A novel method for the core loss calculation in the surface-mounted permanent magnet synchronous machine

Abstract

The core loss is of great significance in the machine designing and optimization. In the paper, an analytical method for calculating the flux density of surface-mounted permanent magnet synchronous machine is first presented based on the equivalent magnetic circuit and conformal mapping method. The model takes into account the slotting effect and stator saturation. Subsequently, the core loss was calculated by improved Bertotti’s formula in terms of the analytical model of flux density. Finally, experiment and finite element analysis (FEA) are used to validate the accuracy of the proposed analytical method for core loss. It shows that the analytical results for core loss can keep excellent agreement with the experimental results and FEA results. Therefore, the analytical model provides a great potential to the machine designing and optimization.

Keywords

Equivalent magnetic circuit conformal mapping core loss permanent magnet synchronous machine stator permeance

1. Introduction

The core loss calculation has a significant effect on the surface-mounted permanent magnet (SPM) synchronous machine design and optimization, in order to determine the efficiency. With the wide working range of electrical machine, the core loss calculation at each operating point is complicated and time-consuming due to the nonlinearity of flux density and core loss. Thus, the core loss analysis over the full range of operation is the key technology in the machine designing [1, 2].

In general case, it is difficult to accurately calculate the core loss in the first step of machine designing. In relative literatures, both the analytical and numerical methods have been developed to predict the core loss. In Zhang and Doppelbauer [3] analyzed the numerical calculation of core loss on a novel axial flux machine with segmented-armature-torus topology based on the Bertotti’s formula. The waveform of real flux density was obtained using 3D FEA, and the harmonic flux density waveform in each finite element was obtained by Fourier decomposition. The accurate results can be calculated. However, the computation is complicated and time-consuming. The literatures [4, 5] presented a simplified voltage model based on the superposition of magnetizing and demagnetizing loss component to calculate the core loss within a field orientated controlled brushless AC permanent magnet machine, and test results taken from a concentrated wound brushless AC traction motor are used to validate the technique. However, the simplified voltage model ignored the stator resistance, so the core loss will cause some errors.

The core loss has been greatly influenced by the flux density. Because of the changing flux density at various operating point, the accurate calculation of air gap flux density is of great significance. At present, the air gap flux density is mainly calculated by the numerical and analytical method. The accurate air gap flux density distribution with considering the stator saturation can be calculated by FEA [6]. However, it is time-consuming and complicated. The analytical method mainly includes equivalent magnetic circuit method, conformal mapping (CM) and Fourier’s series method. The literatures [7, 8] analyzed the magnetic characteristics of SPM, and then established the analytical method using the magnetic circuit analysis, which could directly calculate the average air gap flux density and no-load leakage coefficient. However, the accuracy of analytical model is limit without considering the saturation effect of iron material. Later, Han et al. used the magnetic circuit method to model the electric machine with taking into account the stator saturation in [9, 10]. The magnetic circuit method can calculate the average air gap flux density, but it cannot accurately obtain the air gap flux density waveform.

Unlike the magnetic circuit method, CM can accurately predict the air gap flux density waveform. It was firstly used by carter [11] to calculate the carter’s factor in 1926. Then, CM was used to calculate the relative permeance of the air gap [12, 13]. Subsequently, Boughrara et al. used CM to analyze the slotted air gap flux density distribution of inset permanent magnet synchronous motor. Then, the cogging and electromagnetic torque characteristic was obtained by integrating the Maxwell stress tensor. The accuracy of CM was verified by using FEA, but CM doesn’t consider the stator saturation [14, 15]. An extended usage of the method that takes into account the saturation was given in the literatures [16, 17, 18, 19, 20]. The literatures [16, 17] presented the CM to analyze the air gap flux density and complex relative permeance with considering the stator saturation. However, the method doesn’t consider the effect of saturation on the stator permeability. In literature [18], Hafner et al. presented a method to re-parameterize the CM approach by single FE computations so as to consider saturation in the model over a wide operation range of the electrical drive. However, it doesn’t take into account the no-load flux density and it is FE-dependent. The literature [19] modeled the air gap length as a function of the position and level of the air gap flux to consider the saturation effect of induction machine. The saturation effect of stator teeth with induction machine was considered by Ojaghi and Nasiri in the literature [20]. However, it assumed the slot of stator and rotor is open.

