Split ratio-based performance analysis method of PM-assisted reluctance machine

Abstract

The split ratio is one of the most important design parameters of the machines and has a significant effect on the electromagnetic performance. Using the split ratio, this paper presents a performance analysis method for permanent magnet (PM)-assisted reluctance machine is presented. An accurate analytical model, which considers the saturation of outer iron ribs and the asymmetric saturation of rotor islands, is firstly proposed. In order to reduce the numbers of the adjustable structural parameters during the initial design process and ensure the independence among the parameters, a geometric structure corresponding to the split ratio is then established. The effectiveness of the proposed analytical model based on the split ratio is verified by FEM, and applied to analyze the electromagnetic performance for the three-flux-barrier PM-assisted reluctance machines. On the basis of the analysis results, the design scheme corresponding to the optimal split ratio, which makes the machine obtain a higher efficiency and guarantees the PMs without irreversible demagnetization, is determined rapidly.

Keywords

Permanent magnet (PM)-assisted reluctance machine the gradient flux barrier split ratio analytical model

1. Introduction

Permanent magnet (PM)-assisted reluctance (PMAREL) machines are achieved when PMs (generally Ferrite PMs) are inserted into each rotor flux-barrier with the aim to obtain the higher torque density and the power factor, which is quite low in the reluctance (REL) machines [1,2]. Due to the merits such as, small size, wide speed range, high power density, high efficiency and high power factor with respect to the synchronous REL and induction machines [3,4], the PMAREL machines are widely used in locomotive traction, electric vehicle, and washing application etc.

As the basis for accurate performance analysis and optimization design of the REL and PMAREL machines, accurate calculation of air-gap magnetic field is very important. In the majority of the cases the REL and PMAREL machines are analyzed by using analytical method and finite element method (FEM). The air-gap density distribution and electromagnetic performance are computed precisely by FEM [5,6], however, the process is time consuming. In particular, it is difficult to determine the reasonable design schemes through analyzing the influences of various structure parameters on the electromagnetic performance. The FEM results only refer to the particular solution and do not satisfy the requirement of generality. Compared with FEM, the analytical method is not only useful for obtaining the mapping relationships between the structure parameters and the electromagnetic performance, but also it is easy to find general rules to design the REL and PMAREL machines [7,8]. Therefore, the analytical method is of great significance for predicting the electromagnetic performance with the various structure parameters changing in advance and determining the reasonable design schemes.

The precise analytical model is the prerequisite for the accurate calculation of air-gap magnetic field. However, the analytical models of the REL and PMAREL machines are difficult to be established due to the variety and complexity of magnetic circuit and many structure parameters. To simplify derivation of the analytical models, it is generally assumed that the core permeability is infinite, the stator magneto-motive force (MMF) is sinusoidal distribution and the distance between the flux-barrier ends is distributed evenly along the rotor periphery [9,10]. Actually, the core permeability will decrease when the REL and PMAREL machines operate in the saturation state, which causes the increase of the magnetic voltage drops in iron core and the decrease of the air-gap flux density. Generally, there are two main methods to consider the influence of the saturation in the iron core on the accurate prediction of the air-gap magnetic field distribution of the REL machine and PMAREL machine. One method, namely magnetic equivalent circuit (MEC), is based on a lumped-parameter magnetic network which is composed of the magnetic flux source, the saturating element and non-saturating element. In MEC, the saturation of the iron core is considered as the saturated reluctances and the magnetic scalar potential of each element node is computed in order to calculate the flux through each element [11,12]. The other method regards the magnetic voltage drop associated with each flux line in the stator and rotor iron paths as an additional equivalent magnetic voltage drop in the air gap. In other words, the saturation of the iron core is considered by means of the air-gap length increased [13]. Meanwhile, because of the winding configurations, all the stator slot MMF space harmonics should be considered for the air-gap magnetic field calculation. At present, the stator magnetic scalar potential is generally expressed as the sum of all space harmonic orders in the analytical models of the REL and PMAREL machines [14].

In the preliminary design of the PMAREL machines, many structure parameters such as air-gap length, stator outer diameter, rotor inner diameter, split ratio, active length, dimensions of rotor islands, PMs and flux barriers at the left and right sides of the PMs etc, are needed to be determined. The more the numbers of flux barriers are, the more the numbers of the structure parameters to be determined are. For example, there are more than twenty crucial structure parameters needed to be determined in the three-flux-barrier PMAREL machine. Therefore, the complexity of establishing the analytical model is increased and it will spend a lot of time adjusting repeatedly the dimensions of the structure parameters to consider the design requirements of various electromagnetic performance such as torque density, efficiency and the demagnetization of the PMs etc.

Due to the large numbers of the adjustable structural design parameters, it will spend a lot of time obtaining a reasonable design scheme through evaluating the electromagnetic performance with all combinations of the structural design parameters. In addition, it is more difficult to find the general guidelines to select the structural design parameters because of the coupled relations among them. This leads to a certain blindness and could cause attend to one thing and lose another in the preliminary design stage requiring adjusting repeatedly the dimensions of the parameters. Therefore, it is more important to reduce the numbers of the adjustable structural design parameters and guarantee the independence among them in the process of establishing the analytical models of the PMAREL machines.

The main dimensions in the PMAREL machines such as air-gap length, stator outer diameter, rotor inner diameter and active length, are determined according to the machines with the similar power capacity and the variation range of their dimensions are small. The split ratio is defined as the ratio between stator inner and outer diameters, and it is demonstrated by the mark “𝜒”. It has a significant impact on the average torque, stator copper loss, stator iron loss, magnetic leakage coefficient and efficiency etc [15,16]. Compared with other structure parameters, the variation ranges of the split ratio 𝜒 are large and many dimensions are changed with it. Therefore, in order to calculate the air-gap magnetic field and determine the design schemes, it is necessary to establish a geometric structure that the split ratio 𝜒 is regarded as the main design variable.

This paper presents a performance analysis method based on the split ratio 𝜒 for PMAREL and takes the gradient three-flux-barrier PMAREL machine as research object [17]. An analytical model based on the split ratio 𝜒 is proposed and its effectiveness is verified by FEM. In order to consider the saturation of outer iron ribs and improve the computational accuracy of the rotor magnetic voltage drop in the analytical model, the outer iron ribs are regarded as the separate iron core regions. Meanwhile, considering the asymmetric saturation of rotor islands, the distribution coefficients is used to make the flux flowing through the rotor island equivalent to the synthetic flux of the island d-axis and q-axis flux flowing through the left and right sides of rotor island. The method is further applied to analyze the influences of the split ratio 𝜒 on the average torque, stator loss, efficiency, the utilization rate of PMs and the demagnetization of PMs. And based on the above research, the electromagnetic performance of 48-slot 8-pole PMAREL machine and 36-slot 4-pole PMAREL machine are analyzed and the design schemes, which make the two machines obtain a higher efficiency and guarantee the PMs without irreversible demagnetization, are determined rapidly.

2. Modeling of magnetic field

2.1. Saturation factor

In order to obtain a higher torque density and the power factor for the gradient flux-barrier PMAREL machines, the PMs are inserted into each rotor flux-barrier or the larger stator current is passing through the windings. However, the saturation of the rotor iron and stator iron could occur. In general, the saturated areas are mainly consisted of the stator yoke, the stator teeth, the rotor islands, the iron ribs and the rotor channel. The various areas are presented in Fig. 1(a).

Fig. 1.

Sketch of the machine structure, the distribution of the flux lines and flux density map (48 slot/8 pole). (a) The various areas of the iron core. (b) Actual flux lines. (c) Flux density map.

In Fig. 1(b), the two half-island areas have different saturation. The reason is that the average flux flowing through the air-gap in front of the rotor island bored by flux barriers Φ_gi is separated into two parts: one is the flux Φ_g1i from the left side of the rotor island enters the next layer flux barrier, the other flux Φ_g2i is flow passing through the right side of the rotor island. The flux density map on the rotor is shown in Fig. 1(c), and it is concluded that the saturation at the left and right sides of rotor islands is different.

