# Sensorless Posture Detection of Reluctance Spherical Motor Based on Mutual Inductance Voltage
## 2 Basic Principle of Detecting Rotation Angle from Mutual Inductance Voltage
### Numerical Relationship between Mutual Inductance Voltage and Single-DoF Position Angle  
The reluctance spherical motor studied in this paper adopts a structure with 24 stator poles arranged in three layers and six rotor poles distributed in one single layer, as shown in Figure 2. The stator cores are made of DT4 iron. The stator poles of the middle layer are on the spherical equator, and the stator poles of the upper and lower layers are, respectively, located on the 33\({}^{\circ}\) north and south latitude lines. Eight magnetic poles of each layer are distributed at equal intervals of 45\({}^{\circ}\) longitude. A centralized stator winding with 350 turns is placed on the stator pole. The length of unilateral air gap between stator and rotor is 1 mm;  
The block diagram of the experimental platform is shown in Figure 4. The excitation coil and response coil are two of the spherical motor coils; the others are not illustrated. Due to the signal generator load limitation, in order to stabilize the amplitude of the excitation signal when the rotor rotates, a signal amplification circuit is installed to amplify the sinusoidal signal generated by the signal generator. The signal source \(IN\) comes from a function signal generator, through an amplification circuit composed of a power operational amplifier and is output as an excitation signal \(u_{A}\) to a stator coil of the spherical motor. Figure 3: Mutual inductance voltage experimental platform. Figure 2: (**a**) Stator structure of \(24/6\) switched reluctance spherical motor, (**b**) rotor structure. The excitation signal \(u_{A}\) used in the experiment is a sine wave signal with a frequency of 50 Hz, and the mutual inductance voltage \(u_{B}\) is also a sine wave signal with the same frequency. \(u_{B}\) will be sampled by a signal sampling circuit. The sampling results will be converted into digital data and sent to a single-chip microcomputer. The single-chip microcomputer will process this digital data and give the amplitude of \(u_{B}\). In order to conveniently judge the starting point position of the misalignment angle between the stator and rotor magnetic poles, in accordance with generating the same torque, 24 stator coils are divided into 12 groups according to the spherical symmetry position. The code of each group is arranged in Figure 1a. A coil of one group located in the Eastern Hemisphere is used to inject excitation coils for \(u_{A}\), and the mutual inductance voltage \(u_{B}\) will be generated by the other coil of this group located in the Western Hemisphere. In this way, the mutual inductance voltage value corresponds to the angle between the axis of this coil group and the closest one of the six rotor teeth to this axis. Because of the symmetrical structure of six rotor poles, the influence of the rotor yawing on the mutual inductance voltage in each group of coils is periodic every 60\({}^{\circ}\). Taking the coil group located at 22.5\({}^{\circ}\) east and west longitude on the spherical surface in the middle stator magnetic pole as an example, the excitation coil injects a sinusoidal AC signal with a frequency of 50 Hz and an amplitude of 0-5 V. Rotating the rotor vertically by 60\({}^{\circ}\) while keeping the pitch angle at 0\({}^{\circ}\) is one period of coil mutual inductance voltage. The finite element calculation and actual measurement are carried out on the coil mutual inductance voltage, changing with rotor position angle and excitation signal amplitude, respectively, and the results are shown in Figure 5. Among them, the red and blue waveforms shown in Figure 5a are the finite element calculation results, and the line graph shown in Figure 5b is the experimental test results. The comparison between the two results is shown in Figure 5c. It can be seen that the results measured by the experiment are highly consistent with the calculation results, and they conform to the prediction of Equation (6).