The motor whole domain can be divided into several regions in the Fourier’s series method. Then, the equation was solved by the methods of Fourier’s series and separating variables. An exact analytical subdomain model for the open-circuit magnetic field in SPM machines with any pole and slot number combinations was established by Dubas and Espanet [21, 22]. The coefficients of the Fourier’s series can be determined by the boundary conditions. Then the air gap flux density and cogging torque was obtained. Zhu et al. applied the scalar magnetic potential as the solved variable of equation to solve the no-load flux density in the literatures [23, 24], and applied the vector magnetic potential as the solved variables to analyze the no-load flux density in the literatures [25, 26]. All above the methods of Fourior’s series analyze the air gap flux density without considering the stator saturation.

Each analytical method for calculating the air gap flux density and core loss has its own properties. Therefore, the novelty of the paper is to propose an accurate analytical method of calculating magnetic field based on the equivalent magnetic circuit and conformal mapping, in order to improve the efficiency and accuracy of magnetic field. The stator permeability can be obtained by equivalent magnetic circuit method, and the equivalent magnetic circuit method takes into account magnetic saturation. Then, the permeability has been applied to conformal mapping to analyze the magnetic field distribution. On the basis of the analytical model of air gap flux density, the core loss is calculated by the improved Bertotti theory.

This paper is organized as follows: Section 2 introduces the theory of CM, which maps the model of SPM to the canonical domain. The stator and rotor permeance is calculated by using the equivalent magnetic circuit in Section 3. Section 4 analyzes the analytical model for calculating the core loss with saturated and unsaturated condition. The calculation results, discussion and verification of core loss are presented in Section 5. Section 6 gives the conclusion.

2. Conformal mapping method

Table 1
Motor design parameters

Parameters	Values	Parameters	Values
Rated power (kW)	450	Rated speed (rpm)	3232
Stator outer diameter (mm)	490	Stator inner diameter (mm)	360
Number of poles	12	Number of slots	72
Air gap length (mm)	3	Magnet thickness (mm)	10
Pole arc coefficient	0.7	Axial length of stator (mm)	206

CM is an analytical and numerical method for calculating the air gap flux density of SPM. The basic theory of method is to map the operating field of complicated and unknown air gap magnetic field into a domain of simple and known magnetic field distribution. Then the unknown magnetic field can be solved by the correspondence relationship. Finally, the air gap flux density distribution of motor can be obtained. In the paper, three CMs are applied to reach the simple and known annular domain.

2.1 Conformal mapping in the motor magnetic field

In the paper, SPM has been chosen to demonstrate the CM method. The model of the machine structure is shown in Fig. 1, and the basic machine parameters are listed in Table 1. In the SPM, a three phase, star connected, double-layer distributed winding construction is used. The laminated core packs are made of SiFe (DW310-35), and the magnets are formed from the rare earth-cobalt (XG160/120) grade with $B_{r}=$ 0.86 T and $H_{c}=$ 607131 A/m.

Figure 1.

The analysis model of SPM.

Figure 2.

The motor model in $z$ plane.

Let the motor geometry shown in the Fig. 1 be $s$ plane, each point can be expressed as $s=re^{j\theta}$ with the polar coordinates. Let the right side of graph as the zero axis, the range of angle $\theta$ is 0 $\sim$ 2 $\pi$ /p, where $p$ is the number of pole pairs. A logarithmic complex function is used to transform the geometry in the $s$ plane into the multilateral geometry in the $z$ plane shown in Fig. 2. That is expressed by the following form:

$\displaystyle z=\textit{log(s)}=\textit{log(r)}+j\theta$ (1)

The second CM is the Schwarz-Christoffel mapping. The SC mapping maps one canonical domain into the interior (exterior) of the respective polygon, and the SC Toolbox provides a library of command-line function and a graphical user interface to obtain the canonical rectangle in the $w$ plane. Figure 3 shows the canonical rectangle in the $w$ plane. The width and height of rectangle is calculated by the Eq. (2).

