*Article* **Improvement of the Thermal and Mechanical Strength of the Starting Cage of Double-Cage Induction Motors**

#### **Jan Mróz 1,\* and Wojciech Poprawski <sup>2</sup>**


Received: 25 October 2019; Accepted: 27 November 2019; Published: 29 November 2019

**Abstract:** This article discusses the thermal and mechanical exposure of the starting cage of a double-cage induction motor rotor during start-up. Damage to the starting cage is the most common cause of failure of a double-cage winding during long start-ups. It has been indicated that the end region of the double-cage winding is a key area in the search for a more damage-resistant solution. Among the available studies on improving the mechanical strength of double-cage windings, which typically focuses on improving the cooling system, modifying the shape of the slots, or altering the bar material, a new concept of improving the mechanical strength through the modification of the structure of the end region has appeared. This is achieved by applying sleeves onto the ends of the starting cage bars, which helps to reduce the temperature of the connection between the starting bars and the end rings. A simulation of the temperature field of a double-cage induction motor with this new design is performed and discussed in this paper. It has been confirmed that the new design solution effectively improves the mechanical strength of the starting cage, making it less prone to damage caused by thermal stresses.

**Keywords:** double-cage induction motor; improvement of motor reliability; cage winding constructions; direct start-up; coupled electromagnetic-thermal model

#### **1. Introduction**

In a cage induction motor with an emergency locked rotor, or under long starting conditions, the element most vulnerable to damage is the cage winding of the rotor. Thermal exposure is particularly high in double-cage rotor motors owing to the relatively low thermal capacity of the starting cage bars. Such motors are typically found in drives requiring high starting torque, which translates directly into higher losses in the starting cage. Examples of damage to the starting cage in double-cage motors are displayed in Figures 1 and 2. The most common cause of double-cage winding failure is damage to the starting cage, while the working cage remains functional. This is a defect that is difficult to detect in its initial phase. There are numerous publications presenting new approaches to detecting starting cage failures in a double-cage motor. Thus, [1] presents a method of detecting outer cage damage in double squirrel cage induction motors. This diagnostic method relies on a discrete wavelet transform optimised for sensitive detection under transient operating conditions. Reference [2] presents a complete on-line condition monitoring system designed to detect incipient broken rotor bar faults in a double-cage induction motor using the stator current signature. It is based on successful combination of one of the latest variants of wavelet techniques, the recursive stationary wavelet packet transform and a tool widely used in quality control, the statistical process control in order to deal with several challenges in the continuous monitoring of the incipient fault. In [3] the possibility of using the

stator phase current waveform as a diagnostic signal to detect faults in a double squirrel cage induction motor is discussed. On the basis of tests conducted on a double cage motor prototype, conditions were formulated for distinguishing faults in cages of both medium and high power machines.

**Figure 1.** Damaged rotor cage, view after cutting off the end ring of the starting cage—visible melting of the starting cage bar.

**Figure 2.** Damaged rotor cage—visible sheared bars of the starting cage.

An important, developing issue is diagnosing large induction motors fed by an inverter. Reference [4] presents the use of a fuzzy-based statistical feature extraction from the air gap disturbances for diagnosing broken rotor bars in large induction motors fed by line or an inverter. The method is based on the analysis of the magnetic flux density variation in a Hall Effect Sensor installed between two stator slots of the motor. Reference [5] provides a diagnosis of broken rotor bars in field oriented controlled double cage induction motors, based on current and vibration signature analysis techniques.

Designers and constructors are making attempts to make the double-cage winding more resistant to damage occurring during long start-ups. One of the lines of action is to improve the cooling system. In [6] the cooling performance of axial fans with forward-swept and inclined blades and a structure with low ventilation resistance in large-capacity open-type motors is studied.

Steps are taken for thermal analysis to be more deeply regarded in electric machine design. In [7] some of these problems are discussed and advice is provided as to dealing with them when developing algorithms for inclusion in design software. In [8], a design of a 115 kW squirrel cage induction motor for electric vehicle applications is presented. In the design procedure, initially, an analytical design of the electric vehicle motor is performed depending on specific design criteria. Then, the analytical design is verified by means of finite element analysis. Reference [9] presents the coupled fluid-thermal analysis for an induction motor with healthy and broken bar rotors. Much attention was paid to developing the fluid model on the basis of the computational fluid dynamic theory.

Another area of work aimed at improving the fault resistance of the double cage winding and the deep bar rotor to the effects of a prolonged start-up focuses on changing the cage winding material and modifying the shape of rotor slots. In [10] three double-cage induction motors have been simulated and their electromagnetic characteristics compared. The study is carried out using finite element method (FEM) analysis. Reference [11] presents dynamic modelling of a series of induction motor squirrel cages with different shapes of rotor deep bars, taking into account the skin effect. Reference [12] focuses on designing and optimizing an induction motor with a lower cost and high performance. The starting torque of the induction motor, which is an important aspect in traction applications, has been improved by applying a rotor with a double cage. Analytical modelling is carried out and it is validated by means of FEM analysis. In [13] a new design of the rotor bar which allows to improve the starting torque without decreasing motor efficiency is presented. Reference [14] deals with the influence of the shape of the cage on double cage induction motor's parameters, mainly the values of starting torque, breakdown torque, as well as the power factor, efficiency and starting current. The optimal shape of the rotor slot has been identified for the maximum size of the relative starting torque and minimum size of the relative starting current while maintaining a favorable power factor and efficiency of the motor.

Another area of work undertaken by constructors to improve the fault resistance of the cage winding are special motor structure designs. In [15] an investigation on the design of a high-power induction motor with special constraints is presented. Direct online start-up and pull-up torque of high value are the two imposed requirements. The proposed solution advances a new rotor structure with two different rotor cages. Reference [16] presents the elaborate design procedure for a double rotor double cage motor. The two rotors can run independently, at an equal or unequal speed, depending on their individual loading.

In [17,18], using mathematical models to consider the electromagnetic and thermal interrelations, the author demonstrated that the end region of the cage is a key location in the search for construction solutions more resistant to damage. These were likely the inspiration for the construction of the starting cage of a double-cage motor as discussed in [19], which, according to the authors, allows for a reduction of the temperature of the end region of the starting cage bars. This is achieved by applying sleeves onto the ends of the starting cage bars. Because [19] does not present any test results for a motor with a double-cage winding constructed in such a manner, it is necessary to test the effectivity of this new design. The aim of the present paper is to perform a simulation of the temperature field of a double-cage induction motor with the new construction solution of the starting cage, during start-up with a locked rotor. The results are compared with those of a motor with a starting cage of conventional structure. All results presented in this paper have been achieved through a simulation and should be verified through experimental research in the future. The problem of experimental research on high-powered motors is related to a number of logistic activities, as they typically must be performed under industrial conditions. This entails high experimental costs. Therefore, there are relatively few publications addressing experimental research related to the heating of high-power cage induction motors [20–22].

#### **2. Method of Analysing the Temperature Field of a Double-Cage Induction Motor**

The analysis of thermal and electromagnetic phenomena occurring in an induction motor is commonly conducted by means of professional software available on the market. There are many publications on this issue [6,8,9,12,23]. A considerable limitation, particularly at an early stage of work on the presented problem, is the cost and a long calculation time for 3D issues, hence the choice of faster and less expensive methods which have been partially verified through experimental research [17,24]. Reference [17] presents a mathematical model and corresponding 3D simulation model that allows

the determination of the temperature field of a double-cage motor in transient electromechanical states. It uses a heat network created by the control volume method [25]. The model considers mutual electromagnetic and thermal dependencies. Because the dynamics of electromechanical phenomena in electric machines are considerably greater than those of thermal phenomena, an electromechanical problem can be solved at a specific moment and for a given temperature field. In the next time step, a new temperature field can be calculated using the solution of the electromechanical problem from the previous time step. This procedure is presented in the block diagram in Figure 3.

