Accuracy of Core Losses Estimation in PMSM: A Comparison of Empirical and Numerical Approximation Models

Abstract

The estimation of core loss in permanent magnet synchronous machines (PMSMs) is a fundamental step for the optimization of the performance of PMSM drives. However, there is a lack of literature which fully demonstrates the goodness of some of the empirical approximations that are commonly used in industrial and automotive sectors. This work investigates how different approximations for the core loss estimation of PMSMs can lead to considerable error across the entire machine operating domain. An interior PMSM (IPMSM) is modeled in finite element analysis (FEA) and used to calibrate the coefficients of the Bertotti equation. Approximations of the Bertotti equation are then made, which are calculated from a simple algebraic expression of measurable states, and these are used to estimate machine core loss in the whole direct-quadrature ($d-q$) domain of operation. The estimated core loss obtained with the approximations are finally compared to FEA core loss results. The approximations are shown to have considerable variability in their accuracy compared to FEA results. The results of this work can be used as guidance during the development of estimation algorithms for PMSM losses and the development of control strategies.

1. Introduction

The efficiency of permanent magnet synchronous machines (PMSMs) plays a crucial role in the performance and viability of several applications, including electrified transportation and renewable energies [1,2,3,4,5,6]. The primary propulsion system of many modern electric vehicles (EVs) is based on PMSM drives due to the advanced performance of PMSMs, such as high power density, superior torque characteristics, and excellent efficiency compared to other motor types [1,2,3,4]. In a similar manner, PMSM drives have gained high interest in wind power generation [7] and other applications. Due to their popularity across several sectors, PMSM efficiency improvements and efficiency tracking are very important. For example, even small efficiency improvements can significantly impact the vehicle range, battery life, and overall energy consumption for EVs [1,2], which only supports the need for optimizing the design and the control of PMSM drives to achieve performance improvements. Even with a clear need to improve PMSM efficiency, quantifying PMSM efficiency can be difficult due to the computational challenges that arise when calculating PMSM electromagnetic losses [4,8,9,10].

Electromagnetic losses of electric machines can be divided into two categories: copper loss and iron loss [11]. Copper loss consists of the losses associated with currents flowing through the stator windings, and it includes ohmic loss [11], the skin effect, and the proximity effect [12,13]. Iron loss, also known as core loss, consists of the losses associated with the magnetic flux-carrying materials of the machine, which include both hysteresis loss and eddy current loss [10,14]. Of all of these types of loss, only ohmic loss is relatively easy to compute for an PMSM since it is proportional to the square of the stator currents [15]. All of the other loss types are dependent on more complicated physics, and mathematical modeling approximations are usually challenging [10,16,17,18].

Core loss can be a non-negligible component of a PMSM’s total loss, thus making it critical for proper calculation when finding the machine’s efficiency. At high rotor speeds, core loss can be a dominant part of the machine’s total loss [6,19]. However, popular control strategies such as Maximum Torque per Ampere (MTPA) often neglect to account for this, seeking only to minimize the ohmic loss component of the copper losses [1,2,20]. It is impossible to confidently know that a machine is operating at its optimal efficiency if only one type of loss is minimized rather than all of them. The machine’s total loss must be known to properly calculate the efficiency of a given operating condition.

Several methods of approximating core loss exist, including the Steinmetz equation [21,22], the Bertotti equation [23,24,25,26], equivalent circuit modeling [9,27], finite element analysis (FEA) modeling [6,10,13,22,27], and machine learning [28,29,30,31,32]. The Steinmetz equation can be used to approximate core loss as a function of the electrical frequency and peak magnetic flux of the machine [21,22]. The Steinmetz equation was originally tested and validated by Steinmetz, however it was only performed under sinusoidal excitation in a limited range of frequency and flux density [22]. Because of these original limitations, several extensions have been made to the Steinmetz equation, including the Bertotti equation [22,23,26]. The Bertotti equation expands on the Steinmetz equation by introducing separate terms for the hysteresis loss, eddy current loss, and excess loss [22,23,26]. There exists a collection of literature describing the different validation methods of empirical core loss models such as the Steinmetz and Bertotti equations [33]. However, both equations still have shortcomings due to their dependence on the peak magnetic flux density. The peak magnetic flux is temporally and spatially dependent throughout the machine [34], so meshed models are preferred over lumped ones [27]. From this, finite element analysis (FEA) modeling can further improved on both the Steinmetz and Bertotti equations through meshing of the elements in the model, but has a trade off of being computationally intensive [6,10,13,22,27]. Because of this computational burden, FEA is time consuming and cannot be implemented in real-time controllers [13]. In recent years, machine learning approaches have been proposed in various literature for the estimation of PMSM losses [28,29,30,31,32]. The literature is very active in this field; however, a large amount of data is still needed for the calibration of these models.

