2.2.3. Iron Losses

For the calculation of the iron losses in the HWM, which includes losses due to higher harmonics, the time-dependent magnetic flux density in the stator and rotor lamination is of interest. The calculation of tooth and yoke flux densities in the HWM is done by an integral approach. Since a local resolution of the iron losses in the stator and rotor laminations is also not possible in the HWM, the flux density in tooth and yoke is assumed to be constant, as in the magnetic ECD used in the FWM, and its local mean value is determined accordingly. For this purpose, it is assumed for the determination of the tooth flux density *B*T, that the entire magnetic flux in the area of a slot pitch *τ*<sup>N</sup> flows through the corresponding tooth.

**Figure 3.** Simulated air gap flux density of an IM with linear stator and rotor magnetic material properties using the T-FEM and HWM.

By integrating the air gap flux density *B<sup>δ</sup>* over a stator or rotor slot pitch, the magnetic flux Φ<sup>T</sup> in the respective tooth can be determined. Since no location dependence in the tooth is considered, the area integral for calculating the tooth flux from the tooth flux density can be replaced by a multiplication with the toot width *b*<sup>T</sup> and active axial length *l*Fe so that the tooth flux density ΦT(*t*) follows from

$$
\pi\_{\rm N} = \frac{2\pi}{N}\_{\prime} \tag{3}
$$

$$\Phi\_{\rm T}(t) = l\_{\rm Fe} \int\_{\tau\_{\rm N}} B\_{\delta}(\Theta, t) r \, d\Theta\_{\prime}$$

$$= l\_{\rm Fe} b\_{\rm T} B\_{\rm T}(t), \tag{4}$$

with *N* being the number of slots and *r* being the middle radius of the air gap.

The assumption underlying the calculation of the yoke flux densities *B*<sup>Y</sup> describes that the magnetic flux is equally distributed over both paths in the yoke via a pole pitch, which is why half the air gap flux is present in each case. By converting the area integral for the determination of the yoke flux from the yoke flux density into a multiplication with the yoke height *h*<sup>Y</sup> and *l*Fe the yoke flux density results in

$$
\pi\_{\mathbb{P}} = \frac{\pi}{p} \tag{5}
$$

$$\Phi\_{\rm Y}(t) = \frac{l\_{\rm Fe}}{2} l\_{\rm Fe} \int\_{\tau\_{\rm Fe}} B\_{\delta}(\Theta, t) r \, d\Theta$$

$$= l\_{\rm Fe} h\_{\rm Y} B\_{\rm Y}(t). \tag{6}$$

#### *2.3. Extended Harmonic Wave Model*

The neglect of the iron saturation in the HWM according to [4–6] represents an essential limitation in the operating point calculation of the IM. Therefore, an E-HWM is introduced in this paper, which provides an approach to account for the iron saturation.

#### 2.3.1. Approach

If the influence of the iron saturation on the harmonics is to be considered, the flattening of the hysteresis curve *B*(*H*) in the nonlinear region and the resulting flattening of the air gap flux density must be modeled, which is shown for the idealized case of a sinusoidal air gap flux density in the middle plot of Figure 4. This can be realized mathematically by a circumferential location Θ dependent description of an effective air gap. Here, as a consequence of the main field saturation, the air gap is increased on average by a saturation factor *k*<sup>h</sup> ≥ 1. In the region of large iron saturation, i.e., at the maximum of the air gap flux density *Bδ*, the air gap is increased by another saturation factor *k*h1 and

decreased at zero crossings. Thus, the time- and location-dependent air gap conductance function shown in the left plot of Figure 4 can be defined. This moves synchronously with the fundamental wave field and therefore results in

$$
\lambda\left(\Theta, t\right) = \frac{1}{k\_{\rm h}} - \frac{1}{k\_{\rm h1}} \cos\left(\frac{2\pi\Theta}{\tau\_{\rm P}} - 2\omega t\right),
\tag{7}
$$

where the factor of two in the cosine argument is a consequence of the simultaneous iron saturation by the north and south poles of the air gap field. The flattened airgap flux density *Bδ*,sat follows from multiplying the original airgap flux density by the airgap conductance function given by

$$B\_{\delta, \text{sat}}(\Theta, \mathbf{t}) = B\_{\delta}(\Theta, \mathbf{t}) \cdot \lambda(\Theta, \mathbf{t}). \tag{8}$$

In the right plot of the Figure 4 exemplary flux density curves simulated by a TH-FEM, a HWM, and an E-HWM are shown. Here the effect of the flattening of the flux density by the application of the air gap conductance function can be seen.

