# Characterization of a Mechanical Antenna Based on Rotating Permanent Magnets
## 3 Materials, Methodology and Experimental Setup
### Equations  
Based on the well-known analogy that can be made between the magnetic field produced by a permanent magnet and a coil, or solenoid, in static mode as well as in LF harmonic mode, the respective analytical equations are used in the article. Thus, as shown in Figure 2, the most obvious similarity is the one between Equations (1) and (2), which express the magnetic field for the solenoid [23] and the CABM [24], respectively.  
\[B=\frac{\mu_{0}IN}{2L}\left[\frac{x_{2}}{\sqrt{x_{2}^{2}+R^{2}}}-\frac{x_{1}}{ \sqrt{x_{1}^{2}+R^{2}}}\right] \tag{1}\]  
\[B(z)=\frac{\mu_{0}M}{2}\left[\frac{z}{\sqrt{z^{2}+R^{2}}}-\frac{z-L}{\sqrt{ \left(z-L\right)^{2}+R^{2}}}\right] \tag{2}\]  
Except for the letter assignment for the geometric quantities, the terms in the brackets of Equations (1) and (2) are identical. In Equation (1), \(I\) is the current and \(N\) is the turn number of the solenoid. The results show that, for a \"efficiency\"-related comparative analysis between the magnetic field produced by a solenoid and that given by a CABM with the same geometry, the ratio of the factors in front of the brackets of the two equations must be analyzed. Thus, from the ratio between expressions (2) and (1), we obtain the following:  
\[\frac{B_{r}L}{\mu_{0}IN} \tag{3}\]  
where \(B_{r}=\mu_{0}M=1.21\) [T] (as shown from the numerical results presented in Section 4.1), with \(M\) being the magnetization. In other words, this ratio shows how many times more efficient the permanent magnet is in producing a magnetic field compared with a solenoid with identical geometry.  
It can be proven analytically that the term in the brackets of Equation (2) has a functional dependence identical with \(1/r^{3}\). Thus, the red curve plotted in Figure 3 represents the brackets of Equation (2).  
The black curve in Figure 3, which represents the simple \(1/r^{3}\) dependence, has a shape similar to the red curve; they become identical if a scaling with the factor \(4\pi 10^{-7}\) is carried out.  
On the other hand, the magnetic field transmitted in the axial direction by a loop-type elementary magnetic dipole is given by the following equation [25; 26].  
\[H_{\theta}=\frac{IA}{4\pi}\text{sin}\,\theta\left(-\frac{\beta^{2}}{r}+j\frac {\beta}{r^{2}}+\frac{1}{r^{3}}\right)e^{-j\beta r}, \tag{4}\]  
where \(A\) is the area of the elementary current loop, \(I\) is the current flowing through the loop, \(r\) is the distance to the observation point, sin \(\theta=1\) because the field is measured in the axial direction, and \(e^{-j\beta r}\) is the characteristic factor of the harmonic regime. Equation (4) is valid for the following conditions: \(r>>\) current loop radius; current loop radius \(<<\lambda\).  
Figure 2: (a) Geometric coordinates for solenoid; (b) geometric coordinates for CABM.  
The above equation includes three terms: \(1/r\) is the far field (plane waves) term; \(1/r^{2}\) and \(1/r^{3}\) are the terms characteristic to the magnetic near field.  
Moreover, we have the well-known relationship derived from the Biot-Savart law, which gives the magnetic field of a circular coil in the axial direction [27]:  
\[H_{z}=\frac{a^{2}IN}{2\big{(}a^{2}+z^{2}\big{)}^{3/2}}, \tag{5}\]  
where \(a\) is the coil radius, \(I\) is the current in the coil, \(N\) is the number of turns, and \(z\) is the distance to the observation point.  
It is easily observed that for \(a<<z\) (and except \(N\)), Equation (5) is quasi-identical with the last term in (4), which is expected because both relationships are derived, in different ways, from [PERSON]'s equations. In conclusion, it can be stated that the analytical relations for CABM as well as for the solenoid/loop antenna in the near field zone show a distance dependence that scales as \(1/r^{3}\).