Multiphysics optimization design flow with improved submodels for salient switched reluctance machines

Abstract

Multiphysical optimization design of electrical machines is essential, in which the submodels, including electromagnetic, thermal and noise/vibration (NV) simulation models are required not only to be accurate to facilitate coupling analysis, but also of small computation to guarantee calculation speed. In this paper, a multiphysics design flow for salient electrical machines is proposed. The flow aims to maximize torque with temperature rise and acoustic guaranteed, in which each submodel of fast and accurate calculation is respectively studied. For electromagnetic analysis, an efficient finite element (FE) based program is proposed to model distribution of magnetic saturation on salient poles. For NV analysis, an analytical frequency estimation approach is proposed. For thermal analysis, a lumped parameter network model with compensation elements is studied. An 8/6 salient switched reluctance machine (SRM) of high power density for vehicular traction application is studied.

Keywords

Switched reluctance machine multiphysics analysis electromagnetic modeling thermal network acoustic analysis optimization flow

1. Introduction

Traditional design flow of an electrical machine drive mainly focuses on electromagnetic analysis. For an efficient design, the multiphysical optimization is sharply increasing [1, 2, 3, 4, 5]. All major aspects, including electromagnetic, thermal and noise/vibration (NV) as submodels, are seen correlated, which aims to an optimized design. The optimization flow serves as medium expected to integrate all multiphysical considerations in an efficient approach. Each submodel, for another hand, is required to have not only high accuracy but small computation, the essential feature to guarantee operation efficiency of the flow.

Electromagnetic analysis serves as the main aspect. The difficult point falls onto the estimation of flux due to magnetic saturation, particularly when the electrical machine has prominent saliency [6, 7, 8, 9], such as switched reluctance machines (SRM). Local magnetic saturation occurs in particular on both stator and rotor salient poles. The tips of poles are highly saturated, especially when poles partially overlap. The distribution and intensity are regarded as a function of both rotor position and phase current level, making modeling of flux characteristics that linked to the machine difficult. The degree of saturation is more obviously shown in high power density operation. Further the model is required to flexibly communicate with other physical aspects. Neither traditional magnetic equivalent circuit (MEC) [10, 11, 12, 13, 14] nor magnetic vector field method [15, 16, 17] enables accurate magnetic saturation calculation. In MEC each pole is modeled by a lumped magnetic reluctance, meaning that magnetic saturation is assumed uniform, which is exclusively not feasible for SRMs of high power density. In magnetic vector field method, the solution is reduced to linear by taking permeability of iron infinite against that of air.

Estimation of nodal temperature rise serves as a guarantee of safe operation. Temperature rise is classically estimated by a thermal network [18, 19], in which each component of the machine is represented by a lumped thermal resistance. However, precise results are obtained only if resistances are fitted by means of characteristic temperature measurements. This fitting procedure is often explained by uncertain knowledge of heat transfer coefficients [20]. Our previous work showed that the calculation accuracy can be enhanced by using compensation elements [21, 22, 23, 24].

Acoustic analysis is another critical ongoing concern, as SRM preserves the excessive torque ripple problem [25, 26]. For vehicular application of wide speed range, acoustic should be simultaneously considered at various operation condition [27, 28]. Shortly, the main mode frequencies are compared with operation condition to avoid resonance. During the design progress, these frequencies are expected to be repeatedly calculated in a fast and accurate way under the variation of machine geometry.

In this paper, a multiphysics optimization flow is proposed. Sections 2–4 are respectively the introduction of electromagnetic, acoustic and thermal submodels and each model serves as a submodel for the optimization design flow. Specifically in Section 2, an electromagnetic modeling method considering magnetic saturation distribution is introduced; In Section 3, a mode frequency estimation method for acoustic analysis is illustrated; In Section 4, a novel thermal network with compensation elements as accuracy enhancement is proposed. Then in the following Section 5, a multiphysical design flow integrating submodels is proposed.

