**2. Principal of operation of magnetic gears, available topologies, and design approaches**

A classic gear, meaning a mechanical one, which relies on the contact of two wheels with different number of iron teeth (see **Figure 3** left), has some important drawbacks. First of all, physical contact is needed, which involves local friction and heat (and consequently losses). Besides, lubrication is needed from time to time, in appropriate quantity. Moreover, the teeth are physically and irreversibly damaged with time due to material friction and fatigue. Thus, the losses increase after several hours/years of operation. All these disadvantages are elimi‐ nated in the case of a magnetic gear (MG). Above all, another important benefit of MG use should be indicated here: in order to obtain high transmission ratio, one should use complex gears, with more than two integrated wheels. Usually, two or more mechanical gears are cascaded (linked) in the transmission chain. Such configurations drastically affect the power density and the efficiency of mechanical transmission. This is not the case for MGs: the most important amount of loss is found in the iron loss component, while the mechanical one (due to friction with the air and on the bearings) is in the same range as that of mechanical gears.

**Figure 3.** The classical (left) and magnetic (right) gears.

The first reference of a gear without mechanical contact dates back to 1916 [20, 34]. Since then, the benefits of MG were highlighted, namely: no need for lubrication, no local heating (no friction losses), and no risk of breaking the elements used to transfer power [20–41]. However, the first built viable solution dates to the 1980s (see a similar configuration of such an MG in **Figure 3** right)—evaluated from the torque capability point of view in [35]—but its efficiency was somewhere in the 25–30% range (only a part of magnetic poles was in active magnetic contact). The first efficient MG solution that fully exploits all the gear magnets dates back to early 2000 [20]—see **Figure 4**. Numerous other configurations of MGs have been proposed with time [21–41]. Some of these variants will be shown here in order to have a clue about the state of the art, but before that, we will see the principal of operation of a regular MG.

**Figure 4.** Cross section of the main elements of the active parts of an efficient MG.

#### **2.1. Principal of operation**

density and the efficiency of mechanical transmission. This is not the case for MGs: the most important amount of loss is found in the iron loss component, while the mechanical one (due to friction with the air and on the bearings) is in the same range as that of mechanical gears.

The first reference of a gear without mechanical contact dates back to 1916 [20, 34]. Since then, the benefits of MG were highlighted, namely: no need for lubrication, no local heating (no friction losses), and no risk of breaking the elements used to transfer power [20–41]. However, the first built viable solution dates to the 1980s (see a similar configuration of such an MG in **Figure 3** right)—evaluated from the torque capability point of view in [35]—but its efficiency was somewhere in the 25–30% range (only a part of magnetic poles was in active magnetic contact). The first efficient MG solution that fully exploits all the gear magnets dates back to early 2000 [20]—see **Figure 4**. Numerous other configurations of MGs have been proposed with time [21–41]. Some of these variants will be shown here in order to have a clue about the

state of the art, but before that, we will see the principal of operation of a regular MG.

**Figure 4.** Cross section of the main elements of the active parts of an efficient MG.

**Figure 3.** The classical (left) and magnetic (right) gears.

786 Modeling and Simulation for Electric Vehicle Applications

The operation of the MG is based on the modulation of the magnetic field created by the rotating magnetic poles of the high-speed rotor within the iron poles of the static part [20, 33]. The main components of such MG are shown in **Figure 4**. The field developed in the static iron will interact with the field created by the magnetic poles of the low-speed (outer) rotor and will force it to run in the opposite direction [20, 33].

It has been shown that the highest torque transmission is obtained with the following equality [20]:

$$p\_{out} = N\_s - p\_{in} \tag{1}$$

where *pin, Ns*, and *pout* are the number of poles pair for the inner (high-speed) rotor, of the fixed iron part and of the outer (low-speed) rotor, respectively. The correspondences between the output (*ωout*) and input (*ωin*) speed and the gear ratio (*gr*) are:

$$
\alpha \rho\_{\rm out} = -\mathbf{g}r \cdot \alpha \rho\_{\rm in} \tag{2}
$$

$$\mathbf{g}\mathbf{r} = -\frac{\alpha\_{\text{out}}}{\alpha\_{\text{in}}} = \frac{p\_{\text{in}}}{p\_{\text{out}}} \tag{3}$$

