# A Novel Method for Modeling the Electromagnetic Characteristics of Switched Reluctance Motors
## 2 Measurement of Flux-Linkage Characteristics  
A practical measuring method is developed to obtain the flux-linkage curves of a three-phase 12/8-pole SRM in [21]. One of the most important features of this method is that no locking device is adopted in the test. As analyzed in [21], there are special rotor positions where the torque is zero, named equilibrium positions. For a three-phase 12/8 pole SRM, there are four equilibrium positions, as shown in Figure 1. Firstly, these equilibrium positions can be found by exciting different phases. Then, the pulsed DC voltage is applied to the corresponding phase. Meanwhile, the phase voltage and current waveforms are stored to calculate the flux-linkage values at these special rotor positions. This measuring method is adopted in this paper. The flux-linkage measurement circuit is shown in Figure 2. A three-Phase 1.5-kW 12/8 SRM is used in the test. The phase current and phase voltage are measured with current and voltage sensors and then recorded in the digital oscilloscope for further processing in MATLAB software. The measurement steps are summarized as follows. 1. Exciting Phase 1, the rotor will be attracted to the aligned position of the stator and rotor, and this position is defined as 22.5\({}^{\circ}\). Figure 1: Cross-sectional view of the SRM with different rotor positions and excitation modes. (**a**) Exciting Phase 1 at 22.5\({}^{\circ}\); (**b**) exciting Phases 2 and 3 at 22.5\({}^{\circ}\); (**c**) exciting Phases 2 and 3 at 0\({}^{\circ}\); (**d**) exciting Phase 1 at 0\({}^{\circ}\). The oscilloscope records the current waveforms of the voltage and current of Phase 1. 3. Exciting both Phases 2 and 3 at the aligned position of Phase 1, the rotor will continue to remain in standstill at 22.5\({}^{\circ}\). 4. The oscilloscope records the current waveforms of the voltage and current of Phase 2. 5. Exciting Phase 2, the rotor will be attracted to the aligned position of Phase 2. 6. Exciting Phases 2 and 3 simultaneously by the same voltage, the rotor will be in standstill at 0\({}^{\circ}\). 7. The oscilloscope records the current waveforms of the voltage and current of Phase 3. 8. Exciting Phase 1 by a pulsed voltage and exciting Phases 2 and 3 simultaneously by the same voltage to maintain the position. 9. The oscilloscope records the current waveforms of the voltage and current of Phase 1. According to the measured phase voltage and current, the flux-linkage is calculated as follows:  
\[\psi(k)\left|{}_{\theta=const}\right.=\sum_{j=1}^{k}\left[u(j)-R_{s}i(j) \right]T_{s}+\psi(0) \tag{1}\]  
where \(\psi(k)\) is the flux-linkage characteristic of the phase winding at a specific position \(\theta\). \(u\), \(i\) and \(R_{s}\) are phase voltage, phase current and stator phase resistance, respectively. \(k\) is the number of sampling points, and \(T_{s}\) is the sampling period. \(\psi(0)\) is the initial value of flux-linkage of phase winding. Because there is no permanent magnet in SRM, the initial value of flux-linkage \(\psi(0)\) is zero. Due to symmetry among the phases, the flux-linkage characteristics of different phases are the same, except for the phase shift, which is 15\({}^{\circ}\) for three-phase 12/8 pole SRM. Note that the definition of rotor position is for Phase 1. Therefore, the relationship among flux-linkage curves of three phases can be derived as:  
\[ \begin{cases}\psi_{2}(22.5^{\circ})=\psi_{1}(7.5^{\circ})\\ \psi_{3}(0^{\circ})=\psi_{1}(15^{\circ})\end{cases}  \tag{2}\]  
Combining Equations (1) and (2), the flux-linkage characteristics of the considered SRM at 0\({}^{\circ}\), 7.5\({}^{\circ}\), 15\({}^{\circ}\) and 22.5\({}^{\circ}\) are obtained and shown in Figure 3.  
It should be noted that more than one phase is excited at the same time for measurement of the flux-linkage curves at 0\({}^{\circ}\), 7.5\({}^{\circ}\) and 15\({}^{\circ}\), which will result in mutual inductance between the excited phases. Although the mutual inductance introduces some errors to the measurement results,  
Figure 2: Flux-linkage measurement circuit.