2.1. Analysis of Design Parameters Using Output Equation
Figure 1a,b shows the torque characteristics resulting from the operation of MTPA (maximum torque per ampere) at the rated speed. The black or gray dotted line shows the parameter combination that satisfies 420 Nm at the rated speed. If the saliency ratio
is large, the inductance that satisfies a torque of 420 Nm in the same no-load flux linkage decreases.
Figure 2a,b shows the voltage characteristics at the rated speed. If the salience ratio is 1.4, the operation is impossible at a rated speed exceeding the voltage limit of 346 Vpeak when the no-load flux linkage of the parameter combination is approximately 0.8 wb or smaller. On the other hand, if the saliency ratio is 2, the operation is possible with all parameter combinations.
Figure 3a,b shows the power-factor characteristics at the rated speed. The power-factor map does not show any significant differences depending on the saliency ratio. However, with the parameter combination that satisfies the rated torque, as the flux linkage decreases, the power factor decreases more rapidly when the saliency ratio is 2 than when it is 1.4.
Analysis of the parameters at the rated speed revealed that a larger no-load flux linkage and lower induction yield a greater increase in the power factor and a greater reduction in the induced voltage. Additionally, the changes in the power factor and the induced voltage resulting from the change in the flux linkage are greater when the salience ratio is smaller. Thus, at the rated speed, a large no-load flux linkage and saliency ratio and low inductance yield a suitable motor parameter for a high-power factor.
At the maximum speed of 10,000 rpm, MTPA operation, flux-weakening operation, and MTPV operation can be performed, depending on the induced voltage, the M point, and the load condition. In this study, maximum torque was generated while satisfying the voltage and current limits by conducting a suitable operation according to the parameter combinations.
Figure 4a,b shows the torque characteristics at the maximum speed. The area surrounded by the closed curve of black and gray dotted lines represents the parameter combination generating the target torque or greater at the maximum speed, and the other dotted line represents the parameter combination that satisfies the rated torque at the rated speed. To generate the maximum torque at the maximum speed while satisfying the output characteristics of the rated speed, a motor should be designed with parameters corresponding to the intersection of the two areas.
Figure 5a,b shows the voltage characteristics at the maximum speed. As a result of conducting a suitable operation in accordance with the parameter combinations, it can be seen that the induced voltage is within the voltage limit for most of the parameter combinations. However, the operation is impossible at the maximum speed in the top-left part, where
, is too small compared with
, as the phase-induced voltage exceeds the voltage limit of 346 Vpeak because flux-weakening control is impossible.
Figure 6a,b show the power-factor characteristics at the maximum speed. With the parameter combinations that satisfy the rated torque at the rated speed, the power factor increases up to a certain
, as in the case of the power-factor characteristics of the rated speed, and rapidly decreases above that, creating an area where operation is impossible.
Similarly, in the case of a traction motor for HEVs/EVs, a parameter combination exists for which operation is impossible at a high speed, as the induced voltage exceeds the voltage limit because of the wide operating range. Whether the operation is possible under a light load depending on the parameter combination at the maximum speed must be checked, and the output characteristics at not only the maximum torque but also the rated torque of 120 Nm at the maximum speed must be analyzed. Thus, in the case of a traction motor for HEVs/EVs, an analysis at the maximum speed must be performed differently from that at the rated speed.
2.2. Analysis and Determination of Design Parameters Using Voltage-Parameter Map (VP-Map)
As previously shown, numerous parameter combinations generate the target torque of 120 Nm or greater at the maximum speed while satisfying the rated-speed target torque of 420 Nm. It was previously explained that owing to the characteristics of the traction motor for HEVs/EVs, an analysis at the maximum speed must be performed differently from that at the rated speed.
To analyze the output characteristics at the maximum speed, the d–q voltage equation explained in the
Section 2 can be expressed in the form of a vector, as follows [
22,
23]:
Here, is the induced voltage vector, excluding the resistance drop, and and are the d–q-axis unit vector.
The right side of Equation (4) can be divided into three sub-terms, as follows:
Here, is the voltage vector induced by d-axis inductance, is the voltage vector induced by q-axis inductance, and is the voltage vector induced by the no-load magnetic flux interlinkage.
Two vector diagrams with different parameter combinations are shown in
Figure 7a,b. The vector diagram in
Figure 7a is for a parameter combination with a small λ
pm and a large L
d,q, and that in
Figure 7b is for a parameter combination with a large λ
pm and a small L
d,q. These vector diagrams comprise a d-axis in place of the x-axis, a q-axis in place of the y-axis, and the three voltage vector components of (2). A limiting circle that indicates the voltage limit and a current-vector combination that satisfies the target torque are shown in
Figure 7a,b, respectively.
