## 2 State-of-the-Art (Method)
### The Use of Electromagnetic Sensors for Environment Monitoring
#### 2.1.1 Background
If the medium in which the magnetic flux goes through is modified, for example, changing the amount of OF diluted in water, the output voltage should change. The mutual inductance, M, of two solenoids, can be described by Equation (4). Where L1 and L2 are the inductances of each coil, and k is the coupling coefficient. L1 and L2 depend on the core (\(\mu_{0}\mu_{r}\)), the amount of turns N, the cross-sectional area A in m\({}^{2}\), and the length of the solenoid l, as shown in Equation (5). Therefore, the mutual inductance depends on the medium of the core. In our case, the water. When the permeability of the water changes, the coupling effect k changes. The value of k is maximum (1) when the coils are perfectly coupled and minimum (0) when there is no inductive coupling. When k is 1, it means that 100% of the lines of flux of PC cuts all the turns of the IC, it is assumed high permeability of the water and coils with perfect geometry. In our experiments, the characteristics of the core (permeability) are one of the studied factors. Moreover, the testing of different prototypes is aimed to evaluate the effect of different geometries. The position of coils is a fixed factor. However, the distance between the PC and IC and the total length of both coils changes from one prototype to another. \[M=k\sqrt{L1\;L2} \tag{4}\]  
\[L1=\;\frac{\mu_{0}\;\mu_{r}\;N\;A}{l} \tag{5}\]  
According to the experiments performed in previous works [26], the induced voltage will depend on the characteristics of the coil, such as N, OW, and the diameter of the induced coil (OIC). Moreover, the induced voltage, also known as the Vout, depends on the B and \(\mu_{r}\). Generally, this principle is used with coils that have a ferromagnetic core, and is the principle of the power transformers. The magnetic flux density is determined by the Biot-Savart law presented in Equation (6) and depends on the permeability \(\mu\), a time varying current denoted by \(i(t)\), the distance to the source \(R\), and the unit vector \(\overset{\rightarrow}{R}\)[27]. \[d\overset{\rightarrow}{B}=\;\frac{\mu}{4\pi}\frac{i(t)\overset{\rightarrow} {dl}\times\overset{\rightarrow}{R}}{R^{3}} \tag{6}\]  
Furthermore, it is possible to obtain the induced electric field using [PERSON]'s law, represented by Equation (7). [PERSON]'s law states that a time varying magnetic flux, \(d\Phi\) induces an electric field around a closed path. Taking into account [PERSON]'s theorem, the induced electric field can be written in terms of the number of turns, N, in a non-varying surface, \(dS\)[28]. \[\oint_{C}\overset{\rightarrow}{E}.d\overset{\rightarrow}{l}=\;-N\;\frac{- d\Phi}{dt} \tag{7}\]  
In Figure 1, we show a summary of the considered variables in our experiments. According to the polarisation of the PC and its relative position to the IC, when the generated magnetic field increases, the Vout can increase or decrease. In our experiments, we maintain the same polarisation in each coil. Furthermore, to limit the number of variables in our tests, the input signal (the signal to power the PC) will have two fixed parameters, intensity and voltage; and one variable parameter (frequency). This is done based on the results of [26] when the authors showed that each combination of coils has different peak frequencies. This type of sensor has been revealed as a suitable sensor for monitoring water conductivity. In previous papers [26], it was demonstrated that the variation of different parameters, such as NS (Number of Spires), OW, and OIC or OPC is vital to find the correct configuration of coils to sense a specific parameter. In this case, the parameter will be the amount of fertiliser in the water. As said before, \(\mu_{0}\)\(\mu_{r}\) are modified by the conductivity of water.