Modeling of the Fourth-Generation Toyota Prius Traction Machine as the Reference for Future Designs

Abstract

The automotive market is very competitive and demands consistently improving the technologies used and reducing the product cost and dimensions with each product model iteration. Hence, it is important to have access to well-defined reference designs of high-quality products to evaluate new ideas and technologies. This paper provides readers with a numerical model of such a high-quality product, i.e., the IPMSM-type traction motor from the fourth generation of the Toyota Prius hybrid transaxle. The presented results also serve for a discussion regarding the design decisions of the Toyota engineers and the applicability of the linearized machine model for the approximated torque calculations. In the introductory section, a brief history of the Prius model and references to the reverse engineering reports are given. Afterward, the machine dimensions, material properties, and winding configuration are described. Then, the model is validated with the torque measurements at constant speed. The simulation results are presented in the next chapters, and the numerical source data are supplied to the reader. Finally, the design philosophy of the Toyota drive is briefly discussed in comparison with the BMWi3 drive and the results are concluded in the last section.

1. Introduction

In the last two decades, a clear trend for developing an emissions-free and sustainable society has been observed, and transportation electrification plays an essential role in this process [1,2,3,4]. Along the way of pursuing constantly growing goals in emissions reduction, it is also important to develop energy-efficient and economically justified technologies. It is crucial in the automotive industry, as it is a highly competitive market, and vehicle manufacturers are constantly forced to push the limits of their technology. It presents a significant challenge to achieve this and maintain the product’s price so that it is affordable for customers.

It causes a demand for the everlasting optimization of the electrical drives in pursuit of cost and dimension reduction. In this process, both academia and industry should have access to highly optimized reference designs. One of the good examples of why it is important is the need to compare newly developed technologies with the current state-of-the-art solutions [5]. In the course of these actions, the latest solutions and improvements must be compared with the possibly competitive reference design, which also needs to be well defined, i.e., the information about the product parameters and its performance should be reliable and validated.

This paper aims to provide the readers with a reference design model for the traction machine of the hybrid road vehicle, i.e., the traction motor from the fourth generation of the Toyota Prius. The Toyota Prius model series is one of the most impressive success stories in the electrified vehicles market. It is also very interesting to observe the consistent evolution and optimization of the consecutive versions of the hybrid transaxle from Toyota [6,7].

Since the first-generation Prius was produced in 1997–2003, the Interior Permanent Magnet Motor (IPMSM) has been used as the traction machine. Unfortunately, there is not much public information regarding the first model. A significant milestone for the researchers was the second Prius generation produced in 2003–2009, as the detailed finite element method (FEM)-based electromagnetic model of its traction machine was made public thanks to the report [8]. It aided many researchers worldwide, helping them to evaluate their designs [5] or to re-shape this construction to meet different system requirements [9].

Other examples of the trend for supplying the academic community with the complete numerical model of the successful industrial product are the induction machine from the Tesla Model S60 [10] and IPMSM from the BMWi3 [11].

The third-generation Prius produced in 2009–2015 is the especially well-known design, thanks to the comprehensive reverse engineering report from the Oak Ridge National Laboratory [6]. This report contains detailed information about the power electronics and machine and the efficiency maps measured under different voltage levels. Nevertheless, the numerical data and the detailed machine geometry were not made public, so it is hard to use this machine as the reference design.

The fourth-generation Prius was produced in 2015–2023, and since 2023, the newest fifth-generation model has been on the market (according to the authors’ best knowledge, there are currently no reverse engineering reports available for the fifth-generation model).

The Oak Ridge National Laboratory reverse-engineered the drivetrain from the fourth-generation model, and the results were published in [7]. Unfortunately, the level of published details was nowhere near as deep as for the previous model [6].

Since the authors used the traction machine from the fourth-generation of the Toyota Prius during their research activities, they gained detailed knowledge about its parameters, and they decided to share this knowledge with the community providing the well-defined and validated machine model and its parameters in various forms. Besides providing the reference design, the authors have also used it as the starting point for some additional in-depth analyses and discussions.

The presented model was successfully validated using the induced voltage and torque measurements (see Section 2 and Section 3.1). Moreover, the influence of the temperature and machine losses on the measurement results was discussed. The measurement procedure was designed to consider the influence of the machine’s thermal behavior on the results.

The modeled flux linkage surfaces served for the calculation of the optimal working points of the drive (see Section 3.2 and Section 3.3). The results were used to calculate the maximal torque vs. speed curve of the drive and, as a result, to validate the model results further. Using this opportunity, an excessive discussion regarding the design decisions made by the drive constructors was conducted, and the obtained non-linear machine model was compared with the linearized one in Section 3.4.

Finally, the efficiency and loss maps of the machine were presented in Section 3.5, followed by a brief analysis of the results.

The consequences of the obtained results were additionally discussed in detail and presented in Section 4, followed by the brief conclusions in Section 5.