The core permeability will decrease when the PMAREL machines operate in the saturation state, which causes the magnetic voltage drops in the rotor iron and stator iron are not neglected. In order to improve the accuracy of the analytical model for the gradient flux-barrier PMAREL machine, the areas of magnetic voltage drops in the iron core are divided into the stator yoke, the stator teeth, the left and right sides of the rotor islands, the iron ribs and the rotor channel. In this paper, the magnetic voltage drops in rotor iron and stator iron are regarded as the additional equivalent magnetic voltage drops associated with the same flux paths in air gap. Since the saturation of the rotor and stator iron is different in each flux path, the equivalent length of the air gap associated with the same flux path is different. The g ^′ = [g ₁, g ₂, …, g _{N
_p}]_{1×N
_p} represents the equivalent air gap length of the N _p discrete points lied in the flux paths on the circumference of the air gap. N _p is the total numbers of the discrete points along the circumference of air gap. Therefore, the total air gap magnetic voltage drop is presented as $\begin{eqnarray}\displaystyle \boldsymbol{H}_{\text{g}}\boldsymbol{g}^{\prime }=\boldsymbol{H}_{\text{g}}g+\boldsymbol{F}_{\text{s}}+\boldsymbol{F}_{\text{r}} & & \displaystyle\end{eqnarray}$ (1) where g is the actual air-gap length, H _g, F _s and F _r are the air-gap magnetic field, the stator and rotor magnetic voltage drops of the discrete points lied in the flux paths, respectively. The dimensions of matrix H _g, F _s and F _r are both 1 × N _p. It should be noted that the meaning of the main variables in the analytical model are listed in appendix, and it can increase the readability of this paper.

The air-gap magnetic field H _g is expressed as $\begin{eqnarray} \displaystyle \boldsymbol{H}_{\text{g}}=\frac{\boldsymbol{B}_{\text{g}}}{{{\mu}}_{0}} & & \displaystyle \end{eqnarray}$ (2) where B _g is the air gap flux density of the discrete points lied in the flux paths and its dimension is 1 × N _p, μ₀ is the permeability of vacuum.

In fact, the g ^′ = [g ₁, g ₂, …, g _{N
_p}]_{1×N
_p} can be expressed as a equivalent saturation factor multiplied the actual air-gap length g. Therefore, the total air gap magnetic voltage drop can be expressed further as $\begin{eqnarray}\displaystyle \boldsymbol{H}_{\text{g}}\boldsymbol{g}^{\prime }=\boldsymbol{k}_{\text{sat}}g\boldsymbol{H}_{\text{g}} & & \displaystyle\end{eqnarray}$ (3) where k _sat = [k _sat1, k _sat2, …, k _{sat(N
_p)}]_{1×N
_p} represents the saturation factor of the N _p discrete points lied in the flux paths.

Substitution of ((2)) and ((3)) into ((1)) gives, $\begin{eqnarray} \displaystyle \boldsymbol{k}_{\text{sat}}=\frac{\boldsymbol{H}_{\text{g}}g+\boldsymbol{F}_{\text{s}}+\boldsymbol{F}_{\text{r}}}{\boldsymbol{H}_{\text{g}}g}=\frac{\boldsymbol{B}_{\text{g}}g+{{\mu}}_{0}\boldsymbol{F}_{\text{s}}+{{\mu}}_{0}\boldsymbol{F}_{\text{r}}}{\boldsymbol{B}_{\text{g}}g} & & \displaystyle \end{eqnarray}$ (4)

The distribution of the rotor magnetic voltage drops F _r in flux paths is shown in Fig. 2. The rotor d-axis is regarded as the initial position, and in the anticlockwise direction the rotor magnetic voltage drop F _r is written as $\begin{eqnarray}\displaystyle \boldsymbol{F}_{\text{r}} & = & \displaystyle [\mathop{\underbrace{F_{\text{c}}\ldots }_{}}_{N_{\text{c}}},\mathop{\underbrace{F_{\text{rib}(n_{\text{w}})}\ldots }_{}}_{N_{\text{b}(n_{\text{w}})}},\mathop{\underbrace{F_{\text{R}(n_{\text{w}})}\ldots }_{}}_{N_{\text{a}(n_{\text{w}})}},\ldots ,\mathop{\underbrace{F_{\text{rib}1}\ldots }_{}}_{N_{\text{b}1}},\mathop{\underbrace{F_{\text{R}1}\ldots }_{}}_{N_{\text{a}1}},\mathop{\underbrace{F_{\text{L}1}\ldots }_{}}_{N_{\text{a}1}},\mathop{\underbrace{F_{\text{rib}1}\ldots }_{}}_{N_{\text{b}1}},\ldots ,\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \mathop{\underbrace{F_{\text{L}(n_{\text{w}})}\ldots }_{}}_{N_{\text{a}(n_{\text{w}})}},\mathop{\underbrace{F_{\text{rib}(n_{\text{w}})}\ldots }_{}}_{N_{\text{b}(n_{\text{w}})}},\mathop{\underbrace{F_{\text{c}}\ldots }_{}}_{N_{\text{c}}}]_{1\times N_{\text{p}}}^{\text{T}}\end{eqnarray}$ (5) where n _w is the numbers of the flux barriers, F _c is the magnetic voltage drop in each flux path of the rotor channel [13]. The calculation of F _Li, F _Ri and F _ribi will be introduced in subsequent content. The formulas for the N _a _i, N _bi and N _c are respectively written as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}N_{\text{a}i}={\displaystyle \frac{N_{\text{p}}p({\theta}_{\text{f}i}^{\prime \prime }-{\theta}_{\text{f}(i-1)}^{\prime })}{2{\pi}}}\\[10.0pt] N_{\text{b}i}={\displaystyle \frac{N_{\text{p}}p({\theta}_{\text{f}i}^{\prime }-{\theta}_{\text{f}i}^{\prime \prime })}{2{\pi}}}\\[10.0pt] N_{\text{c}}={\displaystyle \frac{N_{\text{p}}p\left({\displaystyle \frac{{\pi}}{2p}}-{\theta}_{\text{f}(n_{\text{w}})}^{\prime }\right)}{2{\pi}}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where p is the numbers of the pole pairs, θ_fi is the ith flux-barrier angle, ${\theta}_{\text{f}i}^{\prime }$ is the angle between the low boundary line (perpendicular to q axis) of the ith flux barrier and q axis, ${\theta}_{\text{f}i}^{\prime \prime }$ is the angle between the upper boundary line (perpendicular to q axis) of the ith flux barrier and q axis. The angle ${\theta}_{\text{f}i}^{\prime \prime }$ , ${\theta}_{\text{f}i}^{\prime }$ , and θ_fi are shown in Fig. 1(b).

Fig. 2.

Sketch of the magnetic voltage drops in the rotor.

Similarly, the distribution of the stator magnetic voltage drops F _s in flux paths are discrete along the circumference of the air gap and the calculation method has been presented in [13]. According to (4), the air-gap density B _g is modified by the saturation factor k _sat and the distribution of the air-gap density will be obtained until the convergence is realized.

2.2. The rotor island magnetic voltage drop

The flux flowing through the ith rotor island is divided into the d-axis flux Φ_di and the q-axis flux Φ_qi. And the d-axis flux Φ_di of the ith rotor island is defined as $\begin{eqnarray}\displaystyle {\Phi}_{\text{d}i}=\frac{DL}{2}\int _{\frac{{\pi}}{2p}-{\theta}_{\text{f}i}^{\prime \prime }}^{\frac{{\pi}}{2p}-{\theta}_{\text{f}(i-1)}^{\prime }}B_{\text{g}}({\theta}_{\text{r}},t)\text{d}{\theta}_{\text{r}},\quad i=1,2,3,\ldots ,n_{\text{w}} & & \displaystyle\end{eqnarray}$ (7) where B _g(θ_r, t) is the air gap flux density.

