*4.1. Introduction*

Precise calculation of the working of the generator can only be achieved with the finite elements method. However, here we use analytic relations with the aim of making many calculations and making some comparisons between the Vernier structure and synchronous machines with many poles. Most of these analytic relations can be found in [10]. In particular, for the calculation of the torque, we use the relation:

$$T\_{em} = 4 \cdot \pi \cdot R\_c^3 \cdot K\_f \cdot F\_S \tag{10}$$

*Re*: bore radius (m) and *Kf*, form-factor, defined as the ratio: *L*/(2*Re*), *L* being the laminations' length.

For the synchronous machine, if the machine is driven so that electromagnetic force *FS* is in phase with current:

$$F\_S = \frac{\sqrt{2}}{2} \cdot \mathcal{M} \cdot \lambda\_0 \tag{11}$$

*M*: remanence of rotor magnets (T); λ0: length density of current (A/m), connected to the armature magnetic field.

For the Vernier structure:

$$F\_S = M \cdot \frac{a}{a+e} \cdot H \cdot K\_S \tag{12}$$

*a*: thickness of magnets (m); *e*: air gap (m); *H*: magnetic field density of armature reaction (A/m); *KS*: coupling coefficient, function of the waveform of feed currents and of the geometric proportions of the machine structure.

We can define an elementary domain, as in Figure 7. This last one is a repetitive cell of the structure. It contains a stator slot and one alternate magne<sup>t</sup> couple [11].

**Figure 7.** Magnet–slot interaction in the elementary domain.

This elementary domain is defined by five dimensional parameters which have L = domain length (from slot to slot), a = magne<sup>t</sup> thickness, lf = slot width, e = air gap thickness, and h = domain height. We have the normalized domain with α, ε, 2 and s. It has been demonstrated that the average tangential force for an elementary domain can be written as:

$$F\_S = M \cdot \frac{\alpha}{\alpha + \varepsilon} \cdot H \cdot K\_S \tag{13}$$

We have the characteristics on Figure 8.

**Figure 8.** Coupling coefficient *KS*.

#### *4.2. Estimate of Losses and Temperature Rise Product*

To design the generator driven by the vertical axis wind turbine, we performed analytical sizing for a rapid overview of the different feasible PMVM with the rated values TTn, NTn and PTn. The calculations are made by taking into account thrusts deduced from geometric, electromagnetic, and thermal constraints.

Copper losses are calculated with the relation (14)

$$P\_{copper} = \rho \cdot \frac{m\_{copper}}{\rho\_{copper}} \cdot J^2 \tag{14}$$

*ρ*: copper resistivity (2 × 10−<sup>8</sup> Ω·m); *mcopper*: copper mass in the machine (kg); *ρcopper*: copper density (8.9 × 10<sup>3</sup> kg/m3); *J*: surface density of current in stator winding (A/m2).

Core losses are calculated on the basis of data given for laminations designed to work at 400 Hz, by using the relation given in [12], valid with sinusoidal waveforms:

$$P\_{\rm iron} = \left(4 \cdot k \cdot B\_{\rm max}^2 \cdot f + 2 \cdot \pi^2 \cdot a \cdot B\_{\rm max}^2 \cdot f^2\right) \cdot \frac{m\_{\rm iron}}{\rho\_{\rm iron}} \tag{15}$$

*miron*: laminations mass; *ρiron*: alloy density (7.6 × 10<sup>3</sup> kg/m3); *f*: rated frequency (Hz); *B*max: maximum flux density in the laminations (1.5 T).

As the rated frequency is often next to 200 Hz, we will use 0.2 mm thick laminations. With this thickness: α = 6.7 × 10−<sup>3</sup> and *k* = 58.

#### *4.3. Results of Analitical Sizing*

In terms of the permanent magnet, Neodynium–Iron–Boron Magnets were chosen because of their costs, lower specific weight and a mechanical strength much higher than samarium cobalt.

The PMVM has straight teeth while the PMSM has slot isthmus, so to compare the two machines we estimated the equivalent air gap for a PMSM without slot isthmus at a value of 0.6 mm.

From the rated values of the generator, we calculate the possible solutions, by varying:


The evolution of the torque/weight ratio as a function of the rated torque for a rated power of 1.5 kW confirms that the performances of the Vernier machine stands out of those of the synchronous machine all the more because the rated speed of the turbine is low (Figure 9). The same dimensioning can be done for a horizontal axis wind turbine or a different power output.

**Figure 9.** Torque/weight ratio versus rated torque for a 1.5 kW-rated power.

To analyze the results thus obtained, we plot the efficiency of the feasible solutions as a function of their torque-to-weight ratio (Figure 10) for the PMVM, these two parameters being of prime importance for a direct drive.

**Figure 10.** Feasible solutions with TTn = 270 Nm and NTn = 54 rpm (analytical sizing).

With the chosen two objectives, the Pareto front links the solution with the highest efficiency and the solution with the highest torque/weight ratio. As seen in Figure 7, there is a slight difference of efficiency between the solution presenting the highest torque/weight ratio and the solution with the highest efficiency. On the other hand, a small improvement in efficiency leads to a significant decrease in the torque/weight ratio. As a consequence, if the same weighting is given to both objectives, the solution with the highest torque/weight ratio is the more interesting one to implement in our wind energy system. The main characteristics of the resulting machine are listed in Table 1.


**Table 1.** Characteristics of the solutions with the highest torque-to-weight ratio (analytical sizing).

For comparison, we performed sizing on a conventional synchronous machine with the same rated values, also maximizing the torque/weight ratio. The torque/weight ratio is twice as high with the SPMVM. However, we can note the low power factor of the Vernier machine, when it is close to one for the conventional machine.

As a reminder, the calculations are made for a torque TTN = 270 N.m and a rotation speed NTN = 5 rpm.

A characteristic of Vernier machines is that they have a lower power factor than conventional synchronous machines. During motor operation it allows us to dimension the power system. The over-sizing of the power system is largely compensated by the high torque of the actuator.

With the SPMVM behaving externally like a synchronous machine without saliency. The electrical model for this machine is the same as that of a classical synchronous magne<sup>t</sup> machine. We have E electromotive force (e.m.f.), Rs stator resistance, Xs synchronous reactance and V single voltage at the stator terminals.

The Fresnel diagrams obtained with a diode rectifier and with a Power Wave Modulation (PWM) rectifier are presented in Figure 11. The phasors shown in the diagrams are fundamental quantities.

**Figure 11.** Fresnel diagram of the generator when associated with a rectifier.

These diagrams show that a low power factor will require oversizing of the PWM rectifier to achieve the same transmitted power. Concerning the diode rectifier, the torque/weight ratios presented above were obtained by assuming that the e.m.f. was in phase with the stator current, so the expected performance will not be reached at the rated current.

As the low power factor of the generator is a drawback with the two usual types of rectifiers, we want to increase its value. With the efficiency and torque/weight ratio, we then have three optimization objectives. To potentially identify others, we will now consider the energy conversion system as a whole.

#### **5. Study of the Vernier Generator and Rectifier Association**