There exists an extensive amount of other literature regarding the efficiency and efficiency improvement of PMSMs through various methods. Many different pieces of literature focus on better understanding the sources of loss [10], more accurate parameter measurement and estimation [8], different loss modeling techniques [9], control strategy comparisons [2], and other trends relating to machine performance [4]. While all of these efficiency improvement methods are beneficial, this work focuses only on comparing two different methods of core loss calculation for a modeled PMSM.

This work presents a core loss approximation framework for a PMSM that has been adapted from [24,25] based on the Bertotti equation [23,26]. The approximation is used to estimate the PMSM core loss of a simulated machine across its entire healthy operating domain within a direct-quadrature ($d-q$) reference frame. The estimated core loss is then compared to core loss calculated by FEA for the simulated PMSM. This work seeks to validate the goodness of the framework presented in [24,25] by generalizing and comparing it to the core loss calculations obtained by FEA, which is a popular but computationally involved method [6,10,13,22,27]. The core loss calculations of the simulated machine are obtained using an extensive number of FEA simulations over many possible operating conditions of a machine.

While there are studies in the literature that estimate PMSM core loss according to the method presented [24,25,33], there does not appear to be a significant number of sources that generalize the approach across the entire healthy operating region of an electric machine. It is important to have a generalized framework in order for the machine to operate at its most efficient state; to do this, the efficiency of all operating points must be known. Additionally, [24] contains experimental analysis within their work to analyze the accuracy of the core loss approximation to the measured core loss of a 160 kW traction motor. In their work, they find that the accuracy of the proposed model is in general agreement with the experimental results. However, the tests in [24] are limited to no-load and field-weakening conditions and are not generalized to all healthy operating conditions of the machine.

In this paper, Section 2 provides the background and justification of the proposed core loss approximation based on the empirical Bertotti equation. Section 3 describes the creation of an FEA model of an interior PMSM (IPMSM) based on the motor of a Nissan Leaf. This section also provides the fitting of the Bertotti equation of the created FEA model. Section 4 compares the demonstrated core loss approximation to the core loss prediction of FEA modeling. This core loss comparison is generalized over a wide range of possible operating conditions of the IPMSM. Finally, conclusions are provided in Section 5.

2. Empirical Models for Core Losses of PMSM

Using the methods proposed in [24,25], the magnetic flux path through the stator core of a PMSM for any arbitrary condition can be viewed as a superposition of two different flux paths. The two unique flux paths are generated when the PMSM is operating in either the open or short circuit condition. The flux paths of the open and short circuit conditions are then related to the voltage induced in the stator windings when the machine is operating in either of these conditions.

In the open circuit condition, the rotor permanent magnets induce a voltage in the stator windings. In this operating condition, the magnetic flux paths from the north pole to south pole rotor magnets travel along some teeth of the machine and through the back iron, traveling around the stator windings. This magnetic flux will then be related to the induced stator magnetizing voltage.

When the machine is operating in the short circuit condition, there is ideally no voltage drop across the stator windings. In this operation condition, the magnetic flux paths ideally do not travel through the back iron but instead cross through the air gap between stator teeth. The induced stator demagnetizing voltage is then related to the flux path of the short circuit case.

Using the Bertotti equation [23,26], the core loss of the PMSM, $P_{tot}(f,B)$, can be approximated for any arbitrary operating condition, where f is the electrical frequency and B is the peak magnetic flux density. The Bertotti equation [23,26] is shown in (1), where $k_h,k_{ed},k_{ex}$ are fit coefficients of hysteresis, eddy current, and excess loss terms, respectively, within the total core loss.

$$P_{tot}(f,B)=k_{h}fB+k_{ed}{f}^2{B}^2+k_{ex}{f}^{1.5}{B}^{1.5}$$

(1)