The q-axis flux Φ_qi of the ith rotor island is computed by the equivalent magnetic circuit of one pole of the PMAREL machine in Fig. 3. The average between the flux which exits from the (i −1)th flux barrier and the flux which enters in the ith flux barrier is regarded as the q-axis flux Φ_qi [13]. The formula is written as $\begin{eqnarray}\displaystyle {\Phi}_{\text{q}i}=\left\{\begin{array}{@{}ll@{}}{\Phi}_{\text{b}1}, & i=1\\[5.0pt] {\displaystyle \frac{{\Phi}_{\text{b}i}+{\Phi}_{\text{b}(i-1)}}{2}}, & i=2,3,\ldots ,n_{\text{w}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (8) where $\begin{eqnarray}\displaystyle {\Phi}_{\text{b}i}=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{U_{\text{r}i}-U_{\text{r}(i+1)}}{R_{\text{b}i}}}-{\Phi}_{\text{pm}i}, & i=1,2,\ldots ,n_{\text{w}}-1\\[11.0pt] {\displaystyle \frac{U_{\text{r}i}}{R_{\text{b}i}}}-{\Phi}_{\text{pm}i}, & i=n_{\text{w}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (9) where U _ri and U _r(i+1) are the magnetic potentials of the ith and (i +1)th rotor island in the non-saturation state respectively, R _bi is the equivalent reluctance of the ith flux barrier which the reluctance of the ith PM R _mi is in parallel with the leakage reluctance of the flux barrier at the left and right sides of the ith PM R _airi(𝜒)∕2. Φ_pmi is the remanence flux of ith PM.

Fig. 3.

Magnetic equivalent circuit of the PMAREL in the non-saturation state.

In this paper, it is assumed that the d-axis and q-axis flux flowing through the left side of the ith rotor island is expressed as Φ_Ldi = w _diΦ_di(0 ≤ w _di ≤ 1) and Φ_Lqi = w _qiΦ_qi(0 ≤ w _qi ≤ 1), respectively. And the d-axis and q-axis flux flowing through the right side of the ith rotor island is expressed as Φ_Rdi = (1 − w _di)Φ_di and Φ_Rqi = (1 − w _qiΦ_qi) in Fig. 2, respectively. The average flux density in the left side of the ith rotor island is written as $\begin{eqnarray}\displaystyle B_{\text{L}i}=\sqrt{\left(\frac{w_{\text{d}i}{\Phi}_{\text{d}i}}{Lw_{\text{iron}i}}\right)^{2}+\left(\frac{2w_{\text{q}i}{\Phi}_{\text{q}i}}{LL_{\text{s}i}}\right)^{2}} & & \displaystyle\end{eqnarray}$ (10) where L _si is the flux path length of the ith rotor island, w _ironi is the width of the ith rotor island. w _di and w _qi are the distribution coefficients of the d-axis and q-axis flux flowing through the ith rotor island respectively, and they are determined by the iteration process. The average flux density in the right side of the ith rotor island is shown as $\begin{eqnarray} \displaystyle B_{\text{R}i}=\sqrt{\left(\frac{(1-w_{\text{d}i}){\Phi}_{\text{d}i}}{Lw_{\text{iron}i}}\right)^{2}+\left(\frac{2(1-w_{\text{q}i}){\Phi}_{\text{q}i}}{LL_{\text{s}i}}\right)^{2}} & & \displaystyle \end{eqnarray}$ (11)

Form the iron B-H curve, the magnetic field H _Li and H _Ri are obtained. The magnetic voltage drops in the left and right sides of the ith rotor island are expressed as F _Li = H _Li L _si∕2 and F _Ri = H _Ri L _si∕2, respectively.

2.3. Outer iron rib magnetic voltage drop

The flux flowing through the ith outer iron rib is divided into the d-axis flux Φ_ribdi and the q-axis flux Φ_ribqi. The d-axis flux density of the ith outer iron rib B _ribdi is written as $\begin{eqnarray}\displaystyle B_{\text{ribd}i}=\frac{D}{2L_{\text{rib}i}}\int _{\frac{{\pi}}{2p}-{\theta}_{\text{f}i}^{\prime }}^{\frac{{\pi}}{2p}-{\theta}_{\text{f}i}^{\prime \prime }}B_{\text{g}}({\theta}_{\text{r}},t)∼\text{d}{\theta}_{\text{r}} & & \displaystyle\end{eqnarray}$ (12)

The q-axis flux density of the ith outer iron rib B _ribqi is written as $\begin{eqnarray}\displaystyle B_{\text{ribq}i}=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{{\mu}_{\text{rib}i}{\mu}_{0}(U_{\text{r}i}-U_{\text{r}(i+1)})}{L_{\text{rib}i}}}, & i=1,2,\ldots ,n_{\text{w}}-1\\[11.0pt] {\displaystyle \frac{{\mu}_{\text{rib}i}{\mu}_{0}U_{\text{r}i}}{L_{\text{rib}i}}}, & i=n_{\text{w}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (13) where μ_rib _i is the relative permeability of the ith outer iron rib.

The flux density of the ith outer iron rib B _ribi is written as $\begin{eqnarray} \displaystyle B_{\text{rib}i}=\sqrt{B_{\text{ribd}i}^{2}+B_{\text{ribq}i}^{2}} & & \displaystyle \end{eqnarray}$ (14)

Form the iron B-H curve, the magnetic field H _ribi is obtained. The magnetic voltage drops in the ith outer iron rib is expressed as F _ribi = H _ribi L _ribi.

2.4. The air gap density and average torque computation

To speed up the iterative calculation of air-gap density B _g, the adopted iteration scheme is expressed as $\begin{eqnarray}\displaystyle \boldsymbol{k}_{\text{sat}}^{(m+1)}=w_{\text{sat}1}\boldsymbol{k}_{\text{sat}}^{(m)}+w_{\text{sat}2}\boldsymbol{k}_{\text{sat}}^{(m-1)} & & \displaystyle\end{eqnarray}$ (15) where m is the numbers of the iteration, w _sat1 and w _sat2 are the convergence factors.

The air-gap density $\boldsymbol{B}_{\text{g}}^{(m+1)}$ is modified by the saturation factor $\boldsymbol{k}_{\text{sat}}^{(m+1)}$ , and the formula is shown as $\begin{eqnarray} \displaystyle \boldsymbol{B}_{\text{g}}^{(m\mathit{+}1)}({\theta}_{\text{r}},t)={\mu}_{0}\frac{-U_{\text{r}}({\theta}_{\text{r}},t)+U_{\text{s}}({\theta}_{\text{r}},t)}{\boldsymbol{k}_{\text{sat}}^{(m+1)}g} & & \displaystyle \end{eqnarray}$ (16)

The air-gap flux density $\boldsymbol{B}_{\text{g}}^{(m+1)}$ is used to the next iteration, and the adopted convergent condition of iteration is written as $\begin{eqnarray}\displaystyle \left|\frac{\boldsymbol{k}_{\text{sat}}^{(m+1)}-\boldsymbol{k}_{\text{sat}}^{(m)}}{\boldsymbol{k}_{\text{sat}}^{(m)}}\right|\leq \boldsymbol{e}_{\text{sat}} & & \displaystyle\end{eqnarray}$ (17) where e _sat is the iterative relative error and its dimension is 1 × N _p.

The air-gap flux density B _g can be computed till the convergence occurs. And the average torque T _avg is derived as $\begin{eqnarray}\displaystyle T_{\text{avg}}=-\frac{D^{2}L}{4}\int _{0}^{2{\pi}}B_{\text{g}}({\theta}_{\text{r}},t)K_{\text{s}}({\theta}_{\text{r}},t)\text{d}{\theta}_{\text{r}} & & \displaystyle\end{eqnarray}$ (18) where K _s(θ_r, t) is the electric loading when the space harmonic order is equal to 1. The formula for K _s(θ_r, t) is given as $\begin{eqnarray}\displaystyle K_{\text{s}}({\theta}_{\text{r}},t)=\frac{6\sqrt{2}Nk_{1}I}{{\pi}D}\sin (p{\theta}_{\text{r}}-{\alpha}) & & \displaystyle\end{eqnarray}$ (19) where N is the numbers of winding turns in series of stator phase, I is the stator current, 𝛼 is the current phase angle, k ₁ is the fundamental winding factor.

3. The function relationships between structure parameters and split ratio

In order to reduce the numbers of the adjustable structural design parameters and ensure the independence of the various structure parameters in the proposed analytical model, the split ratio 𝜒 is regarded as the main design variable at first. And then, the dimensions of rotor islands, PMs, flux barriers at the left and right sides of the PMs, outer iron ribs, rotor channel, stator body height and the flux-barrier angles are expressed as the function relationships with the split ratio 𝜒 as variable according to the actual flux paths in the different iron paths and the dimensional relations among various structural parameters.

