The remainder of this paper is organized as follows: Section 2 overviews the related theory for the application of the algorithms. The proposed algorithms are presented in Section 3 Section 4 presents a use case example to compare the computational time by a typical process to obtain a motor characteristic and the process using the algorithms. Section 5 is the validation section where the speed characteristics results for the three types of brushless motors are computed with the algorithms and compared with its reference value. Finally, Section 6 concludes the paper.

The goal of this paper is to develop new functions that reduce the time needed to obtain the key performance operation points of any candidate machine for a future iterative design function. Nowadays, we have computers capable of making large amount of data computations that have resulted in the availability of several commercial FEM tools. Most of them offer the possibility to code within the tool to automate tasks. We take advantage of that fact and offer nine simple algorithms as an alternative for the evaluation of electromagnetic operation points in motor models in FEM within acceptable computation times of around twenty minutes for complex tasks such as obtaining the CPSR of the motor for given converter limits and below five minutes for simpler calculations such as the maximum torque for a given current. The algorithms apply three numerical approximation methods, Newton’s method, secant method and a curve fitting natural cubic spline interpolation. The convergence of these methods applied in the algorithms have been well studied in the past [ 11 12 ]. The algorithms presented in this paper were developed with the objective of finding specific characteristic motor points of interest during the design process or during control design that cannot be targeted directly with a FEM tool such as maximum torque per Ampere (MTPA), minimum Ampere per torque (MAPT), maximum torque per voltage (MTPV), the speed for a voltage and current limit, characteristic or short circuit current (), the maximum power point during flux weakening, the CPSR and the boundary transition point from FW to MTPV control. Their logic intention was developed to work for the three types of brushless AC motor obtaining reliable approximations with the least number of simulation step points and iterations possible, therefore, in shorter times compared to a full simulation process, yet still flexible as to whether or not more steps are needed. A module with the algorithms was developed in Jython, which is the language in Altair Flux2D, the FEM tool used in this paper. Validation of the algorithms is done by the results obtained for three already validated motor model designs, a non-salient permanent magnet motor, an IPM and a SynRM [ 13 ]. The results are compared with the reference values of each motor and the time of execution in seconds for specific inputs are shown.

FEM approach: it has high fidelity and any geometry can be directly studied [ 1 5 ]. The key design parameters are not directly obtained from FEM simulations. Additional equations should be added, and normally FEM simulations are done intensively and after interpolation functions are applied. These techniques are not adequate for a high iterative process with a high number of machine candidates.

Functions using reluctance based circuits [ 9 10 ]: this approach can reach satisfactory levels of precision, but any specific geometry will have a specific reluctance network. The time can be reduced, but when the precision is increased the complexity of the development of the network is increased and the computation time is also increased. A very high accurate reluctance circuit could finally require the same computation time as a FEM approach.

Analytical sizing functions [ 7 8 ]: this approach is fast but not a high fidelity model. Then, it is possible that some discarded machines candidates could have better performance than the selected ones.

This paper is focused on developing fast algorithms to obtain the motor key operation points performance in order to compare a high set of machine candidates. For this purpose, different approaches were considered:

The main drawback of FEM is its high computational cost, and the literature shows some practical solutions to reduce the simulation time by applying simplifying techniques limiting the number of simulation points [ 4 6 ]. However, in any case there is a manual involvement of repetitive steps of the designer for performing simulations, exporting data, post-processing the data (usually with some other software tool, e.g., MATLAB) in order to compute specific performance points of the motor. These tasks can be tedious and time consuming, especially in the conceptual phase of the design where some parameters are usually still not defined (e.g., geometry, number of turns per phase, poles, slots, magnets, etc.) and only particular points are of interest to check whether the design meets a given requirement to consequently decide if changes are needed.

Three-phase brushless drive systems, especially those based on permanent magnets motors, have proven to be a good solution for high efficiencies and compactness requirements. The three common types of brushless ac motors are synchronous reluctance (SynRM), non-salient permanent magnet and interior permanent magnet (IPM) ones (throughout this paper the last two will be abbreviated as permanent magnet synchronous motors—PMSMs). In addition, given the converter limits these motors, in particular the IPM type, also present the advantage of a wider constant power speed ratio (CPSR) which is desirable for operation above the corner speed, e.g., in traction applications. Brushless AC synchronous motors are designed using two common types of tools, numerical tools based on the finite element method (FEM) and analytical tools, and also combinations of both. Despite the large amount of simulation time required, FEM is the most adopted and preferred solution since it considers a multi-physics environment taking into account dependency of all parameters, e.g., cross-coupling, magnetic saturations and slotting effects, which may be neglected or assumed constant in analytical models. Analytical models based on the dq model are usually applied during the conceptual phase for initial sizing and when the performance behavior of the specific motor wants to be modeled typically for control purposes [ 1 3 ].