In both the open and short circuit cases, the peak magnetic flux density in the $d-q$ reference frame is assumed to be constant and thus can be combined with the loss coefficients found in the Bertotti equation [24]. From this assumption, the Bertotti Equation (1) is reduced to a function of only the electrical frequency, not the peak magnetic flux density. According to [24,25], the core loss in the open circuit ($P_{OC}$) and short circuit ($P_{SC}$) conditions can now be written as

$$\begin{array}{c}{P}_{OC}\left(f\right)=a_{h}f+a_{ed}{f}^2+a_{ex}{f}^{1.5}\end{array}$$

(2)

$$\begin{array}{c}{P}_{SC}\left(f\right)=b_{h}f+b_{ed}{f}^2+b_{ex}{f}^{1.5}\end{array}$$

(3)

where $a_h,a_{ed},a_{ex},b_h,b_{ed},b_{ex}$ are fit coefficients of the loss terms for the two operating conditions. Considering that the stator voltage can be related to the electrical frequency by the flux linkage of the machine, the following equations describe $P_{OC}$ and $P_{SC}$ in terms of the magnetizing and demagnetizing voltages [24,25]:

$$\begin{array}{c}{P}_{OC}\left(V_m\right)={\displaystyle \frac{a_h}{{\lambda }_{pm}}}{V}_m+{\displaystyle \frac{a_{ed}}{{\lambda }_{pm}^2}}{V}_m^2+{\displaystyle \frac{a_{ex}}{{\lambda }_{pm}^{1.5}}}{V}_m^{1.5}\end{array}$$

(4)

$$\begin{array}{c}{P}_{SC}\left(V_{dm}\right)={\displaystyle \frac{b_h}{{\lambda }_{pm}}}{V}_{dm}+{\displaystyle \frac{b_{ed}}{{\lambda }_{pm}^2}}{V}_{dm}^2+{\displaystyle \frac{b_{ex}}{{\lambda }_{pm}^{1.5}}}{V}_{dm}^{1.5}\end{array}$$

(5)

where ${\lambda }_{pm}$ is the permanent magnet flux linkage, $V_m$ is the magnetizing voltage in the open circuit condition, and $V_{dm}$ is the demagnetizing voltage of the short circuit condition.

In the open circuit condition, it is assumed that all of the magnetic flux travels through the back iron and around the stator windings. In the short circuit condition, it is assumed that none of the magnetic flux travels through the back iron, and it instead travels through the air gaps between stator teeth. Based on these assumptions, according to [24,25], the magnetic flux path of any arbitrary operating condition can then be viewed as a superposition of the flux paths generated in the open and short circuit conditions. Furthermore, refs. [24,25] claim that this then allows for the total core loss to be viewed as a superposition of (4) and (5), as shown in (6).

$$\begin{array}{c}{P}_{tot}(V_m,V_{dm})=P_{OC}\left(V_m\right)+P_{SC}\left(V_{dm}\right)\end{array}$$

(6)

According to [25], the magnetizing and demagnetizing voltages are solved as follows:

$$\begin{array}{c}{V}_m=f\sqrt{{\lambda }_d{\left(i_d,i_q\right)}^2+{\lambda }_q{\left(i_d,i_q\right)}^2}\end{array}$$

(7)

$$\begin{array}{c}{V}_{dm}=-f\left({\lambda }_d\left(i_d,i_q\right)-{\lambda }_{pm}\right)\end{array}$$

(8)

where ${\lambda }_d,{\lambda }_q$ are the magnetic flux linkages on the $d-q$ axes, and $i_d,i_q$ are the currents on the $d-q$ axes. The definitions of the magnetizing and demagnetizing voltages are consistent with the vector diagram of a PMSM in the $d-q$ reference frame, as shown in Figure 1. In the $d-q$ reference frame, the magnetizing voltage is proportional to the magnitude of the sum of the $d-q$ flux vectors. Meanwhile, the demagnetizing voltage is only related to the q-axis voltage that opposes the permanent magnet flux.

It is worth noting that the assumption of constant peak magnetic flux density for the open and short circuit conditions may cause errors for machine operating conditions that experience high magnetic saturation. In general, magnetic saturation effects are nonlinear with respect to the $d-q$ currents; thus, these saturation effects may not be fully captured by the assumptions made in this analysis.

3. Model Implementation

Using an FEA model, a PMSM is simulated to collect core loss data of the machine and compare it with the core loss prediction based on (6). This comparison is extended to the entire healthy operating region of the $d-q$ plane to investigate the goodness of the empirical model approximation for all operating conditions.

