IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO. 8,AUGUST 2002 1187 Modeling the Thermal Response of Semiconductor Devices Through Equivalent Electrical Networks Lorenzo Codecasa, Dario D’Amore, and Paolo Maffezzoni Abstract—The paper presents a general approach for modeling the effects of thermal response in semiconductor devices as they are seen at the electrical terminals. It is shown that this can be achieved by properly connecting at the electrical ports an equivalent elec- trical network representing the transformed thermal impedance. The equivalent model is employed to investigate electrothermal in- teractions in MOSFETs and bipolar junction transistors. Precise conditions for which electrothermal resonant oscillations arise are deduced and an experimental technique for thermal-impedance extraction is presented. Index Terms—Circuit modeling, electrothermal analysis, thermal instability. I. INTRODUCTION T HE CLASSICAL theory of electrical circuits does not consider the heating phenomena which take place inside electronic devices, regarding temperature as a fixed physical property and self heating as a disturbance of normal electrical operation. From this point of view, the portion of electrical power that is transformed into heat corresponds to an irreversible lost of electrical power toward the external environment. In the last few years, the influence of heating effects on elec- trical performances has attracted more attention due to the rele- vance that it has assumed in many applications, such as in power electronics and in high miniaturized integrated circuits [1], [2]. Lumped circuit models, or thermal networks, of the heat dif- fusion equation have been proposed and employed in numer- ical electrothermal simulations [3]–[7]. Thermal networks are composed of thermal resistors, representing the heat diffusion mechanism and of thermal capacitors, accounting for heat ac- cumulation in the silicon device and its external interconnec- tion [8]–[10]. The electrothermal network allows the study of new kinds of phenomena which are neither purely electrical nor thermal, but instead, that arise from the interaction of the two physical processes. However, electrothermal interaction being inherently non- linear, a pure numerical analysis is not enough to achieve a complete understanding of all possible electrothermal effects and a qualitative analysis is important to gain more insight. The first approach to the qualitative analysis of electrothermal dynamic effects has been presented in [11]–[13]. In those works, a first-order thermal model composed of a single cell RC network was employed to investigate the coupling of Manuscript received February 16, 2001; revised February 3, 2002. This paper was recommended by Associate Editor Y. Park. The authors are with the Dipartimento di Elettronica e Informazione, Politec- nico di Milano, Milan I-20133, Italy. Publisher Item Identifier 10.1109/TCSI.2002.801279. electrical and thermal dynamics of bipolar junction transistor (BJT) and MOSFET devices operating in a particular electrical configuration. The model was accurate enough to predict the occurrence of an electrothermal resonance phenomenon that was then confirmed by experimental evidences. While interesting in principle, the previously presented ap- proach suffered from a series of limitations since it considered only a well specific device condition and employed a too simple model for describing the complete thermal response of electrical devices. In order to overcome these drawbacks, this paper approaches the analysis of electrothermal interaction in a very general and systematic way. The derivation here developed is based on very general electrical equations and can be applied to a wide class of electrical devices while the thermal response is described in its full complexity through the thermal-impedance concept. Following an approach similar to [14], we first derive a rig- orous small-signal model of a generic -port electrical device in which the thermal impedance is reported, through a transfor- mation matrix, at the electrical terminals. In this way a compact purely electrical model, embedding also the thermal effect is obtained. The transformation matrix concept is systematically employed to derive compact models of the BJT and MOSFET devices and to establish the precise conditions for which elec- trothermal instability can be induced. This phenomenon is then studied in details for the case of MOSFET devices. It is proved that electrothermal instability may lead to the existence of a stable limit cycle and electrothermal oscillations may appear, corresponding to a resonance condition between the electrical and thermal dynamics. When electrothermal oscillation settles, we can say that in a fraction of the oscillation period, in which temperature increases, electrical energy is transformed into heat accumulation, while in the remaining fraction of the period, in which temperature decreases, a portion of accumulated heat is transformed back into electrical energy. (Let us observe that this mechanism does not violate thermodynamic laws since the above energetic exchange is superimposed to a greater contin- uous flux of power that from the dc power sources diffuses, as dissipated heat, toward the external environment.) Results given by the theoretical analysis here developed are then employed to experimentally investigate on a commercially available MOSFET. In this paper, it is shown that electrothermal measurements carried out in a set of different resonance con- ditions allow the extraction of the complex values of thermal impedance over a wide range of frequencies. This information is then employed in an optimization procedure to identify the parameters of a thermal model able to accurately approximate the thermal response of the MOSFET device. 1057-7122/02$17.00 © 2002 IEEE