Fig. 4.

Sketch of the cross section of the rotor.

3.1. The function relationships between the dimensions of the PMs and split ratio 𝜒

The permeability of the outer iron rib is no longer regarded as a constant since it is changed with the saturation. On the basis of the proposed method which calculates the width of PMs in [18], the saturation of the outer iron ribs is considered by means of the saturated reluctances instead of constant flux sources. The function relationship between the width of the ith PM and the split ratio 𝜒 is written as $\begin{eqnarray}\displaystyle w_{\text{pm}i}({\chi})=w_{\text{pm}i}^{\prime }({\chi})+\frac{2{\Phi}_{\text{g}i}({\chi})R_{\text{g}i}({\chi})w_{\text{rib}i}}{B_{\text{r}}R_{\text{rib}i}({\chi})} & & \displaystyle\end{eqnarray}$ (20) where $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}w_{\text{pm}i}^{\prime }({\chi})={\displaystyle \frac{B_{\text{r}}w_{\text{ins}i}}{B_{\text{r}}w_{\text{ins}i}-B_{\text{avg}i}g}}\left[w_{\text{pm}(i-1)}^{\prime }({\chi})+({\theta}_{\text{f}i}({\chi})-{\theta}_{\text{f}(i-1)}({\chi})){\chi}D_{1}{\displaystyle \frac{B_{\text{avg}i}}{B_{\text{r}}}}\right]\\[15.0pt] R_{\text{rib}i}({\chi})={\displaystyle \frac{L_{\text{rib}i}({\chi})}{{\mu}_{0}{\mu}_{\text{rib}i}Lw_{\text{rib}i}}},\quad w_{\text{pm}0}^{\prime }=0,∼w_{\text{ins}0}=0\end{array}\right. & & \displaystyle\end{eqnarray}$ (21) where $w_{\text{pm}i}^{\prime }({\chi})$ is the effective width of the ith PM. The formulas for calculating Φ_gi(𝜒), R _g _i(𝜒), B _avg _i have been presented in [18], and the formulas for calculating w _ribi, L _ribi(𝜒), θ_fi(𝜒) will be introduced in subsequent content.

In this paper, the length in the magnetization direction of the ith PM h _mi is equal to the length of the ith flux barrier w _ins _i. The function relationship between the length of the zth flux barrier w _ins _z and the length of the (z +1)th flux barrier w _ins(z+1) is written as [17] $\begin{eqnarray}\displaystyle w_{\text{ins}z}={\alpha}_{\text{b}}w_{\text{ins}(z+1)},\quad z=1,2,\ldots ,n_{\text{w}}-1 & & \displaystyle\end{eqnarray}$ (22) where 𝛼_b is the gradient coefficient.

3.2. The function relationships between the dimensions of the different structures in rotor iron core and split ratio 𝜒

(1) Dimensions of the rotor island

It is observed that the width of each rotor iron path in rotor island results to be proportional to the average flux density in the d-axis flux path. And thus, the function relationship between the width of the kth rotor island and the split ratio 𝜒 is given by $\begin{eqnarray} \displaystyle w_{\text{iron}k}({\chi})=\left[ \begin{array}{@{} c@{}}{\displaystyle \frac{({\chi}D_{1}-2g)-D_{\text{sh}}}{2}}\\[10.0pt] -\displaystyle \mathop{\sum }_{q=1}^{n_{\text{w}}}\left({\displaystyle \frac{w_{\text{ins}1}}{{\alpha}_{\text{b}}^{(q-1)}}}\right) \end{array} \right]\overline{B_{\text{g}k}}({\chi}),\quad k=1,2,\ldots ,n_{\text{w}}+1 & & \displaystyle \end{eqnarray}$ (23)

The formula for calculating B _gk (𝜒) is written as $\begin{eqnarray} \displaystyle \overline{B_{\text{g}k}}({\chi})=\frac{{\displaystyle \frac{1}{p({\theta}_{\text{f}k}({\chi})-{\theta}_{\text{f}(k-1)}({\chi}))}}\displaystyle \int _{p{\theta}_{\text{f}(\text{k}-1)}({\chi})}^{p{\theta}_{\text{f}k}({\chi})}\sin (p{\theta})\text{d}{\theta}}{\displaystyle \mathop{\sum }_{i=1}^{n_{\text{w}}+1}{\displaystyle \frac{1}{p({\theta}_{\text{f}k}({\chi})-{\theta}_{\text{f}(\text{k}-1)}({\chi}))}}\displaystyle \int _{p{\theta}_{\text{f}(\text{k}-1)}({\chi})}^{p{\theta}_{\text{f}k}({\chi})}\sin (p{\theta})\text{d}{\theta}} & & \displaystyle \end{eqnarray}$ (24)

The average length of the ith flux barrier l _pmi(𝜒) and the (i +1)th flux barrier l _pm(i+1)(𝜒) is regarded as the length of ith rotor island. And the function relationship between the length of the ith rotor island and the split ratio 𝜒 is given by $\begin{eqnarray}\displaystyle L_{\text{s}i}({\chi})=\left\{\begin{array}{@{}ll@{}}{\lambda}_{i}({\chi}), & i=1\\ {\lambda}_{i}({\chi})+{\lambda}_{i-1}({\chi}), & i=2,3,\ldots ,n_{\text{w}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (25) where the formula for calculating 𝜆_i(𝜒) is written as $\begin{eqnarray}\displaystyle {\lambda}_{i}({\chi})=\frac{2L_{\text{air}i}({\chi})+w_{\text{pm}i}({\chi})}{2} & & \displaystyle\end{eqnarray}$ (26) where L _airi(𝜒) is the length of ith flux barrier at one side of the ith PM, and the formula for calculating L _airi(𝜒) will be introduced in subsequent content.

(2) Dimensions of the outer iron ribs

The arc length of rotor circumferential surface corresponding to the position of each outer iron rib is considered as the equivalent length of the ith outer iron rib L _ribi(𝜒). The function relationship between L _ribi(𝜒) and the split ratio 𝜒 is shown as $\begin{eqnarray} \displaystyle L_{\text{rib}i}({\chi})=({\theta}_{\text{f}i}^{\prime }({\chi})-{\theta}_{\text{f}i}^{\prime \prime }({\chi}))\frac{({\chi}D_{1}-2g)}{2} & & \displaystyle \end{eqnarray}$ (27)

The width of the ith outer iron rib w _ribi is first set an initial empirical value and then the structural strength of the rotor is assessed by the structural FEM. On the premise of ensuring that the rotor could sustain the centrifugal force and the magnetic force, the widths of the ith outer iron ribs w _ribi is the smallest.

(3) Dimensions of the rotor channel

The average length of AB and CD is regarded as the width of the rotor iron path in rotor channel since the flux flows through the rotor channel are non-uniform. The function relationship between the width of the rotor iron path in rotor channel and the split ratio 𝜒 is written as $\begin{eqnarray}\displaystyle w_{\text{b}}({\chi})=\frac{\left({\displaystyle \frac{{\chi}D_{1}-2g}{2}}\right)\sin \left({\displaystyle \frac{{\pi}}{2p}}-{\theta}_{\text{f}(n_{\text{w}})}^{\prime }({\chi})\right)}{2}+\frac{|w_{\text{b}1}({\chi})|\sin \left({\displaystyle \frac{{\pi}}{2p}}-\text{Arg}(w_{\text{b}1}({\chi}))\right)}{2} & & \displaystyle\end{eqnarray}$ (28) where $\begin{eqnarray}\displaystyle w_{\text{b}1}({\chi})=\left(\frac{D_{\text{sh}}}{2}+w_{\text{iron}(n_{\text{w}}+1)}({\chi})\right)+j\frac{w_{\text{pm}(n_{\text{w}})}({\chi})}{2}. & & \displaystyle\end{eqnarray}$ (29)

3.3. The function relationships between the dimensions of the flux barriers at the left and right sides of the PMs and split ratio 𝜒