2. Materials and Methods

2.1. Basic System and Machine Information

The machine under consideration is the traction machine from the forth generation of the Toyota Prius Hybrid Transaxle called P610 according to the terminology of Toyota Motor Corporation (Aichi, Japan). The whole transaxle’s construction is described in Toyota’s documents [12,13,14]. The video [15] showcases the inner construction of the transaxle and explains its manner of operation.

The hybrid drive of Toyota consists of the internal combustion engine, electrical generator, and traction motor, all connected via a sophisticated power split device. The traction machine is connected to the planetary gear’s ring gear (output) and drives the vehicle wheels through the differential gear.

The maximal values of traction motor parameters are given in [12,13,14]. They include maximal power, torque, and speed values. The machine is supplied by an IGBT-based, two-level voltage inverter. The DC-link of the inverter is supplied from the battery through the boost DC/DC converter, which boosts the battery voltage from 200 V to 600 V [7]. The set of the most important drive parameters is summarized in

Table 1. Basic traction drive parameters (according to the literature).

Parameter Name	Symbol	Value
Maximal Power	$P_{max}$	53 kW
Maximal Torque	$T_{max}$	163 Nm
Maximal Speed	$n_{max}$	17,000 rpm
DC-link voltage	$U_{DC}$	600 V
Peak Phase Current	$i_{max}$	250 A

.

The traction machine is of the Interior Permanent Magnet Motor (IPMSM) type. The pictures of the machine are depicted in Figure 1. In contradiction to the previous transaxle versions, the hairpin winding was utilized for the first time in this model (see [6] for a comparison of different motor constructions from Toyota).

2.2. Machine Geometry and Materials

The machine geometry and the materials used are depicted in Figure 2. The main dimensions are summarized in

Table 2. Main machine dimensions.

Parameter Name	Symbol	Value
Stator Outer Diameter	$D_{so}$	215.0 mm
Stator Inner Diameter	$D_{si}$	141.7 mm
Air Gap Length	$l_{ag}$	0.65 mm
Rotor Outer Diameter	$D_{ro}$	140.4 mm
Rotor Inner Diameter	$D_{ri}$	47.0 mm
Stator Lamination Length	$L_s$	59.7 mm
Rotor Lamination Length	$L_r$	60.0 mm

. The machine has $p=4$ pole pairs. The stator has 48 rectangular slots, and the winding is a distributed winding type made of segmented rectangular rods of the hairpin type. The current direction in each coil side is depicted in Figure 2. Letters A, B, and C are phase names. ‘+’ means current flowing into the drawing plane. ‘−’ means current flowing out of the drawing plane.

Both stator and rotor cores are made of steel sheets of 0.27 mm thickness. It was found out that the steel MOTOR-MAX 27HF1500 from Cleveland-Cliffs (Cleveland, OH, USA) was utilized in both stator and rotor [16]. The $BH$-curve of this steel is depicted in Figure 3. The detailed steel parameters in numerical form are supplied to the reader as the Supplementary Materials attached to this paper (File S1).

The more detailed geometry zoomed at one pole is depicted in Figure 4. The detailed geometry in vector format is supplied to the reader in the Supplementary Materials to this paper (Files S2 and S3).

The characteristics of the magnet were found based on the induced voltage test. The motor was run via an external drive with a constant speed of $n=477$ rpm (corresponding to the 2862 rpm of the load drive), and the open-winding line voltages were measured. The test results are depicted in Figure 5 (dashed lines).

Afterward, the established simulation model (Motor-CAD [17]) was run with many different values of remanent flux density $B_r$. Via trial and error technique, it was found that the value of the $B_r=1.2843$ T provides the best matching between the induced voltage values obtained in simulation and the experimental results. Since the experiment was run with motor windings disconnected from the inverter and at no current state, the motor was not heating up during that test, and the temperature of the magnet was equal to the temperature of the environment, i.e., 20 °C. Hence, the authors searched for the magnet, which has the obtained value of the remanent flux density $B_r$ for that temperature. Finally, the magnet N40 was chosen as the model candidate, as the remanent flux density’s searched value lies within its tolerance band [18], which spreads between the minimal value of the $B_r=1.25$ T and the maximal value of the $B_r=1.29$ T.

2.3. Winding Configuration

The winding for each phase consists of 8 groups (8 turns each), and all groups within one phase are connected in series. The coil span equals $y=6$ slots (1$\to$7). The number of slots per phase and pole equals $q_s=2$. The winding diagram is depicted in Figure 6. The three phases are connected in a star configuration with isolated neutral point. One phase’s resistance was found to be equal $R_{ph}=43.6$ m$\Omega$.

2.4. Finite Element Method Model of the Machine

Based on the information described above, the finite element method (FEM)-based simulation model of the machine was built in the Ansys Motor-CAD software (version 2023.2.2) [17]. The model is supplied to the reader in the Supplementary Materials attached to this paper (File S4).