**Figure 3.** Algorithm for calculating rotor thermal field.

The motor model used in calculations is designed for simulations of short operation periods (operating with a locked rotor or during start-up). Research shows that under those operating conditions the heating of the stator components may be examined regardless of the rotor heating [17,24]. The operation periods analysed here are too short for the stator to exert a noticeable influence on the rotor heating and vice versa. Therefore, the heat exchange between the stator and the rotor can be omitted in the analysis. The temperature field is calculated for the area indicated in Figure 4, which encompasses a half of the rotor's length and half of the rotor's slot pitch. The analysed area has three types of boundary conditions: the Dirichlet boundary condition for the *A*<sup>1</sup> surface, the Neumann condition for the *A*2, *A*3, and *A*<sup>4</sup> surfaces, and the Newton condition for the areas that are in direct contact with a cooling fluid.

**Figure 4.** Analysed area of double-cage motor's rotor.

In the control volume method, the analysed object is subdivided into a number of homogeneous elements. The energy balance is calculated for the entire system. If it is assumed that in the analysed object, the separate control element with volume *Vc(i)* is sufficiently small that the temperature in its entire area is the same and has the value *<sup>T</sup>*(*i*), and that the point heat source . *q*ν(*i*) is located at the centre of gravity of this element, then the transient temperature field in a motor analysed using the heat network created by the control volume method [25] is described by the following system of Equation (1) together with the initial condition (2):

$$\sum\_{j} \Lambda\_{\{i,j\}} T\_{\{j\}} - \left(\sum\_{m} \Lambda\_{\mathfrak{a}(i,m)} + \sum\_{j} \Lambda\_{\{i,j\}}\right) T\_{\{i\}} + \sum\_{m} \Lambda\_{\mathfrak{a}(i,m)} T\_{\mathfrak{a}(i)} + V\_{\mathfrak{c}(i)} \dot{q}\_{\mathfrak{v}(i)} = V\_{\mathfrak{c}(i)} \varepsilon\_{\{i\}} \rho\_{\{i\}} \frac{\mathrm{d}T\_{\{i\}}}{\mathrm{d}t},\tag{1}$$

where *i* = 1, 2, ..., *zn* is the number of elements into which the considered area is divided, *zn* is the number of area elements, Λ(*i*,*j*) is the thermal conductivity for the heat flowing from node *i* to node *j*, Λ*a*(*i*,*m*) is the thermal conductivity for the heat flowing from node *i* to the external surface *m* and the heat transferred from surface *<sup>m</sup>*, *<sup>c</sup>* is the specific heat, <sup>ρ</sup> is the density, *<sup>t</sup>* is the time, . *qv*(*i*) is the volumetric density of the heat sources, and *T* is the temperature:

$$\left.T(\mathbf{r},t)\right|\_{t=0} = \left.T\_0(\mathbf{r}),\tag{2}$$

where *T*<sup>0</sup> is the initial temperature and **r** is the positional vector describing the position of the element in question. To determine . *qv* in the area of the windings, it is necessary to calculate the distribution of the current density *J*(**r**, *t*). Then:

$$\dot{q}\_v = J^2(\mathbf{r}, t) \frac{1}{\gamma(T)},\tag{3}$$

where γ-conductivity.

For a double-cage induction motor with a soldered cage, the equations describing the transient electromechanical state in a two-axis coordinate system rotating at the speed ω*<sup>x</sup>* have the form [26]:

$$\mathbf{U} = \frac{\mathbf{d}}{\mathbf{d}t}\Psi + \Omega\Psi + \mathbf{R}\mathbf{I},\tag{4}$$

$$\mathbf{\hat{H}} = \mathbf{L}\mathbf{I},\tag{5}$$

$$J\_{\rm m} \frac{1}{p} \frac{\mathbf{d}\omega}{\mathbf{d}t} = \mathcal{P} \text{Re} \{ j \underline{\mathbf{Y}}\_1 \underline{\mathbf{I}}\_1^\* \} - T\_{\rm L} \tag{6}$$

where **<sup>U</sup>** = *<sup>U</sup>*1, 0, 0<sup>T</sup> , **<sup>I</sup>** = *I*1, *I*2(1), *I*2(2) T , and <sup>Ψ</sup> = Ψ1, Ψ2(1), Ψ2(2) <sup>T</sup> are the voltages, currents, and linkage fluxes, respectively, **<sup>Ω</sup>** <sup>=</sup> diag[jω*x*, j(ω*<sup>x</sup>* <sup>−</sup> <sup>ω</sup>), j(ω*<sup>x</sup>* <sup>−</sup> <sup>ω</sup>)]T, **<sup>R</sup>** and **<sup>L</sup>** are the resistance and inductance matrices, ω is the electric rotor angular velocity, *U*<sup>1</sup> is the amplitude of the supply voltage, *I*1, *I*2(1), and *I*2(2) are the stator and rotor two-axis vector currents (complexor), respectively, *J*<sup>m</sup> is the moment of inertia, *T*<sup>L</sup> is the load torque, and *p* is the number of pole pairs. Knowledge of the currents flowing in the motor windings allows the determination the volumetric density of heat sources . *qV* necessary to determine the temperature field of the motor winding. This method allows the calculation of the temperature field for the double-cage induction motor with the parameters presented in Table 1.


**Table 1.** Chosen motor parameters.

The winding of the cage consists of bars short-circuited with copper end rings. The bars of the starting cage are made of brass, whereas the bars of the working cage are made of copper. The dimensions of the rotor and stator slots are shown in Figure 5.

**Figure 5.** The shape and dimensions of rotor and stator slots of double-cage motor with a welded cage.

The heat transfer coefficient from the areas in direct contact with cooling air was calculated on the basis of paper [18]. It provides an equation for calculating the equivalent heat transfer coefficient through natural convection and radiation, in the following form:

$$
\alpha = \nu\_p K\_p (T + T\_0) \left( T^2 + T\_0^2 \right) + C\_k \sqrt{\varphi} (T - T\_0)^{0.25}, \tag{7}
$$

where: ν*<sup>p</sup>* is the emissivity coefficient of the surface, *Kp* is the Boltzman constant, *T* is the surface temperature, *T*<sup>0</sup> is the temperature of the air surrounding the surface, *Ck* is the coefficient included within the limits (2.79 ... 3.39) W/m2K, φ is the relative air humidity. For *T* = 300 ◦C, *T*<sup>0</sup> = 20 ◦C, φ = 0.9, ν*<sup>p</sup>* = 0.85 (varnished surface), *Ck* = 3 W/m2K1.25 the heat transfer coefficient is α = 13.2 W/m2K. For a rotating rotor (forced convection):

$$a\_v = a \begin{pmatrix} 1 + k \ \sqrt{v} \end{pmatrix} \tag{8}$$

where *v* is the speed of the cooling air thrown over the surface, *k* the coefficient included within the range 0.5 ... 1.3, in the analysis *k* = 1 was assumed for the end region. For the rated speed: α*<sup>v</sup>* = 58 W/m2K.