$\displaystyle\left\{{\begin{array}[]{l}\Delta x=\textit{real}\left[{w(2)}% \right]-\textit{real}\left[{w(1)}\right]\\ \Delta y=\textit{imag}\left[{w(100)}\right]-\textit{imag}\left[{w(1)}\right]\\ \end{array}}\right.$ (2)

Figure 3.

The canonical rectangle in the $w$ plane.

The third CM maps the canonical rectangle in the $w$ plane to the annular domain in the $\Psi$ plane. Figure 4 shows the annular domain in the $\Psi$ plane, and the PM equivalent line current and A-phase winding current mapping to the $\Psi$ plane are also marked in the Fig. 4.

$\displaystyle\left\{{\begin{array}[]{l}r_{\psi}=e^{-\pi\frac{\Delta y}{\Delta x% }+y_{w}\frac{2\pi}{\Delta x}}\\ \theta_{\psi}=\left({\pi-\frac{2\pi}{\Delta x}x_{w}}\right)\frac{180}{\pi}\\ \end{array}}\right.$ (3)

Figure 4.

The annular domain in the $\Psi$ plane.

Where, $x_{w},y_{w}$ is the real and image component of rectangle vertices in the $w$ plane, respectively.

2.2 The air gap flux density analysis in the annular domain

The annular domain in the $\Psi$ plane is obtained by three CMs. The annular is the air gap of a concentric cylinder smooth air gap machine, and Hague’s field solution in the domain is available. Figure 5 shows a single line current of magnitude $I$ , located at $\Psi=ce^{j\theta c}$ . The stator and rotor radii in the $\Psi$ plane is respectively $a$ and $b$ . The relative permeability of the rotor, air gap and the stator are respectively $\mu_{1}$ , $\mu_{2}$ , and $\mu_{3}$ in the $\Psi$ plane. In addition, the relative permeability of magnet is assumed equal to that of air gap. The air gap scalar magnetic potential in the annular domain, as a function of ( $r$ , $\theta$ ) is given as [17]:

$\displaystyle\Omega_{v}\left({r,\theta}\right)=\left\{{\begin{array}[]{l}\frac% {I}{2}+\sum\limits_{n=1}^{\infty}{\left({\left({A_{n}-\frac{I}{2n\pi c^{n}}}% \right)r^{n}+B_{n}r^{-n}}\right)\textit{sin}\left({n\left({\Delta\theta}\right% )}\right)---r\prec c}\\ \frac{I\left({\Delta\theta+\pi}\right)}{4\pi}+\sum\limits_{n=1}^{\infty}{\left% ({A_{n}r^{n}+B_{n}r^{-n}}\right)\textit{sin}\left({n\left({\Delta\theta}\right% )}\right)-----r=c}\\ \frac{I\Delta\theta}{2\pi}+\sum\limits_{n=1}^{\infty}{\left({A_{n}r^{n}+\left(% {\frac{Ic^{n}}{2n\pi}+B_{n}}\right)r^{-n}}\right)\textit{sin}\left({n\left({% \Delta\theta}\right)}\right)---r\succ c}\\ \end{array}}\right.$ (4)

Figure 5.

The canonical annular domain in the $\Psi$ plane.

Where

$\displaystyle\left\{{\begin{array}[]{l}A_{n}=\frac{-I\left({\mu_{1}-\mu_{2}}% \right)\left\{{b^{2n}\left({\mu_{3}-\mu_{2}}\right)+c^{2n}\left({\mu_{3}+\mu_{% 2}}\right)}\right\}}{c^{n}2n\pi\left\{{b^{2n}\left({\mu_{1}-\mu_{2}}\right)% \left({\mu_{2}-\mu_{3}}\right)+a^{2n}\left({\mu_{1}+\mu_{2}}\right)\left({\mu_% {2}+\mu_{3}}\right)}\right\}}\\ B_{n}=\frac{-b^{2n}I\left({\mu_{3}-\mu_{2}}\right)\left\{{c^{2n}\left({\left({% \mu_{1}-\mu_{2}}\right)}\right)+a^{2n}\left({\mu_{1}+\mu_{2}}\right)}\right\}}% {c^{n}2n\pi\left\{{b^{2n}\left({\mu_{1}-\mu_{2}}\right)\left({\mu_{2}-\mu_{3}}% \right)+a^{2n}\left({\mu_{1}+\mu_{2}}\right)\left({\mu_{2}+\mu_{3}}\right)}% \right\}}\\ \Delta\theta=\theta-\theta_{c}\\ \end{array}}\right.$ (5)