The function relationship between the width of the ith flux barrier at the left and right sides of the ith PM and the split ratio 𝜒 is given by $\begin{eqnarray} \displaystyle w_{\text{air}i}({\chi})=\frac{({\theta}_{\text{f}i}^{\prime }({\chi})-{\theta}_{\text{f}i}^{\prime \prime }({\chi}))\frac{({\chi}D_{1}-2g)}{2}+w_{\text{ins}i}}{2} & & \displaystyle \end{eqnarray}$ (30)

The function relationship between the length of ith flux barrier at one side of the ith PM and the split ratio 𝜒 is given by $\begin{eqnarray}\displaystyle L_{\text{air}i}({\chi})=\sqrt{(x_{\text{air}i}({\chi})-x_{\text{r}i}({\chi}))^{2}+(y_{\text{air}i}({\chi})-y_{\text{r}i}({\chi}))^{2}}-w_{\text{rib}i} & & \displaystyle\end{eqnarray}$ (31) where $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} l@{}}x_{\text{r}i}({\chi})={\displaystyle \frac{({\chi}D_{1}-2g)}{2}}\cos \left({\displaystyle \frac{{\pi}}{2p}}+{\theta}_{\text{f}i}({\chi})\right)\\ [10.0pt] y_{\text{r}i}({\chi})={\displaystyle \frac{({\chi}D_{1}-2g)}{2}}\sin \left({\displaystyle \frac{{\pi}}{2p}}+{\theta}_{\text{f}i}({\chi})\right)\\ [10.0pt] x_{\text{air}i}({\chi})=\sqrt{\left({\displaystyle \frac{w_{\text{pm}i}({\chi})}{2}}\right)^{2}+(r_{\text{f}i}({\chi}))^{2}}\\ [10.0pt] \hspace{48.0pt}\cos \left({\displaystyle \frac{{\pi}}{2p}}+\arctan \left(\frac{\frac{w_{\text{pm}i}({\chi})}{2}}{r_{\text{f}i}({\chi})}\right)\right)\\ [10.0pt] y_{\text{air}i}({\chi})=\sqrt{\left({\displaystyle \frac{w_{\text{pm}i}({\chi})}{2}}\right)^{2}+(r_{\text{f}i}({\chi}))^{2}}\\ [10.0pt] \hspace{48.0pt}\sin \left({\displaystyle \frac{{\pi}}{2p}}+\arctan \left({\displaystyle \frac{\frac{w_{\text{pm}i}({\chi})}{2}}{r_{\text{f}i}({\chi})}}\right)\right) \end{array} \right. & & \displaystyle \end{eqnarray}$ (32)

3.4. The function relationships between the stator structure and split ratio 𝜒

In order to reduce the numbers of stator structure parameters, the shape of the stator slot is simplified as shown in Fig. 5. The function relationships between the structure parameters of the two stator slot shapes are written as $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} l@{}}b_{\text{s}1}={\displaystyle \frac{{\pi}(D_{1}+2h_{\text{t}})}{Q}}-w_{\text{t}}\\ [15.0pt] r={\displaystyle \frac{{\displaystyle \frac{{\pi}}{Q}}(D_{1}-2h_{\text{y}})-w_{\text{t}}}{2\left(1+{\displaystyle \frac{4{\pi}}{3Q}}\right)}}\\ [30.0pt] h_{\text{s}}({\chi})=\left({\displaystyle \frac{D_{1}-{\chi}D_{1}-2h_{\text{y}}-2h_{\text{t}}}{2}}\right)-{\displaystyle \frac{r}{6}} \end{array} \right. & & \displaystyle \end{eqnarray}$ (33)

The values of h _t, w _t and h _y are determined by the slot-fill factor, the flux density in stator yoke and teeth.

Fig. 5.

Stator slot shapes.

3.5. The function relationships between the dimensions of flux-barrier angles and split ratio 𝜒

The average torque strongly depends on the flux-barrier shapes which are often designed to be parallel to the d-axis flux lines, and perpendicular to the q-axis flux lines. The higher the reluctance torque is, the larger the difference between d-axis inductance and the q-axis inductance is. Thus, the boundary lines l _A and l _B are parallel to the d-axis flux lines as far as possible.

The distribution of the rotor flux lines can be considered as a closed-form expression and it can be derived from the conformal mapping theory and the Joukowski air-flow potential formulation [19]. Therefore, the angle between each flux line and the q-axis is calculated as $\begin{eqnarray}\displaystyle {\theta}_{\text{f}}({\chi})=\frac{{\pi}}{2p}-\frac{1}{p}\arcsin \left(\frac{C_{\text{f}}({\chi})\left({\displaystyle \frac{{\chi}D_{1}-2g}{D_{\text{sh}}}}\right)^{p}}{\left({\displaystyle \frac{{\chi}D_{1}-2g}{D_{\text{sh}}}}\right)^{2p}-1}\right) & & \displaystyle\end{eqnarray}$ (34) where C _f(𝜒) is the potential factor and it can be expressed as $\begin{eqnarray}\displaystyle C_{\text{f}}({\chi})=\left(\frac{\left({\displaystyle \frac{2r_{\text{f}}({\chi})}{D_{\text{sh}}}}\right)^{2p}-1}{\left({\displaystyle \frac{2r_{\text{f}}({\chi})}{D_{\text{sh}}}}\right)^{p}}\right) & & \displaystyle\end{eqnarray}$ (35) where r _f(𝜒) is the distance from point O to a point on any flux line.

The shortest distance from point O to the boundary line l _A, the boundary line l _B and the center line l _pm between the boundary line l _A and l _B can be expressed as the functions with the split ratio 𝜒 as variable and are written as $\begin{eqnarray} \displaystyle \left\{ \begin{array}{@{} l@{}}r_{\text{f}i}^{\prime }({\chi})={\displaystyle \frac{D_{\text{sh}}}{2}}+\displaystyle \mathop{\sum }_{l=i+1}^{n_{\text{w}}+1}w_{\text{iron}l}({\chi})+\displaystyle \mathop{\sum }_{q=i+1}^{n_{\text{w}}}{\displaystyle \frac{w_{\text{ins}1}}{{{\alpha}_{\text{b}}}^{(q-1)}}}\\ [12.0pt] r_{\text{f}i}^{\prime \prime }({\chi})={\displaystyle \frac{D_{\text{sh}}}{2}}+\displaystyle \mathop{\sum }_{l=i+1}^{n_{\text{w}}+1}w_{\text{iron}l}({\chi})+\mathop{\sum }_{q=i}^{n_{\text{w}}}{\displaystyle \frac{w_{\text{ins}1}}{{{\alpha}_{\text{b}}}^{(q-1)}}}\\ [12.0pt] r_{\text{f}i}({\chi})={\displaystyle \frac{D_{\text{sh}}}{2}}+\displaystyle \mathop{\sum }_{l=i+1}^{n_{\text{w}}+1}w_{\text{iron}l}({\chi})+\mathop{\sum }_{q=i+1}^{n_{\text{w}}}{\displaystyle \frac{w_{\text{ins}1}}{{{\alpha}_{\text{b}}}^{(q-1)}}}+{\displaystyle \frac{w_{\text{ins}1}}{2{{\alpha}_{\text{b}}}^{(q-1)}}} \end{array} \right. & & \displaystyle \end{eqnarray}$ (36)

In addition, according to (23) and (34), the flux-barrier angle θ_f and the width of the rotor island w _iron are mutually involved in the calculation. Therefore, in order to compute the parameter θ_f and w _iron, the initial value is first given to the angle θ_f [20], and then combing (23), (34)–(36), the parameter θ_f and w _iron which satisfy the functions with the split ratio 𝜒 as variable are determined.

4. Modeling procedures of magnetic field and FEM validation

4.1. Modeling procedures of magnetic field based on the split ratio

In the preliminary design of the gradient PMAREL machines, it will adjust repeatedly the dimensions of the structure parameters to give consideration to the design requirements of various electromagnetic performance. Theoretically, it needs to evaluate all the possible schemes by combining the variable parameters one by one in their ranges and the tasks are arduous. Therefore, in the modeling of air-gap magnetic field for the gradient PMAREL machines, the geometric structure corresponding to the split ratio 𝜒 is necessarily established according to the function relationships in Section 3. Based on the geometric structure, it just gives the value of the split ratio 𝜒 and then other structure parameters can be computed by the function relationships in Section 3. It can make the numbers of the adjustable structure parameters and the possible schemes greatly reduced. Thus, it is beneficial to determine the initial design schemes of the gradient PMAREL machines rapidly. Figure 6 shows the comparison of two flow charts for the modeling of the air-gap magnetic field. One is based on the multi-parameter combination, and the other is based on the split ratio 𝜒.