Figure 7a shows the vector diagram for a case where the induced voltage is not satisfied, and
Figure 7b shows the vector diagram for a case where the induced voltage is satisfied. Among the many current vectors shown in
Figure 7a, I
2 has the lowest induced voltage. The voltage induced by I
2 is V
2, and the angle between the two vectors can be described as the power-factor angle if the resistance drop is ignored. The angle β between the q-axis and the current vector is the current phase angle. A current vector that satisfies the voltage limit exists in
Figure 7b differently from
Figure 7a. Among the induced voltages, the voltage induced by the current vector I
2 almost equals the voltage limit. As the current is the lowest at this time, this current vector is suitable for maximum-speed operation because its efficiency characteristics are most advantageous.
The vector diagram of
Figure 7 indicates that although a high voltage of
is induced by the PM, the total voltage of the q-axis is reduced by the
resulting from the d-axis inductance, which is a flux-weakening control characteristic.
is the voltage vector related to the flux-weakening control that lowers the voltage induced by the PM, and that in
Figure 7a is larger than that in
Figure 7b. As
in (a) is smaller than that in (b), the total voltage of the q-axis in Vector Diagram (a) shows a negative value. Although such a result means that the flux-weakening control is better performed in (a) than in (b), while (a) fails to satisfy the voltage limit, (b) satisfies the voltage limit. This is because the
of
Figure 7a has exceeded the voltage limit regardless of the flux-weakening control performance. Whether operation is possible at the maximum speed is determined by the
in
Figure 7 rather than by the flux-weakening control performance.
Figure 8 shows the results of analyzing parameter combinations 1–3 by using the previously explained vector diagram to analyze the output characteristics at the maximum speed as the high-speed torque characteristics when the saliency ratio is 2 in the previous paragraph.
Each parameter and the output characteristics at the maximum speed are shown in
Table 2. At the rated speed, the power-factor characteristics of Point 1 are the best. However,
Table 2 shows that at the maximum speed, the power-factor characteristics of Points 2 and 3 are the best at the maximum torque and the rated torque, respectively. Although the power factor and torque characteristics of Points 1 and 3 at the maximum torque are almost similar, there is a large difference in the input current and power-factor characteristics at the rated torque. To analyze this result, the d–q vector diagrams at the maximum torque and at the rated torque are examined, as shown in
Figure 9 and
Figure 10, respectively. In the case of
Figure 9a, the voltage vector is closer to the q-axis than to the d-axis, as the flux linkage is large and the inductance is small. For Point 2,
Figure 9b and
Table 3 show that the power factor is close to 1 because the current and the voltage vectors are close to each other.
Figure 9c shows a decreased flux linkage and an increased inductance compared with
Figure 9a, making the voltage vector almost parallel with the d-axis. As the d-axis currents of
Figure 9a–c are almost equal, the flux-weakening control performance of (c), which has the largest
, can be said to be the best.
The voltage and current vectors at the time when the rated torque is generated at the maximum speed are shown in
Figure 10. The rated torque decreases by
as the q-axis current decreases from the maximum torque, and the voltage vector approaches the q-axis. The greatest difference between
Figure 10a–c is that the size of the current vector decreases as the point moves from (a) to (c). The efficiency characteristics of (c), for which the current is the smallest when the rated torque is generated, is most advantageous for generating the same rated torque. The phase difference in the voltage and current of the vector diagram indicates that the power factor of (c) is higher than those of (a) and (b).
Although the q-axis currents, which are the torque currents of
Figure 10a–c, do not exhibit a large difference, the d-axis currents for flux weakening differ considerably. The levels of
, i.e., which represents the flux weakening at the rated torque, do not exhibit a large difference; however, the case of (c), for which the inductance is the highest, exhibits the smallest d-axis current, which is the flux-weakening current. As the
of
Figure 10c is smaller than those of (a) and (b), the required for flux weakening becomes smaller than those for (a) and (b) as the voltage vector approaches the q-axis. Thus, the input current of (c) becomes smaller than those of (a) and (b) at torques smaller than the rated torque.
The power factor and the maximum torque characteristics improve up to a certain flux linkage when the maximum torque is generated at the maximum speed, and above this value, the characteristics deteriorate, making operation impossible. However, when the rated torque is generated at the maximum speed, the more the flux linkage grows and the inductance decreases, the more disadvantageous the flux-weakening characteristics become. Thus, at the maximum speed, the parameters should be determined by considering not only the characteristics of the maximum input current but also the characteristics under the rated load or lower. To analyze the characteristics under all loads at the maximum speed, the current and the current phase angle that satisfy the voltage limit under each load should be known.