2. Electromagnetic analysis

The electromagnetic analysis requires a fast and accurate modelling approach of magnetic flux curves in which identification of distribution of magnetic saturation on poles is essential. Conventionally when accuracy is satisfied using FE by discretizing the machine into fine elements, computation amount is considerable. Reversely when computation amount is reduced by using magnetic circuit methods, the accuracy is reduced due to inaccurate estimation of magnetic saturation distribution. To solve such contradiction, the open source FEMM (Finite Element Method Magnetics) program [29] is applied, which is flexibly commanded by the Matlab m-file console, is utilized.

The magnetic flux variation under small increment of phase current and rotor position as variables is simulated. The FEMM implements automatic adaptive meshing that the airgap region is finely discretized while the machine body is with fewer meshes in number, resulting in small computation [30]. Incremental phase current is applied and the excited pole pair moves with 1 ${}^{\circ}$ step. The flux paths concentrate at pole tips where poles begin to overlap, causing severe local magnetic saturation, indicating high accuracy of FE approach. The magnetic curves are shown in Fig. 1, where the curves are at an interval of 5 ${}^{\circ}$ rotor movement.

Figure 1.

Electromagnetic analysis by FEMM program, showing magnetic curves from unaligned position 0 ${}^{\circ}$ till aligned position 30 ${}^{\circ}$ with an interval of 5 ${}^{\circ}$ .

Figure 2.

Iron loss curves, (a) in linear scale, (b) in logarithm scale.

The iron loss as input for the following thermal analysis is calculated. As each mesh element in Fig. 1a has characteristic flux density variation, each is respectively computed. Electrical steel characteristics are taken to calculate loss density. Figure 2 shows the transformation of steel data [31]. In Fig. 2a, loss per volume for different frequencies as a function of flux density is presented. The curves is now re-plotted using logarithm scales to linearize the trends shown in Fig. 2b, where each line is approximated by

$\log\left(P\right)=a\log\left(B\right)+b$ (1)

where $P$ and $B$ are respectively loss and flux density of each mesh element, $a$ and $b$ are coefficients respectively the slope of each line and interception value on Y-axis.

3. Structural analysis

The radial component of electromagnetic force has ovalisation deformation and thus acoustic on stator. For the studied 8/6 machine, the ordinal modes $m=$ 0, 2, 4 take effect. The mode frequencies are calculated from Lagrange energy equation, which is

$\frac{{\rm d}}{{\rm d}t}\left({\frac{\partial L}{\partial q_{i}}}\right)-\frac% {\partial L}{\partial q_{i}}=0$ (2)

where $L$ is the Lagrange equation, $L=T-V$ , in which $T$ is the total kinetic energy and $V$ is the total potential energy, $i=1,2,\ldots,n$ , and $q_{i}$ is the generalized coordinates.

To solve Eq. (2), the energy expression of each stator part should be clearly demonstrated. Specifically, $T$ and $V$ are respectively written as

$\displaystyle T=T_{tr,f-c}+T_{tr,p-w}+T_{rot,f-c}+T_{rot,p-w}$ (3) $\displaystyle V=V_{be,f-c}+V_{sh,c}+V_{ex,c}+V_{ex,p}$ (4)

where in Eq. (3) $T_{tr,f-c}$ , $T_{tr,p-w}$ , $T_{rot,f-c}$ , $T_{rot,p-w}$ respectively mean the translatory motion energy of stator frame and yoke, the translatory motion energy of stator poles and armature windings, the rotational energy of stator frame and yoke, and the rotational energy of stator poles and windings, while in Eq. (4) $V_{be,f-c}$ , $V_{sh,c}$ , $V_{ex,c}$ , $V_{ex,p}$ respectively mean the bending strain energy of stator frame and yoke, the shear strain energy of stator bore, the extension strain energy of stator bore, and the extension strain energy of stator poles.