**Figure 4.** Approach of the consideration of saturation in the E-HWM.

2.3.2. Calculation of the Air Gap Conductance Function

The derivation of the saturation factors *k*<sup>h</sup> and *k*h1 is done using the FWM. By dividing the amplitude of the air gap flux density *B*!*δ*,FWM of the FWM by the fundamental wave *B*!*δ*,1,HWM of the HWM the scaling factor

$$s\_{\text{max}} = \frac{B\_{\delta, \text{FWHM}}}{\widehat{B}\_{\delta, 1, \text{FWHM}}} \tag{9}$$

at maximum saturation can be calculated. This factor thus represents the minimum of the air gap conductance function. Assuming a cosine fundamental wave, the amplitude <sup>1</sup> *k*h1 can be expressed as

$$\frac{1}{k\_{\text{h1}}} = \arccos(s\_{\text{max}}) \frac{4}{2\pi} s\_{\text{max}} \tag{10}$$

and thus describes the part of a period that the magnitude of the fundamental wave of the HWM is above the amplitude of the air gap flux density of the FWM, related to the scaling factor at maximum saturation. The larger this part, the greater the difference between the saturated and unsaturated regions, and thus the amplitude of the air gap conductance function. The mean value <sup>1</sup> *<sup>k</sup>*<sup>h</sup> follows from the addition of the minimum and amplitude. Therefore *k*<sup>h</sup> can be expressed as

$$k\_{\rm h} = \frac{1}{1 + \arccos(s\_{\rm max})\frac{4}{2\pi i}} \cdot \frac{1}{s\_{\rm max}}.\tag{11}$$

To consider the saturation effects on the curves of the flux densities in the HWM, the calculated time and local curves of the air gap flux density, as well as the time curves

of the tooth and yoke flux densities, can be multiplied by the function resulting from (7). For the latter progressions, the scaling factors at maximum saturation must thereby be calculated with the amplitudes of the tooth and yoke flux densities of the FWM and HWM, respectively, resulting in different scaling functions than in the air gap.

The introduction of saturation phenomena in the iron loss calculation in the E-HWM is based on the scaling of existing harmonics as well as the consideration of additional saturation harmonics [7]. From (8), with the help of trigonometric relations, the scaling of existing harmonics can be converted to

$$
\hat{B}\_{1, \text{sat}} = \left(\frac{1}{k\_{\text{h}}} - \frac{1}{2k\_{\text{h}1}}\right) \cdot \hat{B}\_{1} \tag{12}
$$

$$
\hat{B}\_{\rm n,sat} = \frac{1}{k\_{\rm h}} \cdot \hat{B}\_{\rm n} \tag{13}
$$

where the fundamental is scaled differently from the other harmonics. The derivation of the additional saturation harmonics is done analogously. The flattening of the air gap flux density results in particular in a dominant third harmonic, which is calculated differently from the other saturation harmonics. In general, the *n*-th harmonic results in two additional saturation harmonics with the order *n* ± 2. Their amplitudes result in

$$
\widehat{B}\_3 = \frac{1}{1 - 2\frac{k\_{\rm h1}}{k\_{\rm h}}} \cdot \widehat{B}\_{1,\rm sat} \tag{14}
$$

$$
\hat{B}\_{\text{n}\pm 2} = -\frac{1}{2k\_{\text{h}1}} \cdot \hat{B}\_{\text{n,sat}}.\tag{15}
$$

Since a change in the flux densities also results in new induced currents in the rotor, an iterative adaptation in the E-HWM is required. For this purpose, starting from the scaled flux densities, the rotor current is updated, which in turn changes the flux densities. The updated flux densities are scaled again, and so on. To adjust the induced rotor current, a constant scaling factor is applied according to

$$s\_{\rm ind} = \overline{\lambda} = \frac{1}{k\_{\rm h}} \tag{16}$$

by which the inductances are multiplied. Based on the scaled inductances, the rotor current can be updated. By integrating a relaxation factor, the convergence behavior of the successive substitution can also be improved. In Figure 5 the same operating point of an exemplary IM as in Figure 3 is simulated using non linear magnetic material properties. The air gap flux density compared to the linear material in Figure 3 differs strongly. Nevertheless, the use of the E-HWM makes it possible to simulate the saturationdependent flux density approximately well.

**Figure 5.** Simulated air gap flux density of an IM with non-linear stator and rotor magnetic material properties using the T-FEM and E-HWM.