The exemplary simulation results obtained with the model are depicted in Figure 7. In this example, the flux density distribution is shown for two cases. The first case (Figure 7a) is the no-load operation of the drive; i.e., both the d-axis current $i_d$ and q-axis current $i_q$ are equal to zero. The second case (Figure 7b) is the motor operation with the maximal torque, i.e., $i_d=-202.7$ A and $i_q=146.3$ A.

The flux density distribution should be analyzed together with the $BH$-curve of the steel (see Figure 3). The ’knee’ of the steel characteristic lies about 1.3 T. Hence, the yellow areas at Figure 7 indicate highly saturated regions (see the legend, yellow color starts above 1.75 T). Only the ribs around the flux barriers are highly saturated during no-load operation (see Figure 7a). The saturation of this region is a necessary feature of each IPMSM as it is able to push the magnetic field outside of the rotor through the air gap. At maximal torque (see Figure 7b), the bulk saturation can be observed in numerous machine regions, i.e., in the rotor core, stator teeth, and stator yoke. It indicates an extreme utilization of the steel material, and the consequences of that are discussed in the following sections of this paper.

2.5. Test Setup

To perform the experimental investigation of the machine, some manipulations had to be made in its mechanical construction. Normally, the machine is an internal part of the transaxle, with no direct access to its shaft from the outside. Additionally, the machine bearings are part of the transaxle enclosure, so it is impossible to easily take the machine out and run it at the test bench. Moreover, the machine was originally cooled using oil. The stator is cooled with the oil mist sprayed over it from the outside. The rotor is cooled from the inside using the oil pumped through the channel inside the rotor and its shaft.

For the above reasons, the machine was dismounted from the transaxle, and some additional mechanical parts needed to be designed. First of all, a new shaft and bearing shields were manufactured. Additionally, a cooling water jacket with a spiral channel was designed and manufactured.

The picture of the motor closed in the homemade enclosure is depicted in Figure 8. The motor shaft is connected with the auxiliary load drive via the torque measurement sensor DATAFLEX 42/200 from KTR Systems GmbH (Rheine, Germany). The auxiliary drive controls the speed at a constant value during experiments. This drive is connected to the examined machine using the reduction gear of ratio 1:6 to raise its torque to the level required by the Toyota Prius machine. It results in the maximal speed limitation of the test bench at the level of $n=600$ rpm (corresponding to the 3600 rpm of the load drive). For this reason, the tests described in the next chapter are performed at a speed of $n=550$ rpm (corresponding to the 3300 rpm of the load drive), i.e., slightly under the maximal limit.

The motor is supplied from the two-level voltage inverter. The original power stage of the Toyota Prius was used, but it was complemented with the homemade control interface to control the drive freely. The control interface was built using the TMS320F28335 digital signal processor from Texas Instruments (Dallas, TX, USA). The authors have implemented their own Field-Oriented Control (FOC) algorithm to set the desired current values in d- and q-axes.

Additionally, the machine was equipped with temperature sensors inside. Thanks to that, it is possible to monitor the temperature of the cooling liquid and internal machine parts.

3. Results

3.1. Model Validation—Torque Measurements

The torque measurement test was conducted for the model validation/tuning purposes. The speed of the examined machine during the test was held constant at a value of $n=550$ rpm (corresponding to the 3300 rpm of the load drive). The current was varied so that its amplitude was constant, and the angle of the current space vector was varied between 0 and 180 degrees in 10° increments. Such a test was conducted for many different values of the current amplitude. For each measurement point the d- and q-axis current components were calculated as:

$$i_d=i_{mag}·cos\left(\theta \right),i_q=i_{mag}·sin\left(\theta \right),$$

(1)

where $i_d$ and $i_q$ are d- and q-axis current components, respectively [A], $i_{mag}$ is the current amplitude [A], and $\theta$ is the current space vector angle [°].

Contrary to the induced voltage test described in Section 2.2, the machine parts do heat up during the torque measurements due to the high current amplitude values. It is a well-known fact that automotive drives are designed for an excessive overload operation, i.e., continuous operation is provided to cover the working points of the most common road driving conditions, and the drive is working under the thermal overload conditions for short periods during the especially dynamic drive conditions (this topic is explained in detail in [9]). At this point, it should be emphasized that in the used test setup, the original cooling system of Toyota was replaced with a much simpler and less performant cooling jacket, which also does not directly cool the machine’s rotor. It allows the conducting of laboratory tests, but it should be expected that the thermal overload conditions at the test bench will be more severe than during the normal operation of the transaxle.

