FEA was used to analyze inductance and torque of IPMSM. Torque and inductance are analyzed on the dqaxis. It was shown that Ld and Lq have harmonic components, and magnitude as well as phase of the harmonics varies according to the current values. This paper shows the relationship between these inductance harmonics and the 6th harmonic component of torque.
1. Introduction
Interior permanent magnet (IPM) motors can use various control methods because reluctance torque is utilized by field oriented control (FOC), but accurate parameters, such as the flux linkage of the permanent magnets and the difference between the d and qaxis inductance, are required. Therefore, precise inductance measurements and calculations have been researched for many years
[1

2]
. However, salient motors, such as IPM motors, have nonlinear parameter characteristics generally, so there is also ongoing research which models theses nonlinearities and applies these models to calculate the torque
[3

4]
. Some papers show that a modified coordinate transformation is used to control the torque because of the nonlinear characteristics
[5]
. This method improves the torque calculation after modifying the transformation matrix, but it is difficult to consider the complicated inductance variation due to the saturation phenomena.
However, it is very rare to consider the parameter nonlinearities when utilizing FOC even though some research has been done. That is, the torque, T
_{e}
, uses equation 1. In this equation, the parameters, L
_{d}
and L
_{q}
, are assumed to be constants, and the PM flux linkage, ψ
_{f}
, is also assumed to be a constant generally. Thereby, the torque is calculated to be constant for constant i
_{d}
and i
_{q}
currents.
On the other hand, the inductances and flux linkage are not linear in real cases shown in
Fig. 1
, which shows that the torques have ripple.
Fig. 2
depicts the torque FFT analysis, which has high 6
^{th}
harmonic components in the torques due to the nonlinear material and motor structure.
Torque under the loads
Torque FFT Analysis
This paper presents the flux linkage and inductance variations though the FEA results of the IPM Motor in
Table 1
, and the effect in the dq axis and the relationship with the torque are studied.
Specification of IPM motor
Specification of IPM motor
2. Flux linkage Calculation
 2.1 Flux linkage calculation in stationary coordinate
As shown in equation 1, the torque is proportional to the amplitude of the magnet flux linkage, so the accurate calculation of flux linkage is very important. Conventionally, the flux linkage value used is the average value of the magnet flux linkage without applying the phase currents to the windings. However, the applied currents induce core saturation and the amplitude of the magnet flux linkage will be different for real IPM motors. IPM motors have more a complicated shape for the flux linkage because of the saliency. FEA can analyze the flux distribution more precisely.
There are two steps necessary to calculate the flux linkage value of only the permanent magnets when applying the phase currents.
1
^{st}
step: While applying the phase currents, the permeability of the each element from the calculated results needs to be stored after performing the nonlinear analysis.
2
^{nd}
step: After changing the current to 0[A] from the state of the first step and making the source of the flux only from the permanent magnets, the flux linkage is reanalyzed and calculated.
Fig. 3
shows the Aphase flux linkage caused by the rated current powered on the daxis, and the graphs show total flux linkage of the permanent magnets and current, flux linkage of only the current, excluding the impact of the permanent magnets, and flux linkage of only the permanent magnets. The daxis current implies the direction which reduces the flux linkage of the permanent magnets. The total flux linkage decrease, as shown in
Fig. 3
, and the current includes harmonic components causing partial iron saturation, but the flux linkage is symmetric.
Flux linkages of daxis current
Fig. 4
shows the Aphase flux linkage when applying the qaxis current. There is 90 degree phase difference between the flux linkages of the qaxis current and of the permanent magnets. Moreover, the armature reaction causes asymmetric saturation, which gives rise to the asymmetry of the flux linkage, which is not sinusoidal and contains harmonic components. The 5th harmonic component is relatively large as indicated in
Fig. 5
.
Flux linkages of qaxis current
qaxis flux linkage FFT analysis
Fig. 6
shows flux linkage caused by only the permanent magnets when the rated current flows through both daxis and qaxis. The graphs represent flux linkage of the permanent magnets at no current, flux linkage of the permanent magnets when the daxis current flows, and flux linkage of the permanent magnets when the qaxis current flows.
Flux linkages of the magnets
Compared to the flux linkage at no load current (that is, when the current is at 0A), the pattern of the flux linkage of the permanent magnets with the qaxis current has harmonics and is asymmetric because of the saturation. Moreover, the amplitude of the flux linkage with the daxis current is bigger than the 0A current case.
This implies that the daxis current is powered in the direction to demagnetize the permanent magnets, lowering the total level of saturation, as shown in
Fig. 7
, which leads to the increase in the flux linkage of the permanent magnets. Thus, it is shown that the backEMF due to the permanent magnets is increased when applying the daxis current.