For this work, the IPMSM of the Nissan Leaf is selected for FEA modeling, representing the state-of-the-art PMSM for automotive applications. Starting from the parameters found in [35], an FEA model is created using ANSYS Maxwell 2D, part of ANSYS Electronics Desktop 2021 R1. A cross-section of a segment of the designed machine is shown in Figure 2. Not all of the physical dimensions necessary to recreate the Nissan Leaf motor are found in [35], so some were estimated in accordance with the best knowledge of the authors. The testing parameters of the designed machine are listed in

Table 1. Test parameters used for the designed FEA model.

Test Parameter	Parameter Value
DC bus voltage, $V_{DC}$ [37]	360 V
Peak phase current 1, $I_{max}$	300 Apeak
Max speed, ${\omega }_{max}$ [37,38]	10 kRPM
Max torque, $T_{max}$ [38]	280 Nm
Max Power, $P_{max}$	92 kW
Pole pairs, p [36]	4
Phase resistance 2, R	13 mΩ

¹ The current limit was reduced in simulation to match the torque–speed curve of the FEA model to that of the Nissan Leaf [35,38]. ² Estimated based on winding dimensions.

. The parameters of the designed machine are in line with the ones reported for the Nissan Leaf motor in [35,36,37,38].

With this model, the performance of the machine is simulated in both the open and short circuit conditions at different rotor speeds from 0 to 10 kRPM with a step of 1 kRPM. The torque–speed curve of the FEA model is shown is Figure 3.

Figure 4 shows the magnetic flux lines through the IPMSM in the open and short circuit conditions. As predicted, the flux lines travel through the back iron of the stator in the open circuit case, and they do not travel through the stator slots. On the other hand, there are very few flux lines in the stator back iron in the short circuit condition. Many of the short circuit flux lines in the stator do travel through the stator slots, and only a few travel through the back iron in limited areas.

In order to calculate the core loss data of the simulated machine, transient simulations are performed. For the open circuit condition, the transient simulation runs for 30 ms, which is equal to two electrical periods for the slowest (non-zero) simulated rotor speed. For the short circuit condition, the simulations run for 90 ms, which is equal to six electrical periods for the slowest (non-zero) simulated rotor speed. The short circuit simulations run longer due to having a larger transient time before approaching a steady state. For both the open and short circuit conditions, the first electrical period of the simulated core losses solution data is removed, because hysteresis loss requires a full electrical cycle in order to reach its steady-state value. Once the steady-state solution for core losses is obtained and post-processed for each test speed, the mean value of each transient solution is used to create fits for (4) and (5).

Different fits have been created considering the impact of the excess loss coefficients on the accuracy of the models for both the open and short circuit conditions [24,25]. A two-term fit is made that excludes an excess loss component in (2) and (3), while a three-term fit is made to include them.

Table 2. Fitted coefficients of (4) and (5) accounting for both the inclusion and exclusion of excess loss terms.

	${\mathit{a}}_{\mathit{h}}$	${\mathit{a}}_{\mathit{ed}}$	${\mathit{a}}_{\mathit{ex}}$	${\mathit{b}}_{\mathit{h}}$	${\mathit{b}}_{\mathit{ed}}$	${\mathit{b}}_{\mathit{ex}}$
Open circuit, excess loss	0.516	0.00129	0.00706	-	-	-
Open circuit, no excess loss	0.587	0.00146	-	-	-	-
Short circuit, excess loss	-	-	-	0.124	0.00094	0.02571
Short circuit, no excess loss	-	-	-	0.383	0.00156	-

lists the fit coefficients, $a_h,a_{ed},a_{ex},b_h,b_{ed},b_{ex}$, for both the open and short circuit conditions, both including and excluding a third term for excess losses.

Figure 5 shows the different fits of the Bertotti equation in comparison with the core loss data from ANSYS for both the open and short circuit conditions. Figure 5a shows the Bertotti equation fit to the open circuit core loss data, and Figure 5b demonstrates the fit with respect to the short circuit core loss data. It is observed that the fits of (2) and (3) closely match the simulation data from ANSYS regardless of the inclusion or exclusion of an excess loss term in the Bertotti equation.