Based on the scalar magnetic potential, the air gap flux density due to the PM and winding currents can be calculated by

$\displaystyle B_{\psi}=-\mu_{0}\nabla\Omega_{v}=-\mu_{0}\left({\frac{\partial% \Omega_{v}}{\partial r}\overrightarrow{r}+\frac{1}{r}\frac{\partial\Omega_{v}}% {\partial\theta}\overrightarrow{t}}\right)$ (6)

Conformal maps preserve the scalar magnetic potential from $s$ plane to the $\Psi$ plane. Because the link between vector field potential and scalar magnetic potential is established through the coordinate-system-dependent gradient derivative, it is not hold true to conserve the vector field potential. Therefore, the air gap flux density in the desired $s$ plane can be obtained by

$\displaystyle B_{s}=B_{\psi}\left(\lambda\right)^{*}$ (7)

Where $*$ is the conjugate function, $\lambda$ is the complex relative permeance with the slotted air gap [17, 18, 19, 20, 21, 22, 23, 24].

3. Stator permeability analysis

The saturation effect has great influence on the properties of the stator composed by silicon steel sheet. The stator permeability decreases seriously with the increase of saturation degree. Meanwhile, it can be seen from Eq. (5) that the stator permeability has a significant effect to the magnetic field. However, CM doesn’t consider the influence of stator saturation on the permeability. In the paper, the stator permeability was analyzed by the equivalent magnetic circuit method.

3.1 The equivalent magnetic circuit theorem

The flux of SPM can be divided into three sections. It includes the flux source for one magnet pole, the air gap flux for one magnet pole and the leakage flux which cross through one magnet pole and two magnet poles. Given the flux distribution and the Ohm’s law equivalent of magnetic circuit, the equivalent magnetic circuit is shown in Fig. 6a.

Figure 6.

The simplifying magnetic circuit model. (a) The equivalent magnetic circuit; (b) The simplifying magnetic circuit model by simplifying (a); (c) The simplified magnetic circuit model by simplifying (b); (d) The consequently magnetic circuit model.

The variables of Fig. 6a are defined as follows:

$G_{\Lambda}$ is the leakage permeance for magnet pole, $G_{a}$ is the permeance for air gap, $G_{s}$ is the permeance for stator, $G_{\delta m}$ is the leakage permeance through one magnet pole, $G_{\delta}$ is the leakage permeance through the two magnet poles nearby and air gap, $F_{c}$ is the magnetomotive force (MMF) for one magnet pole, $F_{a}$ is the winding MMF per phase.

Where:

$\displaystyle\left\{{\begin{array}[]{l}G_{\Lambda}=\frac{\mu_{0}\mu_{r}b_{m}L_% {ef}}{h_{m}}\\ G_{a}=\frac{\mu_{0}\left({\tau+2\delta_{e}}\right)L_{ef}}{\delta_{e}}\\ G_{\delta m}=\frac{\mu_{0}\delta_{e}L_{ef}}{w_{f}}+\frac{\mu_{0}\frac{\delta_{% e}}{2}L_{ef}}{{2\delta_{e}+l}_{t}}\\ G_{\delta}=\frac{\mu_{0}L_{ef}}{\pi}\ln\left({1+\frac{\pi\delta_{e}}{h_{m}}}% \right)\\ \end{array}}\right.$ (8)

Where, $\mu_{0}$ is the air permeability and $\mu_{r}$ is relative permeability of magnet, $b_{m}$ is width of one magnet pole, $L_{ef}$ is the axial length of stator, $h_{m}$ is magnet thickness, $\tau$ is the pole pitch, $\delta_{e}$ is the effective air gap length, $w_{f}$ is the distance between magnets and $l_{t}$ is average slot width of air gap within the stator slot.