Hence, care needs to be taken during the torque measurements because the machine can heat up substantially during the test. This is one of the reasons why the authors have equipped the machine under tests with temperature sensors. Each measurement is conducted so that the current flow duration is possibly short. It was found that the duration of 3 s is sufficient for the torque sensor output and speed to settle (there is some transient due to the sensor’s elasticity and limited dynamics of the load drive). Afterward, the measured value is averaged for the last 1 s of the pulse, which results in the overall current pulse duration of 4 s. The temperature is monitored during the tests, and there is a pause after each current pulse to cool the machine down. The pause duration was chosen manually by the test engineer conducting the measurements in such a way that a temperature of the hottest winding’s part remains in the range of ca. 40–47 °C.

In the beginning, it was found that the simulated torque values were much higher than the measured ones. The reason for that is the fact that the remanent flux density of the magnet decreases with rising temperature. Unfortunately, the temperature of the magnets cannot be directly measured as they are attached to the rotating part of the machine. Still, it is safe to assume that it reached temperatures higher than the 20 °C. It was found that the best matching between the experimental and simulation results is achieved for the remanent flux density value of $B_r=1.2534$ T. As the N40 magnet has the reversible temperature coefficient of $\alpha \left(B_r\right)=-0.12$%/°C, it corresponds to the magnet temperature of the 40 °C [18], which seems to be a reasonable value. Hence, all the following analyses assume the machine temperature is equal to 40 °C.

The comparison of the experimental results and the model output is depicted in Figure 9. The left column (Figure 9a,c) depicts the simulation results when machine losses are neglected. In that case, the model inaccuracy reaches 2.7% of the official maximal torque value of 163 Nm. The right column (Figure 9b,d) depicts the results after including the losses in the simulation model. In that case, the matching is better since the remanent flux density value of the magnet was tuned based on these results. In that case, the model accuracy is always better than 2%. Based on these results, an excellent match of the model can be stated.

It is worth mentioning that the highest torque value obtained during the measurements equals 169.7 Nm, which is 4% higher than the value published by Toyota [14]. It should be emphasized that the torque sensor was sent to the calibration right before the measurements, and the presented test results were obtained directly after the calibration. Hence, the measured value can be considered reliable. Toyota probably considered additional mechanical losses inside the transaxle or very high magnet temperature.

3.2. Flux-Linkage Maps

As the simulation model is established and validated, it can now be utilized to obtain data needed by drive control engineers. It is a common practice to describe the machine dynamics using the space-vector-based equations in the $dq$ rotor reference frame. The flux-linkage-based model can be described as [1,2]:

$$\begin{array}{cc}\hfill u_d& =R_{ph}·i_d+{\displaystyle \frac{d}{dt}}{\psi }_d-{\omega }_{el}·{\psi }_q,\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill u_q& =R_{ph}·i_q+{\displaystyle \frac{d}{dt}}{\psi }_q+{\omega }_{el}·{\psi }_d,\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill T_{el}& ={\displaystyle \frac{3}{2}}·p·\left(i_q·{\psi }_d-i_d·{\psi }_q\right),\hfill \end{array}$$

(2c)

where $u_d$ and $u_q$ are d- and q-axis voltages, respectively [V], ${\psi }_d$ and ${\psi }_q$ are d- and q-axis flux linkages, respectively [Wb], $R_{ph}$ is the phase resistance [$\Omega$], ${\omega }_{el}$ is electrical angular speed [rad/s], p is the number of pole pairs [-], and $T_{el}$ is the electromagnetic torque [Nm].

The state-of-the-art control algorithm for IPMSMs is the FOC, which facilitates current controllers [19]. Hence, the model expressed in (2) needs to be converted into the current one. For the highly saturated machine, the flux linkages should be considered as functions of current:

$${\psi }_d=f(i_d,i_q),{\psi }_q=f(i_d,i_q).$$

(3)

Considering the exact integral of (3) in (2) leads to the following current-based machine dynamics model:

$$\begin{array}{cc}\hfill u_d& =R·i_d+L_{dd}·{\displaystyle \frac{d}{dt}}{i}_d+L_{dq}·{\displaystyle \frac{d}{dt}}{i}_q-{\omega }_{el}·{\psi }_q,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill u_q& =R·i_q+L_{qq}·{\displaystyle \frac{d}{dt}}{i}_q+L_{qd}·{\displaystyle \frac{d}{dt}}{i}_d+{\omega }_{el}·{\psi }_d,\hfill \end{array}$$

(4b)

with differential inductances defined as:

$$\begin{array}{c}\hfill L_{dd}={\displaystyle \frac{\partial {\psi }_d}{\partial i_d}},L_{qq}={\displaystyle \frac{\partial {\psi }_q}{\partial i_q}},L_{dq}={\displaystyle \frac{\partial {\psi }_d}{\partial i_q}},L_{qd}={\displaystyle \frac{\partial {\psi }_q}{\partial i_d}},\end{array}$$

(4c)

where $L_{dd}$ is a d-axis differential self-inductance [H], $L_{qq}$ is a q-axis differential self-inductance [H], and $L_{dq}$ and $L_{qd}$ are cross-saturation inductances [H].