Thermal conductivity values: for the bars of the working cage 372 W/m·K, for the end rings of the working cage 372 W/m·K, for the bars of the starting cage 103 W/m·K, for the end rings of the starting cage 372 W/m·K. For the rotor core—50 W/m·K (in axial direction), 10 W/m·K (in radial direction). A slight change due to the temperature of the thermal conductivity coefficient was omitted (within the expected temperature range within 20 ... 400 ◦C).

#### **3. Influence of the End Region Structure of the Double-Cage Winding on Mechanical Stresses**

In a double-cage motor with a soldered cage, the mechanical stresses associated with the skin effect in the rotor bars are not as important as in a deep bar motor. In a double-cage winding, the forces due to the thermal expansion of the end ring and centrifugal forces originating from the mass of the end region of the winding are more significant. The associated stresses can be dangerous, especially during long start-up periods, when the end rings experience maximum heat.

In double-cage motors with a soldered winding, the end rings are typically moved away from the core. There are many construction solutions for soldered bar connections with rings. Insert connections, as displayed in Figure 6, are frequently used.

**Figure 6.** Insert connections in double-cage motor.

The length of the overhang of the bars beyond the core is limited mainly by implementation considerations. An excessive increase of the overhang is meaningless owing to the length of the entire machine. This problem was discussed, among others, in [18]. The model of the phenomena occurring in the external heated part, rotating at a constant winding speed, assumes that the bar is rigidly fixed in the packet, and a radial force and circular-symmetric moment act in the location of the rigid connection of the bar to the ring. They cause the displacement of the external part of the cage as indicated by the dashed line in Figure 6. During the motor start-up, the centrifugal forces and forces originating from the thermal deformations of the ring act together on the cage bars.

Using this model, simulation calculations were conducted for a double-cage induction motor with parameters presented in Table 1. The results are presented in Figure 7 as the total stress value in the bars during a prolonged motor start-up.

**Figure 7.** Stresses in starting-cage bar during motor start-up.

As can be observed in Figure 7, a change in the outward extension of the bars within the limits of 23% around the value of *lcr* = 45 mm causes a significant change in the total stresses when the bar emerges from the core during a long-term startup. When reducing the size of *lcr*, the stress increases by approximately 56%; when the value of *lcr* increases, the stresses decrease by approximately 36%. In double-cage motors with bars of a circular cross section, the highest temperature occurs in the end part of the cage. The natural method to avoid a large unevenness in temperature distribution along the bar would be to eliminate the areas with elevated temperature from the cage structure, i.e., to shorten the bars. However, the reduction of this overhang leads to a significant increase in the mechanical stresses in the bars due to the thermal deformations of the end ring.

#### **4. Possibilities of Equalising the Temperature Distribution along the Axis of the Motor Starting Cage Bar**

In certain cases, the temperature in the end region of the cage is sufficiently high to cause a loss of elasticity of the material, whereas in the remainder of the winding, the temperature does not significantly reduce the mechanical strength of the bar material. These highly heated end regions of the rotor winding determine the durability and reliability of the entire motor, despite the fact that their share in the volume of the cage is small. Therefore, we must attempt to obtain a more uniformly heated structure. This applies, in particular, to work in conditions of prolonged start-up or operation with a locked rotor. This issue is discussed in [18], where the possibilities of equalising the temperature distribution along the axis of the starting cage bar of a double-cage motor were considered. The influence of the slot clearance was examined (Figure 8) for a double-cage induction motor with parameters presented in Table 1. The conditions of the heat transfer from these parts to the temperature field of the cage, in the state of working with the rotor locked, were also examined (Figure 9).

**Figure 8.** Influence of slot clearance on temperature distribution along starting-cage bar.

Figure 8 displays the temperature distribution along the starting cage bar with different fits of the bar to the slot (slot clearance 50, 100 or 200 μm). From the calculation results displayed in Figure 8, it can be observed that improving the fit of the bar to the slot leads to a significant reduction in the bar temperature in the core part; however, the temperature remains high in the end region and this costly treatment does not eliminate the cage damage discussed above. Because only somewhat exceeding a certain temperature (dependent on the bar material) decreases the tensile strength sharply, even a marginal reduction in the temperature of the end regions is beneficial from the point of view of its durability. To obtain similar conditions for heat exchange in the end region and slot part of the cage, an equality of thermal resistance should be ensured by means of heat transfer between the external part of the bar and the surrounding air, and between the cage bar and the rotor core. For the motor considered, the value of the heat transfer coefficient in the external parts of the cage was calculated, providing heat transfer conditions similar to those in the slot part, i.e., 254 W/m2K. The temperature distribution along the bars of the starting cage after 12 s of operation of the motor with the rotor locked supplied with the rated voltage is displayed in Figure 9. It is not possible to obtain a coefficient of this value in ordinary construction solutions because the values of this coefficient encountered in practice are many times smaller than required.

**Figure 9.** Influence of heat of transfer coefficient from end region of cage to temperature distribution along bar axis.

#### **5. New Solution for the Construction of a Starting Cage of a Double-Cage Motor**

The presented review of the results of calculations of the temperature of the double-cage winding indicates that in the end region of the cage construction, the possibility of further increasing the resistance of the cage to the related effects of working with the locked rotor and during a prolonged start-up should be investigated. The end region of the starting cage is most exposed to destructive activities under these working conditions, and it is necessary to investigate the possibility of increasing the resistance of the double-cage winding to the effects of a prolonged start-up.

The authors of [19] presented a proposal for an innovative construction solution of the starting cage that allows a limitation of the temperature of the end region of the bars. The starting cage displayed in Figure 10, based on this solution, is characterised by sleeves (3) of the same material as the bars, applied to the ends of all the bars (1) protruding from the core. The sleeves (3) adjacent to the end rings (2) are preferably permanently connected to the rings using a hard solder, welding, or sealing.

**Figure 10.** Innovative construction solution of starting cage: 1—starting cage bar, 2—end ring of starting cage, 3—sleeve, 4—core, 5—working cage bar, 6—end ring of working cage, *dp*—diameter of starting bar.

The method of building the starting cage is based on the fact that after inserting the bars (1) into the slots, the sleeves (3) are heated to a temperature at which the inner diameter of the sleeve is larger than the diameter of the bar *dp*. Before the sleeves are heated, the inner diameter of the sleeve *dr* is less than the diameter of the rotor cage bar. Then, the hot sleeves (3) are applied to the ends of the bars (1). Finally, to the ends of the bars (1), the end rings (2) are inserted and joined with the bars and sleeves with hard solder, welding, or sealing.

The sleeves (3) superimposed on the ends of the bars (1) increase the cross section and thermal capacity of the external part of the bars (1); thus, in the ends of the bars (1), the current density is reduced and the temperature decreases. Hence, a starting cage prepared in this manner is characterised by greater start-up durability.

#### **6. Simulation Tests of the Temperature Field of a Double-Cage Motor with a New Design Solution**

Using Equations (1)–(6), simulation tests were performed for a double-cage motor with the parameters presented in Table 1, with an emergency locked rotor (ω = 0). The calculations were performed for both a motor with a starting cage of conventional structure and a double-cage induction motor with the new construction solution for the starting cage presented in [19].