Each term concerning motion energy $T_{tr,f-c}$ , $T_{tr,p-w}$ , $T_{rot,f-c}$ , $T_{rot,p-w}$ and strain energy $V_{be,f-c}$ , $V_{sh,c}$ , $V_{ex,c}$ , $V_{ex,p}$ are calculated according to theoretical mechanics of rotational cylindrical components. As a specific study, the translatory and rotational kinetic energies of stator frame/yoke are respectively written as

$\displaystyle T_{tr,f-c}=\left({\frac{M_{f}r_{o,f}}{2}+\frac{M_{c}r_{o,c}}{2}}% \right)\int\limits_{0}^{2\pi}{\left({u^{2}+w^{2}}\right)}{\rm d}\theta$ (5) $\displaystyle T_{rot,f-c}=\left({\frac{M_{f}I_{f}}{2∼{}A_{f}r_{o,f}}+\frac{M_{% c}I_{f}}{2∼{}A_{c}r_{o,c}}}\right)\int\limits_{0}^{2\pi}{v^{2}}{\rm d}\theta$ (6)

where $M_{f}$ and $M_{c}$ are the unit circumference quality of stator frame and yoke (kg/m), $r_{o,f}$ and $r_{o,f}$ are corresponding outer radius, $u$ and $w$ are radial and circumferential displacement.

The bending strain energy of stator frame/yoke is written as

$V_{be,f-c}=\left[{\frac{A_{f}E_{f}e_{f}}{2r_{o,f}^{2}\left({1-\mu^{2}}\right)}% +\frac{A_{c}E_{c}e_{c}}{2r_{o,c}^{2}\left({1-\mu^{2}}\right)}}\right]\int% \limits_{0}^{2\pi}{\left({u+\frac{{\rm d}w}{d\theta}+\frac{{\rm d}v}{d\theta}}% \right)^{2}d\theta}$ (7)

where $A_{f}$ and $A_{c}$ are axial sectional area of frame and yoke (m ${}^{3}$ ), $E_{f}$ and $E_{c}$ are corresponding elastic modulus (N/m ${}^{2}$ ), $e_{f}$ and $e_{c}$ are the distance between the neutral layer and the central line of the average radius [32, 33], $\mu$ is the Poisson coefficient, and $v$ is the rotational displacement.

Each energy part of the stator frame/yoke in Eqs (5)–(7) and the rest components of the machine are transformed by Fourier method. The displacement $u$ , $w$ and $v$ are alternatively written as

$\displaystyle u=\sum\limits_{m=1}^{\infty}{\left({a_{m}\cos m\theta+b_{m}\sin m% \theta}\right)}$ (8) $\displaystyle v=\sum\limits_{m=1}^{\infty}{\left({c_{m}\cos m\theta+d_{m}\sin m% \theta}\right)}$ (9) $\displaystyle w=\sum\limits_{m=1}^{\infty}{\left({e_{m}\cos m\theta+f_{m}\sin m% \theta}\right)}$ (10)

All coefficients $a_{m}$ , $b_{m}$ , $c_{m}$ , $d_{m}$ , $e_{m}$ , and $f_{m}$ are time dependent and can be expressed by its amplitude multiplying e ${}^{j\omega t}$ as the generalized coordinate of the system, where $\omega$ is the rotation speed. $m$ is the modal order and $m_{\max}=$ 6 is used, a compromise between calculation amount and accuracy. Equations (8)–(10) are integrated into Eqs (5)–(7) that are further into Eqs (3) and (4). Combining Eqs (2)–(4), the system motion equation in matrix is obtained as

$\left\{{K-\omega^{2}M}\right\}\left\{q\right\}=0$ (11)

where $K$ is the symmetric rigidity matrix and $M$ is the diagonal matrix of mass.

When Eq. (11) has non-zero solution, the following matrix equation holds true as

$\det\left\{{K-\omega^{2}M}\right\}=0$ (12)

The mode frequencies and corresponding modal are obtained based on Eq. (12), where Jacobi method is preferred.