It should be emphasized that this model considers only the DC conduction losses, and all the other machine losses are neglected. Hence, the machine parameters needed for this model are generated using the simulation with losses neglected.

The simulation results are depicted in Figure 10. The numerical data in the spreadsheet form are available to the reader in the Supplementary Materials attached to this paper (File S5).

The extreme magnetic circuit saturation mentioned in the previous section is visible in the flux-linkage surfaces (see Figure 10a,b). It is evident looking at the differential self-inductances in Figure 10c,d as for the linear magnetic circuit, they would be constant in the whole depicted region. The cross-saturation is also visible, as the d-axis flux linkage depends on the q-axis current, and q-axis flux linkage depends on the d-axis current. This is also indicated by non-zero values of the cross-saturation inductances (see Figure 10e). It should also be mentioned that the cross-saturation inductances $L_{dq}$ and $L_{qd}$ are equal. This results from the reciprocity condition [20,21,22]. This condition is derived from the energy conservation rule when the iron losses are neglected. Hence, the results obtained with the lossless FEM model should also fulfill the reciprocity condition, i.e., $L_{dq}=L_{qd}$. For a more detailed explanation of the physical meaning of the surfaces’ shapes depicted in Figure 10, please refer to [21,22].

3.3. Maximal Torque per Current and Field Weakening

The following important task for the drive control is calculating the optimal current reference points for the controllers. This calculation can be performed based on the model (2) and machine parameters obtained with the lossless model (see Figure 10).

The calculation was performed using the numerical search algorithm written by the authors using MATLAB software (version R2023b). The results of the calculation are depicted in Figure 11. The numerical data are available to the reader since they are included in the Supplementary Materials attached to this paper (File S6).

The operational points in the base speed region were chosen using the state-of-the-art solution, i.e., the Maximal Torque per Current (MTPC) strategy [1,2,19,23,24,25]. This strategy assumes the minimization of the current for the given reference torque, as discussed in [24,25]. The constant torque isolines are depicted in Figure 11a as orange lines. For each reference torque isoline, the point of the minimal current space vector’s length was chosen as the reference operational point. The resulting MTPC locus is depicted with green color in Figure 11a. The maximal torque obtained with the given parameters equals 173.4 Nm.

According to (2), the voltage needed to reach some current space vector rises with speed. Above some speed (different for each torque value), the voltage available for the current controllers is not enough for following the MTPC strategy any longer. Assuming Space Vector Modulation (SVM) is applied, the maximal phase voltage on the inverters’ output is limited to the value of:

$$u_{max}=U_{DC}/\sqrt{3},$$

(5)

where $u_{max}$ is the maximal phase voltage amplitude [V], $U_{DC}$ is the DC-link voltage of the inverter [V]. The presented calculation results were obtained with the maximal DC-link voltage of the Toyota Prius, i.e., $U_{DC}=600$ V. The operational points that can be reached using the MTPC strategy under the assumed conditions are marked with green color in Figure 11b.

The maximal voltage isolines drawn in the current $dq$-plane (see purple lines in Figure 11a) have ellipsoidal shapes and converge with the rising speed to a particular point, i.e., the motor characteristic current. This current is defined as the d-axis current amplitude sufficient to weaken the d-axis flux to zero, i.e.,

$${\psi }_d(i_d=-i_{ch},i_q=0)=0,$$

(6)

where $i_{ch}$ is the motor characteristic current [A]. This is a very important quantity, which defines the field weakening performance of the drive [24]. This value equals $i_{ch}=97.35$ A for the examined machine.

The characteristic current of the machine is an important factor impacting the field-weakening performance of the drive. According to the theory derived in [24], it can describe a level of the inverter’s volt–ampere rating utilization. For this purpose, a so-called drive characteristic factor should be calculated first as:

$$k_{ch}=i_{max}/i_{ch}=250.0\mathrm{A}/97.35\mathrm{A}=2.57,$$

(7)

where $k_{ch}$ is the drive characteristic factor [-]. It is well known that the value of $k_{ch}=1$ divides the PMSM drives into two sub-classes [24,25]. For $k_{ch}<1$, the drive has finite maximal speed and poor field-weakening performance. For $k_{ch}>1$, the drive has infinite maximal speed and good field-weakening performance, but it is achieved at the cost of a low power factor and poor inverter volt–ampere rating utilization [24].

It is a common practice to design the drives precisely on the boundary of these two sub-classes, i.e., with $k_{ch}=1$, as it provides a good compromise between the field-weakening performance and inverter utilization. An excellent example of such a practice is the traction machine from the BMWi3 described in [11].