Besides, with a specific configuration of the MG, it is possible to obtain smooth mechanical characteristics—the lowest possible torque ripples are obtained if the ripples coefficient (kr) equals unity (the ideal case) [33].

$$k\_r = \frac{\mathbf{2} \cdot \mathbf{p}\_{\text{in}} \cdot N\_s}{LCM(\mathbf{2} \cdot \mathbf{p}\_{\text{in}}, N\_s)} \tag{4}$$

where *LCM* denotes the 'least common multiple' between the number of poles of the inner rotor and the fixed iron teeth.

These are just a few elements showing the operating principle of an MG in general. The analytical modeling of different MGs found in the literature will be summarized later, after the presentation of a detailed state of the art on the existent configurations.

#### **2.2. State of the art of magnetic gears with fixed or variable transmission ratio**

The first efficient MG, having radial configuration, was proposed by Attalah [20]. Next, other variants have been proposed. For example, in [34] a concentrated flux variant (or the so-called *spoke configuration*) was considered for the inner (high-speed) rotor; for the low-speed and hightorque rotor, the surface-mounted variant is almost exclusively used—because the surfacemounted topology proposes the best power density, and because at low speed the risk of magnet's detaching is reduced. Such a topology, like the one presented in [34], is shown in **Figure 5**. This variant is again of radial flux. Thus, one could imagine all types of configurations for the high-speed rotor, which are usually used for classic permanent magnet synchronous machines (PMSM)—see different rotor configurations in **Figure 6**.

**Figure 5.** Cross section of flux-concentrated MG (spoke variant for the high speed rotor).

**Figure 6.** Possible rotor configurations for the radial flux MG: partially inset (left) and buried magnet (right) variants.

Since in an MG we are not interested in the flux-weakening capabilities of the structure itself, the structure with surface-mounted magnets based on Nd-Fe-B material on the high-speed rotor (shown in **Figure 4**) is almost exclusively used. When cheaper magnet material such as the ferrite is used, the concentrated flux (or spoke) variant can be considered (**Figure 5**). The partially or entirely inset magnet variants in **Figure 6** could be appropriate solutions when very high speeds are to be considered, in order to avoid the use of a consolidating ring of carbon or Titan material—which increases the air gap of the MG and drastically reduces its capability of producing the torque.

Axial flux variants [28, 29], such as the ones shown in **Figure 7**, have also been analyzed, since the axial configuration offers the best power density (due to reduced volume and weight on the passive elements: shaft and housing).

**Figure 7.** Axial flux MG configuration.

mounted topology proposes the best power density, and because at low speed the risk of magnet's detaching is reduced. Such a topology, like the one presented in [34], is shown in **Figure 5**. This variant is again of radial flux. Thus, one could imagine all types of configurations for the high-speed rotor, which are usually used for classic permanent magnet synchronous

machines (PMSM)—see different rotor configurations in **Figure 6**.

808 Modeling and Simulation for Electric Vehicle Applications

**Figure 5.** Cross section of flux-concentrated MG (spoke variant for the high speed rotor).

capability of producing the torque.

the passive elements: shaft and housing).

**Figure 6.** Possible rotor configurations for the radial flux MG: partially inset (left) and buried magnet (right) variants.

Since in an MG we are not interested in the flux-weakening capabilities of the structure itself, the structure with surface-mounted magnets based on Nd-Fe-B material on the high-speed rotor (shown in **Figure 4**) is almost exclusively used. When cheaper magnet material such as the ferrite is used, the concentrated flux (or spoke) variant can be considered (**Figure 5**). The partially or entirely inset magnet variants in **Figure 6** could be appropriate solutions when very high speeds are to be considered, in order to avoid the use of a consolidating ring of carbon or Titan material—which increases the air gap of the MG and drastically reduces its

Axial flux variants [28, 29], such as the ones shown in **Figure 7**, have also been analyzed, since the axial configuration offers the best power density (due to reduced volume and weight on Usually, the MG is attached at the inner rotor side to an electric motor, to transfer the desired speed and torque to a load. In order to reduce the passive elements, some researchers have proposed integrated configurations. Such configurations are usually built on in-wheel motor variants, meaning that the motor has an inner stator and an outer rotor. Next, attached to the outer rotor is the inner rotor of the MG above, the static part with iron teeth, and on top of it the second rotor of the MG. Such compact variants have been studied in [26, 30, 31, 39, 40]. The transition from in-wheel motor to integrated MG-motor is shown in **Figure 8**.