4. Thermal analysis

Nodal temperature rise of critical components of electrical machines is essential and the thermal lumped parameter network model is the most popular. Such a model is based on the hypothesis that the system, under the thermal point of view, can be divided into several components that are connected by means of thermal resistors and capacitors [34]. Hence, the temperature distribution within each component is seen uniform and the numerical value represents the maximum nodal temperature rise.

Figure 3.

The 3D thermal network, (a) the main part, (b) the end part.

The studied machine is accordingly divided into 9 components, namely the frame (abbreviated as fr), stator teeth (ST), stator yoke (SY), rotor teeth (RT), rotor yoke (RY), air gap (AG), shaft (sh) and conductors. A conductor is further divided into windings (wi) and end windings (EW) due to different cooling conditions. The network is shown in Fig. 3, where additional abbreviations in subscript ro and SA respectively stand for rotor and slot air. In Fig. 3a, the network of the main machine body is created. Thermal resistances connecting a couple of components are written as, e.g. between frame and stator yoke $R_{\textit{th\_fr\_SY}}=R_{\textit{th\_SY\_fr}}$ , or between stator teeth and stator yoke $R_{\textit{th\_ST\_SY}}=R_{\textit{th\_SY\_ST}}$ . Such resistances are derived from geometrical dimensions, material thermal properties, and heat transfer coefficients. Heat sources $P$ from each component, including $P_{\textit{ST}}$ , $P_{\textit{SY}}$ , $P_{\textit{RT}}$ , $P_{\textit{RY}}$ and $P_{wi}$ are imposed. In Fig. 3b heat transfer relationship in the machine end part is shown. When the machine is running, turbulent flow occurs, a complicated forced convection to be modeled analytically. As heat flows in the rotor, end windings and frame are correlated, the model is simplified by a triangle connection. $R_{\textit{th\_fr\_EW}}$ , $R_{\textit{th\_fr\_ro}}$ and $R_{\textit{th\_EW\_ro}}$ connecting any couple of components of frame, rotor and end windings are presented, and each resistance is obtained by empirical and curve fitting equations from experiments [35].

Figure 4.

Justification of using compensation elements as accuracy improvement, (a) a cube object with heat source $P$ for analysis, (b) corresponding lumped network, (c) updated by using distributed heat source $P/2$ , (d) comparison of results, where (a) means by FE method.

Figure 5.

The improved network with compensation element $t_{comp}$ .

The network requires small computation. However, there exists a systematic mistake that the temperature is overestimated, which was reported in [20, 21, 22]. The justification is illustrated in Fig. 4 by simulating heat flow of a cube block. In Fig. 4a, the heat source $P$ is assumed uniformly distributed within the block and heat flow goes only along $x$ by isolating lateral four identical surfaces. Simulation results of all the 3 models, including finite element (FE) method, traditional network in Fig. 4b and distributed heat source $P$ /2 in Fig. 4c, are comparatively shown in Fig. 4d. The network in Fig. 4b overestimates the temperature while the nodal temperature from both improved heat source in Fig. 4c and FE in Fig. 4a fit each other, indicating the necessity of separation of heat source.

In Fig. 4c the application of half heat source $P$ /2 is not convenient if the network goes complicated. Instead, compensation element $t_{comp}$ is applied to correct computational deviation while heat source $P$ is collectively concentrated at the middle point. The equivalent network of Fig. 4c is show in Fig. 5, which is seen as an update of the traditional network by adding compensation elements $t_{comp}$ between connecting resistances. The calculation of $t_{comp}$ is illustrated in [21, 23, 24]. For complicated systems of more than one component, compensation elements will be introduced into the thermal network. As a result, the calculation accuracy is enhanced while calculating speed is maintained.