The equivalent magnetic circuit has been established without considering the permeance of the stator and rotor before, but the silicon steel sheet permeability down to a very small value when the motor magnetic field arrivals to saturation. Ignoring the silicon steel sheet permeance will produce the errors. Therefore, stator and rotor permeance is considered as the series connection of stator teeth, yoke and rotor permeance in the paper, which is calculated as:

$\displaystyle G_{s}=\frac{1}{\frac{2}{G_{t}}+\frac{1}{G_{js}}+\frac{1}{G_{jr}}}$ (9)

Where

$\displaystyle G_{t}=\frac{B_{\delta}tL_{ef}}{H_{t}h_{t}};G_{js}=\frac{B_{% \delta}\tau\alpha_{i}L_{ef}}{{2H}_{js}h_{js}};G_{jr}=\frac{B_{\delta}\tau% \alpha_{i}L_{ef}}{{2H}_{jr}h_{jr}}$ (10)

Where, $B_{\delta}$ is the average air gap flux density, $t$ is the teeth pitch, $H_{t}$ , $H_{js}$ and $H_{jr}$ is the magnetic intensity of stator teeth, yoke and rotor, respectively, $h_{t}$ , $h_{js}$ and $h_{jr}$ is the effective length of stator teeth, yoke and rotor, respectively, $\alpha_{i}$ is the effective pole arc factor.

Then thevenin theorem is applied to simplify the two ports A1-B1 and A2-B2 in the circle marked with dotted lines. Only one port need to solve because of the same parameters of two ports. The simplified magnetic circuit is presented and shown in Fig. 6b.

Where

$\displaystyle\left\{{\begin{array}[]{l}F_{c}^{\prime}=\frac{{2F}_{c}G_{\Lambda% }}{(2G_{\delta m}+G_{\Lambda})}\\ G_{h}=\frac{G_{\Lambda}}{4}+\frac{G_{\delta m}}{2}\\ \end{array}}\right.$ (11)

Similarly, thevenin theorem is applied to simplify the port C1-D1 once more. Figure 6c shows the simplified magnetic circuit.

$\displaystyle\left\{{\begin{array}[]{l}F_{z}=\frac{2F_{c}^{\prime}G_{h}}{{2G_{% h}+2G}_{\delta}}-F_{ad}+F_{aq}\\ G=G_{h}+G_{\delta}\\ \end{array}}\right.$ (12)

Where

$\displaystyle\left\{{\begin{array}[]{l}F_{ad}=k_{1}\frac{1.35N_{c}k_{w}k_{ad}i% _{d}}{p}\\ F_{aq}=k_{2}\frac{1.35N_{c}k_{w}k_{aq}i_{q}}{p}\\ \end{array}}\right.$ (13)

Where, $F_{ad}$ is $d$ axis MMF, $F_{aq}$ is $q$ axis MMF, $N_{c}$ is conducts per slot, $k_{w}$ is winding factor, $k_{ad}$ is the $d$ axis armature reaction coefficient, $k_{aq}$ is the $q$ axis armature reaction coefficient, $i_{d}$ is the $d$ axis current, $i_{q}$ is the $q$ axis current, $k_{1},k_{2}$ is the correction coefficient which related to the rated current and the actual current of $d-q$ axis. The coefficients are obtained by the fitting the simulation results and calculation results of average air gap flux density.

Finally, the nortons theorem is applied to the port C2-D2. The consequently magnetic circuit model is shown in Fig. 6(d). The short circuit flux of port C2-D2 is:

$\displaystyle\phi_{z}=F_{z}G$ (14)

Figure 7.

The results of average air gap flux density.