Fig. 6.

The comparison of two flow charts of the modeling of the air-gap magnetic field. (a) Based on the multi-parameter combination. (b) Based on the split ratio 𝜒.

4.2. FEM validation

According to the flow chart of the modeling of the air-gap magnetic field based on the split ratio 𝜒, the geometry structures of a 48-slot 8-pole (𝜒 = 0.619) and a 36-slot 4-pole (𝜒 = 0.68) gradient PMAREL machines are established. And the two machines are considered as the original prototypes. The geometries, diagrams of mesh generation, flux lines distribution maps, and flux density maps of the two machines are shown in Fig. 7. The proposed analytical model with the geometric structure is confirmed by applying a common finite element software-Ansoft Maxwell 19.0. The main dimensions of the two machines are shown in Table 1.

Fig. 7.

The geometries, diagrams of mesh generation, flux lines, and flux density maps of the two machines. (a) 48 slot/8 pole (𝜒 = 0.619). (b) 36 slot/4 pole (𝜒 = 0.68).

Table 1

Dimensions of the two machines

Parameter	Symbol	48-slot	36-slot	Unit	Parameter	Symbol	48-slot	36-slot	Unit
		8-pole	4-pole				8-pole	4-pole
Rated speed	n _N	3000	3000	r/min	Shaft diameter	D _sh	48	25	mm
Rated torque	T _N	12.08	65.20	Nm	Active length	L	140	155	mm
Rated current density	J _N	4.37	5.90	A/mm²	Slot-fill factor	S _f	78	78	%
Pole pair number	p	4	2	–	Remanence	B _r	0.405	0.405	T
Slot number	Q	48	36	–	Numbers of the flux barriers	n _w	3	3	–
Air-gap length	g	1	0.4	mm	Relative permeability of PM	μ_r	1.045	1.045	–
Outer radius of stator	D ₁	210	265	mm	-	-	-	-	-

When t = 0 s and n = n _N, the air gap flux density distributions of the two machines calculated by the proposed analytical method have a good agreement with FEM results at the current densities of J = 40%J _N, J = 60%J _N, J = 80%J _N, and J = J _N, as shown in Fig. 8.

Fig. 8.

Air-gap flux density distributions of the two machines by FEM and the proposed analytical method. (a) 48 slot/8 pole. (b) 36 slot/4 pole.

The fundamental amplitudes relative errors of the air gap flux density calculated by the proposed analytical method and FEM are shown in Table 2, and it is shown that the relative errors are small.

Table 2

Calculation results of air-gap flux density fundamental amplitudes between FEM and the proposed method

	J∕J _N	Proposed	FEM (T)	Relative		J∕J _N	Proposed	FEM (T)	Relative
		method (T)		error (%)			method (T)		error (%)
	0.4	0.1761	0.1782	−1.1785		0.4	0.4663	0.4539	−2.7319
48-slot	0.6	0.2657	0.2670	−0.4869	36-slot	0.6	0.5943	0.5858	−1.4510
8-pole	0.8	0.3552	0.3549	+0.0845	4-pole	0.8	0.6466	0.6363	−1.6347
	1.0	0.4441	0.4405	+0.8172		1.0	0.6824	0.6675	−2.2322

The average torque T _avg with the different current phase angles 𝛽 of the two machines calculated by the proposed analytical method and FEM have a good consistency at the current densities of J = 40%J _N, J = 60%J _N, J = 80%J _N, and J = J _N, as shown in Fig. 9.

Fig. 9.

The average torque T _avg and the relative errors with the different current phase angles 𝛽 of the two machines by FEM and the proposed analytical method. (a) 48 slot/8 pole. (b) 36 slot/4 pole.

In the left part of Fig. 9(a) and Fig. 9(b), it is concluded that the maximum average torque occurs at the current phase angle 𝛽 = 45° with the current density increased in a 48-slot 8-pole PMAREL machine, however, the maximum average torque does not occur at the current phase angle 𝛽 = 45° but at a higher current phase angle with the current density increased in a 36-slot 4-pole PMAREL machine. This is mainly due to the effect of saturation have a significant influence on the 36-slot 4-pole machine.

The relative errors of the T _avg with the different current phase angles 𝛽 of the two machines calculated by the proposed analytical method and FEM at the current densities of J = 40%J _N, J = 60%J _N, J = 80%J _N, and J = J _N are shown in the right of Fig. 9(a) and Fig. 9(b), and the relative errors are small. The relative errors of the rated torque T _N of the two machines calculated by the proposed analytical method and FEM at the current density J = J _N are shown in Table 3 and it is observed that the relative errors are small.

Table 3

Calculation results of the rated torque between FEM and the proposed method

	Proposed method (Nm)	FEM (Nm)	Relative error (%)
48-slot/8-pole	12.20	12.08	+0.99
36-slot/4-pole	65.94	65.20	+1.136

5. Analysis of the optimal split ratio

In Section 4, the air-gap density and the average torque of the gradient PMAREL machines are computed in a certain split ratio 𝜒 which makes the design requirement of the output torque satisfied. And based on this, the proposed analytical model is also verified correctly. In fact, the split ratio 𝜒 which makes the design requirement of the output torque satisfied is not unique and the differences of the electromagnetic performance with different split ratio 𝜒 are large. The 48-slot/8-pole and the 36-slot/4-pole three gradient flux-barrier PMAREL machines are considered and the proposed analytical model with the geometric structure is applied to research the influences of the split ratio 𝜒 on the average torque, stator loss, efficiency, the utilization rate of PMs and the demagnetization of PMs. And based on this, the optimal split ratios which make the comprehensive electromagnetic performance of the two PMAREL machines better are determined.

5.1. Average torque

When J = J _N and n = n _N, the influences of the split ratio 𝜒 on the average torque T _avg of the two machines can be achieved by using (18), and the results are shown in Fig. 10. It can be observed that the average torque T _avg of the two machines are increased to a maximum and then decreased with the split ratio 𝜒 further increased. Then reason is that the stator yoke is more highly saturated when its height is decreased with the split ratio 𝜒 further increased. The average torque of the 48-slot/8-pole PMAREL machine is more than 1.6 times as large as the rated torque when the split ratio 𝜒 is ranged from 0.65 to 0.71. The average torque of the 36-slot/4-pole PMAREL machine is more than 1.05 times as large as the rated torque when the split ratio 𝜒 is ranged from 0.68 to 0.70. Thus, it is concluded that the higher torque can be obtained through the reasonable design of the split ratio 𝜒 in constant current density, winding parameters and the active length.

Fig. 10.

The relationships between the split ratio 𝜒 and the average torque T _avg (J = J _N, n = n _N). (a) 48 slot/8 pole. (b) 36 slot/4 pole.

5.2. Stator loss

The permanent magnet eddy-current loss and rotor iron loss of the PMAREL machines are generally neglected. Thus, the total loss P _loss is regarded as the sum of the stator iron loss P _Fe and the stator copper loss P _cu. The end windings are considered and the extra loss is neglected. The function relationship between the total loss P _loss(𝜒) and the split ratio 𝜒 is described as $\begin{eqnarray}\displaystyle P_{\text{loss}}({\chi})={\rho}(L+l_{\text{e}}({\chi}))QJ^{2}S_{\text{f}}({\chi})A_{\text{f}}({\chi})+(k_{\text{h}}f{B_{\text{m}}}^{2}+k_{\text{c}}f^{2}{B_{\text{m}}}^{2})(V_{\text{y}}({\chi})+V_{\text{t}}({\chi})) & & \displaystyle\end{eqnarray}$ (37) where l _e(𝜒) is the end-winding length, 𝜌 is the resistivity of copper, J is the current density, S _f(𝜒) is the slot-fill factor, A _f(𝜒) is the area of a stator slot. k _h is the coefficient of the hysteresis loss, k _c is the coefficient of the classical eddy current loss, B _m is the maximum flux density in the stator iron, V _y(𝜒) is the volume of stator yoke, V _t(𝜒) is the volume of stator teeth, f is the current frequency. According to the stator structure parameters and the winding parameters, the l _e(𝜒), S _f(𝜒), A _f(𝜒), V _y(𝜒), V _t(𝜒) can be calculated. It should be mentioned that the simple models applied to computed P _Fe(𝜒) and P _cu(𝜒) do not have a significant effect on the optimal split ratio 𝜒 determination [16].