Table 1

List of variables in the thermal network model

Thermal resistance ${{R}}_{\rm{th}}$	Between (&)	Calculated value ( ${}^{\circ}$ C/W)
$R_{\textit{th\_fr\_am}}$	Frame & Ambient	0.0012
$R_{\textit{th\_SY\_fr}}$	Stator Yoke & Frame	0.00359
$R_{\textit{th\_SY\_ST}}$	Stator Yoke & Stator Teeth	0.019
$R_{\textit{th\_wi\_ST}}$	Windings & Stator Teeth	0.0024
$R_{\textit{th\_wi\_SY}}$	Windings & Stator Yoke	0.0058
$R_{\textit{th\_wi\_EW}}$	Windings & End Windings	0.0106
$R_{th\_RT\_RY}$	Rotor Teeth & Rotor Yoke	0.0075
$R_{th\_RY\_sh}$	Rotor Yoke & Shaft	7.39
$R_{th\_wi\_SA}$	Windings & Slot Air	1.256
Compensation element $t$	Towards ( $\to$ )	Calculated value ( ${}^{\circ}$ C/W)
$t_{\textit{RT $\to$ RY}}$	Rotor Teeth $\to$ Rotor Yoke	0.0152
$t_{\textit{RY $\to$ RT}}$	Rotor Yoke $\to$ Rotor Teeth	0.0151
$t_{\textit{RY $\to$ sh}}$	Rotor Yoke $\to$ Shaft	0.048
$t_{\textit{EW $\to$ wi}}$	End Windings $\to$ Windings	0.263
$t_{\textit{wi $\to$ EW}}$	Windings $\to$ End Windings	0.667
$t_{\textit{wi $\to$ ST}}$	Windings $\to$ Stator Teeth	2.316
$t_{\textit{ST $\to$ wi}}$	Stator Teeth $\to$ Windings	0
$t_{\textit{SY $\to$ ST}}$	Stator Yoke $\to$ Stator Teeth	0.0199
$t_{\textit{ST $\to$ SY}}$	Stator Teeth $\to$ Stator Yoke	0.342
$t_{\textit{SY $\to$ wi}}$	Stator Yoke $\to$ Windings	0.342
$t_{\textit{wi $\to$ SY}}$	Windings $\to$ Stator Yoke	0.024
$t_{\textit{SY $\to$ fr}}$	Stator Yoke $\to$ Frame	0.0458
Heat source $P$	Component	Input value (W)
$P_{\textit{SY}}$	Stator Yoke	315
$P_{\textit{ST}}$	Stator Teeth	353
$P_{\textit{wi}}$	Windings	945
$P_{\textit{EW}}$	End Windings	146
$P_{\textit{RT}}$	Rotor Teeth	163
$P_{\textit{RY}}$	Rotor Yoke	112

Figure 6.

The global thermal network with accuracy improvement.

The overall network consisting resistances, heat sources (voltage generator), compensation (voltage potential) and heat flow (current direction) is shown in Fig. 6. Voltmeters as a means of measurement are used. All the elements including heat sources $P$ resistances $R$ and compensation $t$ are marked and specifications are listed in Table 1. The abbreviation in subscript indicates component names. For example, $P_{\textit{ST}}$ means power (heat source) of stator teeth. $R_{\textit{th\_AG\_RT}}$ means thermal resistance between airgap and rotor teeth. At each red dot, nodal temperature of each component $T$ is recorded. The end part convection in Fig. 6b is marked as well.

All variables listed in the network are collectively shown in Table 1, including resistances $R_{th}$ , heat sources $P$ and compensation elements $t$ . Abbreviations and symbols in subscript are explained. Next, all material properties are defined. The windings with coating are considered the worst case as 1/10 of the normal copper material as $k_{c\_wi}=$ 40 W $\cdot$ m ${}^{-1}\cdot$ K ${}^{-1}$ . Others include the machine body alloy $k_{c\_aly}=$ 54 W $\cdot$ m ${}^{-1}$ $\cdot$ K ${}^{-1}$ , and air $k_{c\_air}=$ 0.03 W $\cdot$ m ${}^{-1}$ $\cdot$ K ${}^{-1}$ . The calculated values are based on the studied machine with a specific operation condition to be described in Section 6. Note that in Table 1 due to the cooling difference on the machine end region, copper loss is divided into loss in windings and in end windings. The proportion of ohmic loss into each part is determined according to the actual length of wires.