It can be seen from Fig. 6d that the total magnetic flux of PM and winding current flows through the PM permeance and leakage permeance, and the air gap flux is obtained. Then the air gap flux density can be calculated by the air gap flux and the surface area of air gap. That is,

$\displaystyle\left({4G_{s}+G_{a}}\right)B_{\delta}G+B_{\delta}G_{s}G_{a}=\phi_% {r}G_{s}G_{a}$ (15)

In order to verify the accuracy of equivalent magnetic circuit method, FEA is used. Figure 7 shows the calculation results and FEA results of average air gap flux density when $i_{q}$ varies from 0 $\sim$ 2000 A and $i_{d}$ is $-$ 1000 A, 0 A and 1000 A respectively. The air gap flux density increases when the $i_{q}$ increases, and the armature reaction of positive d axis current is magnetizing. It can be seen from Fig. 7 that the results calculated by analytical method could match well with the FEA results when $i_{q}$ varies, and the differences can’t exceed 5% especially when the current up to the rated current. Therefore, the permeability can be calculated by the average air gap flux density.

Figure 8.

The basic magnetization curve of silicon steel sheet.

3.2 The calculation of stator permeability

The magnetization curve of silicon steel sheet is shown in Fig. 8. Based on the magnetization curve, the magnetic intensity can be obtained with the known flux density, so the permeability can be defined as

$\displaystyle\mu=\frac{B_{i}}{H_{i}}$ (16)

Where $B_{i}=B_{t}$ , $B_{j}$ , $B_{jr}$ is the flux density of stator teeth, yoke and rotor, respectively. That is

$\displaystyle B_{t}=\frac{B_{\delta}t}{b_{t}K_{Fe}}\quad B_{j}=\frac{B_{\delta% }\tau\alpha_{i}}{2h_{js}K_{Fe}}\quad B_{jr}=\frac{B_{\delta}\tau\alpha_{i}}{2h% _{jr}K_{Fe}}$ (17)

Figure 9.

The calculation results of stator and rotor permeability.

According to the Eq. (16), the permeability in various loads can be obtained. Because the flux is mainly influenced by the minimum permeability, the value of stator permeability is the minimum permeability of stator teeth and yoke. Figure 9 is the calculation results of stator and rotor permeability with the variety of q axis current, and the d axis current is 1000 A. The stator permeability drops quickly when the q axis current increases. However, it is difficult to reach saturation for the rotor because of the large size, so its flux density increases with the increasing q axis current. Therefore, the rotor permeability has a small increase.

4. The core loss analytical model

4.1 The air gap flux density analysis

The analytical model of air gap flux density is established based on the CM and equivalent magnetic circuit method .The calculation step is: 1) the permeability of stator and rotor corresponding to the load currents is calculated by the equivalent magnetic circuit method; 2) based on the permeability, the air gap flux density can be analyzed by CM with considering the stator saturation.

According to the two steps, the air gap flux density in the position of $R_{g}=(R_{1}+R_{2})/2$ can be obtained by the analytical calculation, where $R_{1},R_{2}$ is the radii of internal stator surface and outer rotor surface. Figure 10 shows the calculation results and FEA results of air gap flux density when the current is $i_{d}=$ 0 and $i_{q}=$ 966 A. At this point, the stator magnetic field has not yet reach the saturation. The tangential component only has a small fraction because of the high permeability of stator silicon steel sheet. The radial and tangential component of air gap flux density calculated by the analytical model could match well with the FEA results in one complete electrical period. It also illustrates that the q axis current makes the magnetic field to shift a certain degrees along the d axis.

Figure 10.

The calculation results and FEA results of air gap flux density.

The calculation results and FEA results of air gap flux density when the current is $i_{d}=$ 2000 and $i_{q}=$ 1000 A are shown in the Fig. 11. The phase current is more than twice the rated current. The stator material reaches high saturation because of the positive d axis current and the high q axis current. Therefore, the permeability of stator decreases to 10 H/m. Meanwhile, the radial component of air gap flux density increases much more than the current of $i_{d}=$ 0 and $i_{q}=$ 966 A. The radial and tangential component of air gap flux density calculated by the analytical model also could keep consistence with the FEA results in one complete electrical period.

Figure 11.

The calculation results and FEA results of air gap flux density.

4.2 The core loss analytical model

The changing flux density determined by winding current and rotor rotation can produce the core loss. Generally, the Bertotti model is widely applied to calculate the core loss [27]. According to the Bertotti model and the theory of producing the core loss, the core loss is separated into the hysteresis loss, eddy current loss and excess loss.

