Modelling of Spatial Harmonic Interactions in a Modular PM Generator

Abstract

This study analyses the spatial harmonic interactions of the magnetic field with winding currents for a modular PM generator (MPMG) in steady-state operation. We created MPMG mathematical model in which we applied the harmonic balance method (HBM) to an unusual stator winding connection involving parallel path configurations. A star or delta stator winding configuration with two winding paths connected in parallel or in series was introduced. In this way, we obtained four different winding diagrams. However, this approach required some systematization of the analytical model. Using transformations to symmetrical components helped us to make some simplifications of the model, especially for the symmetry of load. We used the proposed model to verify the numerical calculations by performing measurements on a laboratory test stand for a single segment. We performed comparative analyses of the measurement results of the generator-phase currents for four characteristic cases of the stator winding reconfiguration, and the calculations verified the ideal qualitative convergences and satisfactory quantitative convergences.

1. Introduction

The increasing significance of renewable energy sources has driven the widespread implementation of generators excited by permanent magnets (PMs) operating at variable turbine speeds, which replace conventional systems [1,2,3,4,5,6]. The operation of wind turbines at variable rotational speeds is well established. Conversely, in small hydropower plants (SHPs), variations in the flow and head justify the use of generating systems with variable rotational speeds to increase the efficiency of energy conversion. PM-excited generators cooperating with converter systems are widely employed in this field [5,6,7]. Among the wide range of design solutions, Axial Flux PM (AFPM) machines present segmented applications, which typically include generators with a power of several kW [8,9,10]. At higher powers of tens and hundreds of kW, the design of the AFPM generators faces several limitations [11,12,13]. Radial flux PM machines (RFPMs) do not face these challenges; however, their design and construction process are more expensive. Analysis of the European hydropower market presents considerable energy potential in impoundments for head parameters ranging from 1.5 to 2.5 m and flow from 1 to 10 m³/s. For example, approximately 3000 such locations exist in Poland, which corresponds to an installed power ranging from 12 to 200 kW. Designing PM synchronous generators in this power range is very expensive due to the large share of design costs, which account for 60% of the total generator costs [7]. This was the main reason for developing a segmented generator, which helps to obtain a wide range of torques and speeds through the appropriate reconfiguration of stator windings and selection of the number of segments. The use of universal segments (RFPMs) for the construction of the generator enables a cost reduction of approximately 40% when compared with conventional generators designed for individual orders [7].

Consequently, a MPMG was designed to overcome these challenges. Figure 1 depicts a prototype of the MPMG that was built and tested by the authors in a previous study [7]. This paper extends the findings of the previous study by modelling and analysing the interactions of the spatial harmonics of the magnetic field with winding currents.

The accurate analysis of energy generation processes requires a straightforward representation of the operating behaviour of the mathematical models of PM synchronous machines. Circuit-based modelling techniques, which have been extensively used for conventional electrical machines, are well suited for this purpose [1,2,9,13,14]. This study aims to develop modelling methodologies tailored for an MPMG operating under steady-state conditions.

The harmonic balance method (HBM) [15,16,17,18,19,20,21,22] presents an efficient framework for analysing the mathematical models of electrical machines when their parameters exhibit periodic time variation. It helps to extend symbolic techniques and enables the algebraic representation of machine behaviour in steady-state conditions. When compared with the finite element method (FEM), the HBM presents a computationally efficient alternative that still captures the essential electromagnetic interactions and provides clear insight into the underlying physical phenomena [21].

In this study, we apply the HBM to model an MPMG, accounting for the effects of higher-order harmonics in the magnetic flux density, along with spatial harmonics that affect the stator winding currents. Despite the fact that the HBM is a well-established method, it requires some corrections due to the specificity of the MPMG construction. The MPMG requires unique winding connections and reconfiguration; therefore, adapting the HBM approach [15,16,17,18,19,20,21,22] requires modifications corresponding to the stator winding connecting parallel paths in series and realising the star or delta configuration of the windings. In this study, we employ the HBM for the steady-state analysis of an MPMG, as limited research has been conducted in this field. The proposed methodology, which comprehensively considers the possibilities of reconfiguration of the MPMG windings, contributes original novelty to the modelling of PM machines.

We analyse whether spatial harmonic interactions [17,21] can be distinguished and quantitatively evaluated through mathematical modelling. In this work, we assume linear machine parameters [23,24], as MPMG construction [25,26] does not typically involve significant nonlinearities. We designed the MPMG prototype model and performed a series of FEM analyses [7] which confirmed the absence of significant saturations, which justified the adoption of a linear model. It should be emphasized that the analysed MPMG structure [7] differs significantly from commonly known modular solutions [27,28,29].

Under steady-state conditions, the mathematical representation of an MPMG is reduced to a system of linear differential equations with periodically varying coefficients [15,17,21]. Solving this system using HBM presents qualitative (frequency-based) and quantitative (amplitude-based) insights into the Fourier spectra of the current waveforms.

The proposed modelling methodology was verified for one MPMG prototype segment [7] on a laboratory test stand for four variants of stator winding reconfiguration.

2. Stator Winding Reconfigurations of the MPMG Segment

A flexible method for segmenting the generation modules requires the design of a generator that satisfies the power and winding requirements tailored for a specific rotational speed. As explained earlier [7], the winding reconfiguration must accommodate a speed variation range of approximately ±50%.

Such reconfiguration can be achieved without modifying the number of pole pairs, and further voltage regulation can be achieved by relying solely using a full-scale electronic converter. To maintain a high overall system efficiency, the generator voltage must be maximised at the target operating speed within the constraints imposed by the voltage limits of the converter.

The winding structure must be designed to simplify the reconfiguration process, thereby minimising the number of required reconnections. Typically, this adjustment is made prior to the final assembly of the generator, based on the rated rotational speed of the turbine. These design criteria are achieved using a winding configuration that utilises two independent paths per phase (Figure 2). Four distinct winding topologies can be realised by connecting these paths in series or parallel and selecting either a star or delta phase configuration, as shown in [7]. For instance, nominal parameters with an output power of 40 kW, current of 57.7 A, and voltage of 400 V were achieved at 300 rpm through a parallel connection of the two branches, along with a star connection of the phases, as listed in

Table 1. Parameters of the MPMG segment based on the configuration and rotational speed.

Winding Paths	Phase Connection	n (rpm)	I (pu)	U (pu)	P (pu)
Series	¦ Y	0	0.5	0	0
		150	0.5	1	0.5
	¦ Δ	150	0.87	0.58	0.5
		260	0.87	1	0.87
Parallel	||Y	260	1	0.87	0.87
		300	1	1	1
	|| Δ	300	1.73	0.58	1
		519	1.73	1	1.73

.

For individual configurations, the rated power and speeds vary, assuming that the generator generates rated voltages not exceeding 400 V for the rated state. For the ||Y configuration at a rated speed of 300 rpm, the frequency of the generated voltage (400 V) is 50 Hz. The generator can be operated in a wide range of rotational speeds from 0 to 519 rpm while maintaining a maximum voltage of 400 V, as listed in Table 1. However, operation in the individual rotational speed ranges requires the reconfiguration of the windings as listed in Table 1. The maximum value of the generator power depends on the winding configuration; it ranges from 50% of the rated power for the lowest speeds to 175% of the rated power for the maximum speed.

