# Computationally Efficient PM Power Loss Mapping for
## 4 Constant-Power Operation Region  
When a PMSM works under a constant-power condition, due to the direct axis current, compared with the constant-torque condition, the stator phase current and its higher harmonic amplitude change. These changes have a certain regularity, so there is a corresponding rule for the PM loss caused by the armature reaction of the harmonic current. It can be known from [18] that the harmonic amplitude of the stator phase current with the d-axis current is \(\frac{i_{q}^{2}}{i_{d}^{2}+i_{q}^{2}}\) times that of the stator phase current without the d-axis current. Therefore, the PM loss caused by the armature reaction of the harmonic current is \(\frac{i_{q}^{2}}{i_{d}^{2}+i_{q}^{2}}\) times of the original. The additional PM loss at the constant-power operation region can be expressed by the following formula:  
\[\begin{split} P_{PM-EL}(n,N,M)=f(N)\cdot g(M)\cdot\left(\frac{n}{ n_{R}}\right)^{2}\cdot\frac{i_{q}^{2}}{i_{d}^{2}+i_{d}^{2}}\\ =\left(a_{1}\frac{1}{N^{2}}+b_{1}\frac{1}{N}+c_{1}\right)\left(a _{2}(M-M_{0})^{2}+b_{2}(M-M_{0})+1\right)\left(\frac{n}{n_{R}}\right)^{2} \cdot\frac{i_{q}^{2}}{i_{d}^{2}+i_{q}^{2}}\end{split} \tag{8}\]  
On the basis of the fitting formula in Equation (2), the additional PM loss in the constant-power region is modified with the alternating and d-axis current as the variable. Therefore, the fitting formula in Equation (2) is a special form of the fitting formula in Equation (8), that is, the formula is the case where \(i_{d}=0\). This is because when the stator winding acts solely on the quadrature axis current, the value of the q-axis current increases and the amplitude of the harmonic current decreases. In general, the q-axis current has little effect on the PM loss, which can be ignored. When the stator winding generates the d-axis current, the change in the harmonic current amplitude is directly related to the q-axis and the d-axis current. The above correction coefficient can be used to obtain a more accurate calculation method of the additional PM loss. The constant coefficient of the fitting formula in Equation (2) is the same as that of the fitting formula in Equation (8). Therefore, the fitting formula in Equation (8) can be determined only by determining the d-axis current and the q-axis current, and there is no need to use the finite element to determine the additional PM loss under special conditions, which can further save the operation time. The fitting formula in Equation (8) takes into account the influence of time harmonics of the stator winding current, but just like the sinusoidal PM loss mentioned above, it ignores the interaction of the q-axis armature reaction, core saturation effect, and slot effect, as well as the influence of the eddy current reaction on the additional PM loss.  
Figure 4 shows the comparison of additional PM losses under different carrier ratios calculated by the finite element method and the fitting formula in Equation (8) at two speeds under constant-power conditions. Smaller errors appear in the calculation results, because the error of the additional PM loss under the constant-power condition is inherited by the fitting formula, that is, the error of the sinusoidal PM loss fitting formula under the constant-power condition and the error of the sinusoidal PM loss fitting formula under the constant-torque condition. However, the error is small. In general, the fitting formula can fit the additional PM loss under the constant-power condition well.