## 3 Global Loss Model of PMSM
### Fundamental and Harmonic Components of PWM Output Voltage  
The PWM output voltage of inverters must induce a harmonic stator current in the PMSM drive system. The harmonic stator current produces a harmonic motor loss which will affect the system efficiency of the PMSM drive system. To build an accurate harmonic loss model of a PMSM, it is necessary to create an analytic method to calculate the harmonic motor current quickly and exactly. By sine-wave pulse-width modulation (SPWM), the PMSM can obtain a sinusoidal current from the three-phase half-bridge inverter. The typical working principle of SPWM is shown in Figure 3.  
The three-phase sinusoidal references displaced in time by \(120^{\circ}\), which can be expressed as:  
\[\left\{ \begin{array}{c}U_{zz}^{*}=U_{0}\cos\omega_{0}t=\frac{1}{2}M\text{U}_ {dc}\cos\omega_{0}t\\ U_{bz}^{*}=U_{0}\cos(\omega_{0}t-\frac{2}{3}\pi)=\frac{1}{2}M\text{U}_{dc} \cos(\omega_{0}t-\frac{2}{3}\pi)\\ U_{cz}^{*}=U_{0}\cos(\omega_{0}t+\frac{2}{3}\pi)=\frac{1}{2}M\text{U}_{dc} \cos(\omega_{0}t+\frac{2}{3}\pi)\end{array} \right. \tag{15}\]  
where \(U_{0}\) is the amplitude of three-phase sinusoidal references. To obtain the harmonics of the motor current, the harmonics of the inverter PWM output voltage must be acquired. As it is a complicated problem to measure the harmonics of PWM voltage signal in the SPWM and the harmonic components of PWM signal are always obtained from the time varying simulated signal of PWM output voltage by fast Fourier transform (FFT) analysis. Although the FFT method can conveniently solve the harmonics in a relatively accurate way, the precision of FFT results is limited by the computing capacity and the simulated round-off. The FFT analysis can only obtain the harmonics of PWM signal at a certain working point of the PMSM drive system and cannot acquire the harmonics of the inverter output voltage for the whole operation condition. In contrast, the double Fourier integral analysis can calculate the fundamental component and harmonics of a periodic PWM waveform in an analytic method, which ensures the accuracy of the harmonics and acquires the harmonic voltage of the inverter leg across overall operation condition of the PMSM direct drive system. The double Fourier integral analysis [30; 31] applies two time-varying variables (\(x(t)\) and \(y(t)\)) to define the PWM signal of inverter output voltage. The \(x(t)\) is used to define the carrier signal, which can be shown as:  
\[x(t)=\omega_{c}t+\theta_{c} \tag{16}\]  
Figure 3: Typical working principle of SPWM in the three-phase PMSM drive system. where \(\omega_{c}\) is the carrier angular frequency and \(\theta_{c}\) is the arbitrary phase offset angle for the carrier waveform. \(y(t)\) is applied to delimit the fundamental signal (as the sinusoidal reference signal of SPWM), which can be expressed as:  
\[y(t)=\omega_{0}t+\theta_{0} \tag{17}\]  
where \(\omega_{0}\) is the fundamental angular frequency and \(\theta_{0}\) is the arbitrary phase offset angle for the fundamental waveform. The PWM output voltage of the inverter leg can be presented as:  
\[a_{n}(t)=f(x(t),y(t))=\left\{ \begin{array}{cc}U_{dc}&y(t)>x(t)\\ 0&y(t)\leq x(t)\end{array} \right. \tag{18}\]  
where \(a_{n}(t)\) is defined with respect to the negative DC bus. By applying double Fourier integral analysis theory, the time-varying function \(f(x(t),y(t))\) can be expressed as a summation of the harmonic components:  
\[ \begin{array}{ll}f(x,y)&=\frac{A_{0n}}{2}+\sum\limits_{n=1}^{\infty}\{A_{0 n}\cos[n(\omega_{0}t+\theta_{0})]+B_{0n}\sin[n(\omega_{0}t+\theta_{0})]\}\\ &+\sum\limits_{m=1}^{\infty}\{A_{m0}\cos[m(\omega_{c}t+\theta_{c})]+B_{m0} \sin[m(\omega_{c}t+\theta_{c})]\}\\ &+\sum\limits_{m=1}^{\infty}\sum\limits_{\begin{subarray}{c}n=-\infty\\ (n\
eq 0)\end{subarray}}^{\infty}\{A_{mn}\cos[m(\omega_{c}t+\theta_{c})+n( \omega_{0}t+\theta_{0})]+B_{mn}\sin[m(\omega_{c}t+\theta_{c})+n(\omega_{0}t+ \theta_{0})]\}\}\end{array}