Iron loss is obtained by the electromagnetic model is in Section 3. Copper loss as another form of heat source is seen temperature dependent, and the electrical resistivity value is obtained according to material specifications. However, the temperature rise and copper loss generated are correlated. To solve this problem, an empirical temperature rise is used to get initial copper loss, and the loss in value is repeatedly updated based on simulation from the thermal network.

Figure 7.

Overview and variables in the optimization flow.

Figure 8.

The optimization procedure flow by means of Matlab console.

5. The optimization flow

Before optimization, the machine is presized to have an overall dimension according to the expected output power. The proposed machine is 4-phase 8/6 for traction application [36] in Fig. 8. The 5 variables to be optimized include height and width of stator/rotor pole and yoke, which are represented by $R_{sy}$ , $R_{ry}$ , $R_{g}$ , $\beta_{s}$ and $\beta_{r}$ , where $R_{sy}$ , $R_{ry}$ and $R_{g}$ are respectively radius of stator yoke, rotor yoke and airgap, $\beta_{s}$ and $\beta_{r}$ are stator and rotor pole arc. Constraints include $\beta_{r}\geqslant\beta_{s},\min(\beta_{r},\beta_{s})\geqslant\theta$ and $\beta_{r}+\beta_{s}<2\pi$ , where $\theta$ is the stroke angle.

Before optimization, geometry setup starts first. Specifically the widths of poles $\beta_{r}$ and $\beta_{s}$ are initially set identical and equal to width of stator slots, $\beta_{r}=\beta_{s}=2\pi/N_{s}/2$ , where $N_{s}$ is the number of stator poles. As the outer stator bore diameter $R_{out}$ is given, $R_{sy}$ is set to guarantee that the width of stator yoke is identical to that of stator poles, $R_{sy}=R_{out}-\beta_{s}$ , $R_{g}$ is set to guarantee the identical height and width of stator slots, $R_{g}=R_{sy}-\beta_{s}$ . Likewise $R_{ry}$ is set to have the same height and width of rotor poles.

The optimization criterion is to maximize phase torque with structural and thermal features within expected limit by adjusting the 5-variable combination. The minimum slot area is guaranteed by setting the upper limit of current density when slot fill factor is specified. The calculation flow is shown in Fig. 8. During optimization process, phase current density is limited within a numerical boundary to confine temperature rise. The electromagnetic analysis serves as the central and phase torque is calculated from energy conversion principle by the FEMM program based on magnetic curves in Fig. 1. For auxiliary structural analysis, natural mode frequencies are repeatedly calculated as reference when geometry is changing. In particular, the second mode eigen-frequency is expected higher to avoid resonance. As to thermal analysis, the working temperature is always monitored to avoid overheating, leading to a limit of increase of heat generation.

The optimization starts on the basis of initial setup. Among all variables to be optimized, the yoke thickness determined by $R_{sy}$ is initially designated that other ones are linked with. $R_{sy}$ is considered to determine stator yoke thickness, followed by airgap radius $R_{g}$ , which together determines the depth of stator slot. Then, pole widths are optimized to have a large change of flux, which means the unaligned flux linkage is minimized with sufficient clearance between pole corners. Specifically $\beta_{s}$ is initially assumed constant and $\beta_{r}$ incrementally increases. When $\beta_{r}$ reaches the limit, indicating that the co-energy produced may reversely reduce, $\beta_{s}$ will increase slightly and iteration of $\beta_{r}$ repeats. When widths of poles are determined, both copper and iron losses are calculated and as input into the thermal network. Both FEMM and the network program are simultaneously called by Matlab m-file console. The temperature dependent copper loss is iteratively optimized according to temperature rise of windings. If the value is beyond limit settings, $R_{g}$ will be again adjusted to reduce copper loss.