The examined drive from the Toyota Prius is a counter-example of this practice. The drive characteristic factor of $k_{ch}=2.57$ is an extreme example of the infinite maximal speed drive class. Let us put it in numbers. For the Toyota Prius inverter, the volt–ampere rating of the inverter can be calculated as follows:

$$S={\displaystyle \frac{3}{2}}·i_{max}·u_{max}=129.9\mathrm{kVA},$$

(8)

where S is the apparent power of the inverter [VA]. According to the approximations derived in [24], the asymptotic power reached by the drive at high speed can be approximated by:

$$P_{FW}=S/k_{ch}·{\displaystyle \frac{1\mathrm{W}}{1\mathrm{VA}}},$$

(9)

where $P_{FW}$ is the approximated power reached at high speeds [W]. Substituting $k_{ch}=1$ yields that one could design the motor, which could reach approx. 129.9 kW power at the shaft when supplied from the Toyota Prius inverter. At the same time, substituting the value from the Toyota Prius motor, i.e., $k_{ch}=2.57$, results in the shaft power of $P_{FW}=50.55$ kW, which is very near to the value of 53 kW obtained later in this section (it should be remembered that Equation (9) is only an approximation). It means that only 40.8% of the potential power of the inverter is utilized.

Coming back to the working points calculation, the field-weakening strategy needs to be applied above the speeds where the MTPC points can be reached. The Maximal Torque per Voltage (MTPV) strategy limits the maximum torque for a given speed. This strategy’s current space vector locus lies at the points where the torque isolines are tangential to the voltage isolines at a given speed [24]. The maximal drive torque obtained with the MTPV strategy is depicted with black color in Figure 11b. The maximal power achieved equals 82.2 kW (see constant power hyperbole depicted in Figure 11b). This value is much higher than the official value of 53 kW published by Toyota. Hence, the constant power hyperbole for the value of 53 kW was drawn too. It reveals that the torque curve of the drive converges to this official value. This means that the official value published by Toyota does not refer to the peak power but to the maximal power available in the wide speed range. The fact that the calculated torque curve matches so closely with the official value of the maximal power acts as an additional validation of the presented model.

For the partial load operation, i.e., for torque values lower than the MTPV curve, another field-weakening strategy must be applied. In that case, the space vectors on the intersection of the torque isolines (for the given torque) and maximal voltage isolines (for the given speed) should be chosen. These points are depicted in different colors for each torque value in Figure 11. The colors for each torque value in Figure 11a and in Figure 11b match each other.

3.4. Comparison with the Linearized Model

As mentioned in Section 2.4, the examined motor is an extreme example of a highly saturated machine. It can be seen not only in the flux density plot depicted in Figure 7b, but the shape of the MTPC current locus also indicates it. For the linear motor model, the maximum reluctance torque lies at the angle 45° between the current vector and the q-axis. Hence, the angle of the current vector corresponding to the peak torque lies in the range of 0–45° and converges asymptotically to the value of 45° when the motor saliency goes to infinity [24]. The fact that this angle substantially exceeds the value of 45° for the examined motor indicates an extreme saturation of the machine. It can be directly compared with the value obtained for the BMWi3 motor [11], which is very near the value 45°. This indicates that the Toyota Prius machine has an even more non-linear magnetic circuit than the BMWi3 machine (already considered highly saturated).

This difference can be further investigated by comparing it with the linearized model. Additionally, despite being known that the non-linear models should be used for the highly-saturated machines, it is still interesting to quantify the calculation errors resulting from the model linearization for some real-world examples.

For this purpose, the flux linkages should be described using the linear functions as follows:

$${\psi }_{d,Lin}=L_d·i_d+{\psi }_{PM},{\psi }_{q,Lin}=L_q·i_q,$$

(10)

where ${\psi }_{d,Lin}$ and ${\psi }_{q,Lin}$ are the linearized d- and q-axis flux linkages [Wb], respectively, $L_d$ and $L_q$ are the constant d- and q-axis inductances [H], respectively, and ${\psi }_{PM}$ is the constant permanent magnet flux linkage [Wb].

The linearized model parameters $L_d$, $L_q$, and ${\psi }_{PM}$ can be obtained using the linearization of the non-linear flux linkage surfaces (see Figure 10) at some particular operational point. In this study, a point corresponding to the peak torque of the machine is utilized, i.e., $i_d=-202.7$ A and $i_q=146.3$ A. Hence, the linearized model’s parameters can be calculated as:

$$\begin{array}{}\mathrm{(11a)}& \hfill {\psi }_{PM}& ={\psi }_d(i_d=0,i_q=0)=0.0752\mathrm{Wb},\hfill \mathrm{(11b)}& \hfill L_d& ={\displaystyle \frac{{\psi }_d(i_d=-202.7\mathrm{A},i_q=146.3\mathrm{A})-{\psi }_{PM}}{i_d}}=0.617\mathrm{mH},\hfill \mathrm{(11c)}& \hfill L_q& ={\displaystyle \frac{{\psi }_d(i_d=-202.7\mathrm{A},i_q=146.3\mathrm{A})}{i_q}}=1.221\mathrm{mH}.\hfill \end{array}$$

The flux linkages calculated with the linear model (10) are depicted as black planes in Figure 12a,b. For the comparison, the flux linkages obtained with the non-linear FEM model are depicted as blue surfaces. The electromagnetic torque (2c) calculated using both models is shown in Figure 12c.

