*2.3. Idea of the Polyharmonic Field-Circuit Model Accounting for Nonsinusoidal Supply*

The analysis of the influence of higher harmonics on the operation of the squirrel cage induction motor was the subject of very intensive research that has been carried out by various authors over the last few decades [34–38]. In the context of high-speed machines with solid rotors, this issue is of particular importance due to the significant influence of the magnetic field higher harmonics on the machine operation, both resulting from the power supply and core slotting. As shown in previous works by the authors [30,31], the influence of the permeance (slot) harmonics of the magnetic field can be successfully taken into account by applying a non-linear multi-harmonic field-circuit model. A detailed analysis performed in those works proved that through the appropriate formulation of the so-called effective magnetic permeability, it is possible to use a superposition of several field rotor models associated with the appropriate harmonics of the magnetic field distribution in the air gap of the machine that are strongly coupled with the stator model. The results of this analysis prompted the authors to extend their deliberation over the case of the nonsinusoidal power supply using the same main concepts of the model. As an extension of the models presented in [30,31], the construction of a non-linear multiharmonic model of an induction machine with a solid rotor, taking into account both the higher harmonics of the voltage supply and higher permeance harmonics, can be presented in the following form:


(IV) Calculate the effective magnetic permeability distribution *μeff* for the stator and rotor core using the DC magnetization characteristics and the formula [31]:

$$
\mu\_{eff}(H\_{h1}, H\_{h2}, \dots, H\_{hN}) = \frac{\sqrt{B\_{h1}^2 + B\_{h2}^2 + \dots + B\_{hN}^2}}{\sqrt{H\_{h1}^2 + H\_{h2}^2 + \dots + H\_{hN}^2}},\tag{3}
$$

where: *Hhn*—amplitude of the magnetic field strength related to *hn* supply voltage harmonic, *Bhn*—magnetic flux density amplitude related to *hn* supply voltage harmonic:

$$\begin{aligned} B\_{\text{lin}} &= \frac{2}{\pi} \int \mu\_{\text{DC}} (H\_{\text{h1}} \sin h1 a + H\_{\text{h2}} \sin h2 a + \dots + H\_{\text{h1}} \sin hNa) (H\_{\text{h1}} \sin h1 a \\ &+ H\_{\text{h2}} \sin h2 a + \dots + H\_{\text{hN}} \sin hNa) \sin hn a da \end{aligned} \tag{4}$$

(V) Create and solve *N* of independent multi-harmonic linear field-circuit models, each including *M* of spatial harmonics of the magnetic field strength [30,31]:

$$
\begin{bmatrix}
\mathbf{M}\_{11} \begin{pmatrix} \mu\_{\ell f} \end{pmatrix} & \mathbf{M}\_{12} & \mathbf{M}\_{13} \\
\mathbf{M}\_{21} & \mathbf{M}\_{22} & \mathbf{0} \\
\mathbf{M}\_{31} & \mathbf{0} & \mathbf{0}
\end{bmatrix}
\begin{bmatrix}
\mathbf{0} \\
\mathbf{I}\_{-\mathrm{Im}} \\
\mathbf{A}\_{1} \\
\mathbf{A}\_{2}
\end{bmatrix} = \begin{bmatrix}
\mathbf{0} \\
\mathbf{E} \\
\mathbf{0}
\end{bmatrix},
\tag{5}
$$

where: **M**11—matrix describing the magnetic and electrical properties of materials, **<sup>M</sup>**<sup>12</sup> <sup>=</sup> <sup>−</sup>**M**<sup>T</sup> <sup>21</sup>/(j*ωlz*) —matrix describing the distribution and connection method of the stator winding, **M**<sup>13</sup> = **M**<sup>T</sup> 31—matrices describing coupling between the rotor models and the stator model [23], **<sup>M</sup>**22—stator winding impedance matrix, <sup>ϕ</sup>**\_** *hn* —vector of the nodal values of the complex magnetic vector potential for the model associated with *hn* harmonic of the supplying voltage, **I \_S***hn* —vector of the amplitudes of the loop currents in the stator winding due to *hn* harmonic of the supplying voltage, λ **\_** *hn* —vector of complex circulations of the magnetic field strength vector for the model associated with *hn* harmonic of the supplying voltage, **E \_S***hn* —vector of the complex voltage amplitudes in the loops in the stator winding circuit associated with *hn* harmonic of the supplying voltage.


The above procedure requires some additional comments. Firstly, it is assumed here that the nonlinearity of the stator winding currents must be considered as a superposition of nonlinear effects on the individual harmonic waveforms multiplied or divided by a time-invariant function (impedance related with magnetic permeability dependent only on magnitudes of magnetic quantities). The above means that the proposed method does not account for the saturation harmonics of magnetic flux and thus of current. In a solid rotor induction motor the saturation harmonics are, however, not significant due to the large value of inductance of the stator winding, and as shown in [30,31], the calculation results obtained in this way are very close to the ones coming from comprehensive timedomain computations.

It is still assumed that to determine the effective magnetic permeability, the higher harmonics of the magnetic field strength are to be used, and not the higher harmonics of flux density. Due to the fact that each model associated with a given voltage harmonic is a multiharmonic model that includes *M* of spatial field harmonics, it is assumed that the magnetic field strength calculated in step (VI) is the RMS value of the magnetic field strength from *<sup>M</sup>* considered spatial harmonics, multiplied by <sup>√</sup>2. In addition, each model related to

the corresponding supply voltage harmonic must be formulated for the corresponding slip reference value related to the given voltage harmonic and the symmetrical voltage system it creates (positive, negative or zero). Theoretically, the proposed approach allows for taking into account any number of higher harmonics, because individual models are solved independently and their coupling is established only through their common value of the effective magnetic permeability, calculated after solving all particular models. Thus, the solution can be executed simultaneously on distributed or parallel systems. In [31], the effective magnetic permeability was determined as a multi-dimensional look-up table. This type of approach provides accurate results with a small (less than or equal to three) number of harmonics included. For a larger number of considered harmonics, in order to avoid large sizes of data files or interpolation errors, it is necessary to elaborate more suitable functions that calculate the value of effective magnetic permeability with the use of numerical integration procedures.
