## 3 Harmonic Loss Model of the PMSM  
As the PMSM is fed by a PWM inverter, there must be several harmonic currents in the stator windings caused by the PWM output voltage of the inverter. The harmonic current will generate the harmonic iron loss in the stator core, which must affect the efficiency of the PMSM. To achieve the maximum efficiency of the PMSM, it is important to build an accurate harmonic loss model of the PMSM fed by a three-phase half-bridge inverter. Figure 6 shows the typical three-phase PMSM power system where \(2U_{dc}\) is the DC bus voltage. Figure 4: Fundamental loss comparison between FEA and the analytical method at different speeds and torques. Figure 5: Difference of motor loss between the FEA method and the proposed analytical method at different speeds and torques. Figure 6: Typical topology of three-phase PMSM power system. Although there are a large number of pulse-width modulation modes, SPWM and SVPWM are the most common modulation modes. Therefore, this paper only focuses on the harmonic voltage components caused by SPWM and SVPWM in the PMSM. Determination of the harmonic frequency components of a PWM switched inverter output is quite complex and is often done by using a fast Fourier transform (FFT) analysis of a simulated time-varying switched waveform. This approach could offer the benefits of expediency and reduce the mathematical effort, but it requires considerable computing capacity and always leaves uncertainty as to whether a subtle simulation round-off or error may have slightly tarnished the results obtained. In contrast, an analytical solution using double Fourier integral analysis can exactly identify the harmonic components of a PWM waveform, which ensures the correct harmonics are precise. In double Fourier integral analysis theory, the PWM output voltage of the inverter can be obtained by two time variables, \(x(t)\) and \(y(t)\), where \(x(t)\) is the carrier signal and \(y(t)\) is the fundamental (sinusoid) signal. \[x(t)=\omega_{c}t+\theta_{c} \tag{18}\]  
where \(\omega_{c}\) is the carrier angular frequency and \(\theta_{c}\) is the arbitrary phase offset angle for the carrier waveform. \[y(t)=\omega_{0}t+\theta_{0} \tag{19}\]  
where \(\omega_{0}\) is the fundamental angular frequency and \(\theta_{0}\) is the arbitrary phase offset angle for the fundamental waveform. The output voltage of the inverter leg can be present as  
\[a_{n}(t)=f(x(t),y(t))=\left\{ \begin{array}{ll}2U_{dc}&y(t)>x(t)\\ 0&y(t)\leq x(t)\end{array} \right. \tag{20}\]  
where \(2U_{dc}\) is the DC voltage, which is shown in Figure 5. The output voltage of the inverter leg is defined with respect to the negative DC bus rather than with respect to the midpoint of the DC bus, which could simplify the mathematics of the Fourier solution at the trivial expense of introducing a DC offset of \(+U_{dc}\) into the final solution. With double Fourier integral analysis theory [19], the time-varying function \(f(x(t)\),\(y(t))\) can be expressed as a summation of the harmonic components  
\[ \begin{array}{rl}f(x,y)&=\frac{A_{00}}{2}+\sum\limits_{n=1}^{\infty}\{A_{0n }\text{cos}[n(\omega_{0}t+\theta_{0})]+B_{0n}\text{sin}[n(\omega_{0}t+\theta_{ 0})]\}\\ &+\sum\limits_{m=1}^{\infty}\{A_{m0}\text{cos}[m(\omega_{c}t+\theta_{c})]+B_{ m0}\text{sin}[m(\omega_{c}t+\theta_{c})]\}\\ &+\sum\limits_{m=1}^{\infty}\left.\begin{array}{l}\sum\limits_{n=-\infty}^{ \infty}\\ +B_{mn}\text{sin}[m(\omega_{c}t+\theta_{c})+n(\omega_{0}t+\theta_{0})]\\ (n\
eq 0)\end{array}\right\}\end{array}  \tag{21}\]  
where \(A_{00}\) is the DC offset; \(A_{0n}\) and \(B_{0n}\) are the fundamental component and base-band harmonics; \(A_{m0}\) and \(B_{m0}\) are the carrier harmonics; \(A_{mn}\) and \(B_{mn}\) are side-band harmonics.