In this study, to analyze the characteristics under each load at the maximum speed by determining these two values, the VP-Map, which applies the d–q vector diagram previously explained, is employed.
Figure 11 shows the d–q vector diagram with parameters of λ
pm, L
d, and L
q for currents of
and
.
and
are the current vectors that satisfy the voltage limit at the maximum speed, and the levels of their torques differ. To determine
and
, the voltage vector V can be expressed in three voltage components as follows, by putting it in order as the d–q-axis components:
If
,
, and
at the maximum speed and the phase angle δ of the voltage are known,
and
can be determined using the relationship in (3).
and
at this time are the values for which the induced voltage is equal to the voltage limit, and all the current combinations that satisfy the voltage limit at a high speed can be obtained by determining
and
while increasing δ from 0 degE to 180 degE. The x-axis in
Figure 12 indicates the angle of the voltage vector, and the y-axis indicates the flux linkage.
Figure 12a,b show the d–q-axis inductance that satisfies the rated torque at the rated speed. The values are the same depending on the angle of the voltage vector. These are the values of
,
, and
for analysis of the maximum-speed characteristics while satisfying the output characteristics at the rated speed and, as the values remain the same regardless of the voltage phase angle, which is indicated by the x-axis, it is expressed as flux linkage only in the later maps. That is, although only the flux linkage is indicated on the y-axis in the later maps, it represents the parameter combination of
,
, and
that satisfies the rated torque at the rated speed. Such a map, wherein the x-axis represents the angle of the voltage vector, and the y-axis indicates the parameter combination that satisfies the rated torque at the rated speed, is called a VP-Map. Using a VP-Map, the characteristics under all loads that satisfy the voltage limit at a high speed can be determined, making it easy to determine the high-speed characteristics of a motor.
Figure 13 shows the current and the current phase angles under all loads that satisfy the voltage limit, and
Figure 14 shows the torque and power-factor characteristics of the motor at this time.
The dotted line in
Figure 15a represents the maximum input current of 550 Apeak, and each black dot represents the maximum torque of each y-axis parameter. Although the maximum torque is generated at the maximum input current when the flux linkage is 0.095 wb or larger, the maximum torque is generated at an input current smaller than the maximum input current when the flux linkage is 0.09 wb or smaller. Thus, the maximum torque that can be generated at the maximum speed can be determined via parameter combination. The area marked with diagonal lines in
Figure 15a represents the section in which the current phase angle is larger than 80 degE; in this area, the characteristics in the case where the current phase angle is controlled below 80 degE can be determined. With the parameter combination of 0.1 wb, a torque not greater than 60 Nm cannot be generated in such a condition.
The black dotted line in
Figure 15b represents the rated torque of 120 Nm at the maximum speed, and each black dot represents the intersection point between each y-axis flux linkage and the rated torque. In the case where the flux linkage is between 0.085 (wb) and 0.09 (wb), a current of torque, and the more the flux linkage grows from 0.095 (wb), the more the current required for the generation of the rated torque rapidly increases. The current rapidly increases because as the flux linkage increases, the flux-weakening current increases to satisfy the voltage limit.
The black dots in
Figure 15c represent the maximum torque at the maximum speed, and the gray dots represent the rated torque at the maximum speed. The power-factor characteristic is the highest for the rated torque, exhibiting a value of 0.97 or higher when the flux linkage is between 0.085 (wb) and 0.09 (wb). In the case of the maximum torque, the power factor is the highest when the flux linkage is 0.095 (wb).
The design-parameter combination finally obtained is shown by the gray dotted line in
Figure 15, for which the values of λ
pm, L
d, and L
q are 0.095 wb, 0.1766 mH, and 0.353 (0.2472) mH, respectively. The value inside the parentheses is the q-axis inductance at the rated speed, and the value outside the parentheses is the q-axis inductance at the maximum speed. The design parameters are determined in this manner not only because the power factor is high with these parameters at the rated speed but also because the greatest torque of approximately 260 Nm is generated at the maximum speed, and a torque of 30 Nm or higher, which is a light load, can be generated at the maximum speed. Moreover, the current required for generating the rated torque is not significantly different from 245 Apeak, the lowest case, exhibiting a value of 275 Apeak. Thus, the design parameters suitable for the target HEV are determined by analyzing the torque and the power-factor characteristics at the rated speed, the maximum torque that can be generated using the VP-Map at the maximum speed, the current required for generating the rated torque, and whether the operation is possible under a light load.