3. Modelling of MPMG

3.1. General Mathematical Model of Three-Phase PM Generator

The assumption of linearity of the magnetic circuit serves as the basis for the further analyses conducted in this study. ($B_{\mathrm{m}}=B_{\mathrm{r}}+{\mathsf{\mu }}_0{\mathsf{\mu }}_{\mathrm{rm}}{H}_{\mathrm{m}}$ − PM characteristic). Using Lagrange’s formula, the voltage equations of the mathematical model of the generator can be written in the standard matrix form, as follows [1,21]:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\{[\hspace{0.17em}{L}_{\sigma s}+L_s(\phi )]\cdot {\mathbf{i}}_{\mathrm{G}}\}+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\Psi }_{PM}(\phi )+R_{\mathrm{s}}\cdot \hspace{0.17em}{i}_{\mathrm{G}}={\mathbf{u}}_{\mathrm{G}}$$

(1)

where

$$L_{\sigma s}=\left[\begin{array}{ccc}{L}_{\mathsf{\sigma }\mathrm{s}}& 0& 0\\ 0& L_{\sigma \mathrm{s}}& 0\\ 0& 0& L_{\mathsf{\sigma }\mathrm{s}}\end{array}\right]\hspace{1em}{L}_s(\phi )=\left[\begin{array}{ccc}{L}_{1,1}(\phi )& L_{1,2}(\phi )& L_{1,3}(\phi )\\ L_{2,1}(\phi )& L_{2,2}(\phi )& L_{2,3}(\phi )\\ L_{3,1}(\phi )& L_{3,2}(\phi )& L_{3,3}(\phi )\end{array}\right]$$

(2)

$${\Psi }_{PM}(\phi )=\left[\begin{array}{c}{\psi }_{\mathrm{PM}\hspace{0.17em}1}(\phi )\\ {\psi }_{\mathrm{PM}\hspace{0.17em}2}(\phi )\\ {\psi }_{\mathrm{PM}\hspace{0.17em}3}(\phi )\end{array}\right]\hspace{0.17em}\hspace{1em}{R}_{\mathrm{s}}=\left[\begin{array}{ccc}{R}_{\mathrm{s}}& 0& 0\\ 0& R_{\mathrm{s}}& 0\\ 0& 0& R_{\mathrm{s}}\end{array}\right]\hspace{1em}{i}_{\mathrm{G}}=\left[\begin{array}{c}{i}_{\mathrm{G}1}\\ i_{\mathrm{G}2}\\ i_{\mathrm{G}3}\end{array}\right]\hspace{0.17em}\hspace{1em}{u}_{\mathrm{G}}=\left[\begin{array}{c}{u}_{\mathrm{G}1}\\ u_{\mathrm{G}2}\\ u_{\mathrm{G}3}\end{array}\right]\hspace{0.17em}\hspace{1em}{\displaystyle \frac{\mathrm{d}\phi }{\mathrm{d}t}}=\omega$$

(3)

• $i_a\hspace{0.17em}\mathrm{and} \hspace{0.17em}{u}_a$—stator phase, “a” current and voltage, a = 1, 2, 3;
• $R_{\mathrm{s}}$—stator winding resistance;
• ${\psi }_{\mathrm{PM}\hspace{0.17em}a}(\phi )$—flux linkage of winding “a”, produced by PM, a = 1, 2, 3;
• $L_{\mathsf{\sigma }\mathrm{s}}$,$L_{aa}(\phi )$,$L_{a,b}(\phi )$—inductance of the windings (leakage, self and mutual), a, b = 1,2,3;
• $\phi$—rotor position; $\omega$—rotor angular speed.

Typically, for the three-phase PM machines with symmetrical windings and a regularly shaped magnetic circuit, the inductance matrix can be written as follows [23,24,25]:

$$L_{\mathrm{s}}(\phi )={\displaystyle \sum _{n=0,\pm 2p,\pm 4p,\mathrm{\dots }}\hspace{0.17em}{L}_n}\hspace{0.17em}\cdot \hspace{0.17em}{\mathrm{e}}^{\mathrm{j}n\phi \hspace{0.17em}}$$

(4)

where “p” denotes the number of machine pole pairs. The vector of the PM flux linkages can be presented as follows:

$${\Psi }_{PM}(\phi )\hspace{0.17em}={\displaystyle \sum _{\varsigma =\pm p,\pm 3p,\pm 5p\mathrm{\dots }}{\Psi }_{\varsigma }^{PM\hspace{0.17em}}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}\varsigma \hspace{0.17em}\phi }}\hspace{0.17em}$$

(5)

The transformation into symmetrical components is highly effective, as it enables the machine to be represented within orthogonal reference frames. This approach enables the structured organisation of the inductance matrices and PM flux linkage vectors [15,17,21].

By applying a matrix transformation to Equation (1), we obtain

$$T_3=\frac{1}{\sqrt{3}}{\left[\begin{array}{ccc}1& 1& 1\\ 1& \underset{¯}{\mathrm{a}}& {\underset{¯}{\mathrm{a}}}^2\\ 1& {\underset{¯}{\mathrm{a}}}^2& \underset{¯}{\mathrm{a}}\end{array}\right]}_{(3x3)}\hspace{1em}\mathrm{where}\hspace{1em}\underset{¯}{\mathrm{a}}={\mathrm{e}}^{\mathrm{j}\frac{2\pi }{3}}$$

(6)

and the machine voltage equations in the symmetrical components are given as follows:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\{[L_{\sigma s}+L_s^s(\phi )]\cdot i_{\mathrm{G}}^{\mathrm{s}}\}+{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\Psi }_{PM}^s(\phi )+R_{\mathrm{s}}\cdot \hspace{0.17em}{i}_{\mathrm{G}}^{\mathrm{s}}=u_{\mathrm{G}}^{\mathrm{s}}$$

(7)

where

$$u_{\mathrm{G}}^{\mathrm{s}}=T_3\hspace{0.17em}\cdot \hspace{0.17em}{u}_{\mathrm{G}}\hspace{0.17em};\hspace{0.17em}{u}_{\mathrm{G}}^{\mathrm{s}}={[u^{\mathrm{s}0}\hspace{0.17em}{u}^{\mathrm{s}1}\hspace{0.17em}{u}^{\mathrm{s}2}]}^{\mathrm{T}}\hspace{0.17em}$$

(8)

$$i_{\mathrm{G}}^{\mathrm{s}}=T_3\cdot \hspace{0.17em}{i}_{\mathrm{G}}\hspace{0.17em};\hspace{0.17em}{i}_{\mathrm{G}}^{\mathrm{s}}={[i^{\mathrm{s}0}\hspace{0.17em}{i}^{\mathrm{s}1}\hspace{0.17em}{i}^{\mathrm{s}2}]}^{\mathrm{T}}$$

(9)

$$L_s^s(\phi )=T_3\hspace{0.17em}\cdot \hspace{0.17em}{L}_s(\phi )\hspace{0.17em}\cdot \hspace{0.17em}{T}_3^{-1}={\displaystyle \sum _{n=0,\pm 2p,\pm 4p,\mathrm{\dots }}\hspace{0.17em}{L}_n^s}\hspace{0.17em}\cdot \hspace{0.17em}{\mathrm{e}}^{\mathrm{j}n\hspace{0.17em}\phi }$$

(10)

$${\Psi }_{PM}^s(\phi )=T_3\hspace{0.17em}\cdot {\Psi }_{PM}(\phi )\hspace{0.17em}={\displaystyle \sum _{\varsigma =\pm p,\pm 3p,\pm 5p\mathrm{\dots }}{\Psi }_{\varsigma }^{PM\hspace{0.17em}s}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}\varsigma \phi }}\hspace{0.17em}={[{\psi }^{\mathrm{s}0}(\phi )\hspace{0.17em}{\psi }^{\mathrm{s}1}(\phi )\hspace{0.17em}{\psi }^{\mathrm{s}2}(\phi )\hspace{0.17em}]}^{\mathrm{T}}$$

(11)

3.2. Mathematical Model Adaptation for Stator Winding Reconfiguration of MPMG

We assume that for a star (Y) connection of the machine windings, the natural external load system comprises a four-wire system (with a neutral wire), whereas for a delta (∆) connection, the load system comprises a three-wire system. For a system comprising star (Y)-connected machine windings without a neutral wire, the neutral wire resistance in the subsequent analyses must be assumed to be very high (at least three orders of magnitude greater than the generator-phase impedance).