This paper presents an alternative of automatic computation of these operation points within the same FEM tool and within acceptable times of execution by iterative processes. From this point forwardis the torque computed with Equation (3) with the dq fluxes. Besides,is the line-to-line voltage, i.e., Equation (4) times √3. All the results for a particular current coordinate are called characteristic variables which is illustrated in Figure 2 and depending the context of study the phraseinclude all or the ones that are relevant in the context:

Given the voltage and current limit get the mode 2 limit see Figure 1 , the CPSR point and the maximum power to speed ratio (MPSR) point of the electric motor.

Given the speed and voltage reference and a given d axis current get the current vector where that voltage is reached.

In general and regarding electrical machine design, FEM tools compute a working point targeted with phase currents, however we as designers are also interested in results that cannot be targeted directly such as:

The design of electric machines involves a nonlinear magnetic computation with a large number of design variables that makes it difficult to model them analytically with high precision. Therefore, thanks to hardware and software development of computers many commercial and open source FEM analysis tools are available, having a broad acceptance with engineering designers.

The evolution of inexpensive personal computers have enhanced the use of numerical approximations methods in engineering applications. The numerical methods used in this paper are the Newthon-Raphson method using Equation (5), the Secant method which is a variation of the Newton-Raphson method that makes a derivative approximation as shown in Equation (6) and the well-known natural cubic spline interpolation [ 12 ]:

In the finite case only the first two modes of operation can be accomplished since. The current locus for the modes of operation can be seen on the circle diagram in Figure 1 for the case of an infinite IPM drive. Thorough studies on the behavior of PMSMs and SynRMs in the different modes of control operations can be found in [ 15 17 ].

Mode III: MTPV region, for a given speed the maximum torque is obtained from the voltage limit locus where the constant torque curve is tangent to it.

Mode II: FW region, it is applied above rated speeds maintaining rated current with demagnetization done by decreasing the Id current to keep the maximum voltage limit.

Based on a theoretical speed limit, the brushless synchronous AC motor drives are divided in either infinite or finite drives [ 14 ]. For the ideal infinite case three different regions or modes of control operation are defined in control:

In this section, we discuss the brushless ac motor drive systems theory, the numerical approximation methods that are applied in the algorithms and the setting details on the FEM tool in order to understand the application of the algorithms.

The objective of Algorithms 8 and 9 is to find mode 1 and 2 limit, respectively. Algorithm 8 assumes the limit is given by the voltage, it iterates along the MTPA curve using Algorithm 5 until the voltage is found as illustrated in Figure 1 , in this case we do not take into account the current or torque limit, the user criteria must decide if the point is valid or not. This was done because if the torque or current decide the limit then Algorithm 4 or Algorithm 5 can be used. Finally, Algorithm 9 finds the mode 2 limit, the CPSR and the MPSR point for the case wherebecause for the case wherea simpler analysis using Algorithm 6 can be done. These points can be found in different ways e.g., using Algorithm 7, however, its convergence is slower. Therefore, we use Algorithm 6 due to FEM tools simulate a working point targeted with phase currents, then, converging faster compared to any other way of iteration. The process can be explained with Figure 4 b, here we use Algorithm 6 for different currentstoin this example, and then we automatically process the results using cubic spline interpolation to find different query speeds voltage ellipses (the solid lines crossing the currents) then applying interpolation to find the maximum torque on each ellipse, to finally find the limits of the motor.

In Algorithm 5 we target the MTPA point for a given current, which is a simple process of sweeping different angles for the same current magnitude, then using cubic spline interpolation obtain the maximum torque value and return the characteristic variables for that point. Algorithm 6 uses the Newton’s method to obtain the speed for a given voltage limit, in a specific current vector, the process is illustrated in Figure 4 b, for a current Iand a given set of angles 1 to 11 in the figure the algorithm finds the corresponding speed for the voltage limit for each point. In this case we need to compute the derivative of the voltage with respect to speed, this derivative is given in Equation (7). The MTPV point for a given speed and voltage limit is obtained through Algorithm 7, it uses Algorithm 2 for differentcurrents getting values along the voltage ellipse and finally uses cubic spline interpolation to find the maximum torque:

Using the secant method in Algorithm 3 we get I ch based on Equation (2) when solved for I d . I ch is the value of I d needed to drive ψ d s to zero. An initial value of one was given to flux d to avoid zero division error, this can be changed by computing two initial guesses or give the magnet flux if known for the PMSMs. In Algorithm 4 we target the value of the MAPT point for a given torque, in summary what is done is to compute the constant torque curve for different angles using Algorithm 1, then using cubic spline interpolation obtain the minimum current magnitude. The MAPT point is basically the same MTPA point but targeted with the torque instead of the current.