After the core loss data are obtained for the open and short circuit conditions, polynomial fits are made for ${\lambda }_d\left(i_d,i_q\right)$ and ${\lambda }_q\left(i_d,i_q\right)$. These polynomial fits are used to calculate the magnetizing and demagnetizing voltages according to (7) and (8). The machine performance is simulated at various $(i_d,i_q)$ operating conditions using FEA to obtain the $d-q$ axes flux linkages. These operating conditions are provided in

Table 3. ANSYS simulation operating conditions shown in terms of the bounds of the simulation sweeps.

	Lower Bound	Step Size	Upper Bound
Peak phase magnitude, ${\mathit{I}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}$	0 Apeak	20 Apeak	300 Apeak
Current angle, $\mathit{\gamma }$	90°	10°	270°
Rotor speed, ${\mathit{w}}_{\mathit{m}}$	1000 RPM	1000 RPM	9000 RPM

. For purposes of this work, only healthy operating conditions of the PMSM were considered. The operating conditions listed in Table 3 go up to and include the boundary conditions of the machine, which are constrained by both the current and voltage limits of the simulated PMSM. In this work, temperature effects are neglected, and only sinusoidal current inputs are considered during simulation. It is important to point out that an assumption of sinusoidal current inputs may cause an underestimation of the calculated core loss compared to experimental or industrial results. It is common practice to feed PMSMs with power converters which utilize pulse width modulation (PWM) switching. This PWM switching can cause higher harmonics in the phase currents and magnetic fields, which can cause minor hysteresis loops and increases the complexity of core loss calculations [39,40,41].

It is stated here that instead of polynomial fits for the $d-q$ flux linkages, interpolated fits or lookup tables could likely suffice for flux linkage calculation. Polynomial fits are considered for this work in order to have an algebraic expression of the flux linkages rather than the use of tables. Additionally, polynomial fits are chosen to help account for saturation effects of the machine [25] rather than assume a lumped model of the machine parameters [24].

From the data obtained with FEA, polynomial fits of both ${\lambda }_d\left(i_d,i_q\right)$ and ${\lambda }_q\left(i_d,i_q\right)$ are created. Different orders of polynomial fits of the flux linkages with respect to both $i_d$ and $i_q$ were evaluated to investigate the impact of model complexity on model accuracy. The polynomial approximation order considered for this work ranges from 1 to 5 for both $i_d$ and $i_q$, constituting 25 polynomial approximations for both ${\lambda }_d\left(i_d,i_q\right)$ and ${\lambda }_q\left(i_d,i_q\right)$.

Table 4. $R^2$ scores of ${\lambda }_d\left(i_d,i_q\right)$ polynomials.

${\mathit{\lambda }}_{\mathit{d}}\left({\mathit{i}}_{\mathit{d}},{\mathit{i}}_{\mathit{q}}\right)$		Order ${\mathit{i}}_{\mathit{d}}$
		1	2	3	4	5
	1	0.9536	0.9736	0.9739	0.9745	0.9748
	2	0.9664	0.9891	0.9981	0.9983	0.9984
Order $i_q$	3	0.9735	0.9962	0.9981	0.9983	0.9984
	4	0.9749	0.9968	0.9987	0.9988	0.9998
	5	0.9755	0.9974	0.9995	0.9997	0.9998

and

Table 5. $R^2$ scores of ${\lambda }_q\left(i_d,i_q\right)$ polynomials.

${\mathit{\lambda }}_{\mathit{q}}\left({\mathit{i}}_{\mathit{d}},{\mathit{i}}_{\mathit{q}}\right)$		Order ${\mathit{i}}_{\mathit{d}}$
		1	2	3	4	5
	1	0.9466	0.9538	0.9538	0.9539	0.9541
	2	0.9538	0.9538	0.9538	0.9539	0.9542
Order $i_q$	3	0.9905	0.9935	0.9935	0.9942	0.9942
	4	0.9912	0.9942	0.9942	0.9942	0.9942
	5	0.9959	0.9990	0.9990	0.9991	0.9991

list the $R^2$ scores of the polynomial fits as a function of the order of $i_d,i_q$.