For the generator winding configuration connected to an external load depicted in Figure 3 and Figure 4, we employed a universal approach for writing the equations of the mathematical model. We employed the current and voltage constraint matrices for individual star and delta configurations of the generator windings to describe the generator external circuits.

$$u_{\mathrm{G}}=c_{\mathrm{u}}{e}_{\mathrm{L}}-c_{\mathrm{u}}{R}_{\mathrm{L}}\hspace{0.17em}{c}_{\mathrm{i}}{i}_{\mathrm{G}}-c_{\mathrm{u}}{L}_{\mathrm{L}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(\hspace{0.17em}{c}_{\mathrm{i}}{i}_{\mathrm{G}})-c_{\mathrm{N}}{R}_{\mathrm{N}}{c}_{\mathrm{i}}{i}_{\mathrm{G}}$$

(12)

where

$$R_{\mathrm{L}}=\hspace{0.17em}\left[\begin{array}{ccc}{R}_{\mathrm{L}1}& 0& 0\\ 0& R_{\mathrm{L}2}& 0\\ 0& 0& R_{\mathrm{L}3}\end{array}\right];\hspace{1em}{L}_{\mathrm{L}}=\hspace{0.17em}\left[\begin{array}{ccc}{L}_{\mathrm{L}1}& 0& 0\\ 0& L_{\mathrm{L}2}& 0\\ 0& 0& L_{\mathrm{L}3}\end{array}\right];\hspace{1em}{e}_{\mathrm{L}}=\left[\begin{array}{c}{e}_{\mathrm{L}1}\\ e_{\mathrm{L}2}\\ e_{\mathrm{L}3}\end{array}\right]$$

(13)

The current constraints for star and delta connection of generator windings are as follows:

$$c_{\mathrm{i}}=c_{\mathrm{i}}^{\mathrm{Y}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right];\hspace{1em}{c}_{\mathrm{i}}=c_{\mathrm{i}}^{\Delta }=\left[\begin{array}{ccc}1& 0& -1\\ -1& 1& 0\\ 0& -1& 1\end{array}\right]$$

(14)

The voltage constraints for the star and delta connection of the generator windings are given as follows:

$$c_{\mathrm{u}}=c_{\mathrm{u}}^{\mathrm{Y}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right];\hspace{1em}{c}_{\mathrm{u}}=c_{\mathrm{u}}^{\Delta }=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right]$$

(15)

The connecting constraints for the four-wire (Y) and three-wire (∆) systems are given as follows:

$$c_{\mathrm{N}}=c_{\mathrm{N}}^{\mathrm{Y}}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right]; c_{\mathrm{N}}=c_{\mathrm{N}}^{\Delta }=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(16)

Following the transformation to symmetric components, Equation (12) can be written as follows:

$$u_{\mathrm{G}}^{\mathrm{s}}=e_{\mathrm{L}}^{\mathrm{s}}-R_{\mathrm{L}}^{\mathrm{s}}\hspace{0.17em}{i}_{\mathrm{G}}^{\mathrm{s}}-L_{\mathrm{L}}^{\mathrm{s}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_{\mathrm{G}}^{\mathrm{s}}$$

(17)

where

$$e_{\mathrm{L}}^{\mathrm{s}}=T_3{c}_{\mathrm{u}}{e}_{\mathrm{L}}={[e_{\mathrm{L}}^{\mathrm{s}0}\hspace{0.17em}{e}_{\mathrm{L}}^{\mathrm{s}1}\hspace{0.17em}{e}_{\mathrm{L}}^{\mathrm{s}2}]}^{\mathrm{T}}$$

(18)

$$R_{\mathrm{L}}^{\mathrm{s}}=T_3\hspace{0.17em}{c}_{\mathrm{u}}{R}_{\mathrm{L}}{c}_{\mathrm{i}}{T}_3^{-1}+T_3{c}_{\mathrm{N}}{R}_{\mathrm{N}}{c}_{\mathrm{i}}{T}_3^{-1}$$

(19)

$$L_{\mathrm{L}}^{\mathrm{s}}=T_3\hspace{0.17em}{c}_{\mathrm{u}}{L}_{\mathrm{L}}{c}_{\mathrm{i}}{T}_3^{-1}$$

(20)

For the star configuration of the generator windings matrices, $R_{\mathrm{L}}^{\mathrm{s}}$, $L_{\mathrm{L}}^{\mathrm{s}}$ can be represented as follows:

$$R_{\mathrm{L}}^{\mathrm{s}}=R_{\mathrm{LY}}^{\mathrm{s}}=\left[\begin{array}{ccc}{\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+\hspace{0.17em}{R}_{\mathrm{L}2}+\hspace{0.17em}{R}_{\mathrm{L}3})+3R_{\mathrm{N}}& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}\hspace{0.17em}}{R}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{R}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}3})\\ {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+\underset{¯}{\mathrm{a}\hspace{0.17em}}{R}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}3})\hspace{0.17em}& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+R_{\mathrm{L}2}+R_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{R}_{\mathrm{L}3})\\ {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}\hspace{0.17em}}{R}_{\mathrm{L}3})\hspace{0.17em}& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{R}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{R}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(R_{\mathrm{L}1}+R_{\mathrm{L}2}+R_{\mathrm{L}3})\end{array}\right]$$

(21)

$$L_{\mathrm{L}}^{\mathrm{s}}=L_{\mathrm{LY}}^{\mathrm{s}}=\left[\begin{array}{ccc}{\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\hspace{0.17em}{L}_{\mathrm{L}2}+\hspace{0.17em}{L}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{L}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{L}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}3})\\ {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\underset{¯}{\mathrm{a}\hspace{0.17em}}\hspace{0.17em}{L}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}3})\hspace{0.17em}& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\hspace{0.17em}{L}_{\mathrm{L}2}+\hspace{0.17em}{L}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}}\hspace{0.17em}{L}_{\mathrm{L}3})\\ {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}2}+\underset{¯}{\mathrm{a}\hspace{0.17em}}\hspace{0.17em}{L}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\underset{¯}{\mathrm{a}\hspace{0.17em}}\hspace{0.17em}{L}_{\mathrm{L}2}+{\underset{¯}{\mathrm{a}}}^2{L}_{\mathrm{L}3})& {\displaystyle \frac{1}{3}}(L_{\mathrm{L}1}+\hspace{0.17em}{L}_{\mathrm{L}2}+\hspace{0.17em}{L}_{\mathrm{L}3})\end{array}\right]$$

