# A Novel Dual-Permanent-Magnet Mechanical Antenna for Pipeline Robot Localization and Communication
## 2 Theoretical Analysis of the Novel Mechanical Antenna  
### Theoretical Fundamentals  
As shown in Figure 1, a cylinder-shaped permanent magnet with uniform magnetization along the radial direction is placed in infinite space. Its diameter and height are \(D\) and \(h\), respectively, and its magnetic moment \(m_{0}=M_{0}V\) where \(M_{0}\) is the magnetization intensity and \(V\) is the volume. A single stationary magnet can be equivalent to a magnetic dipole with the moment \(m_{0}\) when \(r>>\max(D,h)\). If it rotates counterclockwise at a uniform angular velocity \(\omega\), the magnet can be equivalent to two time-varying dipoles orthogonal in both time and space, and this moment \(m_{0}(t)=m_{x}(t)e_{x}+m_{y}(t)e_{y}\) and \(\dot{m}_{x}=\dot{m}y\). In a spherical coordinate system, the magnetic flux density generated by the dipole \(m_{y}\) can be expressed as follows:  
\[ \begin{array}{rl}\dot{B}_{m_{y}}(r,\theta,\varphi,\dot{m}_{y})&=\frac{\mu_{0} \dot{m}_{y}k^{3}}{2\pi}e^{-jkr}\left[\frac{j}{(kr)^{2}}+\frac{1}{(kr)^{3}} \right]\sin\theta\sin\varphi e_{r}\\ &-\frac{\mu_{0}\dot{m}_{y}k^{3}}{4\pi}e^{-jkr}\left[-\frac{1}{kr}+\frac{j}{( kr)^{2}}+\frac{1}{(kr)^{3}}\right]\cos\theta\sin\varphi e_{\theta}\\ &-\frac{\mu_{0}\dot{m}_{y}k^{3}}{4\pi}e^{-jkr}\left[-\frac{1}{kr}+\frac{j}{( kr)^{2}}+\frac{1}{(kr)^{3}}\right]\cos\varphi e_{\varphi}\end{array}  \tag{1}\]  
where \(k\) is the propagation constant and \(k=\omega\sqrt{\varepsilon_{0}\mu_{0}}\) in which \(\mu_{0}\) and \(\varepsilon_{0}\) are the permeability and dielectric constant of the vacuum, respectively. The magnetic flux density generated by the dipole \(m_{x}\) can be expressed as follows:  
\[\dot{B}_{m_{x}}(r,\theta,\varphi,\dot{m}_{y})=\dot{B}_{m_{y}}(r,\theta,\varphi +\frac{\pi}{2},\dot{m}_{y}) \tag{2}\]  
Therefore, the magnetic flux density generated by a single rotating permanent magnet is as follows:  
\[\dot{B}_{s}(r,\theta,\varphi)=\dot{B}_{m_{x}}(r,\theta,\varphi)+\dot{B}_{m_{y }}(r,\theta,\varphi) \tag{3}\]  
Figure 1: Equivalent model of a single rotating permanent magnet. Based on the magnetic field distribution of a single rotating permanent magnet, the dual-permanent-magnet mechanical antenna model is presented, as shown in Figure 2. In this model, the permanent magnets A and B rotate uniformly around the \(z\) axis with the same angular velocity \(\omega\), and their centers are both at a distance \(d\) from the coordinate origin \(O\). The dimensions of the permanent magnets are same as those in Figure 1, and the angle between the magnetic moments is \(\alpha\). Assume that the magnetic flux densities generated by \(A\) and \(B\) are expressed by \(\dot{B}_{A}\) and \(\dot{B}_{B}\), and then the magnetic flux density generated by the model shown in Figure 2 is as follows according to Equation (3):  
\[ \begin{array}{ll}\dot{\mathbf{B}}_{d}(x,y,z)&=\dot{\mathbf{B}}_{A}+e^{i\alpha}\dot{ \mathbf{B}}_{B}\\ &=\dot{\mathbf{B}}_{s}(x,y,z-d)+e^{i\alpha}\dot{\mathbf{B}}_{s}(x,y,z+d)\end{array}  \tag{4}\]  
In particular, when \(r>>\max(DJ,d)\), the impact of \(d\) is ignored and \(d\approx 0\); thus,  
\[\dot{\mathbf{B}}_{d}\Big{(}x,y,z)=(1+e^{i\alpha})\dot{\mathbf{B}}_{s}(x,y,z). \tag{5}\]  
From Equation (5), the amplitude of \(\dot{\mathbf{B}}_{d}\) changes when \(\alpha\) changes and amplitude modulation is achieved. Suppose that a message comprising code elements 0 and 1 must be sent. When element 0 is sent, let \(\alpha\) = 0 and \(B_{0}=\left|2\dot{\mathbf{B}}_{s}\right|\). When element 1 is sent, let \(\alpha\) = \(\alpha_{1}\), and \(B_{1}=\left|(1+e^{i\alpha_{1}})\dot{\mathbf{B}}_{s}(x,y,z)\right|\). Figure 3 shows the waveforms of the ideal amplitude  
Figure 2: Dual-permanent-magnet mechanical antenna model. modulation signal in the time domain. If the difference between \(B_{0}\) and \(B_{1}\) is not significant and the ambient background magnetic field noise is large at the receiver, the quality of the amplitude modulation signal will become poor, which increases the bit error rate during demodulation.