## 1 Introduction
Sine modulation functions are adopted to represent the duty cycle and this function is substituted by a function of time, modulation and phase [22; 23]. [PERSON] proposes that three sine functions are utilized to represent the duty cycle [22]. [PERSON] adopts one sine function and studies the conduction losses under different modulation ratios [23]. However, the number of sine modulation functions and the choice of the modulation ratio are not investigated in these mentioned papers, as they are closely related to the model accuracy and the computation demands. The parameters of IGBTs and FWDs in the losses model depend on the junction temperature, and the electro-thermal interaction needs to be studied. An accurate temperature-dependent switching power loss model is presented through referring to a look-up table obtained by the experiment measurements [24]. The switching power loss is considered proportional to the dc-link voltage and other two inputs of the table are the current and the junction temperature. [PERSON] analyses the thermal interaction between components in the power converter [25]. The electrical model considering temperature-dependent power loss model is coupled to a thermal model. The converter circuit is built in PLECS simulation software and it can simulate accurate estimation of losses in the power converter. The lumped dynamic electro-thermal model of IGBT module is proposed and it points out that the coupling between the electronic and thermal part is made by the power losses [26]. The energy estimated for a given temperature is linearly extrapolated and the linear extrapolation of the semiconductor losses is accomplished using a look-up table. Therefore, the means by which one can find the accurate and simple relationship between the temperature and losses needs to be researched, taking in to account the electro-thermal interaction. For the automobile electric drive system, the FOC (field oriented control) algorithm is widely adopted due to the high efficiency and torque control accuracy, and the voltage and current signals in the two-phase synchronous rotating coordinate need to be filtered [27]. In this paper, the proposed losses model is built based on the FOC algorithm. Based on the mentioned difficulties above, this paper proposes a simple and feasible power losses model to calculate the switching and conduction losses. The contribution of this paper is concluded as follows:  
1. Considering the waveform of the collector-emitter voltage drop and the collector current at the instant of switching during the inverter operation, a piecewise linear model is proposed to simplify the real waveform to avoid the complex fitting and interpolation process. As for the proposed method, only one set of data under the rated operating condition should be measured, and the switching losses at other operating points can be calculated proportionately. 2. To solve the piecewise integration of conduction losses, the piecewise periodic functions of the duty cycle are derived and studied, and the equivalent three-order harmonic model of the duty cycle is proposed based on Fourier series expansion. The simplified conduction losses model has the advantages of the lower computation requirements and the easy engineering implementation. 3. The proposed losses model provides a simple and practical loss model based on SVPWM modulation in the automobile electrical drive application, and it provides a basis for the accurate junction temperature prediction of IGBTs. This paper is organized as follows: Section 2 introduces the traditional power loss model. In Section 3, the piecewise linear switching losses model and the conduction losses model based on the equivalent three-order harmonic model of duty cycle are proposed. Test setup and experimental results are presented in Section 4. Conclusions are drawn in Section 5.