**Figure 8.** Transition from in-wheel motor to integrated MG-motor configuration [40].

Other types of MGs, such as the cycloid [27] and even the ones with variable transmission ratio [24, 29] have been proposed. Thus, we can emphasize that this topic is of real interest in the field of EVs (for the majority of applications) and power generation for wind turbines [39]. Next, the reader's attention will be focused on the existent analytical approaches that can be used for MG's design and modeling.

#### **2.3. Analytical approaches for the design of MGs**

There are three main design techniques for the analytical modeling of magnetic gears, similar to the case of electrical machines: the harmonic approach [25, 27, 30, 36], the magnetic reluc‐ tance equivalent circuit [22, 38], and vector potential algorithm [26, 28, 37, 41].

#### *2.3.1. Analytical design of MG through harmonic approach*

For the solution presented in [20], later we had the formularization of the operating principal, given by Atallah [36] and others [25, 27 , 30]. In order to express the torque produced by the two rotors rotating in opposite direction, we first need to express the flux density in the air gap. This flux density has two components: a radial and a tangential one. Also, the influence of the stationary part (the iron teeth) has to be evaluated as a modulating component.

The two components of flux density, the radial and tangential one, produced by the inner rotor of the MG, are:

$$\begin{cases} B\_{\nu\dot{n}} = \sum\_{n=1,3,5}^{\nu} a\_n \cdot \cos(n \cdot p\_{\dot{n}} \cdot (\theta - \alpha\_{\dot{n}} \cdot t + \alpha\_0)) \\ B\_{\nu\dot{n}} = \sum\_{n=1,3,5}^{\nu} b\_n \cdot \sin(n \cdot p\_{\dot{n}} \cdot (\theta - \alpha\_{\dot{n}} \cdot t + \alpha\_0)) \end{cases} \tag{5}$$

where *an, bn*, and *α<sup>0</sup>* are the Fourier coefficients and the initial shift angle of the rotor. Next, the modulating component of the fixed iron teeth, similarly obtained, is function of Fourier coefficients and initial shift angle of this armature, meaning *an, bn*, and *α0*:

$$\mathcal{Z} = \mathbf{c}\_0 + \sum\_{m=1}^{n} \mathbf{c}\_m \cdot \cos(m \cdot N\_s \cdot (\theta + \beta\_0)) \tag{6}$$

Similar to (5), one can get the flux density components to the outer rotor of the MG. In [30] some detailed formulation was given for the harmonic components with the pole pair number equal to (*n ⋅ pin*) and (*m ⋅ Ns - n ⋅ pout*), or equal to (*n ⋅ pout*) and (*m ⋅ Ns - n ⋅ pin*).

The torque produced by the inner and outer rotor is a function of the pull-out torque, *Tmin* and *Tmout* (*γ0* is the shift angle of the outer rotor of the MG) [30]:

$$\begin{cases} T\_{in} = T\_{\text{min}} \cdot \sin\left(\frac{N\_s \cdot \beta\_0 - p\_{\text{out}} \cdot \gamma\_0 - p\_{\text{in}} \cdot \alpha\_0}{p\_{\text{in}}}\right) \\\\ T\_{\text{out}} = T\_{\text{max}} \cdot \sin\left(\frac{N\_s \cdot \beta\_0 - p\_{\text{out}} \cdot \gamma\_0 - p\_{\text{in}} \cdot \alpha\_0}{p\_{\text{out}}}\right) \end{cases} \tag{7}$$

#### *2.3.2. Analytical design of MG through magnetic reluctance equivalent circuit*

Based on the literature research, we have found a first approach on MG modeling through magnetic reluctance equivalent circuit in [22]. This approach is interesting because it also takes into account the steel material characteristic (through curve fitting). Actually, this paper was an adaptation of the method presented in [38] where the analytical design approach was applied to a brushless permanent magnet machine.