(22)

By assuming a symmetrical load and introducing the notations, $R_{\mathrm{L}1}=R_{\mathrm{L}2}=R_{\mathrm{L}3}=R_{\mathrm{L}}$ and $L_{\mathrm{L}1}=\hspace{0.17em}{L}_{\mathrm{L}2}=\hspace{0.17em}{L}_{\mathrm{L}3}=L_{\mathrm{L}}$, we obtain the following equations:

$$R_{\mathrm{L}}^{\mathrm{s}}=R_{\mathrm{LY}}^{\mathrm{s}}=\left[\begin{array}{ccc}{R}_{\mathrm{L}}+3R_{\mathrm{N}}& 0& 0\\ 0\hspace{0.17em}& R_{\mathrm{L}}& 0\\ 0\hspace{0.17em}& 0& R_{\mathrm{L}}\end{array}\right];\hspace{1em}{L}_{\mathrm{L}}^{\mathrm{s}}=L_{\mathrm{LY}}^{\mathrm{s}}=\left[\begin{array}{ccc}{L}_{\mathrm{L}}& 0& 0\\ 0& L_{\mathrm{L}}& 0\\ 0& 0& L_{\mathrm{L}}\end{array}\right]$$

(23)

For the delta configuration of the generator windings matrices, $R_{\mathrm{L}}^{\mathrm{s}}$, $L_{\mathrm{L}}^{\mathrm{s}}$ can be represented as follows:

$$R_{\mathrm{L}}^{\mathrm{s}}=R_{\mathrm{L}\Delta }^{\mathrm{s}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& R_{\mathrm{L}1}+R_{\mathrm{L}2}+R_{\mathrm{L}3}& -R_{\mathrm{L}1}{\underset{¯}{\mathrm{a}}}^2+\underset{¯}{\mathrm{a}}\hspace{0.17em}{R}_{\mathrm{L}2}-R_{\mathrm{L}3}\\ 0& -R_{\mathrm{L}1}\underset{¯}{\mathrm{a}}-R_{\mathrm{L}2}{\underset{¯}{\mathrm{a}}}^2-R_{\mathrm{L}3}& R_{\mathrm{L}1}+R_{\mathrm{L}2}+R_{\mathrm{L}3}\end{array}\right]$$

(24)

$$L_{\mathrm{L}}^{\mathrm{s}}=L_{\mathrm{L}\Delta }^{\mathrm{s}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& L_{\mathrm{L}1}+L_{\mathrm{L}2}+L_{\mathrm{L}3}& -L_{\mathrm{L}1}{\underset{¯}{\mathrm{a}}}^2+\underset{¯}{\mathrm{a}}\hspace{0.17em}{L}_{\mathrm{L}2}-L_{\mathrm{L}3}\\ 0& -L_{\mathrm{L}1}\underset{¯}{\mathrm{a}}-L_{\mathrm{L}2}{\underset{¯}{\mathrm{a}}}^2-L_{\mathrm{L}3}& L_{\mathrm{L}1}+L_{\mathrm{L}2}+L_{\mathrm{L}3}\end{array}\right]$$

(25)

Assuming a symmetrical load, we obtain

$$R_{\mathrm{L}}^{\mathrm{s}}=R_{\mathrm{L}\Delta }^{\mathrm{s}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 3R_{\mathrm{L}}& 0\\ 0& 0& 3R_{\mathrm{L}}\end{array}\right];\hspace{1em}{L}_{\mathrm{L}}^{\mathrm{s}}=L_{\mathrm{L}\Delta }^{\mathrm{s}}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 3L_{\mathrm{L}}& 0\\ 0& 0& 3L_{\mathrm{L}}\end{array}\right]$$

(26)

Therefore, the equations of the external circuits of the generator are written such that they contain the currents of the generator windings for various winding configurations (star and delta). This procedure enables a general and uniform notation of the voltage equations of the mathematical model of the working of the generator connected to a load, given as follows:

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\{[L_{\mathsf{\sigma }\mathrm{s}}+L_{\mathrm{s}}^{\mathrm{s}}(\phi )+L_{\mathrm{L}}^{\mathrm{s}}]\cdot i_{\mathrm{G}}^{\mathrm{s}}\}+(R_{\mathrm{s}}+R_{\mathrm{L}}^{\mathrm{s}})\cdot \hspace{0.17em}{i}_{\mathrm{G}}^{\mathrm{s}}=e_{\mathrm{L}}^{\mathrm{s}}-{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\Psi }_{\mathrm{PM}}^{\mathrm{s}}(\phi )$$

(27)

Similarly, the voltage vector of the external circuit can be represented using a Fourier series expansion, assuming that the signals are periodic. Furthermore, assuming that the load-side electromotive forces form a balanced three-phase voltage system (an assumption that, while convenient, is not strictly necessary), this setup corresponds to a generator connected to the electrical grid. Under these conditions, the voltages expressed in terms of the symmetrical components are presented as follows:

$$e_{\mathrm{L}}^{\mathrm{s}}=T_3\hspace{0.17em}\cdot \sqrt{2}\hspace{0.17em}{E}_{\mathrm{Sph}}\left[\begin{array}{c}\cos ({\mathsf{\omega }}_0\hspace{0.17em}t+{\beta }_0)\\ \cos ({\mathsf{\omega }}_0\hspace{0.17em}t+{\beta }_0-\raisebox{1ex}{$2\mathsf{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.)\\ \cos ({\mathsf{\omega }}_0\hspace{0.17em}t+{\beta }_0-\raisebox{1ex}{$4\mathsf{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.)\end{array}\right]={\displaystyle \sum _{\eta =\pm 1}}{E}_{\eta }^s\hspace{0.33em}\cdot {\mathrm{e}}^{\mathrm{j}\eta \hspace{0.17em}{\omega }_0\hspace{0.17em}t}=E_1^s\cdot {\mathrm{e}}^{\mathrm{j}\hspace{0.17em}{\omega }_0\hspace{0.17em}t}+E_{-1}^s\cdot {\mathrm{e}}^{-\mathrm{j}\hspace{0.17em}{\omega }_0\hspace{0.17em}t}=\left[\begin{array}{c}0\\ \underset{¯}{E}\\ 0\end{array}\right]\cdot {\mathrm{e}}^{\mathrm{j}\hspace{0.17em}{\omega }_0\hspace{0.17em}t}+ \left[\begin{array}{c}0\\ 0\\ \stackrel{\vee }{\underset{¯}{E}}\end{array}\right]\cdot {\mathrm{e}}^{-\mathrm{j}\hspace{0.17em}{\omega }_0\hspace{0.17em}t}$$

(28)

where $\underset{¯}{E}=E_{\mathrm{S}}\cdot {\mathrm{e}}^{\mathrm{j}\hspace{0.17em}{\beta }_0\hspace{0.17em}}=\sqrt{\frac{3}{2}} E_{\mathrm{S}\hspace{0.17em}\mathrm{ph}}\cdot {\mathrm{e}}^{\mathrm{j}\hspace{0.17em}{\beta }_0\hspace{0.17em}}$ and $E_{\mathrm{ph}}$ denotes the RMS value of grid phase voltage (line-neutral).

