4.2. Steady-State Characteristics
Based on the parameters given in
Table 3, the base speed, the boundary speed, and the critical speed can be calculated as described in
Section 3. The calculated values for the nominal machine parameters are as follows:
[rad/s],
[rad/s], and
[rad/s]. To present the effect of
on these speed values, the calculated base, boundary, and critical speeds are given in
Table 4 for five different iron loss resistance values. It can be concluded that the base speed slightly increases while the boundary speed slightly decreases if the value of
starts to decrease, that is, if iron loss starts to become significant. The effect of
is more obvious on the value of critical speed: this speed value can increase by several percent if the resistance representing the iron loss is taken into account.
Figure 8a shows the numerically calculated trajectories of
reference current pairs by sweeping the
reference torque from zero up to its maximum value (see
in
Table 3) at different constant
mechanical speed values. The
values were varied between 0 and 1000 rad/s, in steps of 25 rad/s. For better visibility, the current limit circle and the MTPV curve belonging to the critical speed have been marked with a dashed line. The algorithm can be seen to provide the reference currents in the MTPC, MC, FW, and MTPV regions.
Figure 8b shows the achievable electric torque of the machine as a function of mechanical speed. The base speed, boundary speed, and critical speed are also marked in the figure. As can be seen in the figure, below the base speed, the machine can produce its maximum torque
. Above the base speed, the achievable electric torque decreases with increasing mechanical speed. The dashed line in
Figure 8b presents the allowable torque of the machine for continuous operation as a function of the mechanical speed. Below the base speed, the machine can generate nominal torque (constant torque region) while it provides nominal power above the base speed (constant power region) for continuous duty.
As previously mentioned, in the literature, the effect of iron loss resistance is typically neglected during reference current calculation. To demonstrate the effect of iron loss resistance and why it is important to consider in the calculations,
Table 5 shows how
and
current components as well as the electric torque differ if iron loss is not taken into account during the reference calculations, even though the machine has a certain amount of
resistance. The calculations were carried out for six different working points (denoted by A, B…F) and for five different
values. For better comparability, these working points are examined in subsequent studies as well. The second column of
Table 5 (titled
) shows the calculated
and
reference current components by neglecting the effect of iron loss. The achievable electric torque is also calculated from (
3) by assuming that the current controllers work properly, and in the steady state,
and
track their reference signals. If the machine has a non-negligible iron loss (expressed with
resistance as given in
Figure 1), then
and
. As can be seen in
Table 5, in the case of columns for different
values, as the effect of iron loss becomes more and more significant (
decreases), the difference between the controlled
and
stator current components and the actual
and
current components, which are important in terms of torque generation, becomes considerable. These differences mean that the value of the achievable electric torque is reduced and the current controller cannot provide the reference torque. From the table, it can be concluded that, in certain operation ranges (MTPC, MC, MTPV), these differences are in the range of a few percent, even in the case of a low
. However, in the FW operation region, a much more dominant difference can be seen. For example, the relative reduction in electric torque at working point A is 1% (
) and 2% (
), respectively. At the same time, at working points C and E, this value is 4% (
)/7.9% (
)/5% (
)/10% (
), respectively.
Based on
Table 5, it is clear that, if the machine has a non-negligible iron loss, its effect should be considered in the reference current calculation using the proposed method introduced in the current manuscript.
Table 6 shows the calculated
d and
q axis reference currents and the achievable electric torque for the same working points (denoted by A, B…F) and
values using the proposed reference current calculation algorithm. These working points are also denoted in
Figure 8a,b. It should be noted that the calculated values given in the second column titled
are the same as the second column given in
Table 5. Based on the values given in
Table 6, it can be concluded that, for working points found in the MC or MTPV regions (such as working points B, D, and F), taking the effect of
into account only slightly modifies the calculated
d and
q axis reference currents, compared to the case when the effect of iron loss is neglected. However, for working points located in either the MTPC or FW regions (such as working points A, C, and E), considering the effect of
modifies the reference currents much more significantly. As can be seen, in these points, the machine can provide the reference torque, even when the effect of iron loss is considerable. It is a clear advantage of the proposed reference current calculation method as, in these working points, the machine cannot guarantee the reference torque if
is not considered (compare values belonging to working points A, C, and E in
Table 5 and
Table 6).
4.3. Closed-Loop Operation
In order to verify the proposed reference current calculation strategy in different operation modes, a closed control loop was constructed in the MATLAB/Simulink R2023a environment similar to the block diagram in
Figure 3. During the simulation study, the nominal machine parameters given in
Table 3 were used. The current reference calculation algorithm was run at a sampling rate of 2 kHz, while the sampling time of the inner current loop was selected to be 20 kHz. The switching frequency of the VSI was 10 kHz. The sampling of the phase currents was synchronized to the positive and negative peaks of the carrier signal used for space vector modulation. The gains of the current controllers were calculated using the modulus optimum method, considering a one-step delay as well as
and
for the
d axis current controller and
and
for the
q axis current controller.
As a first step, the outer speed controller was not considered, and the actual value of the
reference torque was generated externally.
Figure 9 shows the simulated waveforms of the torque, mechanical speed,
d and
q axis current components, and one phase current. The
values were changed from 0 up to
with a ramp signal, and the loading torque was zero. The mechanical speed was increasing, and the machine worked along the MTPC curve. At
, the mechanical speed exceeded the base speed, and the machine entered the MC region, where both the current and the voltage reached their limit values. As can be seen, the IPMSM could not provide the reference torque anymore, so the machine accelerated further with a smaller rate. After that, the mechanical speed exceeded the
boundary speed, and then, at
s, it exceeded the
critical speed as well. From this point on, the machine operated along the MTPV trajectory. At
s, the reference torque was reduced to 4 Nm. The machine started to operate in the FW region. At
s, a loading torque of 4 Nm was applied, and the mechanical speed settled down.
Figure 10 shows the trajectory of
d and
q axis current in the
–
plane. The trajectories of the MTPC, MC, MTPV, and FW regions are clearly visible.
Based on the simulated results, it can be concluded that the control requirements are fulfilled, and the current signals follow the reference signals. The IPMSM can work in the MTPC, MC, MTPV, and FW regions, and the voltage and current limits were not violated.
As a second step, the external speed controller was also considered in the simulation. The PI-type speed controller forms the
reference torque based on the difference between the
reference and the actual mechanical speed
. The discrete PI speed controller ran at a sampling rate of 2 kHz, similarly to the current reference generation algorithm. The gains of the controller were calculated using the symmetrical optimum method as
and
.
Figure 11 shows the simulated waveforms of the torque, mechanical speed,
d and
q axis current components, and one phase current. The trends for the reference mechanical speed and the load torque were determined so that, during the simulation, the drive worked at the working points (see points A, B…F in
Figure 8) given in S
Section 4.2. As can be seen, the current reference generation algorithm works properly for sudden changes in both the reference speed and loading torque. The simulated values are consistent with the calculated quantities (see
Table 6,
).