The yoke thickness $R_{sy}$ is optimized according to the mode frequency calculation. The yoke thickness is expected small as long as acoustic requirement can be satisfied. To get the $R_{sy}$ value, the mode frequency is compared with the working frequency in the last step in the flow, regardless of magnetic saturation of yoke that is actually unlikely to happen. When the yoke thickness is minimized while the acoustic is satisfied, the program flow will finish with optimized airgap radius and teeth widths.

Table 2
The 8/6 traction machine overview

$T$ ( ${}^{\circ}$ C)	Improved	Conventional	FE
$T_{\textit{fr}}$	54.81	54.81	55.03
$T_{\textit{SY}}$	67.83	67.91	67.82
$T_{\textit{ST}}$	82.91	84.53	83.11
$T_{\textit{wi}}$	88.05	81.71	88.02
$T_{\textit{EW}}$	88.81	86.16	–
$T_{\textit{RT}}$	94.51	95.89	94.21
$T_{\textit{RY}}$	94.36	95.71	94.04

Table 3

Calculation results and comparison

6. Simulation and discussion

The geometry and performance specification of the studied machine are shown in Table 3. The machine has a large speed range and therefore both copper and iron losses vary significantly. Due that there is not a specific working point for the traction machine, the output performance should be judged at different speeds. Here one representative operation condition is 6.5 krpm with reduced phase current is studied.

For thermal analysis, the enhanced accuracy by adding compensation elements means efficiency improvement of the multiphysics optimization flow, which is proved by comparing with FE and the traditional network. The ambient temperature is set 50 ${}^{{\circ}}$ C as the ambient worst case. The cooling environment is set frame convection 6000 W $\cdot$ m ${}^{-2}$ $\cdot$ K ${}^{-1}$ , natural convection or equivalent contact between rotor yoke and shaft, and contact between stator and frame 6000 W $\cdot$ m ${}^{-2}$ $\cdot$ K ${}^{-1}$ . Calculation results in terms of nodal temperature $T$ of each component is shown in Table 3. For FE, the value is averaged from all discretized elements within a component. A numerical agreement between FE and improved network is shown, indicating the importance of adding compensation elements. Compared with conventional lumped model, it is shown that the improved model has higher calculation accuracy. In particular in some machine sections, such as coil sides and end windings, the values have a relatively large discrepancy. This is because of the regarded compensation elements from coils that are much numerically higher than other counterparts. Due that both copper loss $P$ and electrical resistance of copper $R$ of windings are higher, the value of $t_{wi\to ST}$ is considerable. Consequently, nodal temperature on windings is more compensated, leading to calculation difference compared with the conventional model. Such calculation deviation amended by compensation elements is vital to optimization accuracy.

Table 4
The Optimized Geometry and Performance Data (D0: Initial Design, D1: Optimized Design 1, D2: Optimized Design 2)

Parameter	D0		D1		D2
$\beta_{s}$ (deg)	20		18	.95	18	.95
$\beta_{r}$ (deg)	20	.5	21	.15	21	.15
$R_{sy}$ (mm)	142	.3	146	.8	146	.8
$R_{g}$ (mm)	97	.1	101	.9	101	.9
$R_{ry}$ (mm)	62	.2	66	.6	66	.6
$R_{bore}$ (mm)	175		175		175
$L_{stk}$ (mm)	125		125		125
$T_{avg}$ (Nm)	141	.5	148	.8	148	.8
$B_{\max}$ (T)	2	.05	2	.08	2	.03
$T_{wi}$ ( ${}^{\circ}$ C)	135		141		135
$f_{m=0}$ (Hz)	5736		5621		5621
$f_{m=2}$ (Hz)	1051		951		951
$f_{m=4}$ (Hz)	3344		3102		3102

Figure 9.

The optimized design, (a) geometry variation, (b) magnetic curves.

Figure 10.

Dynamic flux density distribution by the FEMM program, when one phase is excited at partial overlap position, (a) conventionally designed machine, (b) multiphysically optimized machine.

