## 2 Fundamental Loss Model of PMSMs
### Fundamental Copper Loss  
The mathematics model of the PMSM decoupled into the \(d\)-axis and \(q\)-axis is shown in Figure 1[15; 16; 17].  
From Figure 1, the voltage equation of the PMSM can be described as  
\[\left\{\begin{array}{l}u_{d}=L_{d1}\frac{di_{d}}{dt}+R_{s}i_{d}+L_{dm}\frac{di_ {im}}{dt}-n_{p}\omega_{r}\vb_{q}\\ u_{q}=L_{q1}\frac{di_{q}}{dt}+R_{s}i_{q}+L_{qm}\frac{di_{im}}{dt}+n_{p}\omega_{ r}\vb_{d}\end{array}\right. \tag{1}\]  
where \(L_{d1}\) and \(L_{q1}\) are the stator's leakage inductance in the \(d\)-axis and \(q\)-axis, and \(L_{dm}\) and \(L_{qm}\) are the stator's self-inductance in the \(d\)-axis and \(q\)-axis, respectively; \(R_{s}\) is the stator's resistance, and \(n_{p}\) is the pole pairs of the PMSM; \(\omega_{r}\) is the motor rotation speed; \(u_{d,q}\) and \(i_{d,q}\) are the stator's voltage and the stator's current, respectively; \(i_{dm}\) and \(i_{qm}\) are the magnetizing currents of the PMSM in \(d\)-axis and \(q\)-axis; \(\Psi d\) and \(\Psi q\) are the flux linkage in \(d\)-axis and \(q\)-axis.  
The flux linkage in the \(d\)-axis and \(q\)-axis can be described as  
\[\left\{\begin{array}{c}\vb_{d}=\vb_{f}+L_{dm}i_{dm}\\ \vb_{q}=L_{qm}i_{qm}\end{array}\right. \tag{2}\]  
where \(\vb_{f}\) is the flux linkage of the permanent magnet.  
Additionally, the current equations of the PMSM in Figure 1 are expressed as  
\[\left\{\begin{array}{c}i_{di}=\frac{1}{R_{i}}\Big{(}L_{dm}\frac{di_{im}}{dt} -n_{p}\omega_{r}\vb_{q}\Big{)}\\ i_{d}=i_{dm}+i_{di}\end{array}\right. \tag{3}\]  
\[\left\{\begin{array}{c}i_{qi}=\frac{1}{R_{i}}\Big{(}L_{qm}\frac{di_{im}}{dt} +n_{p}\omega_{r}\vb_{d}\Big{)}\\ i_{q}=i_{qm}+i_{qi}\end{array}\right. \tag{4}\]  
Figure 1: Decoupled mathematics model of PMSM: (**a**) Dynamic mathematical model in \(d\)-axis; (**b**) Dynamic mathematical model in \(q\)-axis.  
where \(i_{di}\) and \(i_{qi}\) are the iron loss current in the \(d\)-axis and \(q\)-axis. Ri is the iron loss resistance of the PMSM.  
From Equation (2), the fundamental copper loss of the PMSM can be obtained as  
\[P_{f\_Cu}=\frac{3}{2}R_{s}\left(i_{d}^{2}+i_{q}^{2}\right) \tag{5}\]  
where \(P_{f\_Cu}\) is the fundamental copper loss of the PMSM. As the \(d\)-axis current \(i_{d}\) and \(q\)-axis current \(i_{q}\) are calculated by the CLARKE transmission and PARK transmission using the principle in which the flux value remains invariable, there is the transformation coefficient of \(\frac{3}{2}\) in the power calculation.  
The electromagnetic torque of the PMSM is described as  
\[T_{e}=\frac{3}{2}n_{p}\left[\psi f_{iqm}+\left(L_{dm}-L_{qm}\right)i_{dmi}i_{ qm}\right] \tag{6}\]  
The fundamental copper loss can be derived from Equation (5) as  
\[\begin{array}{rl}P_{\text{Cu\_f}}&=\frac{3}{2}R_{s}(i_{d}^{2}+i_{q}^{2})\\ &=\frac{3}{2}R_{s}\Bigg{\{}\left(i_{dm}+\frac{L_{dm}-n_{p}\omega_{r}L_{qm}i_{ qm}}{R_{i}}\right)^{2}+\left[i_{qm}+\frac{L_{qm}\frac{i_{qm}}{d}+n_{p}\omega_{r} \left(\psi_{f}+L_{dm}i_{dm}\right)}{R_{i}}\right]^{2}\Bigg{\}}\end{array} \tag{7}\]  
From the electromagnetic torque of the PMSM, the fundamental copper loss in the steady state can be simplified as  
\[\begin{array}{rl}P_{\text{Cu\_f}}&=\frac{3}{2}R_{s}(i_{d}^{2}+i_{q}^{2})\\ &=\frac{3}{2}R_{s}\Bigg{\{}\begin{array}{l}\left(i_{dm}+\frac{-n_{p}\omega_{ r}L_{qm}}{R_{i}}\cdot\frac{2T_{e}}{3n_{p}\omega_{f}+3n_{p}\left(L_{dm}-L_{qm} \right)i_{dm}}\right)^{2}+\\ \left[\frac{2T_{e}}{3n_{p}\omega_{f}+3n_{p}\left(L_{md}-L_{qm}\right)i_{dm}}+ \frac{n_{p}\omega_{r}\left(\psi_{f}+L_{dm}i_{dm}\right)}{R_{i}}\right]^{2} \Bigg{\}}\\ &=f(i_{dm},T_{e})=f(i_{d},T_{e})\end{array} \tag{8}\]  
From Equation (8), the fundamental copper loss is a function of the \(d\)-axis current and electromagnetic torque.