3.3. Application of HBM for Modelling Spatial Harmonic Interaction in Three-Phase MPMG

The steady state is considered when the angular velocity of the rotor is constant, i.e., if $\omega =\Omega$ then $\phi =\Omega \cdot t+{\phi }_0$. If the synchronous steady-state dependence, ${\omega }_0=p\Omega$, is satisfied, the inductance matrix (10) and vector of PM flux linkages (11) become periodic. Subsequently, the solutions for the set of Equation (27) can be assumed as follows:

$$i_{\mathrm{G}}^s={\displaystyle \sum _{\nu =\pm p,\pm 3p,\pm 5p\mathrm{\dots }}}{I}_{\nu }^s\cdot \hspace{0.33em}{\mathrm{e}}^{\mathrm{j}\nu \Omega t};\hspace{1em}{I}_{\nu }^s={[{\underset{¯}{I}}_{\nu }^{\mathrm{s}0}\hspace{0.33em}{\underset{¯}{I}}_{\nu }^{\mathrm{s}1}\hspace{0.17em}{\underset{¯}{I}}_{\nu }^{\mathrm{s}2}]}^{\mathrm{T}}$$

(29)

The angle value, ${\phi }_0$, corresponds to the generator load. This angle is essential only when the generator cooperates with the power grid (for standard monoharmonic models $p{\phi }_0-{\beta }_0=\vartheta +\frac{3}{2}\mathsf{\pi }$, where $\vartheta$ denotes a generator power angle).

3.3.1. General Case (Possible External Asymmetry of the Generator)

According to the HBM [15,17,21], solution (27) satisfies an infinite-dimensional system of algebraic equations and determines the currents in the steady state:

$$\begin{gathered} \mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}3p\Omega \hspace{0.17em}{E}_{(3x3)}\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}{E}_{(3x3)}\\ -\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}{E}_{(3x3)}\\ -\mathrm{j}\hspace{0.17em}3p\Omega \hspace{0.17em}{E}_{(3x3)}\\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{cccccc}\ddots & & ⋮& ⋮& ⋮& \\ & L_{\sigma s}+L_0^{ss}+L_{\mathrm{L}}^{\mathrm{s}}\hspace{0.17em}& L_{2p}^{ss}& L_{4p}^{ss}& L_{6p}^{ss}& \dots \\ \dots & L_{-2p}^{ss}& L_{\sigma s}+L_0^{ss}+L_{\mathrm{L}}^{\mathrm{s}}\hspace{0.17em}& L_{2p}^{ss}& L_{4p}^{ss}& \dots \\ \dots & L_{-4p}^{ss}& L_{-2p}^{ss}& L_{\sigma s}+L_0^{ss}+L_{\mathrm{L}}^{\mathrm{s}}\hspace{0.17em}& L_{2p}^{ss}& \dots \\ \dots & L_{-6p}^{ss}& L_{-4p}^{ss}& L_{-2p}^{ss}& L_{\sigma s}+L_0^{ss}+L_{\mathrm{L}}^{\mathrm{s}}\hspace{0.17em}& \\ & ⋮& ⋮& ⋮& & \ddots \end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ I_{3p}^s\hspace{0.17em}\\ \hspace{0.17em}{I}_p^s\hspace{0.17em}\\ \hspace{0.17em}{I}_{-p}^s\hspace{0.17em}\\ I_{-3p}^s\\ ⋮\end{array}\right]+ \\ +\mathrm{diag}\left[\begin{array}{c}⋮\\ R_{\mathrm{s}}+R_{\mathrm{L}}^{\mathrm{s}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}^{\mathrm{s}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}^{\mathrm{s}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}^{\mathrm{s}}\\ ⋮\end{array}\right]\hspace{0.17em}\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ I_{3p}^s\hspace{0.17em}\\ \hspace{0.17em}{I}_p^s\hspace{0.17em}\\ \hspace{0.17em}{I}_{-p}^s\hspace{0.17em}\\ I_{-3p}^s\\ ⋮\end{array}\right]=\left[\begin{array}{c}⋮\\ 0\\ E_1^s\\ E_{-1}^s\\ 0\\ ⋮\end{array}\right]-\mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}3p\Omega \hspace{0.17em}{E}_{(3x3)}\\ \hspace{0.17em}\mathrm{j}p\Omega \hspace{0.17em}{E}_{(3x3)}\\ -\mathrm{j}p\Omega \hspace{0.17em}{E}_{(3x3)}\\ -\mathrm{j}3p\Omega \hspace{0.17em}{E}_{(3x3)}\\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\Psi }_{3p}^{ss}\\ {\Psi }_p^{ss}\\ {\Psi }_{-p}^{ss}\\ {\Psi }_{-3p}^{ss}\\ ⋮\end{array}\right] \end{gathered}$$

(30)

where $L_n^{\mathrm{s}s}=L_n^s\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}n{\phi }_0}$ and ${\Psi }_{\varsigma }^{ss}={\Psi }_{\varsigma }^{PM\hspace{0.17em}s}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}\varsigma {\phi }_0}$.

The elements of winding inductance matrix and vector of the PM flux linkages following the symmetrical component transformation are dependent on the distribution of winding MMF harmonics. Two types of winding should be taken into account, with an integer and a fractional number of slots per pole and per phase.

For windings in which the MMF harmonics of the ${\nu }^{\mathrm{th}}$ order belong to set $P=P^{\mathrm{qc}}=\{\dots -5p,-3p,-p,p,3p,5p\dots \}$, i.e., windings with an integer number of slots per pole and per phase, the stator inductance matrix and vector of the PM flux linkages are given as follows:

$$L_n^s={\displaystyle \sum _{\nu \in P}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot \left[\begin{array}{ccc}\begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\mathrm{\dots }-5p,p,7p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-7p,-p,5p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}\\ \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\mathrm{\dots }-5p,p,7p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-7p,-p,5p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}\\ 1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.& 1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-5p,p,7p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.& 1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-7p,-p,5p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\end{array}\right]$$

(31)

$${\Psi }_{\varsigma }^{PM\hspace{0.17em}s}\hspace{0.17em}={\displaystyle \sum _{m\in M}\hspace{0.17em}\sqrt{3}\hspace{0.17em}{\psi }_{\varsigma ,m}^{\mathrm{PM}\hspace{0.17em}\mathrm{s}}}\cdot \left[\begin{array}{c}1\hspace{0.17em}\forall \hspace{0.17em}\{-(\varsigma +m)=\pm 3p,\pm 9p\mathrm{\dots }\}\hspace{0.17em}\\ 1\hspace{0.17em}\forall \{-(\varsigma +m)=\mathrm{\dots }-5p,p,7p\mathrm{\dots }\}\hspace{0.17em}\\ 1\hspace{0.17em}\forall \hspace{0.17em}\{-(\varsigma +m)=\mathrm{\dots }-7p,-p,5p\mathrm{\dots }\}\end{array}\right]$$

(32)

For windings in which the MMF harmonics of the ${\nu }^{\mathrm{th}}$ order belong to set, $P=P^{\mathrm{qf}}=\{\dots -5p,-4p,-3p,-2p,-p,p,2p,3p,4p,5p\dots \}$, i.e., the windings with a fractional number of slots per pole and per phase, the inductance matrix and vector of PM flux linkages are given as follows:

$$L_n^s={\displaystyle \sum _{\nu \in P}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot \left[\begin{array}{ccc}\begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\mathrm{\dots }-2p,p,4p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-4p,-p,2p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}\\ \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\mathrm{\dots }-2p,p,4p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}& \begin{array}{l}1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-4p,-p,2p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.\\ \end{array}\\ 1\hspace{0.17em}\forall \left\{\left.\begin{array}{c}-v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }\\ m=\mathrm{\dots }-2p,4p,10p\mathrm{\dots }\end{array}\right\}\right.& 1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-2p,p,4p\mathrm{\dots }\\ m=\mathrm{\dots }-4p,2p,8p\mathrm{\dots }\end{array}\right\}\right.& 1\hspace{0.17em}\forall \hspace{0.17em}\left\{\left.\begin{array}{c}-v=\mathrm{\dots }-4p,-p,2p\mathrm{\dots }\\ m=0,\pm 6p,\pm 12p\mathrm{\dots }\end{array}\right\}\right.\end{array}\right]$$