Figure 11 displays the temperature distribution along the centre axis of the starting cage bar for the classic solution and with 1.8 mm thick sleeves made of brass, similar to the starting cage bars. Figure 12 indicates the influence of the thickness of the sleeves (1.8 mm and 1.0 mm) on the distribution of the temperature along the axis of the starting cage bar. Owing to the new construction of the end region of the starting cage, locations previously threatened by overheating are largely eliminated. For the solution proposed in [19], the temperature of the starting bar in the region of the outward reach of the bars is considerably less than for the classical solution. Increasing the thickness of the sleeves further reduces this temperature (Figure 12).

**Figure 11.** Distribution of temperature along axis of starting-cage bar for classic solution and with sleeves.

**Figure 12.** Influence of thickness of sleeves on temperature distribution along axis of starting-cage bar.

#### **7. Conclusions**

The startup process of the induction cage motor is one of the most important phases of the drive operation. During the start-up an electrical motor can be subject to severe electrical and thermal loads. These loads, despite their relatively short durations, significantly affect the motor's lifetime and reliability. The heat generated during a prolonged start-up causes mechanical stresses which may damage the motor cage. By using the double-squirrel-cage soldered winding, large values of starting torques are achieved. This is, however, at the cost of non-uniform heating of the cage, along with the tendency of the cage bars to overheat. A large amount of heat is released during start-up in the bars of the motor's starting cage. The highest temperature is observed in the end region of the starting cage bar. There is a significant difference in the axial temperature distribution in the bars owing to the different types of heat exchange in the core region and the end region of the bars. The limitation of the slot clearance to improve heat transfer into the core does not change the fact that the temperature of the end region of the bars remains high. Moreover, reducing the distance between the end rings and the core is disadvantageous owing to the increase in the bending stress in the bars from the thermal expansion of the rings. Excessive heating of the end region of the starting cage during start-up can lead to motor failure.

The application of sleeves onto the ends of the starting cage bars can significantly reduce the temperature of the connection of the starting cage bars with the rings and thus increase the resistance of the rotor starting cage to thermal exposure during motor start-up. The conducted simulation tests demonstrated that thanks to the new construction of the end region of the starting cage, locations previously threatened by overheating were largely eliminated. However, there arise additional technological difficulties associated with the process of setting the sleeves onto the bars of the starting cage.

The heating of the motor windings is a transient phenomenon which is closely related to the transient electromechanical process. Both processes are mutually related since the distribution of heat sources depends on the temperature of motor windings. The results of the analysis can be massively improved by applying detailed numeric calculation methods (FEM analysis). A considerable limitation, particularly at an early stage of work on the presented problem, is the cost and a long calculation time for 3D coupled problems, hence the choice of faster and less expensive methods which have been partially verified through experimental research.

In view of the relatively limited number of publications in the area of costly experimental research involving high-power motors, it is necessary to plan and conduct such research. This applies in particular to investigating energy aspects in transient states for verifying the simulation models and the further improvement of cage winding constructions.

**Author Contributions:** Conceptualization, J.M. and W.P.; methodology, J.M.; software, J.M.; formal analysis, J.M. and W.P.; investigation, J.M. and W.P.; resources, J.M.; writing—original draft preparation, J.M.; writing—review and editing, J.M. and W.P.; visualization, J.M.; supervision, J.M.; project administration, J.M.; funding acquisition, J.M.

**Funding:** This work is financed in part by the statutory funds of the Department of Electrodynamics and Electrical Machine Systems, Rzeszow University of Technology and in the part by Polish Ministry of Science and Higher Education under the program "Regional Initiative of Excellence" in 2019–2022. Project number 027/RID/2018/19, amount granted 11 999 900 PLN.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Multi-Physics Tool for Electrical Machine Sizing**

#### **Yerai Moreno 1,\*, Gaizka Almandoz 1, Aritz Egea 1, Patxi Madina <sup>1</sup> and Ana Julia Escalada <sup>2</sup>**


Received: 17 February 2020; Accepted: 26 March 2020; Published: 2 April 2020

**Abstract:** Society is turning to electrification to reduce air pollution, increasing electric machine demand. For industrial mass production, a detailed design of one machine is usually done first, then a design of similar machines, but different ratings are reached by geometry scaling. This design process may be highly time-consuming, so, in this paper, a new sizing method is proposed to reduce this time, maintaining accuracy. It is based on magnetic flux and thermal maps, both linked with an algorithm so that the sizing process of an electrical machine can be carried out in less than one minute. The magnetic flux maps are obtained by Finite Element Analysis (FEA) and the thermal maps are obtained by analytical models based on Lumped Parameter Circuits (LPC), applying a time-efficient procedure. The proposed methodology is validated in a real case study, sizing 10 different industrial machines. Then, the accuracy of the sizing tool is validated performing the experimental test over the 10 machines. A very good agreement is achieved between the experimental results and the performances calculated by the sizing tools, as the maximum error is around 5%.

**Keywords:** sizing methodology; electrical machines; thermal model; electromagnetic model; permanent magnet

#### **1. Introduction**

Society is turning to electrification in transportation and industrial processes to reduce emissions, embrace alternative energy, and increase efficiency. In this regard, permanent magnet synchronous machines (PMSM) are mostly used for traction applications, due to their high power density and high efficiencies [1].

However, the main drawback of PMSM and electrical machines in general is their complex and time-consuming design process, as different targets must be reached optimizing various parameters that are cross-coupled. Usually, the final objective of the design is to minimize the cost of the machine while maximizing its efficiency. To fulfill these objectives coping with the growing machine demand, the design process must be improved.

Once the design requirements are defined, there are two designing scenarios, one where the machine is designed from scratch and another one where a 2D computer-aided design plane (2D CAD) is selected from the ones previously designed and the machine is sized setting the appropriate stack length and number of turns for the specific application. Different magnets and sheet types can be also chosen. Usually, in a company with a wide range of electric machines, the second scenario could be more usual, making it more competitive in the market, as the process is faster than designing a machine from zero. In addition, keeping the same electric sheets and the same motor concept reduces the manufacturing cost considerably, as the same manufacturing process can be used for building a wide range of electrical machines.

In this paper, the second scenario is analyzed, where the stator and rotor sheets are selected from a 2D plane database, and after the machine is sized. In Figure 1a, the general process is shown.

**Figure 1.** Description of the sizing methodology using multi-physics tool. (**a**) classical Method; (**b**) integration of the sizing multi-physics tool in the classical method.

Firstly, machine requirements must be defined (1) including working currents, voltages, application temperature, output torque, and speed. These technical requirements come from the customer functional specifications that must be precise, as the design is optimum for the application.

Secondly, the machine is sized (2). The optimum 2D plane and cooling type must be selected for the application. Then, an analytical pre-sizing is made, to have an approximate number of turns, and stack length, together with the wire section. Next, the main step of the sizing process starts, using electromagnetic simulation tools together with thermal ones, to make a more accurate calculation of the machine. Lastly, the obtained performances are automatically checked, and if the machine does not comply the requirements, changes are made to the stack length or the number of turns, making the calculation again, until the design fulfills the requirements.

Finally, the machine is validated experimentally in (3) to check that the machine performance is equal to the simulated one. If the bench test is correct, the sizing of the machine is achieved; otherwise, the process must start again from the beginning.

As mentioned before, PMSM is the most used electric machines for traction, and one of their main characteristics is the nonlinearity, as well as their temperature dependences in the generated magnetic flux. They can also get demagnetized if they continuously suffer thermal overload [1–3]. This is why electromagnetic simulations must be linked to thermal ones, making them more accurate as a whole. However, linking the two simulations causes a longer iterative process, increasing the computational and design time.

