## 2 Basic Structure and Operating Principle of the Three-Phase AC Magnetic Separator
### Force Analysis of Magnetic Particles on the Magnetic Separator  
The magnetic particles are subjected to three forces during the sorting process, namely gravity \(F_{G}\), fluid resistance \(F_{D}\) and magnetic force \(F_{m}\), wherein the gravity \(F_{G}\) of the magnetic particles is:  
\[F_{G}=\frac{\pi}{6}d^{3}\rho\mathrm{g} \tag{2}\]  
in this formula, \(\rho\) is the density of the object; \(\mathrm{g}\) is the acceleration of gravity; \(d\) is material diameter.  
The fluid resistance \(F_{D}\) acting on paramagnetic particles is:  
\[F_{D}=3\pi\eta d\
u \tag{3}\]  
in this formula, \(\eta\) is the viscosity of the fluid; \(\
u\) is the flow rate; \(d\) is the diameter of the material.  
In the traveling wave magnetic field, the magnetic particles are subjected to two kinds of magnetic forces: magnetic suction and hysteresis torque. The suction force \(F\) is:  
\[F=V\frac{(\mu_{r}-1)H_{0}}{[4\pi+D(\mu_{r}-1)]}\mathrm{grad}B_{0} \tag{4}\]  
in this formula, \(V\) is the volume of the magnetic particles; \(\mu_{r}\) is the relative magnetic permeability; \(H_{0}\) is the external magnetic field strength; \(D\) is the demagnetization factor, which is related to the shape of the magnetic particles, \(D=4\pi/3\) to the sphere or square; \(B_{0}\) is the external magnetic field magnetic induction.  
The suction force \(F_{m}\) per unit mass of magnetic particles is:  
\[F_{m}=X_{m}H_{0}\mathrm{grad}B_{0} \tag{5}\]  
in this formula, \(X_{m}\) is the specific magnetic susceptibility of the magnetic particles; \(H_{0}\) is the external magnetic field strength; and \(B_{0}\) is the external magnetic field magnetic induction.  
As shown in Figure 4a, the magnetic particles are magnetized by the magnetic field at the surface of the separator, causing the particles to induce a corresponding polarity in the direction of the magnetic field. As shown in Figure 4b, after a certain period of time, the magnetic field is rotated by a certain angle. Due to the hysteresis effect of the magnetic particle material, the polarity is too late to change, so the magnetic particles generate a counterclockwise torque and cause the magnetic particles to roll in a counterclockwise direction on the separator surface.  
The magnitude of this torque \(T\) can be expressed by the following formula:  
\[T=V\frac{2\pi\mu_{r}{H_{0}}^{2}}{4\pi+D(\mu_{r}-1)}\sin\gamma_{0} \tag{6}\]  
in the formula, \(\gamma_{0}\) is the hysteresis angle, which is related to the material properties.  
The hysteresis torque per unit mass is:  
\[T_{m}=V\frac{2\pi\mu_{r}{H_{0}}^{2}}{\rho{\left[4\pi+D(\mu_{r}-1)\right]}^{2}} \sin\gamma_{0} \tag{7}\]  
When the magnetic particles fall on the surface of the magnetic separator, due to the magnetic attraction force, the magnetic particles are pulled toward the surface of the separator. The hysteresis torque generated by the action of the rotational component in the magnetic field causes the magnetic particles rolling along the surface of the separator to overcome the fluid resistance \(F_{D}\) as well as the friction generated by the component of gravity \(F_{G}\). The nonmagnetic material falls under the action of gravity, thus separating the magnetic material from the nonmagnetic material.