Fig. 11.

The relationships between the split ratio 𝜒 and P _Fe, P _cu and P _loss (J = J _N, n = n _N). (a) 48 slot/8 pole. (b) 36 slot/4 pole.

When J = J _N and n = n _N, the influences of the split ratio 𝜒 on the P _Fe(𝜒), P _cu(𝜒) and P _loss(𝜒) are shown in Fig. 11. In the 48-slot 8-pole PMAREL machine, P _cu(𝜒) and P _loss(𝜒) are decreased and then increased with the split ratio 𝜒 increased, however, P _Fe(𝜒) is decreased with the split ratio 𝜒 increased. It can be concluded that the optimal split ratio 𝜒 is 0.70 which the P _loss(𝜒) is minimum. In the 36-slot 4-pole PMAREL machine, P _cu(𝜒) and P _loss(𝜒) are increased with the split ratio 𝜒 increased, however, P _Fe(𝜒) is decreased with the split ratio 𝜒 increased. It can be concluded that the optimal split ratio 𝜒 is 0.60 which the P _loss(𝜒) is minimum.

5.3. Efficiency

The function relationship between the efficiency 𝜂 and the split ratio 𝜒 is written as $\begin{eqnarray}\displaystyle {\eta}({\chi})=\frac{T_{\text{avg}}({\chi}){\Omega}}{T_{\text{avg}}({\chi}){\Omega}+P_{\text{loss}}({\chi})} & & \displaystyle\end{eqnarray}$ (38) where 𝛺 is mechanical angular velocity.

Fig. 12.

The relationships between the split ratio 𝜒 and efficiency 𝜂 (J = J _N, n = n _N). (a) 48 slot/8 pole. (b) 36 slot/4 pole.

When J = J _N and n = n _N, the influences of the split ratio 𝜒 on the efficiency 𝜂 of the two machines can be achieved by using (38), as shown in Fig. 12. In Fig. 12, the efficiency 𝜂(𝜒) of the two machines are increased to a maximum and then decreased with the split ratio 𝜒 further increased. Compared with the 36-slot 4-pole PMAREL machine, the split ratio 𝜒 has a significant impact on the efficiency 𝜂(𝜒) in 48-slot 8-pole PMAREL machine. In particular, the efficiency 𝜂(𝜒) in 48-slot 8-pole PMAREL machine is sharply decreased when the split ratio 𝜒 is more than 0.70.

5.4. The effective flux coefficient

The effective flux coefficient is acquired by the ratio of main magnetic flux to total magnetic flux and it reflects the effective utilization of total flux provided by PMs to the external magnetic circuit. The larger the coefficient is, the higher the utilization rate of the PM is. Due to the part of the magnetic flux of the PM flows through the iron ribs up to saturate them, the saturated reluctances in the outer iron ribs are considered additionally in Fig. 3. Thus, the function relationship between the effective flux coefficient and the split ratio 𝜒 is computed as $\begin{eqnarray}\displaystyle {\sigma}_{i}({\chi}) & = & \displaystyle \frac{{\Phi}_{\text{m}i}({\chi})-{\Phi}_{{\sigma}i}({\chi})}{{\Phi}_{\text{m}i}({\chi})}\nonumber\\ \displaystyle & = & \displaystyle \left\{\begin{array}{@{}ll@{}}1-{\displaystyle \frac{R_{\text{m}i}({\chi})}{{\displaystyle \frac{R_{\text{m}i}({\chi})R_{\text{mb}i}({\chi}){\phi}_{\text{pm}i}({\chi})}{(U_{\text{r}i}({\chi})-U_{\text{r}(i+1)}({\chi}))}}-(R_{\text{m}i}({\chi})+R_{\text{mb}i}({\chi}))}}, & i=1,2,\ldots n_{\text{w}}-1\\[35.0pt] 1-{\displaystyle \frac{R_{\text{m}i}({\chi})}{{\displaystyle \frac{R_{\text{m}i}({\chi})R_{\text{mb}i}({\chi}){\phi}_{\text{pm}i}({\chi})}{U_{\text{r}i}({\chi})}}-(R_{\text{m}i}({\chi})+R_{\text{mb}i}({\chi}))}}, & i=n_{\text{w}}\end{array}\right.\end{eqnarray}$ (39) where Φ_mi is total magnetic flux, Φ_σi is leakage magnetic flux. The difference between the flux Φ_mi and the flux Φ_σi is defined as main magnetic flux. The formulas for calculating R _mbi, R _bi and Φ_pmi are written as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}R_{\text{mb}i}({\chi})={\displaystyle \frac{R_{\text{rib}i}({\chi})R_{\text{air}i}({\chi})}{2(R_{\text{rib}i}({\chi})+R_{\text{air}i}({\chi}))}}\\[15.0pt] R_{\text{b}i}({\chi})={\displaystyle \frac{R_{\text{m}i}({\chi})R_{\text{mb}i}({\chi})}{R_{\text{m}i}({\chi})+R_{\text{mb}i}({\chi})}}\\[15.0pt] {\phi}_{\text{pm}i}({\chi})=B_{\text{r}}w_{\text{pm}i}({\chi})L\end{array}\right. & & \displaystyle\end{eqnarray}$ (40) where R _mbi(𝜒) is the equivalent reluctance which the reluctance R _ribi(𝜒)∕2 is in parallel with the reluctance R _airi(𝜒)∕2.

Fig. 13.

The relationships between the split ratio 𝜒 and the effective flux coefficient σ(𝜒) (J = J _N, n = n _N). (a) 48 slot/8 pole. (b) 36 slot/4 pole.

The influences of the split ratios 𝜒 on the effective flux coefficients of the two machines are shown in Fig. 13. In the 48-slot 8-pole PMAREL machine, it is observed that the effective flux coefficients are decreased with the split ratios 𝜒 increased. In the 36-slot 4-pole PMAREL machine, it is observed that the effective flux coefficients is increased and then decreased with the split ratios 𝜒 increased as a whole, and the optimal split ratios 𝜒 corresponding to the maximum effective flux coefficients are 0.66, 0.72, 0.72 for the first, second and third layer of PM, respectively. Meanwhile, it is noted that the two machines have a common feature, on the whole, which the effective flux coefficient for the first layer of PM is higher than the second and third layer of PMs and it is also shown that the most effective utilization of the flux provided by the first layer of PM to the external magnetic circuit.

5.5. Demagnetization of the PMs

The magnetic network of the gradient three-flux-barrier PMAREL machine at the no load condition is obtained when the average value of the stator magnetic potential in front of the ith flux-barrier U _si is equal to zero in Fig. 3. The no load flux density of the ith PM is derived as $\begin{eqnarray}\displaystyle B_{\text{u}i}({\chi})=\frac{{\Phi}_{\text{u}i}}{Lw_{\text{pm}i}({\chi})}=\left\{\begin{array}{@{}ll@{}}{\displaystyle \frac{{\Phi}_{\text{pm}i}({\chi})R_{\text{m}i}({\chi})+U_{\text{r}0(i+1)}({\chi})-U_{\text{r}0i}({\chi})}{Lw_{\text{pm}i}({\chi})R_{\text{m}i}({\chi})}}, & i=1,2,\ldots n_{\text{w}}-1\\[15.0pt] {\displaystyle \frac{{\Phi}_{\text{pm}i}({\chi})R_{\text{m}i}({\chi})-U_{\text{r}0i}({\chi})}{Lw_{\text{pm}i}({\chi})R_{\text{m}i}({\chi})}}, & i=n_{\text{w}}\end{array}\right. & & \displaystyle\end{eqnarray}$ (41) where Φ_ui is the no load flux flowing through the ith PM, U _r0i(𝜒) and U _r0(i+1)(𝜒) are the magnetic potentials of the ith and (i +1)th in rotor island at no load condition, respectively.

