_r d\_1}{\varepsilon\_r} E\_{1p}' \approx dE\_{1p}' = 3.5 \times 10^{-4} \times 3 \times 10^4 \times 10^2 = 1050 \text{ V} \tag{7}$$

At this time, the plate capacitance *C*<sup>2</sup> is

$$\mathbf{C}\_2 = \frac{4\pi\varepsilon\_l\varepsilon\_rLW}{\varepsilon\_r d\_1 + d\_2} \tag{8}$$

The experimental results show that if there is no crack in the eggshell of the egg rotating in the middle of two electrodes, the equivalent capacitance value would stay basically stable at *C*<sup>1</sup> in the whole process. However, if there is a crack in the eggshell, the equivalent capacitance will jump between *C*<sup>1</sup> and *C*<sup>2</sup> when the electrode passes the cracks of the rotating egg, resulting in a transient current. Setting the egg rotation as an angular velocity of *α*, the time to rotate the width of *W* is *<sup>W</sup> <sup>α</sup><sup>R</sup>* , where *R* is the radius of the egg. Therefore, when the egg rotates from a no crack zone to a crack zone, the current generated is

$$\begin{split} I &= \frac{\Delta Q}{\Delta t} = \frac{\text{ULAC}}{\frac{W}{\pi R}} = \frac{\text{LaR}}{W} \cdot 4\pi \varepsilon\_l \varepsilon\_r L \text{lV} (\frac{1}{d\_1 + d\_2} - \frac{1}{\varepsilon\_r d\_1 + d\_2}) \\ &= \text{lLR} \cdot 4\pi \varepsilon\_l \varepsilon\_r L \cdot \frac{(\varepsilon\_r - 1)d\_1}{(d\_1 + d\_2)(\varepsilon\_r d\_1 + d\_2)} \end{split} \tag{9}$$
  $\text{If } d\_1 \approx d\_2 = d, \text{ then }$  
$$\begin{split} I &= \text{lLR} \cdot \text{R} \cdot 4\pi \text{sec}^2 I \end{split} \tag{9}$$

$$I \approx \frac{\mathcal{U}\varkappa \mathcal{R} \cdot 4\pi \varepsilon\_l \varepsilon\_r^2 L}{2(\varepsilon\_l + 1)d} \tag{10}$$

The following data were obtained in the experiment: the angular velocity was 2 cycles/SEC, *<sup>α</sup>* = <sup>4</sup>*π*, the radius of the shell *<sup>R</sup>* = <sup>3</sup> × <sup>10</sup>−<sup>2</sup> m, *<sup>ε</sup><sup>l</sup>* = 8.85 × <sup>10</sup>−12, the *CaCO*<sup>3</sup> dielectric constant of the eggshell *<sup>ε</sup><sup>r</sup>* ≈ 8.8, the length of the electrode *<sup>L</sup>* = <sup>4</sup> × <sup>10</sup>−<sup>2</sup> m, and the shell thickness was 350 μm. Then, we have

$$I \approx \frac{4\pi \times 3 \times 10^{-2} \times 4\pi \times 8.85 \times 10^{-12} \times 8.8^2 \times 4 \times 10^{-2} \times \mathcal{U}}{2 \times (8.8 + 1) \times 3.5 \times 10^{-4}} = 1.894 \times 10^{-8} \times \mathcal{U} \tag{11}$$

where when *U* = 1500 V, *I* ≈ 28.4 μA.

Figure 4 shows the current curves collected when detecting intact eggs and cracked eggs under the above electrode shapes and experimental parameters. The blue line represents intact eggs, and the red line represents cracked eggs. It is clear that there was a peak in the data for cracked eggs. In the online detection system, the detected current value may be the microcurrent generated by a capacitance jump or microcurrent superposed with that produced in the electric breakdown.

**Figure 4.** Comparison diagram of current measurement curves without cracks or cracked eggs under discharge electric field.

#### 2.2.2. Electric Breakdown Model of Poultry Eggs

According to the basic principle of electric breakdown, if the voltage applied to an insulator is increased, the number of charge carriers in the material will increase sharply under a certain electric field, and its resistivity will decrease, resulting in producing a strong current. For poultry eggs, an intact one is not conductive under normal conditions, but when there is a crack in the eggshell, an air interlayer with low insulation may occur in the eggshell. Because the breakdown voltage of the air dielectric is much less than that of a solid dielectric, when high voltage is applied on both sides of the egg body, an egg with cracks is more likely to cause electrical breakdown, and there will be a significant difference in the current.

Since the width of the crack is much smaller than the size of the eggshell or the electrode, it can be approximated that the electric field in the crack area is uniform. The gap breakdown voltage is subject to Paschen's law when the air pressure is below 1 standard atmosphere (about 0.1 mpa):

$$V = f(pd) \tag{12}$$

where *p* is the air pressure and *d* is the distance between the electrodes.

The breakdown voltage *Ub* can be calculated according to the empirical formula:

$$\mathcal{U}\_b = \frac{B\_{pd}}{\ln(\frac{A\_{pd}}{\ln\frac{1}{\gamma}})} \tag{13}$$

where *γ* is the ionization coefficient and *A* and *B* are constants related to the composition of the air. At standard atmosphere pressure, *A* = 43.66 and *B* = 12.8.

For a static, intact egg, a sudden change in current occurs when solid dielectric breakdown occurs. The breakdown voltage of a solid dielectric is much higher than that of an air dielectric, so if we keep the voltage at both electrodes stable and only allow air dielectric breakdown, we can identify cracked eggs according to the change in current signals. Therefore, the key to the problem is to apply a stable electric field at the crack that can break down the air but not the eggshell. This problem is solved by analyzing the electrode shape and simulation experiments under different voltages. As is shown in Figure 5, there was a tiny crack in the Z direction on top of the egg. U-shaped linear electrodes were applied to the upper and lower sides of the egg to wrap the eggshell to the maximum extent and make the electric field uniform. By adjusting the electrode shape and voltage, the current detection system was optimized in the simulation environment and verified by experiments in the real scene.

**Figure 5.** Simulation analysis of egg electric field distribution with cracks.

At the same time, the conditions of the air in the crack gap, such as the temperature, humidity, and other factors, will affect the ionization tendency of the air and correspondingly affect the breakdown voltage or discharge voltage in the crack gap. When the temperature decreases, the density of the air increases, the mean free path of free electrons in the air is shortened, and it is not easy to cause collision ionization, thus causing the breakdown voltage of the air to increase. As an electronegative gas, water vapor easily captures free electrons and transforms them into negative ions when the humidity of the air increases, which weakens the ionization and decreases the breakdown voltage of the air. Given the potential influence of the high temperature and humidity in the egg production line, special attention should be paid to these factors in the process of the analysis and experiment.

In short, the final current value is usually the superposition value of the current generated by the above two cases. When the electrodes are passing the cracked area of the rotating egg, if the detection voltage is less than the breakdown voltage threshold, the total current in Equation (1) is mainly *I*2; otherwise, the total current is mainly *I*1.