When fitting polynomials of ${\lambda }_d\left(i_d,i_q\right)$, ${\lambda }_q\left(i_d,i_q\right)$, lower-order polynomials are preferred over higher orders. The utilization of lower orders helps avoid any overfitting of data, and it minimizes the computational complexity of the model. Furthermore, lower orders more closely resemble the commonly used linear equations of ${\lambda }_d\left(i_d\right)={\lambda }_{pm}+L_d{i}_d$, ${\lambda }_q\left(i_q\right)=L_q{i}_q$, where $L_d,L_q$ are the inductances of the $d-q$ axes, respectively [1,25,32]. The $d-q$ axes inductances are often assumed to be lumped values, or they are lookup table functions of $i_d,i_q$. Based on the data shown in Table 4 and Table 5, it is found that a polynomial approximation of order 2 with respect to both $i_d$ and $i_q$ provides a good approximation for ${\lambda }_d\left(i_d,i_q\right)$. Additionally, a polynomial of order 1 with respect to $i_d$ and order 3 with respect to $i_q$ works well for ${\lambda }_q\left(i_d,i_q\right)$. Based on Table 4 and Table 5, the $R^2$ scores of ${\lambda }_d\left(i_d,i_q\right)$, ${\lambda }_q\left(i_d,i_q\right)$ do not increase significantly enough to justify an increase model complexity beyond the chosen polynomial orders.

With the calculation of the fit coefficients used in (4) and (5) and the polynomial fits of the flux linkages, the core losses of the machine can be approximated as a function of $i_d,i_q$. These approximations are compared with the FEA simulated data in Section 4.

4. Data Analsyis

Using the FEA model in ANSYS, the IPMSM is simulated to run at various operating conditions. The simulation sweeps through the peak phase current, the current angle, and the rotor speed for the operating conditions shown in Table 3. Without repeated data points for $I_{peak}=0$, this gives 2574 simulated operating conditions.

The core loss data are recorded as described in Section 3 for each simulation. Figure 6 plots the simulated core loss in the left-hand side of the $d-q$ plane for all simulated speeds. The simulated core loss is then compared to the core loss that is predicted using the method described in Section 2. Both the two-term and three-term approximations are compared to the ANSYS data in Figure 6. For all plots in Figure 6, the plotted regions are bound by the current and voltage limits of the machine. We can see that surfaces created by both approximations produce similar predictions of machine core losses. The surfaces of the approximations are generally similar in shape to that of the ANSYS data with the largest differences near the boundaries of the operating region. It can be said that the three-term approximation appears to be marginally better than that of the two-term approximation. This may be explained by the fact that the three-term approximation will be a better fit to the open and short circuit FEA data given that it has an extra polynomial term.

Figure 7, Figure 8, Figure 9 and Figure 10 show contour plots of the absolute and percent errors between the two-term and three-term approximations. These figures also show the trajectory of the $d-q$ command currents of the machine if an MTPA control strategy is used. The trajectory lines help to give a perspective of some of the coordinates that are likely of the most interest for machine operation assuming that core losses are neglected when calculating command values for the $d-q$ currents, as in popular MTPA control. At high speeds, these curves simply follow along the voltage ellipse when the MTPA curve is not available. If core losses are not neglected and a new maximum torque per loss (MTPL) control is implemented instead of MTPA, a new current trajectory would have to be introduced [42,43].

Figure 7 and Figure 8 show the absolute error of the approximations compared to the FEA data in units of Watts for both the two- and three-term approximations, respectively. The absolute error of the core loss values is calculated as

$$P_{a.err}=P_{ANSYS}-P_{approx}$$

(9)

where $P_{a.err}$ is the absolute error of core loss, $P_{ANSYS}$ is the core loss predicted by FEA, and $P_{approx}$ is the core loss predicted by the approximation methods. Once again, the contour plots yield similar results regardless of the inclusion or exclusion of an excess loss term. From Figure 7 and Figure 8, we can see that the error between the approximations and the FEA calculation tends to become more extreme as the rotor speed increases.

Because the absolute error has a wide range of values at high rotor speeds, it is important to view the difference between the approximation and FEA data as a percent error. Figure 9 and Figure 10 show the difference of the approximations compared to the FEA data as a percent error for both the two- and three-term approximations, respectively. The difference of the core loss values is calculated as

$$P_{p.err}={\displaystyle \frac{P_{ANSYS}-P_{approx}}{P_{ANSYS}}}·100\%$$

(10)

where $P_{p.err}$ is the percent error of core loss, $P_{ANSYS}$ is the core loss predicted by FEA, and $P_{approx}$ is the core loss predicted by the approximation methods. It is easy to see that the percent error at low rotor speeds has a much wider range than at higher speeds. It is expected that there is only a small amount of core loss at low rotor speed conditions. As a result, any small deviation in the absolute error of the core loss estimation can lead to a wide range of percent error in the calculation. Inversely, we expect large amounts of core loss at high rotor speeds. Because of this, even if the absolute error of the core loss estimation grows, the absolute error may still only be a small percentage of the core loss predicted by FEA. As a result, it is important to jointly use the absolute and percent error plots provided in this work to assess the impact of the core loss estimation.