(33)

$${\Psi }_{\varsigma }^{PM\hspace{0.17em}s}\hspace{0.17em}={\displaystyle \sum _{m\in M}\hspace{0.17em}\sqrt{3}\hspace{0.17em}{\psi }_{\varsigma ,m}^{\mathrm{PM}\hspace{0.17em}\mathrm{s}}}\cdot \left[\begin{array}{c}1\hspace{0.17em}\forall \hspace{0.17em}\{-(\varsigma +m)=\pm 3p,\pm 9p\mathrm{\dots }\}\hspace{0.17em}\\ 1\hspace{0.17em}\forall \hspace{0.17em}\{-(\varsigma +m)=\mathrm{\dots }-2p,p,4p\mathrm{\dots }\}\hspace{0.17em}\\ 1\hspace{0.17em}\forall \hspace{0.17em}\{-(\varsigma +m)=\mathrm{\dots }-4p,-p,2p\mathrm{\dots }\}\end{array}\right]$$

(34)

The equations used to determine the coefficients for the inductance, $L_{v,m,n}^{\mathrm{ss}}$, and the PM-linked fluxes, ${\psi }_{\varsigma ,m}^{\mathrm{PM}\hspace{0.17em}\mathrm{s}}$, are presented in the Appendix A.

3.3.2. Special Case (External Symmetry of the Generator)

In the case of symmetry of load for the stator winding configuration presented in Y, Equation (30) is reduced to the following simpler form (if connection Y without a neutral wire is considered, a very large $R_{\mathrm{N}}$ value must be assumed, e.g., 1 MΩ).

$$\begin{gathered} \mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}3p\Omega \\ \mathrm{j}\hspace{0.17em}p\Omega \\ -\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}\\ -\mathrm{j}\hspace{0.17em}3p\Omega \\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{cccccc}\ddots & & ⋮& ⋮& ⋮& \\ & L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}0}+L_{\mathrm{L}}& {\underset{¯}{L}}_{2p}^{\mathrm{s}1}& {\underset{¯}{L}}_{4p}^{\mathrm{s}2}& {\underset{¯}{L}}_{6p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-2p}^{\mathrm{s}0}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}1}+L_{\mathrm{L}}& {\underset{¯}{L}}_{2p}^{\mathrm{s}2}& {\underset{¯}{L}}_{4p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-4p}^{\mathrm{s}0}& {\underset{¯}{L}}_{-2p}^{\mathrm{s}1}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}2}+L_{\mathrm{L}}& {\underset{¯}{L}}_{2p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-6p}^{\mathrm{s}0}& {\underset{¯}{L}}_{-4p}^{\mathrm{s}1}& {\underset{¯}{L}}_{-2p}^{\mathrm{s}2}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}0}+L_{\mathrm{L}}& \\ & ⋮& ⋮& ⋮& & \ddots \end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{I}}_{3p}^{\mathrm{s}0}\\ {\underset{¯}{I}}_p^{\mathrm{s}1}\\ {\underset{¯}{I}}_{-p}^{\mathrm{s}2}\\ {\underset{¯}{I}}_{-3p}^{\mathrm{s}0}\\ ⋮\end{array}\right]+ \\ +\mathrm{diag}\left[\begin{array}{c}⋮\\ R_{\mathrm{s}}+R_{\mathrm{L}}+3R_{\mathrm{N}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}\\ R_{\mathrm{s}}+R_{\mathrm{L}}+3R_{\mathrm{N}}\\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{I}}_{3p}^{\mathrm{s}0}\\ {\underset{¯}{I}}_p^{\mathrm{s}1}\\ {\underset{¯}{I}}_{-p}^{\mathrm{s}2}\\ {\underset{¯}{I}}_{-3p}^{\mathrm{s}0}\\ ⋮\end{array}\right]=\mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}3p\Omega \\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}p\Omega \\ -\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}\\ -\mathrm{j}\hspace{0.17em}3p\Omega \\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{\psi }}_{3p}^{\mathrm{s}}\\ {\underset{¯}{\psi }}_p^{\mathrm{s}}\\ {\underset{¯}{\psi }}_{-p}^{\mathrm{s}}\\ {\underset{¯}{\psi }}_{-3p}^{\mathrm{s}}\\ ⋮\end{array}\right]-\left[\begin{array}{c}⋮\\ 0\\ \underset{¯}{E}\\ \underset{¯}{\stackrel{\vee }{E}}\\ 0\\ ⋮\end{array}\right] \end{gathered}$$

(35)

For the symmetry of load for stator winding configuration in ∆, Equation (35) is reduced to a simpler form, as given below:

$$\begin{gathered} \mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}3p\Omega \\ \mathrm{j}\hspace{0.17em}p\Omega \\ -\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}\\ -\mathrm{j}\hspace{0.17em}3p\Omega \\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{cccccc}\ddots & & ⋮& ⋮& ⋮& \\ & L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}0}& {\underset{¯}{L}}_{2p}^{\mathrm{s}1}& {\underset{¯}{L}}_{4p}^{\mathrm{s}2}& {\underset{¯}{L}}_{6p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-2p}^{\mathrm{s}0}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}1}+3L_{\mathrm{L}}& {\underset{¯}{L}}_{2p}^{\mathrm{s}2}& {\underset{¯}{L}}_{4p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-4p}^{\mathrm{s}0}& {\underset{¯}{L}}_{-2p}^{\mathrm{s}1}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}2}+3L_{\mathrm{L}}& {\underset{¯}{L}}_{2p}^{\mathrm{s}0}& \dots \\ \dots & {\underset{¯}{L}}_{-6p}^{\mathrm{s}0}& {\underset{¯}{L}}_{-4p}^{\mathrm{s}1}& {\underset{¯}{L}}_{-2p}^{\mathrm{s}2}& L_{\mathsf{\sigma }\mathrm{s}}+{\underset{¯}{L}}_0^{\mathrm{s}0}& \\ & ⋮& ⋮& ⋮& & \ddots \end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{I}}_{3p}^{\mathrm{s}0}\\ {\underset{¯}{I}}_p^{\mathrm{s}1}\\ {\underset{¯}{I}}_{-p}^{\mathrm{s}2}\\ {\underset{¯}{I}}_{-3p}^{\mathrm{s}0}\\ ⋮\end{array}\right]+ \\ +\mathrm{diag}\left[\begin{array}{c}⋮\\ R_{\mathrm{s}}\\ R_{\mathrm{s}}+3R_{\mathrm{L}}\\ R_{\mathrm{s}}+3R_{\mathrm{L}}\\ R_{\mathrm{s}}\\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{I}}_{3p}^{\mathrm{s}0}\\ {\underset{¯}{I}}_p^{\mathrm{s}1}\\ {\underset{¯}{I}}_{-p}^{\mathrm{s}2}\\ {\underset{¯}{I}}_{-3p}^{\mathrm{s}0}\\ ⋮\end{array}\right]=\mathrm{diag}\left[\begin{array}{c}⋮\\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}3p\Omega \\ \hspace{0.17em}\mathrm{j}\hspace{0.17em}p\Omega \\ -\mathrm{j}\hspace{0.17em}p\Omega \hspace{0.17em}\\ -\mathrm{j}\hspace{0.17em}3p\Omega \\ ⋮\end{array}\right]\cdot \hspace{0.17em}\left[\begin{array}{c}⋮\\ {\underset{¯}{\psi }}_{3p}^{\mathrm{s}}\\ {\underset{¯}{\psi }}_p^{\mathrm{s}}\\ {\underset{¯}{\psi }}_{-p}^{\mathrm{s}}\\ {\underset{¯}{\psi }}_{-3p}^{\mathrm{s}}\\ ⋮\end{array}\right]-\left[\begin{array}{c}⋮\\ 0\\ \underset{¯}{E}\\ \underset{¯}{\stackrel{\vee }{E}}\\ 0\\ ⋮\end{array}\right] \end{gathered}$$