Since the operating points of the drive during the motor operation lie solely in the second quadrant of the current $dq$-plane, this quadrant was depicted as a separate plot in Figure 12d for improved readability. The points corresponding to the maximal current limitation are drawn with thick solid lines, and the MTPC loci calculated for both models are depicted as white filled dots. For the low d-axis current values, the reluctance torque is low, and the majority of the torque is the permanent magnet torque, i.e., torque depends mainly on the d-axis flux linkage. It can be seen in Figure 12a that this flux linkage component of the non-linear model substantially drops in that region due to the cross-saturation. Hence, a relatively big difference in calculated torques can be observed in the low d-axis current region. On the other hand, this region corresponds to the MTPC locus of the linear model (current vector angles to q-axis lower than 45°) and results in a high difference between the MTPC loci calculated with both models.

The MTPC loci are also depicted at the current $dq$-plane in Figure 13a for better readability, and the resulting torque vs. current magnitude plots are shown in Figure 13b. As the results obtained with the non-linear model can be considered as the reference for the comparison, the linearized model’s calculation error can be expressed as the difference between the values calculated with both models expressed in percents of the nominal torque 163 Nm (see gray curve in Figure 13b). The torque calculation error for the linearized model varies in the range of −12.1% to +15.5%, which is almost twice as big as the error obtained in the similar analysis for the BMWi3 [11]. This is another indicator that the saturation of the Toyota Prius machine’s magnetic circuit is higher.

To complement the analysis, the torque vs. speed curves for different current limitation values were calculated with both models, and the results are depicted in Figure 14.

It can be seen that the current vector loci at low speeds (see Figure 14a) are very different for both models as they result from the MTPC calculation presented before. As a result, the torque vs. speed curves calculated with both models differ substantially in the base speed operational region (see Figure 14b).

On the other hand, the torque vs. speed curves calculated at high speeds, i.e., during the field-weakening operation, are very similar for both models (see Figure 14b). This is caused by the fact that the MTPV loci of both models lie in the region where the difference between the torque surfaces calculated with different models do not differ much from each other (compare MTPV loci in Figure 14a with torque surfaces in Figure 12d). A similar phenomenon was observed when analyzing the results for the BMWi3 machine [11].

3.5. Efficiency and Losses

The previous analysis was carried out using the lossless model. Hence, it should be complemented by the results obtained with the model where losses are included. For this purpose, the dedicated sub-module of the Motor-CAD software (version 2023.2.2) was utilized [26].

The simulation results are depicted in Figure 15. The numerical source data are available in the Supplementary Materials attached to this paper (Files S7–S9).

After considering the losses, the maximal torque of the drive equals 171.8 Nm, and the peak power drops to 80 kW compared to the results obtained with the lossless model. The torque curve at high speed again converges to the officially published value of 53 kW.

The peak machine efficiency is very high and equals 98%, and the high-efficiency operational region is vast. Compared to the previous generation of Toyota Prius, the efficiency at high speeds rose by approximately 2% [6].

The total losses in the machine (see Figure 15b) reach a peak value of 5.5 kW. Most of the total machine losses consist of copper losses (see Figure 15c). In comparison, the iron losses (see Figure 15d,f) are much lower, and the magnet losses are almost negligible (see Figure 15e).

Since the primary source of the losses is the joule losses in the conductors, the machine’s total losses map is similar to the copper losses map. The losses in the base speed region are mostly torque-dependent (since higher torque means higher current). On the other hand, the iron losses are mostly speed-dependent. It can also be observed that there is a slight increase in the copper losses with rising speed. This is caused by the AC component of the copper losses, which is frequency-dependent.

In deep field-weakening regions, all the losses seem rather power-dependent.

4. Discussion

This paper aims to provide researchers from different fields with valuable data for their particular research fields.

The machine design engineers are provided with the full numerical FEM model of the machine developed in the Ansys Motor-CAD software [17]. The full data set required to build that model in different software applications is provided for those not using this particular tool. The data include detailed machine geometry in vector file format and material properties.

It is important to emphasize that the presented model was successfully validated with the measurement results, and the matching of the results was excellent. It is also important to point out that the experimental tests of the machines containing permanent magnets present a great challenge due to the susceptibility of the results to the magnets’ temperature. The authors have proposed a method to overcome this problem by monitoring the machine temperature and trying to keep the temperature of the parts, which are available by measurement, in a narrow range. It is also worth mentioning that the running rotor test was used during the torque measurements to obtain the average torque values, which helped to eliminate the errors caused by the spatial harmonics. This additionally improved the quality of the results compared to the state-of-the-art solution, which is the locked rotor test. As described in [11], the results of the locked rotor test are corrupted by the fact that they depend on the particular rotor position due to the existence of the aforementioned spatial harmonics of torque.