The hysteresis loss is produced by the frictional interactions among the magnetic domains of iron material when the electromagnetic field varies. The theory of eddy current loss is the eddy current in the iron material induced by the changing flux density because of the winding current and rotation of rotor. The excess loss is produced in anisotropic iron material. Most research hold that it is related with the magnetic domains. Based on the principle of three loss model, the Bertotti’s formula is:

$\displaystyle\left\{{\begin{array}[]{l}p_{Fe}=p_{h}+p_{e}+p_{ecx}=\sum\limits_% {i=1,3,5\cdots}{k_{h}fB_{si}^{\alpha}}+\sum\limits_{i=1,3,5\cdots}{k_{e}f^{2}B% _{si}^{2}}+\sum\limits_{i=1,3,5\cdots}{k_{ecx}f^{1.5}B_{si}^{1.5}}\\ P_{Fe}=k_{a}p_{Fe}G_{a}\\ \end{array}}\right.$ (18)

Where, $p_{Fe}$ is the core loss per weight, $p_{h}$ is the hysteresis loss, $p_{e}$ is the eddy current loss, $p_{ecx}$ is the excess loss, $k_{h}$ is the hysteresis loss coefficient, $k_{e}$ is the eddy current loss coefficient, $k_{ecx}$ is the excess loss coefficient, $f$ is the frequency, $B_{si}$ is the i ${}^{th}$ harmonic amplitude of the stator flux density, $\alpha$ is the coefficient, $k_{a}$ is the empirical coefficient, $G_{a}$ is the weight of motor.

It is assumed in the Bertotti’s formula that the real waveform of flux density is far from Sinusoidal because of the changing flux density determined by winding current and rotation of rotor. All the fundamental and harmonic flux density produces the eddy current loss and excess loss. Therefore, the eddy current loss can be expressed in the form of time-domain due to the formation theory of eddy current loss. The eddy current loss can be defined as:

$\displaystyle p_{e}=\frac{1}{2\pi^{2}}k_{e}\frac{1}{T}\int_{0}^{T}{{\left({% \sum\limits_{i=1,3,5\cdots}^{\infty}{\frac{\partial B_{si}\left({\omega t}% \right)}{\partial t}}}\right)}^{2}dt}$ (19)

According to the core loss performances of iron materials, the coefficient of hysteresis loss, the eddy current loss and excess loss can be determined with the curve fitting method. The core loss at various operating point can be calculated by the Bertotti’s formula. The flowchart for calculating core loss is shown in the Fig. 12.

Figure 12.

The flowchart for calculating core loss.

5. Results and verify

5.1 The core loss analysis

Based on the air gap flux density analysis, the flux density distribution of stator teeth and yoke at different moments can be obtained. Then, the fundamental and harmonic amplitude can be calculated by the Fourier decomposition of the stator teeth and yoke flux density. Finally, the stator core loss can be obtained by the Eqs (18) and (19).

Figure 13.

The flux density of stator teeth.

Figure 13a shows the analytical and FEA results of stator teeth flux density in one complete electric period when the winding current is $i_{d}=$ 0 and $i_{q}=$ 966 A in the rated speed. Figure 13b shows the fundamental and harmonic amplitude calculated by the Fourier decomposition of the stator teeth flux density. Figure 14a shows the analytical and FEA results of stator yoke flux density in one complete electric period when the winding current is $i_{d}=$ 0 and $i_{q}=$ 966 A in the rated speed. Figure 14b shows the fundamental and harmonic amplitude calculated by the Fourier decomposition of the stator yoke flux density.

Figure 14.

The flux density of stator yoke.

Because the analytical calculation of stator teeth and yoke are not take into account the pole-to-pole leakage flux and its tangential flux density, there are some errors between the analytical results and FEA results, and the differences is below than 5%. It also illustrates the teeth harmonic amplitude of third and fifth is much more than the corresponding harmonic amplitude of stator yoke due to the slotting effect.

According to the analytical method shown in the Fig. 12, the core loss of external characteristic operating point is calculated. The external characteristic curve of SPM divides into constant torque and constant power area, as shown in the Fig. 15.