To avoid this, a map interpolation is proposed, to obtain magnet and copper temperature based on Joule and Core losses. It will obtain the steady-state temperatures by a fast iteration taking into account loss change with temperature. Finally, working temperature performances can be obtained with a new magnetic flux interpolation and some simple calculations.

With respect to electromagnetic simulations, there are different tools to make it, with diverse accuracy and time consumption ranks—for instance, analytic equations, lumped parameter models, Fourier series models, and Finite Element Analysis (FEA).

The fastest way of sizing a machine is using fundamental machine sizing equations as shown in [4,5]. This analytical method is mostly used for preliminary machine sizing, in the first stages of the process due to its speed. Nevertheless, it does not provide a high accuracy compared to FEA solutions.

Another analytical method is the one based on Fourier series. In this case, the accuracy and computational time depend on the number of spatial harmonics considered in the different machine regions. The higher the harmonic orders are, the higher the accuracy is, but the longer time the computation takes. If magnetic saturation plays an important role in the designed machine, this method should be avoided using FEA, where local magnetic saturation may be considered [6].

With the aim to be more accurate, Lumped-Parameter (LP) models are developed. This method takes into account magnetic saturation and it can be more accurate than other analytical methods, still being faster than FEA. However, the more accurate is the result, the slower is the calculation, so the equilibrium between speed and accuracy is the key [1].

FEA is considered the standard tool for electric machine analysis, as it has a detailed magnetic field solution, including saturation, providing an accurate result of the machine performance according to the density of the mesh. Despite this, FEA needs a massive computational effort and time consumption [7].

In order to reduce FEA computation time maintaining its accuracy, different methods are found in the literature that mix initial FEA simulations with different scaling methods to size different machines. In [8,9], dimensional and number of turns scaling techniques are used after a base machine FEA simulation is carried out. This simulation generates magnetic flux, loss, and torque maps that are used to generate other machine models with the mentioned scaling techniques. For instance, second-order polynomial functions are used in [3] to describe magnetic flux linkage variations respect to current, using FEA models just to calculate parameters of the functions. Then, combined with other analytical models, other machines can be sized based on the reference. Moreover, in [10], FEA is also combined with analytical models to obtain a rapid tool of induction machine mapping in dq axes. The results of these methods are rather accurate, as they are based on FEA models.

As mentioned before, the thermal model must also be simulated, to evaluate working temperatures and their distribution along the machine, linking it to the electromagnetic model. As in electromagnetic calculations, there are various methods with different performances such as FEA, LP models, and Computational Fluid Dynamic (CFD), shown in [11,12].

As in the previous case, the objective is to simulate as accurately and as quickly as possible. The speediest way is to simulate the machine with LP thermal networks as they are quite accurate as shown in [4,13], but they can become as slow as FEA if many nodes are introduced. Therefore, some reduced node models are found in literature, reducing computation time considerably, maintaining good accuracy [14,15].

Gaining accuracy, there is CFD software, used for modelling cooling systems, calculating flow rates, and heat transfer. The main asset is that it can be used to predict the flow in complex regions, such as around the end windings, with great precision. Moreover, the data obtained from CFD can be used to improve analytical algorithms in analytic thermal networks. However, its big disadvantage is its huge computation needs and time consumption, making it unsuitable for fast sizing process, but might be used in big machines with a high cost of prototypes [11,12].

In thermal modelling, FEA is used to accurately calculate the conduction heat transfer in complex geometric shapes, such as heat transfer through strands of copper in a slot. Nevertheless, it has an important limitation as the software uses analytical/empirical-based algorithms for convection boundaries, exactly as in the lumped circuit analysis. As a result, the accuracy is dependent on the same factors for the thermal network; just making a difference when the solid component conduction must be calculated precisely. Some authors used a 3D reduced-order FEA model to reduce computational time maintaining a rather good accuracy [2,11].

Taking into account all the simulation methods found in literature, magnetic flux and temperature maps are used in the proposed sizing method. Magnetic flux maps are obtained from FEA simulations and temperature maps are generated from lumped parameter network simulations. In this way, a fast and accurate method can be defined.

In this article, a fast sizing method is proposed. It sizes a machine in about one minute, maintaining the accuracy of the Finite element models. The major novelty of this method is the coupling between thermal and electromagnetic fields. The sizing is done by an iterative algorithm.

Finally, the proposed method is validated in a real case study, sizing 10 industrial machines used for people transportation systems.

This paper is organized as follows: Section 2 describes the proposed method and explains the theoretical base of the method. The initial simulation process and the sizing algorithm are presented. Section 3 compares the obtained data with 10 industrial machine experimental results to validate the algorithm and the proposed process. In Section 4, the novelty of the work is presented, with the obtained results and their conclusions.

#### **2. Description of the Proposed Sizing Procedure**

In this section, the procedure for obtaining the magnetic flux maps and the thermal maps is explained. As it is shown in Figure 2, this procedure consists of two different simulations. Magnetic flux maps are obtained by electromagnetic FEA simulations performed using Altair Flux-<sup>R</sup> (Troy, MI 48083, USA), whereas the temperature maps are obtained by thermal simulations carried out using Motor-CAD-<sup>R</sup> (Wrexham LL13 7YT, UK).

**Figure 2.** Map database creation.

Concerning the magnetic flux maps, d-q axis magnetic flux is computed as a function of d-q axis currents and magnet temperature. Regarding the temperature maps, two maps are obtained as well—one map for the average winding temperature and another one for the average magnet temperature. Both maps are computed as a function of the Joule and magnetic losses, accounting for different stack lengths.

These maps make up a database in which different magnetic circuit 2D geometries and cooling types are included. This way, during the sizing process of electrical machines, different magnetic flux and temperature maps are used according to the chosen 2D geometry of the lamination and the cooling solution. In the next sections, more details are given about the map creation procedure.

#### *2.1. Magnetic Flux Map Creation*

D-Q axis magnetic flux maps are obtained performing FEA magneto-static simulations. As the electrical machine analysis involves magnetic and electric domains, the magnetic circuit of the motor is coupled to the electric circuit.

In case the machine geometry is simple, FEA 2D simulations could be suitable. However, complex motor geometries might require FEA 3D simulations. The proposed procedure in this paper is suitable for both cases, FEA 2D and FEA 3D.

The objective is to generate magnetic flux maps depending on current (in d-q axis) and temperature, so *ϕ*<sup>q</sup> - *i*q, *i*d, *T*magnet and *ϕ*<sup>d</sup> - *i*q, *i*d, *T*magnet are obtained. As the map depends on three variables, magnet temperature, d, and q currents, many simulations must be performed varying these three variables.

The first step is to define the currents and temperature variation ranges to generate the solving scenario for the model, as shown in Figure 3.

**Figure 3.** Magnetic flux map generation process.

One criterion for defining the currents' variation range could be the saturation of the magnetic material. For instance, for a given *N*, the maximum supplying current value could be defined according to the linearity of the q axis magnetic flux as a function of the q axis current, from 0 A up to the point at which the linearity of the q axis magnetic flux decrease in a given value. The effect of the magnet temperature is accounted for changing the remanence value of the magnets field. The maximum value of the remanence value can be set according to the standard data-sheets of commercial magnets. For instance, nowadays, the maximum remanence field that can be found in the market for Neodymium magnets is around 1.43 T (N52M from Baker Magnetics (5692 Elson, The Netherlands)). Concerning the minimum remanence value, it could be defined also considering standard properties of the magnets in the market. For example, a suitable criterion could be to consider the minimum remanence value at room temperature around 0.98 T (N25 BH from Baker Magnetics) and compute the remanence value at the maximum working temperature of the same magnet, about 240 ◦C. Applying this criterion, the magnet remanence value should be varied in the range of 0.74 T–1.43 T.

