**2. Operation principle of FRPMM**

To clearly exhibit the operating principle, a three-phase FRPMM with two pole windings, six stator slots, and eight rotor teeth is cited as an example. The flux distributions at different rotor positions are illustrated in **Figure 2**. The magnetic flux field is excited only by the PMs, and the difference of each rotor movement is 11.25 mech. degrees (i.e., 1/4 rotor slot pitch). Taking flux linkage of phase A winding as an example, when the rotor position is 0 degree, the flux linkage is 0; when the rotor position is 11.25 mech. degree (90 elec. degree), the flux linkage reaches the positive maximum value; when the rotor position is 22.5 mech. degree (180 elec. degree), the flux linkage is 0; when the rotor position is 33.75 mech. degree (270 elec. degree), the flux linkage reaches the negative maximum value. Therefore, in the duration of one rotor slot pitch (360 elec. degrees), the winding flux linkage reverses the polarity, thus it is called "flux reversal machine." Then,

**3. Sizing equation of FRPMM**

*DOI: http://dx.doi.org/10.5772/intechopen.92428*

flux density, back-EMF, and torque will be deduced.

polarity in these two cases is just opposite to each other.

In order to derive the sizing equation of FRPMMs, the magnetic circuit model should be built at first; then, based on the model, the analytical equations of airgap

The equivalent magnetic circuit model can be plotted as **Figure 4**. At No.1 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(b)**, that is, the rotor tooth is closer to the S-pole magnet. The S-pole magnetic generates two paths of magnetic flux, one is pole leakage flux *Φpl*, which goes through the adjacent N-pole magnet, the other is main flux *Φm*, which goes through the stator tooth, stator yoke, rotor tooth, and rotor yoke, thus can provide winding flux linkage and back-EMF. At No. 2 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(c)**, that is, the rotor axis is at the same distance from the S-pole and N-pole magnets. Thus, at this time, the two magnets can only generate one magnetic flux path, that is, the pole leakage flux *Φpl*. At No. 3 stator tooth, its magnetic field distribution corresponds to the position shown in **Figure 2(d)**, that is, the rotor tooth is closer to the N-pole magnet. The N-pole magnetic generates two paths of magnetic flux, one is pole leakage flux *Φpl*, which goes through the adjacent S-pole magnet, the other is main flux *Φm*, which goes through the stator tooth, stator yoke, rotor tooth, and rotor yoke, thus can provide winding flux linkage and back-EMF. It should be noted that the magnetic flux path of No. 1 stator tooth is just opposite to that of No. 3 stator tooth, so winding flux

As mentioned above, **Figure 4** provides the magnetic circuit of FRPMMs, which

can help analyze the flux distribution of FRPMMs at different rotor positions. However, the magnetic circuit requires the establishment of the whole FRPMM

**3.1 Magnetic circuit model**

*Flux Reversal Machine Design*

**Figure 4.**

**71**

*Equivalent magnetic model of FRPMMs.*

**Figure 2.**

*No-load flux lines of the FRPMM excited by the PMs: (a) rotor position =0 elec. degree; (b) rotor position =90 elec. degree; (c) rotor position =180 elec. degree; (d) rotor position =270 elec. degree.*

**Figure 3.** *Variation of flux linkage of phase a winding at different rotor positions.*

after obtaining the bipolar flux linkage, as shown in **Figure 3**, the winding can produce a bipolar back-electromagnetic motive force (EMF). If the armature windings are injected with currents having the same frequency and phase with the back-EMF, a steady torque can be yielded.
