## 2 Nonlinear Loss Model of Three-Phase Inverter  
The most common three-phase inverter for PMSM system is the three-phase half-bridge voltage source inverter, which is shown in Figure 1. The main inverter losses of the PMSM system are conduction loss and switch loss within power devices. To optimize the inverter loss, it is important to build an accurate loss model for the power device. Although there is strongly nonlinearity in the conduction characteristics and switch characteristics of power device, the most common mathematical model of power device loss is a linear model, which is shown in Figure 2.  
Figure 1: Typical topology of three-phase voltage source inverter. Figure 2: Linear model and nonlinear model. The linear model of conduction characteristics in IGBT can be expressed as:  
\[U_{cesat}=U_{l0}+R_{ce}\cdot I_{ce} \tag{1}\]  
where \(U_{cesat}\) is the saturation voltage of power device, \(U_{l0}\) is the equivalent conduction voltage of power device in rated current \(I_{n}\), \(R_{ce}\) is equivalent conduction resistance of power device in rated current \(I_{n}\). It can be seen that the traditional linear model of power device can only obtain accurate power device loss at the rated current \(I_{n}\). The prediction error will increase when the current of power device deviates from the rated working point. However, the motor current in PMSMs is a sinusoidal current, whose value is changing all the time. Therefore, the traditional linear model will not calculate the inverter loss of PMSM drive system exactly. To acquire the inverter loss accurately, this paper builds a novel nonlinear model of power devices, which can fit the nonlinear conduction and switch characteristics in each operation condition of a PMSM system. The nonlinear conduction model of IGBT can be shown as:  
\[U_{ce}=a_{c}+b_{c}\cdot I_{ce}+c_{c}\cdot I_{ce}^{2} \tag{2}\]  
where \(a_{c}\), \(b_{c}\) and \(c_{c}\) is fitting coefficients of the nonlinear conduction characteristics. From Figure 2, it can be seen that the nonlinear model can accurately fit the power device nonlinear conduction characteristics in the whole operation range of a PMSM system. As the stator current in PMSM system is changing in a sinusoidal way, the inverter loss will also change at any moment. Therefore, the average conduction loss of IGBT in on current period can be calculated as:  
\[P_{conduction\_IGBT}=\frac{1}{T_{current}}\int_{0}^{T_{current}}u_{ce}\cdot i_{ce}dt \tag{3}\]  
where \(T_{current}\) is the period of sinusoidal stator current. By substituting Equation (2) into Equation (3), the conduction loss of IGBT can be shown as:  
\[P_{conduction\_IGBT}=\frac{a_{c}}{\pi}I_{0}+\frac{2c_{c}}{3\pi}I_{0}^{3}+\frac {b_{c}}{4\pi}I_{0}^{2}+\frac{a_{c}}{4}M\cos\varphi\cdot I_{0}+\frac{2b_{c}}{3 \pi}M\cos\varphi\cdot I_{0}^{2}+\frac{3c_{c}}{16}M\cos\varphi\cdot I_{0}^{3} \tag{4}\]  
where \(I_{0}\) is amplitude of stator current, \(M\) is the modulation ratio of sine-wave pulse-width modulation (SPWM), and \(\varphi\) is power factor of the PMSM:  
\[M=\frac{U_{0}}{\frac{U_{dc}}{2}}=\frac{2U_{0}}{U_{dc}}=\frac{2\sqrt{u_{d}^{2}+ u_{q}^{2}}}{U_{dc}} \tag{5}\]  
The nonlinear conduction model of free-wheeling diode (FWD) can be also expressed as:  
\[U_{f}=a_{f}+b_{f}I_{f}+c_{f}I_{f}^{2} \tag{6}\]  
where \(a_{f}\), \(b_{f}\) and \(c_{f}\) is fitting coefficients of the nonlinear conduction characteristics of FWD. Similar to the average conduction loss of IGBT, the average conduction loss of FWD in one current period can be shown as:  
Figure 2.0  
From Equations (4) and (7), the average conduction loss of three-phase half-bridge inverter in one current period can be calculated as:  
Figure 2.1The switch characteristics of power devices can also be fitted by proposed nonlinear model. The nonlinear switch-on and switch-off model of IGBT can be expressed as:  
Figure 2.2  
Figure 2.3  
where \(E_{on}\) and \(E_{off}\) are the energy consumption in a switch -on and switch-off process, respectively.