(36)

where

$${\underset{¯}{L}}_n^{\mathrm{s}0}={\displaystyle \sum _{v=\pm 3p,\pm 6p,\pm 9p\mathrm{\dots }}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}$$

(37)

For windings with an integer number of slots per pole and phase, we obtain

$${\underset{¯}{L}}_n^{\mathrm{s}\hspace{0.17em}1}={\displaystyle \sum _{v=\mathrm{\dots }-5p,p,7p\mathrm{\dots }}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}n{\phi }_0}$$

(38)

$${\underset{¯}{L}}_n^{\mathrm{s}2}={\displaystyle \sum _{v=\mathrm{\dots }-7p,-p,5p\mathrm{\dots }}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}n{\phi }_0}$$

(39)

For windings with a fractional number of slots per pole and phase, we obtain

$${\underset{¯}{L}}_n^{\mathrm{s}\hspace{0.17em}1}={\displaystyle \sum _{v=\mathrm{\dots }-2p,p,4p\mathrm{\dots }}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}n{\phi }_0}$$

(40)

$${\underset{¯}{L}}_n^{\mathrm{s}2}={\displaystyle \sum _{v=\mathrm{\dots }-4p,-p,2p\mathrm{\dots }}\hspace{0.17em}{\displaystyle \sum _{m\in M}3L_{v,m,n}^{\mathrm{ss}}}}\hspace{0.17em}\cdot {\mathrm{e}}^{\mathrm{j}n{\phi }_0}$$

(41)

For PM flux linkage with windings, we obtain

$$\hspace{0.17em}{\underset{¯}{\psi }}_{\mathsf{\varsigma }}^{\mathrm{s}}=\sqrt{3\hspace{0.17em}}\hspace{0.17em}{\psi }_{\varsigma }^{\mathrm{PMs}}\cdot {\mathrm{e}}^{\mathrm{j}\varsigma {\phi }_0}$$

(42)

Based on the symmetry properties of system Equations (35) and (36), the following can be concluded:

$${\underset{¯}{I}}_{\nu }^{\mathrm{s}0}=\stackrel{\vee }{{\underset{¯}{I}}_{-\nu }^{\mathrm{s}0}}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{\underset{¯}{I}}_{\nu }^{\mathrm{s}1}=\stackrel{\vee }{{\underset{¯}{I}}_{-\nu }^{\mathrm{s}2}}$$

(43)

Considering the phase coordinates, the time form of the first phase generator current is given as follows:

$$i_1(t)=\frac{2}{\sqrt{3}}\mathrm{Re}\hspace{0.17em}\{\hspace{0.17em}{\displaystyle \sum _{n=0}^{\infty }}\hspace{0.17em}{\underset{¯}{I}}_{(2n+1)p}^{\mathrm{s}\hspace{0.17em}(2n+1)\hspace{0.17em}\mathrm{mod}\hspace{0.17em}3}\hspace{0.33em}\cdot {\mathrm{e}}^{\mathrm{j}(2n+1)p\Omega t}\hspace{0.17em}\}$$

(44)

4. Laboratory Tests and Model Verifications

4.1. Description of Tested Generator

The developed models were verified for the MPMG segment with the following parameters: power of 40 kW, speed of 300 rpm, current of 57.7 A, and voltage of 400 V [7].

Figure 5 depicts the cross-section of the generator and the characteristic dimensions. The main dimensions and parameters of the MPMG segment are summarised as listed in

Table 2. Design data of the MPMG depicted in Figure 5b.

Parameters and Dimensions
Axial length of stator and rotor core $l_{\mathrm{c}}$ = 120.0 mm; Number of pole pairs p = 10;
Stator outer radius 345.0 mm; Stator internal radius $r_{\mathrm{s}}$ = 265.0 mm;
Rotor outer radius $r_{\mathrm{r}}$ = 261.5 mm; Air gap thickness $l_{\delta }=r_{\mathrm{s}}-r_{\mathrm{r}}$ = 3.5 mm;
Number of stator slots $z_{\mathrm{s}}$ = 114; Winding type: double layer;
Number of stator parallel paths— 2 or 1; Winding coil span—4 slots
Layout of first phase (1 2 -7 -8 13 14 -18 -19 24 25 -30 -31 36 -41 -42 47 48 -53 -54 58 59 -64 -65 70 71 75 -76 81 82 -87 -88 93 -98 -99 104 105 -110 -111);
Total number of phase winding turns $w_{\mathrm{s}}$ = 2 × 190; Equivalent slot opening $b_{\mathrm{s}}$ = 4 mm
Dimensions of a single magnet: 30.0 mm × 8.0 mm; lm = 8.0 mm;
PM residual flux density Br = 1.38 T; PM coercive force Hc = 963 kA/m; Magnet type N45SH
Opening angle between magnets $\gamma$ = 50°; $\frac{1}{2}$PM pole span $\beta$ = 6.8°;

.

For the MPMG tests, we developed a laboratory stand and setup (Figure 6) to help obtain the appropriate test parameters (speed and power). The MPMG being analysed was driven using a squirrel cage induction motor (110 kW, 740 rpm) powered by a voltage inverter (132 kVA). The generator was loaded using three-phase resistance heaters with a varying power of 20 kW, 12 kW, and 24 kW. All the data were obtained using an NI-DAQ 16 bit measurement card. The measurements were recorded over a period of 10 s, with a sampling frequency of 100 kHz.

The parallel paths of the winding were connected to obtain the four connection configurations discussed earlier. Figure 7 depicts the arrangement of the ends of the winding paths in the terminal box, enabling their easy reconfiguration.

4.2. Parameters of the Mathematical Model

The parameters of the analytical model (inductance and PM flux linkages harmonics) were determined based on the analytical equations presented in Appendix A and the previous study conducted by the authors [25]. The analytical calculations of these parameters require the determination of the machine permeance function, the permeance function of the air gap, PM flux density distribution in the air gap for a slotless cylindrical machine, and the stator winding factor for the assumed MMF harmonics based on design data (Table 2).

Based on the results from Figure 8 and Figure 9, the main MPMG mathematical model parameters used for one segment were determined.

Table 3. Main parameters of the analytical models.