It is also important to define *N I* properly, to obtain data in the machine working range. If a working point of a machine exceeds the maximum value of the map range, the map will not be suitable for sizing the desired machine because it would need to extrapolate and extrapolation may generate incorrect results.

Flux-<sup>R</sup> software performs the electromagnetic analysis solving Maxwell's equations with a magnetic vector potential, finally solving Equation (1) by finite element methods:

$$\nabla \times \left( v\_0 \left[ v\_r \right] \nabla \times \vec{A} \right) + \left[ \sigma \right] \left( \frac{\partial \vec{A}}{\partial t} + \nabla V \right) = 0 \tag{1}$$

where [*v*r] is the tensor of the reluctivity of the medium, *<sup>v</sup>*<sup>0</sup> is the reluctivity of the vacuum, −→<sup>A</sup> is the magnetic vector complex potential, [*σ*] is the tensor of the conductivity of the medium, and *V* is the electric scalar potential.

Once the solving scenario is defined, the simulation is carried out to get *ϕ*a - *i*q, *i*d, *T*magnet , *ϕ*<sup>b</sup> - *i*q, *i*d, *T*magnet and *ϕ*<sup>c</sup> - *i*q, *i*d, *T*magnet . Then, these variables are post-processed to get *ϕ*<sup>q</sup> - *i*q, *i*d, *T*magnet and *ϕ*<sup>d</sup> - *i*q, *i*d, *T*magnet . This will be done with Clark–Park transformation [16], as shown in the second step of Figure 3. Finally, the obtained magnetic fluxes shown in Figure 3 will be saved in a magnetic flux map database for future use.

The resolution of the maps must be properly chosen as it might affect the accuracy of the final results given by the sizing tool. Defining at least 10 computation points in the variation range of each variable could be a criterion. This leads to at least 1000 different simulations to be performed by FEA. Another key point affecting the accuracy is the resolution of the simulations. A criterion could be to consider at least 100 points in a single electric period, so 100 points are performed for each simulation, leading to a total amount of 100,000 simulation points. Using an average computer (16 GB RAM, 64 bytes—3.41 GHz Microprocessor), the solving of a single point could take around 5 s, leading to a total simulation period of five days. It might not be too much considering that the magnetic flux maps are obtained once and then no more FEA simulations are required for a given magnetic circuit geometry. Nevertheless, in case the data-sheet must be made-up with many different magnetic circuit geometries, this task could take a lot. Thus, in this paper, a proposal is presented to reduce the computation load of the magnetic flux mapping process.

As shown in Figure 4, during one full electric period of *ϕ*a, *ϕ*b, and *ϕ*c, there are six *ϕ*<sup>d</sup> periods, so it is enough to simulate 1/6 of the period to obtain *ϕ*d, reducing significantly the computation time.

**Figure 4.** Magnetic fluxes during a full electrical period.

#### *2.2. Temperature Map Generation*

As mentioned in Section 1, there are different methods and software to model the thermal performance of electrical machines. In this paper, the commercial software Motor-CAD-<sup>R</sup> is used to obtain the required temperature maps.

Motor-CAD-<sup>R</sup> uses a three-dimensional lumped circuit model that can be used to calculate the steady-state and transient thermal characteristics of several motor types. One of the most complex aspects of motor thermal analysis is the prediction of Convection Heat Transfer mainly relating to the outer surface of the motor, but also for the internal air-gap. An estimation is made by the software using natural and forced convection correlations. Radiation Heat Transfer is also modelled in Motor-CAD.

The process for obtaining these maps is shown in Figure 5. First, the model is defined, using the stack 2D plane and the desired cooling system, generating the thermal network.

Average winding temperature and average magnet temperature maps are obtained as a function of the Joule and Magnetic losses and accounting for different stack lengths. The variation range for the losses and the stack length must be properly established to assure that all thermal situations demanded during the sizing process are covered by the maps.

**Figure 5.** Temperature map generation.

The variation range and the resolution of the stack length will be defined by the user.

The variation range of the losses could be established accounting for the winding temperature. For instance, it does not make sense to consider losses that raise the temperature above the maximum limit for the maximum stack length (keeping the losses constant, the smaller the stack length is, the higher the temperatures are. Thus, for small stack lengths, the temperatures might lead above the maximum limit).

This way, the variation range of the losses and the stack length should be defined for every particular case. The model is simulated for each defined point of power losses and length combination. One of these simulations is done for each cooling type.

Finally, temperature maps are created. *T*magnet - *P*LCu, *P*Lmag and *T*Cu - *P*LCu, *P*Lmag maps are shown in next Section 3, Section 3.2. Each layer represents one stack length, from the smallest machine in the upper layer, to the longest in the lower (and coldest) layer.

#### *2.3. Sizing Method*

In this section, the proposed sizing process of electrical machines is described. As it is shown in Figure 6, before the sizing process begins, the design requirements must be defined. Then, the cooling solution and the 2D magnetic circuit geometry must be chosen. Once these two elements are chosen, the corresponding magnetic flux and temperature maps are uploaded to the sizing program. As it can be appreciated in Figure 6, the sizing process consists of three main stages:


are calculated, such as Joule and magnetic losses. These losses will be used to obtain the working temperature of the winding and the magnets by interpolating in the temperature maps. With the working temperatures, performances are calculated again, based on a new magnetic flux map interpolation. At the end of this process, performances of the machine are obtained at ambient and working temperatures.

• STAGE 3: AUTOMATIC CHECKING. The performances are checked and, depending on the results, the process is finished or a new iteration is started returning to Stage 1.

**Figure 6.** Sizing algorithm.

#### 2.3.1. Stage 1: Definition of *L* & *Z*

Once the main design requirements are defined, the number of turns and the stack length are pre-calculated considering the torque Vs current requirement, and the voltage limitation, applying the next well known torque and voltage analytical Equations (2):

$$\begin{cases} T\_{\rm nom} = 3p\varphi\_{\rm d} L N I\_{\rm n} \\ V\_{\rm max}^2 = \left( -\varphi\_{\rm q} + N^2 L k\_{\rm ov} w\_{\rm e} I\_{\rm n} \right)^2 + \left( 2\rho N^2 n\_{\rm cap} I\_{\rm n} \frac{L + L\_{\rm vrd}}{A\_{\rm r} k\_{\rm f}} + \rho\_{\rm d} N L w\_{\rm e} \right)^2 \end{cases} \tag{2}$$

where *T*nom is the nominal torque (Nm), *p* is the machine pole pairs, *ϕ*<sup>d</sup> is magnetic flux d (Wb), *ϕ*<sup>q</sup> is magnetic flux q (Wb), *L* is the machine stack length (m), *N* is winding number of turns, *I*<sup>n</sup> is the desired nominal current (A), *V*max is the maximum allowed voltage (V), *k*ov is the overlapping factor,

*w*<sup>e</sup> is the machine rotational speed (rad/s), *ρ* is the wire resistivity (Ωm), *n*cap is the number of winding layers, *L*end is the end-winding length (m), *A*<sup>r</sup> is the copper wire area (m2), and *k*<sup>f</sup> is the filling factor.