The optimized parameters and performance characteristics of the machine are shown in Table 4, in which D0, D1 and D2 respectively represent initial design, optimized design 1 and optimized design 2. The initial design D0 depends only on electromagnetic analysis. Both optimized designs are from the proposed multiphysics procedure. In D1, the maximum allowed current density is maintained by keeping stator slot area constant. However due to increase of iron loss, coil temperature exceeds the limit. Hence in D2, an identical temperature rise of 135 ${}^{{\circ}}$ C is realized by reducing current density.

Accordingly the peak flux density is reduced to 2.03 T as well, making lower iron loss. The second modal shape frequency is the most influential. It is shown in the optimized design that the value is reduced to $f_{m=2}=$ 951 Hz, corresponding to 9.5 krpm rotor speed that is even higher than the assumed maximum speed, making safe operation. By maximizing $R_{sy}$ with mechanical resonance avoided in optimized designs, the yoke thickness is optimized.

The optimized machine geometry and torque improvement is shown in Fig. 4. From sketch comparison in Fig. 4a, the airgap radius is enlarged by reducing stator yoke thickness and subsequently increased stator slot area. Meanwhile, modal frequency is analyzed to avoid resonance while temperature rise in terms of current density is checked. It is stressed here that the accurate estimation of nodal temperature of windings by compensation elements enables identifying maximum allowable phase current density, which helps to optimize the airgap radius that possibly leads to a larger change of magnetic resistance from aligned to unaligned rotor position. Also the accuracy of the electromagnetic model in terms of magnetic saturation determination with high power density is essential. Flux linkages at typical positions by FEMM program are shown in Fig. 4b. The curves at unaligned position are identical while aligned position in enhanced, indicating the rate of energy conversion is improved.

Figure 11.

Optimization analysis, (a) phase torque enhancement, (b) temperature variation of coils as constraints.

Dynamic flux density distributions of the traditional and the optimized designs (D2) are shown in Fig. 10 comparatively. One pole pair partially aligned under the same output torque shows the reduced peak flux density on the saliency in the optimized design, due to the reduced phase current and hence less iron loss as well, making a more efficient design.

Phase torque and temperature variations of windings are further shown in Fig. 11, in which the benefits of using a higher airgap radius are further illustrated. In D1, phase torque is enhanced at any rotor position, and the working temperature rises up accordingly. In D2 where excitation current is reduced to follow the same temperature variation, phase torque is still higher than that of the initial design D0. The optimized design not only reduces phase current, but potentially requires lower VA ratings of the power inverter.

7. Conclusions

Electrical machine design requires multiple physical considerations. In each physical aspect, the modeling approach with fast and accurate calculation features is needed. In this paper, an analytical design flow is introduced. The flow effectively integrates multiple physics with three simple and accurate submodels. For electromagnetic analysis, magnetic saturation due to double saliency that determines flux linkage characteristics is modeled by FEMM but with small number of discretized elements. The FEMM program connects Matlab console, facilitating interaction with other modules. For thermal analysis, the network with compensation elements improves calculation accuracy. Temperature compensation especially on coils helps accurate design of stator slot area and airgap radius. For acoustic analysis, a method enabling fast estimation of mode frequency is proposed. The proposed submodels and subsequent design flow enable efficient design of switched reluctance machines and possibly more electromagnetic devices.

Footnotes

Acknowledgments

This work is supported by the Natural Science Foundation of China (51607180). The authors gratefully acknowledge Institute of Electrical Drives and Actuators (EAA), University of Bundeswehr Muenchen, Germany and KSB Aktiengeselschaft, Frankental, Germany, for their support in this research.

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Multiphysics optimization design flow with improved submodels for salient switched reluctance machines

Abstract

Keywords

1. Introduction

2. Electromagnetic analysis

Table 2 The 8/6 traction machine overview

Table 4 The Optimized Geometry and Performance Data (D0: Initial Design, D1: Optimized Design 1, D2: Optimized Design 2)

Footnotes

Acknowledgments

References

Table 2
The 8/6 traction machine overview

Table 4
The Optimized Geometry and Performance Data (D0: Initial Design, D1: Optimized Design 1, D2: Optimized Design 2)