# Fractional Models for Thermal Modeling and Temperature Estimation of a Transistor Junction

- Jocelyn Sabatier
^{1}Email author, - HuyCuong Nguyen
^{1, 2}, - Christophe Farges
^{1}, - Jean-Yves Deletage
^{3}, - Xavier Moreau
^{1}, - Franck Guillemard
^{2}and - Bernard Bavoux
^{2}

**2011**:687363

https://doi.org/10.1155/2011/687363

© Jocelyn Sabatier et al. 2011

**Received: **11 December 2010

**Accepted: **20 January 2011

**Published: **21 March 2011

## Abstract

The thermal behavior of a power transistor mounted on a dissipator is considered in order to estimate the transistor temperature junction using a measure of the dissipator temperature only. The thermal transfers between the electric power applied to the transistor, the junction temperature, and the dissipator temperature are characterized by two fractional transfer functions. These models are then used in a Control Output Observer (COO) to estimate the transistor junction temperature.

## Keywords

## 1. Introduction

Fractional differentiation has been widely used in the modelling of many physical and chemical processes and engineering systems such as electrochemistry and diffusion waves, electromagnetic waves, fractal electrical networks, electrical machines, nanotechnology, viscoelastic materials and systems, quantum evolution of complex systems [1], and heat conduction [2].

Automatic control is also a field in which many applications of fractional differentiations has been proposed. Recently, the authors have demonstrated that the real state of a fractional order system is not exactly observable [3]. However, they also have demonstrated that the pseudostate vector of the pseudostate space description (also defined in this paper) can be estimated using a Luenberger like observer.

In this paper, this theory is applied to the estimation of the junction temperature of a power transistor. Temperature management and control are among the most critical functions in power electronic devices, as operating temperature and thermal cycling can affect device performance and reliability. Transistor junction temperature estimation is a problem that was several times addressed in the literature. However, some of the proposed methods are open loop estimations [4], or cannot be implemented on-line due to the complex model used [5, 6] or also require additional devices [7].

In this paper, a simple fractional model is used to evaluate on-line the transistor junction temperature. The considered transistors are fitted with protection diodes that are used in a preliminary study and after their characterisations as a junction temperature sensor. This substitution sensor is used to characterise the transistor mounted on a dissipator by two transfer functions. The first one links the electric power applied to the transistor to its junction temperature and the second one links the junction temperature to the sensor temperature. Given the link existing between diffusion-based systems and fractional systems, the transfer functions obtained are fractional with orders multiples of 0.5 [2]. These transfer functions are then used to derive a pseudostate space description in which the junction temperature and the dissipator temperature are pseudostate variables. An observer based on a dynamic feedback of the real dissipator temperature is used to estimate the junction temperature.

## 2. Preliminaries

in which and . and denote fractional differentiation operators of orders and , respectively. Such operators are defined in [8–11] and a detailed survey of the properties linked to these definitions can be found in [8].

If orders and verify relations , , , then differentiation orders and are commensurate [12] (multiple of the same rational number ).

where is the pseudostate vector, is the fractional order of the system, and , , , and are constant matrices.

As explained in [3], representation (2.2) is not strictly a state space representation and this is why it is denoted in the sequel *pseudostate space representation*. In the usual integer order system theory, the state of the system,
, known at any given time point, along with the system equations and system inputs, is sufficient to predict the response of the system. That comment can be found in [13].

As demonstrated in [3], and whatever the fractional derivative definition used (excepted Caputo's definition but this last one is not physically acceptable [14]), the value of vector at initial time in (2.2) is not enough to predict the future behavior of the system. Vector in (2.2) is thus not a state vector of the system. However, as also shown in [3], a Luenberger type observer can be used to estimate pseudostate vector .

## 3. Experimental System Description

One can thus conclude that the protection diode in the considered transistors can be used as a temperature sensor through the measure of its forward drop when a forward current of 25 mA is applied.

This electrical system permits to characterize the protection diode forward characteristic and to apply a current to the transistor to heat it. The two switches and and the transistor grid are controlled by a laptop, a National Instruments I/O card and a Virtual Instrument designed for the experiment. With such a system, the following operations are performed.

(i)Protection diode forward characterization is obtained by switching off and . Forward current is measured through the shunt resistor and forward voltage— is measured at the transistor terminals.

(ii)Transistor on-state operation is obtained by switching on and ; transistor on-state current is measured through the difference of the voltage at the shunt resistors and terminals. voltage is also measured.

## 4. System Modeling

## 5. Temperature Estimation

The Controller Output Observer (COO), used in this section has been developed by MARGOLIS at Davis University in California [17]. This observer has been mainly applied in the mechanical domain (estimator of torque, speed, etc.). Here, it is used to estimate the temperature of a MosFet junction using the temperature measurement of the dissipator only.

where and represent, respectively, an estimation of the dissipator and junction temperatures.

## 6. Implementation and Results

An advantage of this estimator with respect to the traditional ones (Luenberger like for instance) is that it only requires one information to estimate the junction temperature : the dissipator temperature (no measure of the input).