In the following sections, drive control engineers are provided with the machine’s flux linkage and inductance maps, which are necessary for the dynamic modeling of the machine and the control system design. Additionally, system engineers are provided with the numerical data for the optimal operational points map.

These results aided some additional analyses. First of all, it was observed that the drive characteristic factor of the examined Toyota Prius drive has a relatively high value, making it an extreme example of the so-called ’infinite maximal speed drive’. It seems extraordinary in comparison with other known automotive drives [11] and leads to a relatively low utilization of the inverter’s volt–ampere rating.

Hence, it is aimful to inquire the reason for this design decision. In the authors’ opinion, the answer lies in the space vector locus for field-weakening operation at partial loads (see Figure 11a). Since the MTPV locus converges to the ($i_d=-i_{ch},i_q=0$) point, the reduction in the characteristic current allows the reduction in the current amplitude during the field-weakening operation and, as a result, the reduction in the conduction losses (which, are the primary source of losses). This is probably the reason for such a good machine efficiency at torques below 50 Nm (see Figure 15).

It should be emphasized that the drive designed for the full inverter utilization, i.e., $k_{ch}=1$, has a different current locus, and current amplitudes rise drastically during the field-weakening operation (see results for BMWi3 published in [11]). Hence, the authors hypothesize that, in general, the rise in the drive characteristic factor $k_{ch}$, besides decreasing the inverter utilization, offers the limitation of losses instead. Nevertheless, it is only a hypothesis at this point, and more analysis is necessary to draw such generalized conclusions. This is another example of why providing researchers worldwide with many well-defined reference designs is so important, as this may be a great way to gain insight into optimal drive design strategies.

Even though it is a well-known fact that the highly saturated machines should be modeled using the non-linear flux linkage surfaces, it is still interesting to analyze the difference between the results obtained with the non-linear and linearized models. The reason is that it can sometimes still be desirable to use the linear approximation of the machine to speed up the higher-level analyses at the vehicle level [27]. The analysis presented in Section 3.4 allows the quantification of a calculation error, which should be expected when linear approximation is used instead of the non-linear model of the highly saturated machine. It is important to emphasize that a similar analysis of the highly saturated machine was already conducted for the BMWi3 and presented in [11]. The conclusions of this analysis said that the linear model could be successfully used for the drive torque characteristics approximation under some circumstances as the calculation error was below 8% of the peak torque. On the other hand, the results obtained for the Toyota Prius machine lead to different conclusions. The torque calculation error reaches over 15%, which, in the authors’ opinion, excludes the usage of the linearized model even for the approximated analyses. The reason for the different conclusions for the two drives is the fact that the Toyota Prius machine is even more saturated than the BMWi3 machine, which is evident from the MTPC current vector locus (as discussed in Section 3.4).

Finally, drivetrain, vehicle, and heat transfer engineers are provided with the machine’s efficiency and loss maps. The interested reader can refer to [28] for an excellent analysis and explanation of the expected shapes of the loss maps for the IPMSMs. Both the efficiency and loss maps obtained for the Toyota Prius traction machine stay in full compatibility with the results presented in [28], which can serve as additional indirect confirmation of the model’s validity.

5. Conclusions

The FEM model of the fourth-generation Toyota Prius traction machine was presented. This model series is an excellent example of the consistent improvement and optimization of the product design with each model iteration. As reported in [7], each next machine design had more compact dimensions and higher power density. The simulation results also showed an increase in efficiency compared to the previous model. The interested reader can refer to [6,7] for comprehensive analyses and comparisons of many different performance indices and design differences between the consecutive Toyota Prius drive models. Additionally, the Toyota materials [12,13,14] explain the improvements made between the third- and fourth-generation models and their purposes.

The simulation results have proven an extreme utilization of the magnetic circuit. First, it is visible in the flux density distribution at high currents. The bulk magnetic saturation of broad machine regions is observed in that state. The saturation is also clearly visible in the machine’s flux maps. Consequently, the MTPC current locus achieves relatively high load angles (i.e., angles between the current space vector and the q-axis) exceeding the 45°. It should be emphasized that for the linear magnetic model, the maximum of the reluctance torque is achieved at angle 45°. Hence, the MTPC current locus exceeding that angle can occur only for extremely saturated machines, which indicates a very high magnetic material utilization, leading to a compact design.

The characteristic current of the machine has a relatively low value, leading to good field-weakening performance, i.e., low current is sufficient for weakening the d-axis flux linkage to zero. It results in relatively low losses at high speeds and very high efficiency of over 97% in a wide speed range. On the other hand, it also leads to a relatively low inverter volt–ampere rating utilization.

The above facts prove the maturity of the Toyota Prius traction machine design and its usefulness as the reference design for new machine designs and improvements.