Magnetic flux density distribution
From the above results, it is confirmed that the magnitude, harmonic component, and phase angle of the flux linkage can change based on the current amplitude. These points are factors when equivalent circuit simulation can consider.
Fig. 7
shows the magnetic flux density distribution when the current flows according to the previous three conditions. The magnetic flux density (
Fig. 7(a)
) caused only by the permanent magnets is greater than the flux density with the daxis current (
Fig. 7(b)
), and the magnetic flux density (
Fig. 7(c)
) with the qaxis current becomes asymmetric. In other words, since the saturation of the permanent magnets becomes asymmetric, the backEMF contains more harmonic components because of the combination of the permanent magnet daxis component and the qaxis component.
 2.2 Flux linkage on the rotating coordinate system by the daxis current
FOC is the control method transforming 3phase variables on the dqaxis. Thus, all the characteristics should be taken into consideration after transforming the flux linkages on threephase system to the dqaxis system.
When applying the daxis current, the flux linkages of only the permanent magnets, of only the current, and of both the permanent magnets and current are decomposed into daxis component and qaxis component in
Fig. 8
, and then the flux linkages are compared to the flux linkages of only the permanent magnets without the load current. The qaxis flux linkage component is, on average, close to zero, which implies it has only the daxis component. The flux linkage of the daxis current (Flux d only D) is negative and has the direction to demagnetize the permanent magnets. Thus, the magnitude of the flux linkage of the permanent magnets (flux d PM at D) gets to be greater than that of the flux linkage without the load current (flux d PM at 0). Therefore, it predicts that the torque calculation will have larger error under field weakening control conditions.
Flux linkages by daxis current
 2.3 Flux linkage on the rotating coordinate system for the qaxis current
When current flows on the qaxis, the flux linkages of only the permanent magnets (PM at Q), of only the qaxis current (only Q), and of both the permanent magnets and qaxis current (PM + Q) are shown in
Fig. 9
.
Flux linkages by qaxis current
The flux linkage of the permanent magnets and the qaxis current is decomposed into the daxis component (flux d) and qaxis component (flux q). These cases are compared to the flux linkage of only the permanent magnets at no current. This shows that a flux linkage component appears on the daxis and the qaxis due to the nonlinearity of saturation caused by the combination of the flux linkages of the permanent magnets on the daxis and of the current on the qaxis. That is, the flux linkage component of the qaxis current (flux d only Q) is detected with the negative value on the daxis partially, and the flux linkage component of the permanent magnets is partly observed with the negative value on the qaxis. Thus, these should be considered in calculating the torque.
3. Inductance Calculation
 3.1 Phase inductance calculation
As shown in the Eq. (1), the torque is directly affected by the difference between L
_{d}
and L
_{q}
. For FOC, this is also used as the average value along with the case of backEMF. However, it is widely known that the dqinductance reduces as the current increases in IPM motors, due to the iron saturation depicted in
Fig. 10
.
Ld and Lq waveforms according to current variation
Since the inductance changes more complexly because of the saliency in IPM motors, FEA is necessary to calculate inductance more accurately.
Fig. 11
illustrates the calculated Aphase inductance using FEA when the qaxis current is applied from 0 to 150%. As the current increases, the inductance decreases because of the iron saturation, and the inductance waveforms are asymmetric because the inductance values are not sinusoidal due to the partial saturation of the iron. FFT analysis of this result is shown in
Fig. 12
. It shows that 2nd harmonic and 6th harmonic are large. The 4th harmonic becomes the 6th harmonic component in dqreference frame.
Phase inductance waveforms according to the current variation
Phase inductance FFT analysis according to the current variation
Fig. 13
describes the asymmetry of the phase inductance.
Fig. 13(a)
shows the flux density distribution at 160 electrical degrees, and
Fig. 13(b)
indicates the flux density distribution at 180 electrical degrees. The saturation is more immense at 160 electrical degrees rather than at 180 electrical degrees, and the flux is asymmetric due to the qaxis current component in
Fig. 13(b)
.
Flux density distribution
Moreover, the inductance includes the harmonic components because the permanent magnet is located in the rotor of IPM motors, the distance from the magnet to the center of the motor is not constant, and the ribs on the edge of the permanent magnet are thin, which makes the saturation level uneven.
 3.2 dqaxis inductance calculation
As mentioned before, the asymmetric saturation of phase inductance is caused by the qaxis current, and this causes harmonic and asymmetric characteristics. In this case, it is expected that harmonic component and phase shift will be changed after dqtransformation is applied.