The influences of the split ratio 𝜒 on the no-load flux density of the ith PM, load flux density of the ith PM and the flux density which makes the ith PM demagnetized of the two machines are shown in Fig. 14. As can be seen from the left part of Fig. 14, the flux density B _u1 is decreased with the split ratio 𝜒 increased, whereas the flux density B _u2 and B _u3 are almost regarded as a different constant with the split ratio 𝜒 increased in 48-slot 8-pole PMAREL machine. However, the flux density B _u1, B _u2 and B _u3 are hardly changed with the different the split ratios 𝜒 in 36-slot 4-pole PMAREL machine.

To compute the flux due to the stator at the worst condition, the PMs are removed and the stator current is producing an MMF completely along the q-axis. The flux density in the region where the PM is inset B _fi(𝜒) can be calculated [21]. Thus, the flux density at the worst load condition of the ith PMs is expressed as $\begin{eqnarray} \displaystyle B_{\text{LO}i}({\chi})=B_{\text{u}i}({\chi})-B_{\text{f}i}({\chi}) & & \displaystyle \end{eqnarray}$ (42)

The influences of the split ratio 𝜒 on the load flux density B _LO of the two machines are shown in the middle part of Fig. 14. It is observed that the larger the split ratio 𝜒 is, the higher B _LO is. Meanwhile, it is concluded that the capability of the PM anti-demagnetization can be improved through increasing the split ratio 𝜒 properly. The influences of the split ratio 𝜒 on the flux density B _f which makes the ith PM demagnetized of the two machines are shown in the right part of Fig. 14, it is noted that the MMF completely along the q-axis has the most impact on the value of B _LO1 in 48-slot 8-pole PMAREL machine.

In this paper, the ferrite is used that the safe area is B > 0.06 T at −20 °C, and Br = 0. 404 T at +20 °C. As can be seen from the middle part of Fig. 14, it is concluded that the flux density B _LO on the all PMs of two machines are more than 0.06T when the split ratio 𝜒 is ranged from 0.60 to 0.75. Therefore, the two machines do not exist the risk of demagnetization.

Fig. 14.

The relationships between the split ratio 𝜒 and B _u, B _LO and B _f. (a) 48 slot/8 pole. (b) 36 slot/4 pole.

5.6. Determination of the optimal split ratio 𝜒

From a comprehensive analysis of the influences of the split ratio 𝜒 on the average torque T _avg, stator loss P _loss, efficiency 𝜂, the effective flux coefficient σ and the demagnetization of the PMs, the method for determining the optimal split ratios which the efficiencies of two machines are highest and the PMs do not exist the risk of the demagnetization is introduced as follows:

Step 1. According to the influences of the split ratio 𝜒 on the load flux density B _LO and the effective flux coefficient σ, the range of the split ratio 𝜒 is determined under the condition that the utilization rates of the PMs are high and the PMs do not exist the risk of the demagnetization.

In Fig. 13, the utilization rates of the PMs are relatively high in the two machines when the split ratio 𝜒 is ranged from 0.65 to 0.71. The influences of the split ratio 𝜒 on the load flux density B _LO of the two machines are shown in the middle part of Fig. 14. It is concluded that the flux density B _LO on the all PMs of two machines are more than 0.06T when the split ratio 𝜒 is ranged from 0.60 to 0.75. Therefore, the PMs of the two machines have a relatively high utilization rate and the PMs do not exist the risk of the demagnetization when the split ratio 𝜒 is ranged from 0.60 to 0.75.

Step 2. The influences of the split ratio 𝜒 on the efficiency 𝜂 of the two machines can be achieved by using (38) and the split ratio 𝜒 corresponding to the highest efficiency is determined according to the performance curve of the efficiency 𝜂 under the different split ratios.

In Fig. 12, the efficiency 𝜂(𝜒) of the two machines are increased to a maximum and then decreased with the split ratio 𝜒 further increased. The efficiency 𝜂(𝜒) of the 48-slot 8-pole PMAREL machine and the 36-slot 4-pole PMAREL machine are the highest when the split ratio 𝜒 is 0.70 and 0.69, respectively.

Step 3. If the split ratio 𝜒 determined in step 2 is within the range of the split ratio 𝜒 in step 1, the split ratio 𝜒 determined in step 2 is regarded as the optimal split ratio 𝜒. If not, the split ratio 𝜒 determined in step 1 corresponding to the highest efficiency can be selected as the optimal split ratio 𝜒.

As can be seen from Fig. 12, the split ratio 𝜒 determined in step 2 of the two machines are both within the range of the split ratio 𝜒 in step 1. Thus, the optimal split ratios for the 48-slot 8-pole and the 36-slot 4-pole PMAREL machine are 𝜒 = 0.70 and 𝜒 = 0.69, respectively.

The geometries, diagrams of mesh generation, flux lines distribution maps, and flux density maps of the two machines are shown in Fig. 15.

Fig. 15.

The geometries, diagrams of mesh generation, flux lines, and flux density maps of the two machines. (a) 48 slot/8 pole (𝜒 = 0.70). (b) 36 slot/4 pole (𝜒 = 0.69).

As can be seen from Fig. 16, it is observed that the flux density on all PMs of two machines are more than 0.06 T when the stator current is producing an MMF completely along the q-axis. Therefore, the two machines do not exist the risk of demagnetization.

Fig. 16.

The flux density maps of the PMs. (a) 48 slot/8 pole. (b) 36 slot/4 pole.

When J = J _N and n = n _N, the comparison of electromagnetic performance between the 48-slot 8-pole PMAREL machine when the split ratio 𝜒 is 0.70 and the original prototype, such as P _cu, P _Fe, P _loss, T _avg and 𝜂, are analyzed by FEM. The results are listed in Table 4. Compared with the original prototype, the average torque T _avg is increased by 3.2 Nm, the total loss P _loss is decreased by 36.11 W, and the efficiency 𝜂 is increased by 1.97%.

Table 4

Performance of the 48-slot 8-pole PMAREL machine

Parameter	𝜒 = 0.619	𝜒= 0.70	Unit
T _avg	12.08	15.28	Nm
P _cu	108.80	104.5	W
P _Fe	154.11	122.30	W
P _loss	262.91	226.80	W
𝜂	93.52	95.49	%

Table 5

Performance of the 36-slot 4-pole PMAREL machine

Parameter	𝜒 = 0.68	𝜒= 0.69	Unit
T _avg	65.20	68.50	Nm
P _cu	420.03	433.6	W
P _Fe	343.52	334.55	W
P _loss	763.55	768.15	W
𝜂	96.41	96.55	%

When J = J _N and n = n _N, the comparison of the electromagnetic performance between the 36-slot 4-pole PMAREL machine when the split ratio 𝜒 is 0.69 and the original prototype, such as P _cu, P _Fe, P _loss, T _avg and 𝜂, are analyzed by FEM. The results are listed in Table 5. Compared with the original prototype, the average torque T _avg is increased by 3.3 Nm, the total loss P _loss is increased by 4.6 W, but the efficiency 𝜂 is increased by 0.14%.

In summary, the proposed analytical model together with the geometric structure is convenient for analyzing the influences of the split ratio 𝜒 on the average torque, stator loss, efficiency, the utilization rate of PMs and the demagnetization of PMs. And on the basis of above research, the initial design scheme corresponding to the optimal split ratio 𝜒 which makes the comprehensive electromagnetic performance of the machine better is determined.

6. Conclusion

This paper presents a performance analysis method for PMAREL is presented. This method is able to avoid the blindness of the preliminary design stage requiring adjusting repeatedly the dimensions of the parameters, and it is capable of reducing the numbers of the adjustable structural design parameters during the initial design process and ensuring the independence among the parameters. More importantly, it is perfect for finding general guidelines to design the PMAREL machines with the variety and complexity of magnetic circuit and many structure parameters. Therefore, the performance analysis method is of great significance for predicting the electromagnetic performance with the various structure parameters changing in advance and determining the reasonable design schemes.

Footnotes

Acknowledgements

This work was supported in part by the Major Program of National Natural Science Foundation of China under Grant 51690183 and in part by the Tianjin College Innovation Team Training program of China under Grant TD13-5039.

Nomenclature

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