Solving the equation system (2), the initial length and number of turns are estimated, providing an accurate starting point of the iterative loop, instead of traditional iteration starting from a particular point every time. Overlapping and filling factor values for the estimation are fixed, but they could be adjusted depending on the number of turns and the final chosen wire section.

#### 2.3.2. Stage 2: Electromagnetic & Thermal Analysis

Once initial number of turns (*N*ini) and initial stack length (*L*ini) are defined, it is time to interpolate in the magnetic flux map. At the first iteration, ambient temperature is taken for the interpolation. Nominal current is set to obtain the required torque, interpolating and obtaining *ϕ*<sup>d</sup> and *ϕ*q.

The interpolated value at a query point is based on linear interpolation of the values at neighbouring grid points in each respective dimension. This method is accurate enough if the map resolution is properly defined. Extrapolation is not recommended as it may result in false values of magnetic flux or temperature in case of thermal maps.

For resistance calculation, standard values of wire diameter are tabulated for each number of turns, to obtain a suitable filling factor depending on the chosen winding type, p.e. around 0.42 for distributed windings and around 0.5 for concentrated windings. The overlapping factor is defined with an experimentally adjusted curve. With this data and motor geometry, end-winding length (*L*end) is calculated with (3), finally obtaining winding resistance (*R*Cu) with (4), where *ρ* is copper resistivity at 20 ◦C. Moreover, q axis inductance is calculated by definition in (5):

$$L\_{\rm{end}} = \frac{\frac{\pi}{2} (\frac{D\pi}{Q\_{\rm{\ast}}} + w\_{\rm{d}}) \ast k\_{\rm{ov}}}{1000} \tag{3}$$

where *D* is stator diameter, *Q*<sup>s</sup> is the number of slots, *w*<sup>d</sup> is the slot width, and *k*ov is the overlapping factor:

$$R\_{\rm Cu} = \frac{\rho (L\_{\rm end} + 2L)(1 + 0.0039(T\_{\rm Cu} - 20))}{A\_{\rm r}} \tag{4}$$

$$L\_{\mathbf{q}} = \frac{\varphi\_{\mathbf{q}}}{i\_{\mathbf{q}}} \tag{5}$$

Then, voltage is calculated, as shown in (6), where *V*<sup>d</sup> and *V*<sup>q</sup> are dq voltages, and *L<sup>σ</sup>* is the leakage inductance:

$$\begin{cases} V\_{\rm d} = R\_{\rm Cu} i\_{\rm d} - L\_{\rm \sigma} w\_{\rm e} i\_{\rm q} - \varphi\_{\rm q} w\_{\rm e} \\ V\_{\rm q} = R\_{\rm Cu} i\_{\rm q} + L\_{\rm \sigma} w\_{\rm e} i\_{\rm d} + \varphi\_{\rm d} w\_{\rm e} \end{cases} \tag{6}$$

Finally, losses are calculated. Joule losses are calculated by Joule's law (7), while Core losses are calculated with the Bertotti's Model (8) shown in [17]:

$$P\_{\rm LCu} = 3R\_{\rm Cu}I^2 \tag{7}$$

$$\begin{cases} p\_{\rm Fe} = p\_{\rm h} + p\_{\rm \varepsilon} + p\_{\rm \varepsilon} = k\_{\rm h} f B\_{\rm s}^{\rm a} + \sum\_{i} k\_{\rm c} f^{2} B\_{\rm si}^{2} + \sum\_{i} k\_{\rm c} f^{1.5} B\_{\rm si}^{1.5} \\\\ P\_{\rm Fe} = k\_{\rm a} p\_{\rm Fe} \mathcal{W}\_{\rm m} \end{cases} \tag{8}$$

where *p*Fe is the core loss per weight, *p*<sup>h</sup> is the hysteresis loss, *p*<sup>c</sup> is the eddy current loss, *p*<sup>e</sup> is the excess loss, *k*<sup>h</sup> is the hysteresis loss coefficient, *k*<sup>c</sup> is the eddy current loss coefficient, *k*<sup>e</sup> is the excess loss coefficient, *α* is an Steinmetz coefficient, *f* is the frequency, *B*si is the *i* th harmonic amplitude of the stator magnetic flux density, *k*a is the empirical coefficient, and *W*m is the weight of motor.

Once Joule and Core losses are obtained at working temperature, they are sent to the thermal analysis. Then, the first interpolation can be made in the temperature maps, obtaining winding and magnet temperatures. Nevertheless, these temperatures are not the steady-state ones; as with temperature change, losses also change. To obtain the steady-state working temperatures, Joule losses are updated with temperature, as the resistance varies with temperature. The thermal analysis block will make this iteration until the steady-state losses and temperatures are obtained, taking about 16 iterations.

Once steady-state temperatures are obtained, they are returned to the electromagnetic analysis block, so the performances are obtained at working temperatures.

#### 2.3.3. Stage 3: Automatic Checking

After obtaining the electrical performances at room and working temperatures, they must be checked, and, if they fulfil all the requirements, the sizing process will be finished, generating a favourable machine design report. Otherwise, the design parameters are changed and the calculus is addressed again. Figure 7 shows which parameters are checked, and the actions adopted (in STAGE 1) if they are not fulfilled. In the figure, L+ refers to increasing the machine length in one step, and Z+ or Z− means increasing or decreasing conductors in each slot.

**Figure 7.** Automatic checking process after each iteration.

Checking minimum voltage is interesting, but it is not mandatory as some machines will not be able to fulfill both maximum and minimum voltages.

This checking is made automatically with the proposed algorithm taking into account designer specifications as the minimum and maximum voltages, maximum current, and maximum temperatures.

These requirements are set by the designer at the beginning of the process. The maximum voltage and current usually are limited by the inverter or the grid and the winding temperature normally is limited by the material or the machine class. If just one of the parameters does not comply, changes are made and another iteration is made, checking the four parameters again at its end. If all parameters are fulfilled, this is the optimum length and number of turns for the machine so the report is favourable, ending the process.

A Graphic User Interface (GUI) is designed to implement the proposed method in an easy and user-friendly way, to save time and effort when sizing machines. This GUI is composed of a database containing magnetic flux and temperature maps, an interface to choose those maps and introduce design requirements, a calculation core to implementing the proposed algorithm and a report generator to show the results after the results are automatically checked.

#### **3. Case Study: Sizing of PMSM for People Transportation**

In this section, the proposed improved sizing methodology is implemented in a real case study. The objective is to validate the multi-physics tool sizing several commercial PMSM for people transport application.

#### *3.1. Description of the Machines*

These machines are based on conventional topology comprising 36 slots in the stator and 30 poles in the rotor (Qs36p15). In total, ten different machines have been sized by the proposed method and tested experimentally. Their performances are shown in Table 1.


**Table 1.** Main machine performances

In Figure 8, the stator and the rotor corresponding to one of the tested motors, and the ID 5 are shown.

**Figure 8.** Rotor and Stator of one motor experimentally tested (Motor ID 5).

#### *3.2. Map Creation*

All of the tested machines have the same 2D magnetic circuit shown in Figure 9. The active length and the number of turns per phase are adjusted to fulfill the requirements of each application.