## 7. Conclusion

In this paper, the efficiency of fractional models for thermal systems modeling is demonstrated through an application to transistor junction temperature estimation. Two fractional models are indeed used to represent the dynamic evolution of the junction temperature and of the dissipator temperature on which the transistor is mounted. These two models are then used to estimate a transistor junction temperature. Such a temperature diagnostic is really important on power electronic devices to optimize the heat management of IGBTs (Insulated Gate Bipolar Transistor). This estimator provides information about junction temperature using a dynamic output feedback-based observer, with only a measure of the dissipator temperature. This limited number of required measures and the simple heat transfer fractional models obtained permit a simple and online implementation of the estimator. The estimator has been implemented and produces very accurate junction temperature estimation. Such an estimator should thus be used to monitor electronic systems such as motor controller, welding, DC/DC and AC/DC converters and to prevent damages in electronic systems on which the considered transistors (or similar components) are used.

## Authors’ Affiliations

## References

- Sabatier J, Agrawal OP, Tenreiro Machado JA (Eds):
*Advances in Fractional Calculus, Theoretical Developments and Applications in Physics and Engineering*. Springer, Dordrecht, The Netherlands; 2007:xiv+552.MATHGoogle Scholar - Sabatier J, Melchior P, Oustaloup A:
**A testing bench for fractional order systems education.***Journal Europeen des Systemes Automatises*2008,**42**(6–8):839-861.View ArticleGoogle Scholar - Sabatier J, Merveillaut M, Fenetau L, Oustaloup A:
**On observability of fractional order systems.***Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE '09), 2009, San Diego, Calif, USA*Google Scholar - Musallamt M, Acarnley PP, Johnson CM, Pritchard L, Pickert V:
**Open loop real-time power electronic device junction temperature estimation.***Proceedings of the IEEE International Symposium on Industrial Electronics (IEEE-ISlE '04), May 2004, Ajaccio, France***2:**1041-1046.Google Scholar - Bruckner T, Bernet S:
**Estimation and measurement of junction temperatures in a three-level voltage source converter.***IEEE Transactions on Power Electronics*2007,**22**(1):3-12.View ArticleGoogle Scholar - Khatir Z, Dupont L, Ibrahim A:
**Investigations on junction temperature estimation based on junction voltage measurements.***Microelectronics Reliability*2010,**50**(9–11):1506-1510.View ArticleGoogle Scholar - Kim Y-S, Sul S-K:
**On-line estimation of IGBT junction temperature using on-state voltage drop.***Proceedings of the 33rd IAS Annual Meeting, 1998***2:**853-859.Google Scholar - Oldham KB, Spanier J:
*The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order*. Academic Press, New York, NY, USA; 1974:xiii+234. Mathematics in Science and Engineering, vol. 11MATHGoogle Scholar - Samko SG, Kilbas AA, Marichev OI:
*Fractional Integrals and Derivatives: Theory and Applications*. Gordon and Breach Science Publishers, Yverdon, Switzerland; 1993:xxxvi+976.MATHGoogle Scholar - Miller KS, Ross B:
*An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication*. John Wiley & Sons, New York, NY, USA; 1993:xvi+366.Google Scholar - Podlubny I:
*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar - Matignon D:
**Stability properties for generalized fractional differential systems.**In*Systèmes Différentiels Fractionnaires (Paris, 1998), ESAIM: Proceedings*.*Volume 5*. Society for Industrial and Applied Mathematics, Paris, France; 1998:145-158.Google Scholar - Lorenzo DF, Hartley TT:
**Initialization in fractional order systems.***Proceedings of the European Control Conference (ECC '01), 2001, Porto, Portugal*1471-1476.Google Scholar - Sabatier J, Merveillaut M, Malti R, Oustaloup A:
**How to impose physically coherent initial conditions to a fractional system?***Communications in Nonlinear Science and Numerical Simulation*2010,**15**(5):1318-1326. 10.1016/j.cnsns.2009.05.070MathSciNetView ArticleMATHGoogle Scholar - Melchior P, Lanusse P:
**CRONE Toolbox : une boîte à outils Matlab pour les systèmes fractionnaires.**In*Colloque sur l'Enseignement des Technologies et des Sciences de l'Information (CETSIS-EEA '01), October 2001, Clermont-Ferrand, France*Edited by: Dancla F, Cois O. 241-244.Google Scholar - Melchior P, Lanusse P, Cois O, Dancla F:
**Crone Toolbox for Matlab: Fractional Systems Toolbox—Tutorial Workshop on "Fractional calculus applications in automatic control and robotics".***Proceedings of the 41st IEEE Conference on Decision and Control (CDC '02), December 2002, Las Vegas, Nev, USA*Google Scholar - Ozkan B, Margolis D, Pengov M: The controller output observer: Estimation of vehicle tire cornering and normal forces. Journal of Dynamic Systems, Measurement and Control 2008,130(6):-10.Google Scholar
- Oustaloup A:
*La Dérivation Non Entière: Théorie, Synthèse et Applications*. Hermès, Paris, France; 1995.Google Scholar

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