An Improved Fourierseriesbased IGBT Model by using PID controller to Mitigate the Effect of Gibbs’ Phenomenon
Yifei Ding
^{a}, Xin Yang
^{a}, Member, IEEE, Jun Wang
^{a}, Senior Member, IEEE, Guoyou Liu
^{b}, Senior Member
^{a} School of Electrical and Information Engineering, Hunan University, Changsha, China, 410082
^{b} CRRC Zhuzhou Electric Locomotive Institute Co., Ltd., Zhuzhou, China, 412000
Introduction
In recent years, IGBT modeling has attracted a lot of attention. To accurately simulate switching waveform, predict switch loss and EMI, an accurate IGBT simulation model is of great importance. Most physicsbased IGBT models can well simulate the electrical characteristics and physical characteristics of the devices[1]. Therefore, the physicsbased IGBT model is widely used in power electronics.
Fourierseriesbased IGBT model is one of the physical models presented by Palmer[2, 3]. The model can well describe the carrier behavior in the CSR that is the key part of bipolar devices. A parameter extraction and optimization method based on the model is proposed in [4], which further improves the accuracy of the model. The input capacitance was corrected in [5]. The switching behaviors of the highspeed buffer layer IGBT is optimized in [6]. These works make the Fourierseriesbased model suitable for IGBT with various structures such as punchthrough (PT), nonpunchthrough NPT and field stop (FS). In addition, the model has been proven to be very efficient in terms of simulation speed.
Motivation
Fig. 2. Illustration of carrier concentration in Fourierseriesbased IGBT model and finite element simulation in the turnon process.

The Fourierseriesbased IGBT model, due to the truncation error of the Fourier series, leads to Gibbs' Phenomenon, which is particularly significant in the turnon process. Gibbs' phenomenon basically has no effect in the turnoff process, because within this process, Gibbs’ phenomenon mainly affects the region with high carrier concentration, but the carrier concentration close to
x2
, i.e. the depletion layer, rapidly drops to 0. Such a sharp slope of
p makes the carrier at
x2
basically unaffected by Gibbs’ phenomenon as shown in Fig. 1. However, in the turnon process, Gibbs’ phenomenon will have a significant impact on the carrier concentration near
x2
where the carrier concentration of the Fourierseriesbased model oscillates greatly as shown in Fig. 2. Due to the existence of such an oscillation, regions with carrier concentration less than zero would emerge in the drift region. Therefore, the boundary between CSR and the depletion region cannot be defined by the position of "minute" carrier concentration. At the same time, the region with carrier concentration less than zero will produce an abnormally large electric field in the CSR.
Fig. 1. Carrier concentration distribution in Fourierseriesbased IGBT model at turnoff.

Due to Gibbs' phenomenon, physical characteristics in regions with low carrier concentration can not be well simulated, which in turn will lead to the inaccuracy of the Fourier seriesbased solution. Therefore, it is necessary to find a way to mitigate the Gibbs’ phenomenon.
prior art
The original Fourierseriesbased IGBT model in the turnon process was proposed by Palmer and Angus. The core program block diagram of solving the ambipolar diffusion equation is shown in the Fig. 3. The original method provides a feedback on the error between
px2
and 0 to obtain the depletion layer voltage Vd2
which is essentially the proportion term in the proportional controller. In the original model, Gibbs’ phenomenon seriously affects the regions with low carrier concentration because
px2
is less than 0. Lu et al. take into account the twodimensional effect of MOS side [7], witch can effectively increase
px2
and reduce the influence of Gibbs’ phenomenon on the low carrier concentration regions as shown in Fig. 3.
Novelty and major achievements in this work
The modified model proposed in this paper replaces the proportional controller in the original model with the PID controller.
Fig. 3. The flow of the original model, Lu’s model and the modified model to get the boundary conditions.

The integral controller is used to make
px2
greater than 0 in the early turnon process which can effectively mitigate the effects of Gibbs’ phenomenon, and the differential controller is used to improve model convergence. In this paper, particle swarm optimization (PSO) optimization algorithm is used to extract parameters of PID controller, which greatly increases the accuracy of the modified model. The preliminary results are shown in Fig. 4. Compared with other models, the voltage and current waveforms of the modified model are closest to the waveforms of finite element simulation.
Few of these Fourierseries based models compares the carrier concentration distribution and the electric field distribution throughout the drift region in IGBT turnon process, which is the key elements of physicsbased IGBT model.
Fig. 4. Turnon comparison of the modified model, Lu's model and the original model with TCAD simulation under resistive load.

In this paper, finite element simulation is used as a reference to compare the modified model, traditional model and Lu' model. Electrical characteristics as voltage and current will be compared, as well as physical characteristics as carrier concentration distribution and electric field strength.
Reference
[1] S. Ji, Z. Zhao, T. Lu, L. Yuan, and H. Yu, “HVIGBT physical model analysis during transient,” IEEE Trans. Power Electron., vol. 28, no. 5, pp. 2616–2624, May 2013.
[2] P. R. Palmer, E. Santi, J. L. Hudgins, X. Kang, J. C. Joyce, and P. Y. Eng, “Circuit simulator models for the diode and IGBT with full temperature dependent features,” IEEE Trans. Power Electron., vol. 18, no. 5, pp. 1220–1229, Sep. 2003.
[3] X. Kang, A. Caiafa, E. Santi, J. L. Hudgins, and P. R. Palmer, “Characterization and modeling of highvoltage fieldstop IGBTs,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 922–928, Jul./Aug. 2003.
[4] A. T. Bryant, X. Kang, E. Santi, P. R. Palmer, and J. L. Hudgins, “Twostep parameter extraction procedure with formal optimization for physicsbased circuit simulator IGBT and PIN diode models,” IEEE Trans. Power Electron., vol. 21, no. 2, pp. 295–309, Mar. 2006.
[5] X. Yang, M. Otsuki, and P. R. Palmer, “Physicsbased insulatedgate bipolar transistor model with input capacitance correction,” IET Power Electron., vol. 8, no. 3, pp. 417–427, 2015.
[6] P. Xue, G. Fu, and D. Zhang, “Modeling inductive switching characteristics of highspeed buffer layer IGBT,” IEEE Trans. Power Electron., vol. 32, no. 4, pp. 3075–3087, Apr. 2017.
[7] L. Lu, A. Bryant, J. L. Hudgins, P. R. Palmer, and E. Santi, “Physics based model of planargate IGBT including MOS side twodimensional effects,” IEEE Trans. Ind. Appl., vol. 46, no. 6, pp. 2556–2567, Nov./Dec. 2010, doi: 10.1109/TIA.2010.2071190.
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