The method consists in computing the torque, as a function of a flux (*Ψm*) and magneto-motive force (*Fm*) product, produced in each element of the equivalent circuit. Such a magnetic reluctance equivalent circuit is shown in **Figure 9**.

$$T = \frac{n\_{\theta}}{2 \cdot \pi} \cdot \sum\_{j=1}^{n\_{\theta}} \Psi\_{m^j\_j} \times \Delta F\_{m^j\_j} \tag{8}$$

where *nθ* is the number of elements of reluctances on the considered equivalent circuit.

**Figure 9.** Magnetic reluctance equivalent circuit for MG design [22].

*2.3.1. Analytical design of MG through harmonic approach*

8210 Modeling and Simulation for Electric Vehicle Applications

1,3,5

¥ = ¥ =

å

*n*

*n*

c

*Tmout* (*γ0* is the shift angle of the outer rotor of the MG) [30]:

min

*out mout*

*in*

applied to a brushless permanent magnet machine.

sin

sin

*2.3.2. Analytical design of MG through magnetic reluctance equivalent circuit*

ì

ï í

î

1,3,5

å

of the MG, are:

For the solution presented in [20], later we had the formularization of the operating principal, given by Atallah [36] and others [25, 27 , 30]. In order to express the torque produced by the two rotors rotating in opposite direction, we first need to express the flux density in the air gap. This flux density has two components: a radial and a tangential one. Also, the influence

The two components of flux density, the radial and tangential one, produced by the inner rotor

cos( ( ))

qw

sin( ( ))

where *an, bn*, and *α<sup>0</sup>* are the Fourier coefficients and the initial shift angle of the rotor. Next, the modulating component of the fixed iron teeth, similarly obtained, is function of Fourier

0 0

Similar to (5), one can get the flux density components to the outer rotor of the MG. In [30] some detailed formulation was given for the harmonic components with the pole pair number

The torque produced by the inner and outer rotor is a function of the pull-out torque, *Tmin* and

0 00

 ga

*in s out in*

*p*

Based on the literature research, we have found a first approach on MG modeling through magnetic reluctance equivalent circuit in [22]. This approach is interesting because it also takes into account the steel material characteristic (through curve fitting). Actually, this paper was an adaptation of the method presented in [38] where the analytical design approach was

*s out in*

*p*

0 00

 ga

*out*

cos( ( ))

=+ × × ×+ å *m s*

qw

q b 0

(5)

(7)

 a

0

*c c mN* (6)

 a

of the stationary part (the iron teeth) has to be evaluated as a modulating component.

= × × × - ×+ ï

*B a np t*

*rin n in in*

<sup>ï</sup> = × × × - ×+ <sup>ï</sup>

*B b np t*

*tin n in in*

coefficients and initial shift angle of this armature, meaning *an, bn*, and *α0*:

1

equal to (*n ⋅ pin*) and (*m ⋅ Ns - n ⋅ pout*), or equal to (*n ⋅ pout*) and (*m ⋅ Ns - n ⋅ pin*).

b

*Np p T T*

*Np p T T*

<sup>ì</sup> æ ö × - ×- × <sup>ï</sup> = × ç ÷ ï è <sup>ø</sup> <sup>í</sup>

b

ï æ ö × - ×- × <sup>ï</sup> = × ç ÷ <sup>î</sup> è ø

¥ =

*m*

The magnetic flux is a function of the magnetic reluctance, which can be expressed through curve fitting for each specific material. For a certain iron sheet, the magnetic reluctance can be expressed as:

$$R(\Psi\_m) = a\_1 \cdot \frac{h}{S^1} \cdot \Psi^0 + a\_n \cdot \frac{h}{S^n} \cdot \Psi^{n-1} \tag{9}$$

where *h* is the length of the flux for a specific trajectory and the *S* is the area of the flux on a specific element of the geometry—*a* or *n* index varies the function of material. These reluctances (and finally the fluxes) are calculated on radial and tangential direction.

The magneto-motive force, *Fm*, can be computed as well on each specific element, recalling that:

$$F\_m = H\_m \cdot h$$

where *Hm* is the magnetic field intensity calculated on each element of the magnetic circuit.