	Inductances			$\mathbf{PM} \mathbf{Flux} \mathbf{Linkage} \mathbf{Harmonics} {\mathit{\psi }}_{\mathit{\varsigma }}^{\mathbf{PMs}}=\hspace{0.17em}{\mathit{\psi }}_{-\mathit{\varsigma }}^{\mathbf{PMs}}$$; \mathit{\varsigma }=\hspace{0.17em}\mathit{p},3\mathit{p},5\mathit{p},\mathrm{\dots }$
Paths	$L_{\mathsf{\sigma }\mathrm{s}}$	$L_{\mathbf{0}}^{\mathbf{1},\mathbf{1}}$	$L_{\mathbf{2}p}^{\mathbf{1},\mathbf{1}}$	$\hspace{0.17em}{\psi }_p^{\mathrm{PMs}}$	$\hspace{0.17em}{\psi }_{\mathbf{3}p}^{\mathrm{PMs}}$	$\hspace{0.17em}{\psi }_{\mathbf{5}p}^{\mathrm{PMs}}$	$\hspace{0.17em}{\psi }_{\mathbf{7}p}^{\mathrm{PMs}}$	$\hspace{0.17em}{\psi }_{\mathbf{9}p}^{\mathrm{PMs}}$	$\hspace{0.17em}{\psi }_{\mathbf{11}p}^{\mathrm{PMs}}$
||	0.95 mH	1.67 mH	0.22 mH	0.532 Wb	3.26 mWb	0.85 mWb	0.59 mWb	0.36 mWb	0.07 mWb
¦	3.80 mH	6.68 mH	0.88 mH	1.064 Wb	6.52 mWb	1.70 mWb	1.18 mWb	0.72 mWb	0.14 mWb

lists these parameters.

4.3. Verification of the Spatial Harmonic Interaction Model

4.3.1. Verification of the Stator EMF

First, the results for the induced EMF as a comparative measure of the PM flux linkage are presented to compare and indicate the main source of the generation of spatial harmonic interactions in MPMG. The drawing below shows the stator phase separate path EMF for the tested generators at a rotational speed of 300 rpm (50 Hz) (Figure 10). Results were obtained from the analytical model and measurements. The presented spectra (in dB) are for the adopted reference level of 1 mV.

The values of the THD_EMF coefficient and the RMS values for the induced EMF of the generator are listed in

Table 4. Comparison of the results obtained from analytical model and laboratory tests for EMF stator winding parallel path.

MPMG	THDEMF		EMF(RMS)
	Analytical Calculations	Measure	Analytical Calculations	Measure	|ΔEMF(%)|
Separate parallel path	2.25%	2.19%	236.22 V	236.23 V	0.004%

; this comparison demonstrates the relative error of calculations corresponding to the measurements (Figure 10).

The results from the above table indicate a small level of EMF higher harmonics, which was basically to be expected due to the low values of the winding factor (Figure 9b). Notably, the measurement results are in good agreement with the analytical calculations. The difference is very small.

4.3.2. Verification of the Stator Currents

The laboratory tests were conducted for MPMG phase currents at various rotational speeds and winding configurations for the symmetrical three-phase resistance load, R_L. The spectra (in dB) presented in Figure 11 are employed for the adopted reference level of 1 mA.

An ideal qualitative convergence can be observed based on the analysis of the spectra depicted in Figure 11. Conversely, the measurement results and analytical calculations vary from each other by no more than 7 dB. For a star connection, the third-order harmonic is obviously zero, while for a delta connection, it unfortunately appears. It should be added that higher harmonics (above third-order) are at relatively low levels (30–40 dB below the fundamental harmonic).

5. Discussion

Table 5. Comparison of the results obtained from the analytical models and laboratory tests for the phase current, $i_{\mathrm{G}1}$.

MPMG (Configuration and Load)		THDI		IG(RMS)
		Analytical Calculations	Measure	Analytical Calculations	Measure	|∆I(%)|
||Y 300 rpm	$R_{\mathrm{L}}=6.7\Omega$	0.82%	0.80%	31.91 A	31.54 A	1.2%
||Δ 300 rpm	$R_{\mathrm{L}}=3.6\Omega$	42.03%	42.42%	25.11 A	25.19 A	0.3%
¦Y 150 rpm	$R_{\mathrm{L}}=12.9\Omega$	0.86%	0.89%	18.31 A	18.46 A	0.8%
¦Δ 260 rpm	$R_{\mathrm{L}}=9.8\Omega$	32.79%	32.94%	15.06 A	15.95 A	5.6%

lists the values of the THD_I coefficient as an indicator of the content of higher harmonics and RMS value of the currents, including the relative error of calculations corresponding to the measurements (∆I_(%)). The obtained convergences can be considered to be satisfactory; however, the value for the third harmonic is relatively high for the delta configuration of the stator windings. However, the vibration level measurements [7] did not exhibit any adverse levels deviating from the typically accepted standards for rotating machines.

During the MPMG winding design, we aimed to eliminate the third harmonic EMF; however, due to the compromise related to reducing the cogging torques, we replaced the originally planned winding distributed in 120 slots with a configuration of 114 slots. In this case, we achieved only a partial reduction of the third-order harmonics, as seen for the summarized winding factor (Figure 9b), despite assuming a winding coil pitch equal to 2/3 p.u. (120 electrical degrees). As a result, when the winding is connected in a delta configuration, a non-zero EMF can arise around the delta due to the third-order harmonic components in all three phases being in phase. Since the impedance to this EMF is typically low, it may lead to a significant internal circulating current, which is not observable at the load terminals. In the future, it is essential to completely eliminate the third harmonic EMF if we plan to configure the windings in delta. Cogging torque reductions should be achieved using other known methods, such as skewing the stator slots or segmenting the magnets and skewing the magnets.

6. Conclusions

This study analysed the effects of spatial harmonic interactions in MPMGs using the proposed circuit-based analytical modelling approach. Our primary goal was to present a mathematical modelling methodology for this specific single-segment MPMG design. Individual segments can be connected in parallel or treated as separate sources. This approach was validated using a three-phase MPMG configuration with symmetrical load structures. The developed models depend on the parameters expressed in the integral form, which introduces certain constraints on the precision of the simulation outcomes. These limitations are primarily attributed to the inherent simplifications of the mathematical modelling, which cannot fully replicate the complex physical phenomena observed in real machines. The two-dimensional analytic magnetic field distribution model was insufficient to fully account for the influence of end-winding connections and leakage fluxes. Furthermore, minor discrepancies in the results can be attributed to the assembly and construction tolerances in the manufactured MPMG prototypes. In the case of differences between the outcomes of the analytical models and experimental measurements, a maximum deviation of 6% was observed for the generator currents. This level of accuracy is typically considered to be sufficient for most engineering applications.

The analysis results revealed certain shortcomings of the MPMG design [7]. For delta-connected stator windings, significant third-order harmonics occur in the phase currents. However, there are design options to completely eliminate this problem, e.g., by designing a winding with a winding factor equal to zero for the third harmonic or by selecting magnet size and positions to eliminate the third harmonic in the PM flux density distribution.

These results demonstrate the underlying assumption that the HBM, when applied to the MPMG circuit modelling, presents reliable and accurate insights. Therefore, the developed models serve as valuable tools for analysing a wide range of design and operational aspects of the MPMG systems. The proposed methodology can be used as an alternative to FEM analyses in the first stages of design calculations. It can also be used for the analysis of parasitic electromagnetic phenomena and for cooperation with converter systems in steady state operation.