In order to execute the proposed algorithms some auxiliary functions will be necessary, however, these depend upon the specific FEM tool commands or are well known algorithms. Therefore, to make it short and simple, in this paper some steps within the algorithms will be only mentioned as processes and the developer must adjust the tool to execute these respective processes. Among these functions are Park’s transformation, cubic spline interpolation algorithm, and automatic simulation of a point on the FEM tool. In addition, some inputs depend on the motor model e.g., pole pair number, initial alignment angle, phase resistance, etc. and others are specific of the FEM tool e.g., the name given to parameters, step number, initial position, etc., henceforth, other inputs not mentioned for the specific model and tool are called motor model and tool parameters .

4. Use Case Example and Process Comparison

One of the advantages the algorithms can provide is in the conceptual phase of the design, where changes to features of the model such as geometry is repetitive in order to find a valid model that can later be completely characterize. Figure 5 a shows a typical process followed to compute a motor’s characteristics using FEM. Once the model is set and ready, a sweep of simulation points are done by introducing a range of three sinusoidal phase currents, the outputs are (but not limited to) fluxes, currents and electromagnetic torque. After the results are ready, usually the designer exports them for post-processing in order to obtain the characteristics that satisfies requirements and study the performance of the machine at different load points.

The algorithms provide an alternative to compute the motor characteristics within the same FEM tool. It is important to remark if the model design is already valid, the conventional process in Figure 5 a will allow to have the exported results at hand which will make the study of the performance of the motor easier. Then, the algorithms not intend to eliminate any valid design process but to provide an alternative of motor characteristics computation, in addition to help find answers faster when the need of the computation of one or more characteristics exist and the model is still not definite to be completely characterized. Figure 5 b shows the process using the algorithms that can be followed to check if the motor model is valid. As illustrated this can be done in two fewer steps compared to the typical process.

To compare the computation time consumption in both processes with similar conditions an example is now given with a ferrite IPM motor. For comparison only computation time is considered, not any human work time to develop a task is taken into account, although it is evident the conventional process involves more work. Suppose we have just finished a new IPM motor geometry design with the features mentioned in Table 1 . looking to satisfy the rated values specified in Table 2 . The best current phasor that will satisfy these requirements is of course on the MTPA curve. However, we do not know what that current phasor is. So with no prior analysis the current phasor will be found using both processes mentioned in Figure 5 . The application is set to magnetostatic 2D, the tool use is Flux 2D/3D from Altair, it has Jython as the coupled language.

The process “set FEM tool for simulations” only refers to prepare the model to execute simulations like faces, regions, mesh, physics, materials, etc. Because this is common for both processes the time in the comparison is not considered.

The sweep of simulations for the conventional process is launched with the following range of values, assuming the result lies between these values:

(1) Range to search the MTPA current angle: [5°, 65°] in 10 steps. (2) Currents magnitude range for each angle: [8 A, 15 A] in 8 steps. (3) Number of steps in one-sixth of the electrical period (position angle): 3.

The time to finish the simulations was of 1457.37 s. The results were exported to excel, to be used in Matlab for post-processing. The time of exporting the results was of 91.02 s. Applying pre-developed post-processed scripts after the results are imported in Matlab the MTPA for each current is found as it is shown in Figure 6 . The computation time to find the current on the MTPA curve that satisfies the rated torque value was of 0.4315 s.

Now, for the process using the algorithms. Algorithm 4 computes the minimum current vector for a given torque value. For the reference torque of 370 Nm the algorithm inputs were set as follows:

(1) Torque tolerance 1 × 10−2 Nm (2) Current angle range [5°, 65°] in 10 steps. (3) Current magnitude initial guess 10 A (4) Number of steps in one-sixth of the electrical period (position angle): 3.

The process lasted a total of 1076.83 s. The current results for each process and the computation time consumption comparison are summarized in Table 3

The computation time results suggest that in similar conditions using the algorithms will give the advantage to the designer of reducing computational time to obtain a characteristic, besides less work to do it. The results show the current magnitude is above requirements then changes to the model must be made in order to satisfy them and the process should be repeated. If changes to the model are automated with scripts and the algorithms are used inside them, these provide the possibility to generate different machine candidates within the same FEM tool taking into account all multi-physics effects of FEM. This example was given with Algorithm 4 to compute the minimum current phasor for a given torque which is one of the algorithms that takes longer to give a result, nevertheless, Algorithms 1, 2, 3 and 6 are algorithms that usually take less than 5 min to provide an answer. The use and selection of the algorithms must be adjusted by the designer needs to get the most out of it. Validation of the algorithms is presented in the next section by computing the speed characteristics of the three typical brushless motors with already valid motor models designs.