Fig. 14
shows the daxis inductance when the qaxis current varied from 0 to 150% of rated current and
Fig. 15
depicts the qaxis inductance. The angles of the qaxis inductance peak value vary, especially, when the current increases. This is expected to have an impact on the torque calculation. If the phase angle variation is also considered, a more precise interpretation could be possible.
L_{d} according to various currents
L_{q} according to various currents
Fig. 16
indicates the daxis and qaxis inductances when the qaxis current is varied from 50 to 150%. L
_{d}
 L
_{q}
at 150% current tends to be significantly lower than L
_{d}
 L
_{q}
at 50% current, since the iron saturation leads to overall decrease of inductances. Thus, it is expected that the torque magnitude caused by reluctance difference in the torque component will be changed severely. Furthermore, the inductance varies periodically according to the angle because of the saliency of IPM motors, and the reason for the slight deviation of the waveform from the symmetry is the saturation effect by the qaxis current. Such an asymmetrical aspect implies that the torque characteristics will be asymmetric too.
L_{d} and L_{q} waveforms with 50% and 150% currents
Fig. 17
shows the daxis inductance FFT analysis, which has a high 6th harmonic component, and there is no large variation with the changes in current.
L_{d} FFT Analysis
In
Fig. 18
, the qaxis inductance FFT analysis is presented, which reveals significant 6th harmonic components and a large variation with the current changes.
L_{q} FFT Analysis
When the phase of the 6th harmonic is considered, the phase changes as the current varies. These phase differences have an impact on the torque calculation, and considering these differences is expected to result in a more precise torque calculation.
4. Torque Results
The torques calculated in Eq. (1) for FOC and simulated in FEA are compared.
The torque only from applying the qaxis current is equal to Eq. (2) when substituting i
_{d}
=0 into Eq. (1) and assuming ψ
_{d}
= ψ
_{f}
.
Fig. 19
illustrates the calculated torque waveforms from Eq. (2).
Calculated torque waveforms
In
Fig. 19
, the graphs represent the calculated torques from Eq. (2) and the daxis flux linkage of the permanent magnets and the qaxis rated current (PM + Q), of only the permanent magnets without load current (PM at 0), of only the permanent magnets with the rated qaxis current (PM at Q), and the torque simulated by FEA (by FEA). As shown in
Fig. 9
, the daxis total flux linkage is the summation of the daxis flux linkages of the permanent magnets and the armature reaction of the qaxis current. There is a difference in the flux linkage with and without the load current. Therefore, there are corresponding results depending on which of ψ
_{d}
or ψ
_{f}
is applied to the Eq. (2) in
Fig. 19
.
Finally, the calculated torque deriving from the total flux linkage is closely similar to the result from FEA. However, there are differences between these torques in terms of the magnitude and phase because the reluctance torque component from i
_{q}
current is omitted from the torque equation. Accordingly, it is clear that the torque from only the combination of ψ
_{f }
and i
_{q}
can’t be derived due to the armature reaction and saturation effects when applying the qaxis current.
5. Conclusion
Conventionally, the equivalent circuit for FOC assumes that the motor circuit parameters are sinusoidally distributed and linear, but for IPM motors these assumptions must be reconsidered because of the saliency and high saturation effect. This paper presents the nonlinear characteristics of the inductance and flux linkage from permanent magnets. It was shown that the cause of 6th harmonic component was the inductance and backEMF harmonics, the inductance and flux linkage were not symmetric due to the daxis current, and daxis and qaxis values were reviewed after dqtransformation. When applying current, the flux linkage of the permanent magnets changes due to the armature reaction and saturation. Particularly, FEA shows that the qaxis current generates the daxis component. The flux linkage of only the permanent magnets was reviewed on daxis and qaxis, and it was shown that the qaxis current could generate the daxis flux linkage too. These could be considered when calculating the torque, and FEA results proved that the daxis flux linkage of only the permanent magnets when applying the daxis current to demagnetize is larger than the flux linkage of only the permanent magnets without the load current. This can be used effectively when calculating the torque for field weakening control. L
_{d}
and L
_{q}
have harmonic components, and the magnitude as well as phase of the harmonic varies according to the current values. Considering these phase differences is expected to result in a more precise torque calculation.
For future work, more accurate torque prediction will be performed through vector control simulations and confirmed though experiments. This approach is expected to obtain reasonable results in real IPM Motor drives.
Acknowledgements
This research is financially supported by Changwon National University in 2013
BIO
Kibong Jang He received Phd. degree in electrical engineering from Hanyang University. His research interests are analysis and drive of electric machines
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