The load current of external characteristic can be obtained by the motor pharos relationship, as listed in Table 2. It can be seen that the phase current remain unchanged in the constant torque area (0 $\sim$ 3232 rpm). Because the d axis inductance is the same as the q axis inductance, the d axis current is zero and the q axis current is determined by the torque. In the constant power area (3232 $\sim$ 6000 rpm), the phase current firstly decreases with the decrease of the torque. Then, the phase current increases because of the increase of d axis demagnetize current.

Table 2

The load current of external characteristic operating point

Speed (rpm)	Power (kW)	Torque (Nm)	id (A)	iq (A)	Phase current (A)	Line-voltage (V)
1000	139.23	1329.7	0	966	575	368.4
2000	278.47	1329.7	0	966	575	414.9
3000	417.70	1329.7	0	966	575	498.8
3232	450	1329.7	0	966	575	500
4000	450	1074.4	447.2	781.2	520	500
5000	450	859.5	807	624	588.8	500
6000	450	716.25	1206.8	520	758	500

Figure 15.

The external characteristic curve of SPM.

Figure 16.

The core loss of the external characteristic operating point.

Base on the operating point listed in Table 2, the core loss distribution can be obtained by the analytical flowchart shown in the Fig. 12. Figure 16 shows the total core loss and the separating core loss (hysteresis loss, eddy current loss and excess loss) of the external characteristic operating point. In the constant torque area, the flux density of stator teeth and yoke remain unchanged because of the same phase current. The core loss is mainly determined by the motor speed, so the core loss increases with the increases of speed. All the separating loss increases in the constant torque area, and most of the core loss is the hysteresis loss. In the constant power area, the flux density of stator teeth and yoke become smaller because of the effect of d axis demagnetize current. However, the core loss cannot decrease a lot due to the increasing speed. At the time, the hysteresis loss drops due to the decrease of flux density. The eddy current loss and excess loss remain increase because of the increasing of flux density harmonic amplitude and speed. In addition, the rising speed of eddy current loss is higher than that of excess loss.

Figure 17.

The dimension of experimental motor.

Figure 18.

The calculation, FEA and experimental results of core loss.

5.2 Model verify

In order to verify the accuracy of the analytical method, experiment was performed on the SPM. The rated power of experimental machine is 450 kW. The core loss tested by the experiment is obtained by core loss separation of test data. The dimension of experimental motor is shown in the Fig. 17. The core loss obtained by analytical method, FEA and experiment is respectively shown in the Fig. 18.

From Fig. 18, the core loss obtained by experiment could keep consistence with the FEA and calculation results in the external characteristic operating point of SPM. However, the experimental value is slightly smaller than the FEA and calculation results in the constant torque area, and the experimental value is slightly larger than the FEA and calculation results in the constant power area. Therefore, the analytical model can be applied to calculate the core loss.

6. Conclusions

The paper has described a new analytical method for calculating the core loss of saturated SPM using a model based on the equivalent magnetic circuit method and CM. The stator permeability can be obtained by equivalent magnetic circuit method, and the equivalent magnetic circuit method takes into account magnetic saturation. Then, the permeability has been applied to conformal mapping to analyze the magnetic field.

Based on the analytical model, the radial and tangential component of air gap flux density due to the PM magnetization and winding currents can be separately calculated. Regardless of whether the stator material is saturated or unsaturated, the analytical results of air gap flux density keep consistency well with the FEA results. Hence the relations between the d-q axis currents and air gap flux density can be easily analyzed by the analytical method.

Moreover, compared with the FEA results and experimental results of core loss, the analytical results show very excellent agreement. Therefore, the analytical method could be applied to calculate the core loss, so it is helpful for the optimization and designing of the electric machine.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 51677005).

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A novel method for the core loss calculation in the surface-mounted permanent magnet synchronous machine

Abstract

Keywords

1. Introduction

2. Conformal mapping method

Table 1 Motor design parameters

3.1 The equivalent magnetic circuit theorem

4.1 The air gap flux density analysis

5.1 The core loss analysis

6. Conclusions

Footnotes

Acknowledgments

References

Table 1
Motor design parameters