From Figure 7, Figure 8, Figure 9 and Figure 10, we can see that the points of the smallest error are located around $(i_d,i_q)=\{(0,0),(-300,0)\}$. This makes sense given that the coordinates for the open and short circuit conditions are located at $(i_d,i_q)=\{(0,0),(-299.14,-9.99)\}$, respectively. We expect the core loss approximation to be more accurate near the open and short circuit conditions given that the approximation is based on a superposition of these two conditions. A general trend can be seen such that the core loss values that fall closer on the vertical line of $i_d=0$ tend to underestimate the core loss. Meanwhile, the values located near the vertical line of $i_d=300$ tend to overestimate the core loss. This trend of worse performance along these lines can be explained by the distance of the points from the open and short circuit conditions. It is expected that points that are further away from the open and short circuit operation conditions will likely have worse performance since it is harder to approximate these as a superposition of the two states. This is further explained given that the machine experiences magnetic saturation effects nonlinearly, which is difficult to capture from just the open and short circuit conditions.

Table 6. A summary table of the RMSE, maximum error, and mean percent error along the entire MTPA current trajectory for each speed for the 2-term model without an excess loss term.

No Excess Loss Model	RMSE (W)	Maximum Error (W)	Mean Percent Error (%)
1000 RPM	8.49	13.0	8.84
2000 RPM	17.6	36.8	3.34
3000 RPM	32.1	74.5	8.17
4000 RPM	40.4	85.7	−8.21
5000 RPM	51.0	91.0	−14.2
6000 RPM	56.2	89.6	−13.7
7000 RPM	56.4	90.7	−10.7
8000 RPM	52.6	92.6	−7.87
9000 RPM	36.1	67.4	−4.25

and

Table 7. A summary table of the RMSE, maximum error, and mean percent error along the entire MTPA current trajectory for each speed for the 3-term model with an excess loss term.

Excess Loss Model	RMSE (W)	Maximum Error (W)	Mean Percent Error (%)
1000 RPM	11.5	16.4	14.3
2000 RPM	18.4	29.6	6.24
3000 RPM	34.1	69.3	9.56
4000 RPM	38.5	83.4	−6.54
5000 RPM	48.3	91.5	−12.7
6000 RPM	55.2	92.1	−13.0
7000 RPM	57.5	94.0	−10.7
8000 RPM	54.6	95.3	−8.14
9000 RPM	39.9	67.7	−4.57

list the root mean squared error (RMSE), maximum error, and mean percent error along the shown current trajectories for both the two-term and three-term approximations, respectively, for all tested speeds. The values shown in the tables are calculated along the current trajectories shown in Figure 7, Figure 8, Figure 9 and Figure 10, which consist of the MTPA curve at low speeds, and these follow the voltage ellipse at higher rotor speeds due to field weakening. The statistics shown in the tables once again confirm that both approximation models have similar performance. On average, both models are within 15% error and have maximum errors of less than 100 W along the current trajectories for all speeds.

5. Conclusions

Based on the results, it is easy to see the clear tradeoffs when implementing the proposed core loss approximation in place of FEA modeling. The proposed methodology has several advantages such as a drastically reduced amount of FEA simulations and a simple polynomial approximation that can easily estimate the core loss of a PMSM. However, under the assumption that the FEA core loss results are a more accurate modeling method, the approximation model leads to a large error variability over the entire healthy operating region of the machine. These errors vary nonlinearly with rotor speed and $d-q$ current magnitudes, making it difficult to know the severity of the error.

Quantifying error statistics along the current trajectories used in MTPA control, the mean percent error of the model is within 15%. Additionally, the maximum error of the model along the MTPA is always less than 100 W with the RMSE always less than 60 W for all simulated speeds.

The scope of this work was limited to approximating core loss to the healthy operating domain of the simulated machine. Operating conditions beyond the maximum current or voltage constraints were not considered nor were the fault states of the machine. Considerations of thermal effects were neglected when modeling FEA, and only sinusoidal current inputs were utilized in the simulated model. Thermal, harmonic, and fault effects can be investigated in future works.