# Communication System Based on Magnetic Coils for Underwater Vehicles
## 2 Magnetic Field Induction Using Coils  
Any conductor through which an electric current circulates generates a magnetic flux; the relationship between the magnetic flux and the current that circulates is an electrical property known as inductance, which is measured in Henry (H). As the current (amount of moving charge) is increased, the magnetic field increases proportionally. If the observation point is moved away from the conductor, the field decreases inversely proportional to the distance [22]. The analysis of a rectangular magnetic coil that uses thin conductors can start from the physical phenomenon of the generation of a magnetic field (\(\vec{B}\)) due to the flow of electric current through the conductor. [PERSON]'s equations indicate that the divergence \(\vec{\
abla}\cdot\vec{B}=0\), which indicates that \(\vec{B}\) has the solenoidal property (no divergence). \(\vec{B}\) can be presented by using an auxiliary vector function \(\vec{A}\), through the rotational relationship, as in Equation (1). \[\vec{B}=\vec{\
abla}\times\vec{A} \tag{1}\]  
Function \(\vec{A}\) is called a magnetic vector potential field. This field is related to the sources of stable current density responsible for the magnetic field \(\vec{B}\). For the case of a thin linear conductor, \(\vec{A}\) is defined by Equation (2). \[\vec{A}=\frac{\mu_{0}\vec{I}}{4\pi}\int_{l}\frac{dl^{\prime}}{r} \tag{2}\]  
where \(\vec{I}\) is the current flowing through the linear conductor and \(r\) is the distance from the conductor to the analysis point. For a rectangular coil of thin conductor of \(n\) turns, the magnetic field is calculated by adding the contribution generated by each segment of the coil. In Figure 1, the magnetic coil is shown in the plane \(XY\), and the analysis point of the magnetic field \(P(X,Y,Z)\). The calculation of the field \(\vec{B}\) must be done using Equation (2) that implies the integral over the geometry defined by the emitting coil, each of the 4 segments of linear conductor of each of the \(n\) turns considered in Figure 1.  
As an example, the calculation of the field \(\vec{A}_{1}\) is shown, which represents the first segment of the coil on the plane \(XY\) that goes from the coordinates \((a,-b,0)\) to the point \((-a,-b,0)\)[23]:  
\[\vec{A}_{1}=\frac{\mu_{0}\vec{I}}{4\pi}\int_{a}^{-a}d\frac{dx^{\prime}}{\sqrt{( x-x^{\prime})^{2}+(y+b)^{2}+z^{2}}} \tag{3}\]  
The vector magnetic potential field \(\vec{A}\) is calculated as the sum of the fields of all the segments \(\vec{A}_{k}\). Later, the field \(\vec{A}\) can be evaluated with Equation (2). Finally, to find the magnetic field, \(\vec{B}\) Equation (1) is used. Figure 2 shows the simulation results of the magnetic field generated by a coil using the following parameters: \(a\) = 0.12 m, \(b\) = 0.10 m, \(I\) = 70 mA, \(n\) = 17, and height \(z\) = 0.05 m.  
The magnetic field generated by the current that circulates in a conductor (transmitter coil) can be induced to another conductor (receiver coil) that will in turn generate electric current. Figure 1: Square magnetic coil and point of interest to analyze. Figure 2: \(|B|\) in an area of 0.4\(\times\)0.4 m at a height of z = 0.05 m.  
The induced voltage or electromotive force (EMF) can be obtained by taking the derivative with respect to time of the integral of the magnetic field \(\vec{B}\) along the geometry defined by the receiver coil, according to Equation (4). \[EMF=-\frac{d}{dt}\int_{\mathrm{s}}\vec{B}(I(wt))ds \tag{4}\]  
where \(s\) is the trajectory of the receiver coil including all the linear segments. The electrical current is defined in terms of the frequency \(w\) of the periodic signal used. Figure 3 shows the results of the voltage induction simulation are shown using the following parameters: \(n=40\), \(a=0.12\) m, and \(b=0.10\) m for the receiver coil. In Figure 4, the comparison between experimental and simulated values of magnetic field is shown.