**Figure 9.** Geometry of the sized machines (Qs36p15).

In this case, the tested motors have surface-mounted permanent magnets and concentrated windings. In the future, the multi-physics sizing tool can be used with interior magnet motors or distributed windings to broaden the validation.

As mentioned in Section 2, magnetic flux maps must be created for each 2D geometry, generating the maps shown in Figure 10. Figure 10a shows magnetic flux in the d axis, whereas Figure 10b shows magnetic flux in the q axis.

**Figure 10.** Generated magnetic flux maps: magnetic flux vs. Current and Temperature for the Qs36p15 motor. (**a**) D magnetic flux map; (**b**) Q magnetic flux map.

As it can be appreciated in Figure 10, the temperature mainly affects to the d-axis magnetic flux because the remanent field of the magnets decreases linearly as the temperature increases. On the contrary, the q-axis magnetic flux changes slightly with the temperature. In this case, these small variations are due to changes in the saturation of the magnetic circuit. The magnetic flux created by the magnets changes with the temperature leading to variations in the saturation.

Ideally, in universal d-q axis models of the PMSM, it is commonly considered that the d-axis magnetic flux depends on the magnet flux and the d-axis current, while the q-axis magnetic flux depends only on the q-axis current. Nevertheless, there might be a fairly cross-coupling effect between the d-q axis depending on the saturation level of the motor, which might lead to relevant errors in the final results. In this case, this cross-coupling effect is clearly appreciated in Figure 10a as the d-axis magnetic flux changes with the q-axis current. In addition, this relationship is not linear, which makes it more difficult to model. Interpolating the magnetic flux maps, as it is done in the proposed tool, all these nonlinearities are taken into account, making it possible to achieve accurate results to some extent.

With respect to thermal maps, the Q36p15 motor model is shown in Figure 11. This model is used to create the maps, with the geometry and the selected cooling system—natural convection in this case.

**Figure 11.** Motor-CAD-<sup>R</sup> Model of the sized machines.

As mentioned in Section 2, temperature maps must also be created for each 2D geometry and cooling type, generating the maps shown in Figure 12. Figure 12a shows the mean winding temperature, whereas Figure 12b shows the mean magnet temperature. As it can be seen, shorter machines get warmer easier, and they will tolerate lower losses. In conclusion, if a machine exceeds the maximum desired temperature of the winding, or the demagnetization temperature of the magnets, a longer machine may be chosen.

**Figure 12.** Generated Temperature maps: Temperature vs. Magnet Losses, Copper Losses and Stack length in the naturally cooled machine with Qs36p15 configuration. (**a**) winding temperature map; (**b**) magnet temperature map.

#### *3.3. Sizing*

In Figure 13, sizing results are shown. It can be seen that most of the machines have the same EMF constant, as it was expected. In addition, voltage trends are very similar between machines, although some of them reach slightly higher voltage values. The figure also shows that machines are more efficient at nominal currents than at low and high currents.

With respect to time consumption, sizing a machine with the proposed sizing tool takes less than one minute for each machine. Taking into account that sizing a machine with the classical method can take about 8 h, the time reduction is considerable. Naturally, obtaining magnetic flux and temperature maps takes time, but these calculations are carried out only once, so this time is paid off when some machines are sized.

With respect to the mapping process, the computation of a temperature map with a resolution of 13 stack length values, 15 Joule loss values, and 15 Magnetic loss values could take around 2 h. This leads to a 3D matrix with 13 × 15 × 15 dimensions. Magnetic flux map creation for six temperatures varying *iq* with 10 values over a full electric cycle takes approximately 6 h, but, applying the 1/6 reduction mentioned in Section 2, the consumed time is reduced in a 83%, leading to a 6 × 10 2D Matrix.

**Figure 13.** EMF, Torque, Voltage, and Efficiency vs. Current calculated by the Sizing Tool during the sizing process of 10 machines.

#### *3.4. Experimental Validation*

As the final step of the designing process, all sized machines are prototyped and experimentally tested to validate the designs. The test bench is shown in Figure 14.

**Figure 14.** Picture of the test bench consisting of the load motor, the motor under test (MUT), and the torque transducer.

In Figure 15, the main experimental performances, such as electromotive force, torque, supplying voltage, and efficiency are shown for all built prototypes.

As illustrated in the upper-left of Figure 15, speed and EMF are proportional and most of the machines have a similar *ke*. In the bottom-left side, voltage is plotted against the current, and the majority of the machines follow a similar trend, although the values differ slightly. With respect to the efficiency, on the bottom-right side, it can be seen that efficiency decreases in low and high currents, and it increases in medium and nominal currents.

To validate the accuracy of the proposed sizing tool, the obtained results during the sizing process are compared to the experimental measurements. In Figure 16, the difference between sizing tool results and experimental results is plotted for the 10 sized motors.

It can be seen that the maximum difference in the EMF is about 1%, whereas, in the supplying voltage, it is about 2%. Nevertheless, the mean error is about 0.2% and 0.5%, respectively. With respect to the torque, the maximum difference is about 5% at low currents, mainly due to the uncertainty in the measurements at low currents. Nevertheless, it must be emphasized that the error is rather small as it is below 1.5%.

**Figure 15.** Experimental results of EMF, Torque, Voltage, and Efficiency vs. Current measured during the validation tests of the 10 sized motors.

**Figure 16.** Error of the sizing tool computed as the difference in percentage between the experimental measurements and the results given by the sizing tool; error computed for the 10 sized motors.

Then, in Figure 17, the results of a given machine, the M7 (see Table 1), are shown for a close-up view. As illustrated in Figure 17, measurements agree with the sizing tool results with a small difference at some points.

Concerning the efficiency, higher differences can be observed. This could be due to many different factors—on one hand, due to the uncertainty in the calculation of iron losses, mechanical losses, and stray losses; on the other hand, due to the possible measuring errors in the torque transducer and Voltage/Current probes. Even these errors might not be very significant, as it is shown in the torque comparison; for instance, the accumulation of all of them could justify the differences in the efficiency. In any case, it must be stated that these differences are not very significant, as the error is very low, below 6%.

**Figure 17.** Comparison between experimental and calculated results in the M7 motor.

#### **4. Conclusions**

In this paper, a PM machines sizing methodology is developed. The proposed methodology, based on the coupling of magnetic flux and temperature maps, has been put into practice in a real case study. This methodology improves the competitiveness of ten industrial motors, reducing the design time and, consequently, the resources needed for that design. As a result, all motors have been sized accomplishing a very good trade-off between cost and required performances.

This procedure enables to perform sizing in a faster way, using less computational resources. Using magnetic flux and temperature maps enables achieving very good accuracy. As the influence of the temperature is considered, the accomplished results are more realistic. It must be remarked that specialized software is only needed for Map creation, and the sizing algorithm can be run on any computer.

Moreover, a faster procedure is described to obtain magnetic flux maps at different temperatures, just simulating 1/6 portion of the electric period, instead of considering the overall electric period. This method makes it possible to reduce the mapping process time in 83%.

Taking the overall results into account, the proposed sizing method fulfills the desired objectives of time reduction and accuracy in the sizing process, coupling electromagnetic with thermal effects, and sizing machines in less than a minute and with an error below 6%.

**Author Contributions:** G.A., A.E., and A.J.E. conceptualized the research; A.E. and Y.M. developed the methodology and the software; P.M. conceived and performed the experiments; P.M. and G.A. analyzed the data; Y.M. wrote the original draft. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations and symbols are used in this manuscript:


#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
