**3. Intrinsic performance of magnetic speed multiplier**

In a traditional speed multiplier gearbox, which transmits force via the intermediary of a mechanical contact, the sizing, in terms of transmissible torque, is dictated by considerations of wear and material strength. This is a good solution which allows compact structures, but the design must take into account many safety factors to reduce the risk of breakage, which is ultimately quite high [9].

The level of transmissible force without contact, via the intermediary of a magnetic field coupling within a magnetic gear, is obviously much weaker than with a traditional mechani‐ cal solution. On the other hand, it is not necessary to introduce safety factors that penalize the design, because there is no risk of breakage.

In order to quantify the performance level of the magnetic speed multiplier, we shall intro‐ duce tangential magnetic force density, defined by the following equation:

$$F\_{st} = \frac{F\_{\text{max}}}{S\_e} \tag{9}$$

Maximum force on the low speed rotor, Fmax, is obtained when the flux density fields created by the permanent magnets of both rotors are phase shifted at an electric angle of π/2, as shown in the figure 6.

The tangential force density can be calculated from the elementary domain of width τ1 and depth Lf . The air gap surface, Se, is equal to τ1.Lf .

Lf

low speed rotor

Nrs permanent magnets

stator

high speed rotor

: length of the rotor

Nrh permanent magnets

 

0,5 1 1,5 2 2,5 3 

magnetic slot

**Figure 6.** Position at maximum force

0

0,05

**ks**

Figure 4. Variation of the coupling coefficient in terms of α and Λ

Figure 5. Magnetic gear: operating principle

Figure 6. Position at maximum force

τ1.Lf.

It is

0,1

0,15

*NN pN s rs rh* - == (7)

(8)

*<sup>S</sup>* <sup>=</sup> (9)

Operation is possible with Ns > Nr: the high speed rotor will then move in the opposite di‐ rection to the low speed rotor; or Nr > Ns: the two rotors will rotate in the same direction. Previous studies of Vernier structures [2] show however that the first configuration, Ns > Nr,

When the low speed rotor is displaced to the value of one small magnet pitch, τ2, the high speed rotor will displace to the value of one large permanent magnet pitch, τ1. The gear ratio

Operation in synchronous mode, characterized by the above equation, is possible only if the torque on the output shaft does not exceed a maximum value. We also show the consequen‐

In a traditional speed multiplier gearbox, which transmits force via the intermediary of a mechanical contact, the sizing, in terms of transmissible torque, is dictated by considerations of wear and material strength. This is a good solution which allows compact structures, but the design must take into account many safety factors to reduce the risk of breakage, which

The level of transmissible force without contact, via the intermediary of a magnetic field coupling within a magnetic gear, is obviously much weaker than with a traditional mechani‐ cal solution. On the other hand, it is not necessary to introduce safety factors that penalize

In order to quantify the performance level of the magnetic speed multiplier, we shall intro‐

max

*F*

*e*

Maximum force on the low speed rotor, Fmax, is obtained when the flux density fields created by the permanent magnets of both rotors are phase shifted at an electric angle of π/2, as

The tangential force density can be calculated from the elementary domain of width τ1 and

.

duce tangential magnetic force density, defined by the following equation:

*st*

*F*

1 2 / / / *m high low rs rh v k N N NN k* = = = = t t

is far better, giving higher force in relation to the air gap surface.

**3. Intrinsic performance of magnetic speed multiplier**

is simply obtained by calculating the ratio of τ1 and τ2:

ces of a possible stall.

252 Advances in Wind Power

is ultimately quite high [9].

shown in the figure 6.

depth Lf

the design, because there is no risk of breakage.

. The air gap surface, Se, is equal to τ1.Lf

50000 60000 kv = 5 kv = 10 The calculation of Fst is made using a magnetic field, finite element, calculation software. Pa‐ rameters are defined from the relative thickness of the main permanent magnet, β = la/l, and the multiplication ratio, kv. The adimensional parameters of the magnetic pattern, relative to the low speed rotor, are arbitrarily assigned using the following values:

40000 *Λ* = 1, *α* = 0.2, *ε* = 0.05

30000 **Fst (N/m²)**kv =20 These values generally give a good result in the Vernier machines design [1,10].

10000 20000 The air gap on the high speed rotor is defined as equal to one tenth of the thickness of the main magnet. This dimension, however, has less importance.

0 0 0,2 0,4 0,6 0,8 1 The following figure shows the flux lines within a configuration similar to that in figure 6, for a value of kv equal to 5 and β equal to 1/2.

**beta**

**Figure 7.** Flux lines in maximum force position

The tangential force density, Fst, is represented in figure 8 versus the relative thickness, β. These are trend curves in linear mode.

It is important to note that the tangential force density level reached in the speed multiplier, from 40 to 50.103 N/m², is significantly higher than that obtained in a conventional electro‐ mechanical converter, which is closer from 10 to 20.103 N/m², at steady state and with air cooling [10]. This complex phenomenon is created by the fact that in an electromechanical converter, one of the magnetic field components, that interact to create the torque, is created τ1.Lf.

It is

by currents, with less efficiency than with permanent magnets, as is the case in the speed multiplier. This is a general principle observed in all converters using magnets. low speed rotor Figure 6. Position at maximum force

0,5 1 1,5 2 2,5 3 

magnetic slot

 

low speed rotor

Nrs permanent magnets

stator

high speed rotor

: length of the rotor

stator

Lf

permanent magnet

Lf

high speed rotor

: length of the rotor

Nrh permanent magnets

Figure 8. Tangential force density for different values of kv (Bar = 1.2 T) **Figure 8.** Tangential force density for different values of kv (Bar = 1.2 T)

0

0,05

**ks**

Figure 4. Variation of the coupling coefficient in terms of α and Λ

Figure 5. Magnetic gear: operating principle

la

l

0,1

0,15

Figure 9 shows, on the same basis, the evolution of the normal force density, Fsn = F/Se, ver‐ sus , for a value of kv equal to 10.

The force F is perpendicular to the surfaces of the air gap; it produces no motive force but tends to introduce constraints on the bearings, as we shall see later. The normal force densi‐ ty is calculated for the low speed rotor and for the high speed rotor. It is much higher on the low speed rotor side, making it impossible to balance the force on the stator.

The normal force density level, which comes from the following relationship for flux density B, is, in most cases, still well above the level of tangential force density.

0

$$F\_{\rm su} = \frac{B^2}{2 \cdot \mu\_0} \tag{10}$$

1

2' 3' 3

2' 3 3' 1

2

1'

permanent magne

1'

Figure 10. Vernier permanent magnet synchronous generator

2

polyphase winding

Figure 9. Normal force density (kv = 10) **Figure 9.** Normal force density (kv = 10)

power ratio is unrivalled in this kind of application [8].

4.

The

in figure 5, is thus the following:
