630 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

Behavioral Approach to SiC MPS DiodeElectrothermal Model Generation

Łukasz Starzak, Member, IEEE, Mariusz Zubert, Member, IEEE,Marcin Janicki, Tomasz Torzewicz, Małgorzata Napieralska, Member, IEEE,

Grzegorz Jabłonski, Member, IEEE, and Andrzej Napieralski, Senior Member, IEEE

Abstract—A comprehensive approach to generation of elec-trothermal models of silicon carbide (SiC) power Schottky diodesis presented. Both the electrical and thermal parts of the model arebehavioral. The electrical one was developed in order to accuratelyrepresent the nonlinear properties of SiC merged PiN Schottky(MPS) diodes. Its parameters are automatically obtained with adedicated numerical procedure. The thermal model derivationdiffers from the earlier approaches because it is based on theanalysis of thermal constant spectrum of the temperature responseof a modeled device. Simple model generation and parameterextraction procedures are proposed in which no knowledge oftechnological data is necessary.

Index Terms—Compact thermal models (CTMs), electrother-mal models, power semiconductor devices, Schottky diodes, siliconcarbide.

I. INTRODUCTION

COMPACT MODELING consists in describing the opera-tion of electronic devices using a small number of equa-

tions in closed form. Compared to physics-based distributedmodels, they offer faster simulation (up to several orders ofmagnitude) without significant degradation in accuracy. Thedevelopment of compact models is a kind of art; there areseveral approaches, but none of them guarantees success.

Power devices are particularly difficult to model with com-pact models due to their compound structure and large di-mensions. High power dissipation makes it necessary to takeinto account the electrothermal coupling inside as well as theinteraction with the environment. On the other hand, compactmodels are not supposed to be too complex, and they shouldprovide results in a reasonable time for switch-mode converters.

Silicon carbide devices are becoming the most promisingsemiconductor devices for power applications [1]. They offerexcellent thermal properties and low ON-state resistance to-gether with high voltage capability [2]. This has made possiblethe manufacturing of high-voltage unipolar devices reachingvery high operating frequencies [3]. As a consequence, the mostfrequently used SiC devices are merged PiN Schottky (MPS)diodes [4].

Manuscript received July 17, 2012; revised September 13, 2012; acceptedSeptember 28, 2012. Date of publication November 26, 2012; date of currentversion January 18, 2013. This work was supported in part by the University In-ternal Grant K25/DzSt/1/2012. The review of this paper was arranged by EditorM. Miura-Mattausch.

The authors are with the Department of Microelectronics and ComputerScience, Technical University of Lodz, 90-924 Lodz, Poland (e-mail: lukasz.starzak@p.lodz.pl; mariuszz@dmcs.pl; janicki@dmcs.pl; torzewicz@dmcs.pl;mnapier@dmcs.pl; gwj@dmcs.pl; napier@dmcs.p.lodz.pl).

Digital Object Identifier 10.1109/TED.2012.2222887

The MPS diode models provided by device manufactur-ers, the classical SPICE embedded diode model, and physicalmodels are all unable to accurately reproduce temperature-dependent device behavior of these devices in a relativelywide range of operating temperatures [5]. However, accuratemodels are required in order to provide engineers with a reliabletool for the design of robust state-of-the-art power conversionappliances. Consequently, a simple and accurate electrothermalmodel of the MPS diode is necessary.

Behavioral models have great potential in this regard, par-ticularly when unipolar devices are concerned. Electrical phe-nomena in Schottky diodes during conduction and switchingare generally simpler as compared to those in PiN diodes[3], which renders behavioral modeling feasible. However, SiCMPS diodes exhibit nonlinear behavior when temperature influ-ence is involved [6] what requires the development of dedicatedmodels [4]. Moreover, even though switching processes are veryshort as compared to thermal time constants, they may soonbecome more important as switching frequencies are increased.

Until now, compact thermal models (CTMs) were obtainedeither by detailed investigation of the device structure, whosephysical and geometrical parameters are often unavailable, orby application of the Network Identification by Deconvolution(NID) method, based on the analysis of a single transienttemperature measurement processed appropriately [7]. To de-velop the electrothermal MPS model, a novel model generationapproach can be applied, which is based on the NID method andtime-constant spectrum examination. It offers greater simplicityand better numerical properties while preserving the physicalmeaning of the obtained CTM [8].

The thermal model is coupled with the electrical part to formthe complete electrothermal model. The proposed approach isdemonstrated for the CSD20030 common-cathode dual MPSrectifier manufactured by Cree. It is demonstrated that the be-havioral approach enables the derivation of an accurate unipolarpower semiconductor device model without any knowledgeabout its technological parameters. This has a great practi-cal impact as such data are normally not revealed by devicemanufacturers.

II. SiC DIODE ELECTROTHERMAL MODEL

A. Behavioral Electrothermal Model

The research presented in this paper was focused on mod-eling of commercially available SiC MPS diodes. First, be-havioral models made available by device manufacturers were

0018-9383/$31.00 © 2012 IEEE

STARZAK et al.: APPROACH TO SiC MPS DIODE ELECTROTHERMAL MODEL GENERATION 631

Fig. 1. Comparison of (points) measured and (lines) simulated static forwardcharacteristics of the CSD20030 for different junction temperatures using thebuilt-in SPICE diode model. The model parameters have been extracted usingmeasurements for 25 ◦C.

subjected to tests which showed serious inconsistencies be-tween simulated and measured static characteristics. Possibleexplanations of these discrepancies were given in [5] and [9].In addition, the application of very simplified heat conductionmodels was discovered.

Then, using the classical diode model based on Shockley’sequation and widely adopted in SPICE-based simulators hasbeen considered [10], [11]

Id(Vd) = Is ·(

1 − expVd −Rs · Id(Vd)

n · Vt

)(1)

where I and V are the device current and voltage, respec-tively, Is is the saturation current, Rs is the internal parasiticresistance, n is the emission coefficient, and Vt is the thermalvoltage. However, analyses again demonstrated a disagreementbetween this model and measurement data obtained for thesecond- and third-generation SiC Schottky diodes manufac-tured by Infineon and Cree [4]. The inconsistencies between theShockley’s diode model and measurements have been shownin Fig. 1. In the authors’ opinion, these inconsistencies maybe due to these devices being manufactured as MPS diodeswith additional p-islands [12]. This results in electrothermalbehavior involving a combination of nonlinear effects whichbecome particularly pronounced around 60 ◦C–70 ◦C, both forforward and reverse biases. Despite including additional tem-perature relationships, the aforementioned model was shown tobe unable to reproduce those effects.

Considering the aforementioned problems, it was foundpromising to turn to behavioral models derived directly frommeasurement data [4]. In the initial model structure, compo-nents were distinguished related to the voltage drop acrossthe internal resistance Rs, to an intrinsic voltage drop Vintrsc

(thought to be influenced by the p-islands), and to the tem-perature T influence on both former elements. Moreover, arelationship between ln(Ifwd) and Vfwd was assumed, whereIfwd and Vfwd are the forward device current and voltage,respectively. A similar relationship was assumed for the reversecurrent Irev and voltage Vrev.

In a regression-based analysis of the diode model, an exoge-nous variable Vd and endogenous variables ln(Ifwd) (or Ifwd)and T were taken into consideration, with the temperature Tbeing linearly or quadratically related to Rs and Vintrsc. Anal-yses of measured quantity variability (Vfwd,I , Ifwd,I , and Ti)lead to the following relationship describing the static behaviorof the diode under forward bias:

Vfwd(Ifwd, T )=(p1+p2 · T ) · ln Ifwd+(p3+p4 · T )+Ifwd · p5

·[1+p6 ·(T−27 ◦C)+p7 ·(T−27 ◦C)2

](2)

where p1 to p7 represent the empirical constant coefficients.Formula (2) was then converted to the form containing Rs(T )and Vintrsc(T )

Vfwd(Ifwd, T )=Vr(T ) · ln Ifwd+Rs · Ifwd+Vintrsc(T ) (3)

and later to the form suitable for numerical simulations, similarto (1)

Ifwd(Vfwd, T )=expVfwd−Vintrsc(T )−Rs ·Ifwd(Vfwd, T )

Vr(T )(4)

where

Vr(T ) = p1 + p2 · T (5)

Vintrsc(T ) = p3 + p4 · T (6)

Rs(T ) = p5 ·[1 + p6 · (T − 27 ◦C)

+ p7 · (T − 27 ◦C)2]. (7)

The aforementioned model has been validated on the basisof several SiC MPS diodes [4], [13], [14]. In every case, (2)was found to be consistent with the measurement results atconfidence levels of 0.95–0.99, while (1) was unable to fitthese data.

A similar analysis was carried out for the reverse bias,leading to the formula (valid for all the diodes investigatedexcept CSD04060)

Irev(Vrev, T ) = β(T ) · exp [Vrev · α(T )] . (8)

For every diode investigated, the hypothesis of linear characterof α(T ) and β(T ) relationships was confirmed on the groundsof conformance of (8) with the measurement data.

In the case of the CSD04060 diode, reverse bias modelingproved to be more complex due to the asymmetric characterof temperature influence on the behavior of this diode belowand over 75 ◦C. An exponential and a hyperbolic relationshiphad to be used to represent the way the current varies withtemperature

Irev(Vrev, T ) = expa+ b · V75 + c · V 2

75

1 + d · V75 + e · V 275

(9)

632 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

where V75 is the reverse voltage drop Vrev at T = 75 ◦C

V75 =Vrev + f + g · T

1 + h · T (10)

and a to h are empirical constant coefficients.Model derivation has been described in detail in [13] and

[14]. An automated parameter extraction procedure was devel-oped based on the weighted least squares (WLS) method inorder to speed up the modeling process [13].

The aforementioned static electrical model served as thestarting point for extension to include device dynamics andelectrothermal coupling with the external environment whichtakes part in heat extraction out of the device. The latter issueis substantial due to the strong influence of the thermal domainon the electrical one. The thermal model will be presented inSection II-B, and the complete electrothermal coupling will bepresented in Section II-C.

The SiC MPS diode total capacitance is not significant,and its charge can only become pronounced under very highswitching frequencies on the order of several megahertz ormore. However, the junction capacitance value is relativelylarge for low reverse voltages. As the charge present inside anMPS diode can be attributed to the space charge layer of thejunction, the dynamic behavior of the diode was described inthe model using the well-known space charge layer capacitanceequation

Cj(Vd) = Cj0 · |Vd + Vj |1−Mj (11)

where Cj0 is the capacitance of an unbiased junction (Vd = 0),Vj is the junction potential, and Mj is the junction gradingcoefficient. The applied capacitance model is in full agree-ment with the capacitance–voltage characteristic included in themanufacturers’ datasheets.

B. Thermal Modeling Methodology

Electrothermal simulations require relatively simple but ac-curate thermal models of semiconductor devices and coolingassemblies. Thermal behavior can be predicted by CTMs (see[15]), which can be derived adopting the structural or the behav-ioral approach. The first one consists in the classic reduction ofthe system of equations resulting from the detailed distributedthermal model, whereas in the latter approach, the model isfitted by some optimization routine to the observed behaviorof a device [15].

The generation of structural DELPHI style CTMs, recentlystandardized by the JEDEC [16], requires the knowledge of adetailed thermal model which might not be available. Then, thebehavioral approach to the generation of CTMs remains theonly choice. These CTMs are usually generated in the formof ladder RC circuits based on the analysis of the measuredtransient temperature responses of a system, and the values ofmodel elements are found by fitting of multiple exponentialcurves to the thermal responses [17], [18].

The methodology proposed here for the generation ofCTMs is based on the NID method, and it allows straight-forward generation of CTMs based on the analysis of a single

Fig. 2. Recorded heating curves.

transient temperature measurement. The fundamental conceptsbehind the NID method, introduced in [7], can be outlinedas follows.

From the network theory, it is known that the time responsesof a linear circuit to arbitrary excitations can be found as theconvolution of a particular excitation and some time-dependentcharacteristic functions, which are the circuit responses tothe Dirac-delta function or the unit-step function. Because ofthe practical difficulties in the realization of the Dirac-deltafunction, usually, the unit-step function is used, and then, therecorded response is differentiated in time. Then, the time-constant spectra, expressed in kelvins per joule, representingthe spectral density of thermal resistances in the heat flow pathRτ are found performing numerical deconvolution. Finally, thetime-constant spectra of thermal responses can be divided intoindividual RC segments corresponding to the stages in theheat flow path such as the semiconductor chip, package, orheat sink, hence obtaining the Foster RC ladder representation.The Foster network cannot be physically correct since thecapacitances form a direct path between a heat source and theambient, implying the infinite speed of heat diffusion. Thus,the ladder should be converted to the Cauer form, in whichall thermal capacitances are connected to the ambient, whichis already physically correct.

The main problem during the creation of a CTM from thesecurves is how to divide them into individual segments sothat the particular model elements could be given a physicalinterpretation. In [8], the authors demonstrated that the time-constant spectra should be divided at the locations of theirminima. The major advantage of this approach consists in thefact that the spectra are divided into individual regions beforethe conversion of the ladder from the Foster to the Cauer form,which is a numerically unstable operation for a large number ofRC stages.

The CTMs of this type were generated also for the deviceconsidered here. First, the heating curves, shown in Fig. 2,were recorded for the device operating, as described in [19],without any heat sink as well as with a heat sink first screwedloosely and, then, tightly. The measurements were performedon the dual cold plate stand, presented in [20], with the T3Sterequipment. Then, the results were processed with the software

STARZAK et al.: APPROACH TO SiC MPS DIODE ELECTROTHERMAL MODEL GENERATION 633

Fig. 3. Integrated time-constant spectra.

TABLE ICAUER RC LADDER CTM ELEMENT VALUES

provided together with the measurement equipment so as toobtain the respective time-constant spectra. As can be seen, theheating curves of this device were recorded in different coolingconditions described in [19]. They were measured using theT3Ster equipment and processed with the software provided byits manufacturer to obtain the time-constant spectra. As can beseen, all the curves are almost identical during the first secondof heating. Then, they diverge significantly, and the differencesin the steady-state temperature rise values exceed even 40 K.

The time-constant spectra computed for the recorded curvesand integrated over time are shown in Fig. 3. Analyzing thisfigure, one can see that, in all cases, there exists a largemaximum round of 250 μs, which could be attributed to thesemiconductor structure. This is followed by a more blurredregion which ends at some 1–2 s. Then, more distinct peaksreflecting the heat transfer processes from the package to theambient appear in the spectra.

The generation of the Cauer RC ladder CTMs began withthe division of the time-constant spectra into four ranges at thelocations of the distinct local minima visible in the spectra.Then, the spectral density Rτ was integrated in the particularranges to obtain thermal resistances Rth. These resistanceswere then assigned to the time-constant value for which themaximum resistance value in a given range occurred. Finally,thermal capacitances were calculated based on τ and Rth

values.Next, the resulting four-stage Foster RC ladder CTMs were

converted into the respective Cauer. The final element values ofthe Cauer RC ladder CTMs obtained for the device cooled instill air with and without a heat sink are presented in Table I.

Fig. 4. MPS model—electrical domain (general idea).

For a detailed analysis of these values, refer to [20]. Here, it issufficient to know that the data from the table were used for thestatic and the dynamic electrothermal simulations of the devicepresented in the following section of this paper.

C. Model Implementation in HSPICE

The final HSPICE behavioral model implementation for theCSD20030 dual MPS diode is shown in Figs. 4 and 5, andmodel netlists are shown in Figs. 6 and 7. The ideal electricalstatic model (Idiode and the Rs block in Fig. 4) is the mostimportant component block.

The main components of the electrical model includethe controlled current source GIFwdRev, the current probeVIProbe, the series resistances RRs1 and RRs2, and the con-trolled voltage source ERsT. The latter is used for describingthe diode current as function of the voltage drop between theanode (node 100) and the cathode (node 200). This diode be-havior is modeled separately for the forward (EIFwd) and thereverse (EIRev) bias, with the transition region for low voltages(from VdLeft to VdRight) being modeled using interpolationformulas (EIFwdInterp and EIRevInterp).

The series resistance is modeled using resistors RRs1 andRRs2, and the controlled voltage source ERsT according tothe simplified formula

Rs(TempCase) = Rs|Tnom +RsThC(TempCase− Tnom)(12)

where Rs|Tnom represents the value of the diode’s series resis-tance at the nominal temperature (Tnom) of 300 K, TempCaseis the case temperature (node 310), and RsThC determinesthe increase of the series resistance with the diode tempera-ture. Analyses showed that higher order relationships can beneglected.

The thermal and the electrical domain are strongly coupled.Therefore, the current and voltage across the diode are usedto approximate the power dissipation in the device (controlledcurrent sources in Fig. 5; GTh1 component in Fig. 6), whereasthe device case temperature (nodes 1000 and 2000) is usedto compute current values of the electrical model parametersusing the controlled voltage sources: EIFwdInterp, EBetaT,EAlphaT, EIFwd, and ERsT.

The electrical dynamic behavior of the MPS diode is mod-eled by the Cqj component. As stated in Section I, the dynamicsof this block can be mostly neglected in comparison to thethermal domain inertia.

634 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

Fig. 5. CTM of the dual MPS.

Fig. 6. MPS diode model implementation in HSPICE.

The thermal model follows the principles outlined inSection II-A and accounts for the presence of two diodes inte-grated in one package (see Fig. 5). The possibility of assigning aphysical interpretation to the particular RC units simplified thedevelopment of this model topology based on previous work[4], [14] so as to include the thermal coupling between the twointegrated devices.

In the case of the considered CSD20030 diode, the ther-mal and electrical domains are strongly coupled. There-fore, the model was simplified so that electrical parametersare related to the temperature after SiC inertia (Rth1Cth1

and Rth2Cth2). This simplification was made deliberately inorder to improve the numerical properties of the HSPICEmodel.

STARZAK et al.: APPROACH TO SiC MPS DIODE ELECTROTHERMAL MODEL GENERATION 635

Fig. 7. CSD20030 model with free convection cooling implemented in HSPICE.

The proposed behavioral MPS diode model implementedin HSPICE allows very fast simulation in comparison to adistributed model. In addition, simulations can be performed forpower converter circuits that include MPS diodes. Simulationconvergence is dependent on the electrothermal domain cou-pling. MPS diodes with a weak coupling [small thermal inertiaCth1 ·Rth1 + Cth2 ·Rth2, small static thermal resistanceRth1 + · · ·+Rth4, as well as weak dependences in (5)–(10)and (12)] can be implemented in a single precision simulationenvironment kernel (e.g., PSPICE; see [4]). A double precisionsimulation environment (e.g., HSPICE) should be used in thecase of transient simulations involving diodes exhibiting strongelectrothermal coupling (such as some Infineon ones). In thiscase, convergence can be reached using the following simula-tion parameters: ACCT, RELTOL = 1P, METHOD = GEAR,MAXORD = 2, DCON = 1, ITL1 = 10 000.

III. EXPERIMENTAL RESULTS

A. Measurement Setup for Temperature-DependentElectrical Characteristics

In order to obtain measurements for a set of known valuesof junction temperature, an automated measurement stand hasbeen used. The device under test (DUT) is placed between twoPeltier modules and two cold plates in the measurement stand,as described in [20]. The cold plates are heated or cooled withwater circulating in channels inside so as to roughly obtainthe desired temperature both on the top and at the bottomsurfaces of the DUT case. Temperature is then set preciselywith the Peltier modules. The Peltier modules are also usedfor temperature stabilization based on measurements from fourthermocouples. This is done by a digital PID controller capableof stabilizing case temperature in a few dozens of microsecondswith an accuracy of ±1 ◦C. As power dissipation in the DUT is

zero or negligible during experiments, its junction temperatureis equal to the stabilized case temperature.

With the CSD20030 diode mounted on the stand, the mea-surements of static voltage-versus-current (V –I) characteristicswere performed using an analog curve tracer Tektronix 576for reverse bias and for low forward currents up to 50 mAby applying constant voltage or half-sine-wave current, respec-tively. For high forward current values, considerable heatingoccurs inside the diode, which would invalidate the assumptionof case and junction temperature equality. Therefore, in therange of 0.5–15 A, forward characteristics were measuredusing a pulsed-current setup shown in Fig. 8 and a dedicatedmeasurement board. The power MOSFET Q is normally turnedon. When the measurement is started, it is turned off for a periodof about 100 μs, which produces a current pulse of amplitudeJdut in the DUT and enables voltage drop measurement. Inreality, the Jdut source is a dc power supply operating in itsconstant current mode; the dynamic properties of its output(current invariability) are improved with the choke L. Thediode Dbias provides an additional voltage drop required by themeasurement board as well as prevents the DUT being turnedon by the voltage drop across Q. The V –I characteristics weremeasured at three different temperatures of 25 ◦C, 75 ◦C, and125 ◦C. The results are shown in Fig. 9.

The same setup was used to measure voltage-versus-temperature (V –T ) characteristics for several values of forwardcurrent. A constant current source Jaux (see Fig. 8) of 5 mAwas used to measure the V –T characteristic of CSD20030’ssecond diode Daux. Based on these results, V –T relationshipswere represented with the second- and first-order polynomialsfor Ddut and Daux, respectively.

Forward voltage waveforms were also recorded for bothdiodes with the DUT subjected to step current pulses of 13 min,which enabled attaining thermal steady state or shorter if themaximum junction temperature of 175 ◦C was reached. Using

636 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

Fig. 8. Setup for pulsed V –I forward characteristic measurement(simplified).

Fig. 9. Comparison of (points) measured and (lines) simulated static forwardcharacteristics of the CSD20030 diode for different junction temperatures.

Fig. 10. HSPICE test circuit for the MPS diode subjected to a current step.

Fig. 11. Diode junction temperature TJ , case temperature TC , and powerloss Ploss waveforms (lines—simulated; points—measured) after applicationof a 7-A current step to the DUT.

V –T functions established previously, internal temperaturesof the investigated diode were calculated. Selected results areshown in Figs. 11 and 12.

Fig. 12. Diode voltage VF and junction temperature TJ waveforms(lines—simulated; points—measured) after application of a 7-A current step.

B. Electrical Model Parameter Extraction

The parameters were estimated using the WLS method forthe linearized model equations in the current-dependent form.Model linearization was performed using the Gauss–Newtonapproximation method applied to the Taylor series expansion.This procedure has been described in detail in [13] and [14].The parameter values extracted for the CSD20030 diode canbe seen in Fig. 6. Parameters of this model extracted for someother SiC MPS diodes can be found in [4] and [14].

C. Experimental Verification

A comparison of CSD20030 forward static characteristicsmeasured and simulated using the model presented is shownin Fig. 9 for different temperatures. The χ2-test showed that themodel was consistent with measurements at a confidence levelof 0.99. This could not be obtained using models supplied bySiC diode manufacturers [4], [5] (see Fig. 1).

Example results of the diode’s simulated time response to astep current of 7 A (see Fig. 10) are compared to measured onesin Figs. 11 and 12. As can be seen, up to 50 s, a good agreementbetween simulation and measurements was obtained. Afterthis moment of time, the simulated junction temperature risesabove real values. This is due to the simplified ambient modelwhere, for high temperatures, a nonlinear increase in diode casecooling intensity occurs. This effect should be reduced when aheat sink is applied.

IV. CONCLUSION

The proposed method has enabled deriving an electrothermalmodel of an SiC dual MPS diode. It has been validated and hasshown a very good agreement with measurements, in contrastto the models provided by device manufacturers. The mainadvantage of the proposed model is its closed form, whichallows its straightforward implementation in the behavioralextensions of modern SPICE simulators.

A next step will be the improvement of the dynamic electricalmodel so as to accurately render dynamic power loss, which isimportant when high-frequency switching is involved.

STARZAK et al.: APPROACH TO SiC MPS DIODE ELECTROTHERMAL MODEL GENERATION 637

APPENDIX

PARAMETER EXTRACTION PROCEDURE

This procedure has been presented and described in [13] butis included here in brief for paper completeness. All parametersin this paper were estimated employing the WLS method, usingthe objective function

J = (Δz)TR−1(Δz) (13)

where Δz = z − h(x) denotes the estimated measurementresiduals. The measurement vector z is constructed as follows:

z = [z•]n×1

= [Vfwd,1, Ifwd,1, . . . ,

Vfwd,i, Ifwd,i, . . . , Vfwd,n, Ifwd,n]T (14)

where (Vfwd,i, Ifwd,i, Ti) is the ith measurement pair (volt-age, current, and temperature). The nonlinear function h(x) =[h•,•](2n)×(n+7) relates measurements to the system state vectorx, which contains information on the estimated parameters p•and the residuals Δz•

x = [x•](n+7)×1 = [Δz1 · · ·Δzn p1 · · · p7]T . (15)

The estimated parameters are associated with (1), rewrittenhere in a more convenient form without the explicit exponentfunction

Vfwd,i = [(p2Ti + p1) + p3] · log Ifwd,i + p4Ti + Ifwd,ip5

·[1 + p6 · (Ti − Tnom) + p7 · (Ti − Tnom)

2]. (16)

The nonlinearity of (16) is overcome by the Gauss–Newtonmethod applied to the Taylor series expansion, which leads tothe iterative solution of the so-called normal equation

(HTR−1H) ·Δx =HTR−1Δz (17)

xNEW :=x+Δx (18)

where the nonvanishing elements of the Jacobian matrix of h(x)

H = [H•,•]2n×(n+7) =

[∂h(x)

∂x

](19)

are calculated using the following rules:

Hi,n+1 =∂Vfwd,i

∂p1, . . . , Hi,n+7 =

∂Vfwd,i

∂p7(20)

H2i,i =1, H2i−1,1 = Vfwd,i, H2i,1 = xi (21)

for i = 1, . . . , n. The weights R−1 take into account the accu-racy of each measurement and are computed as the reciprocalsof the measurement standard deviation (δ2Vfwd,i, δ

2Ifwd,i)

R = diag{δ2Vfwd,1, δ

2Ifwd,1, . . . , δ

2Vfwd,i, δ

2Ifwd,i, . . . ,

δ2Vfwd,n, δ2Ifwd,n

}. (22)

ACKNOWLEDGMENT

The authors would like to thank Prof. P. Grybos from theAGH University of Science and Technology, Kraków, Poland,who kindly enabled carrying out HSPICE simulations.

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638 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 2, FEBRUARY 2013

Łukasz Starzak (M’05) received the Ph.D. degreein electronics from the Technical University of Lodz(TUL), Lodz, Poland, in 2010.

Since 2002, he has been with TUL. His interestsinclude power semiconductor devices and electronicconverters.

Mariusz Zubert (M’12) received the D.Sc. degreein computer science from the Silesian University ofTechnology, Gliwice, Poland, in 2011.

He is currently with the Technical University ofLodz, Lodz, Poland. He is an author or co-author ofover 110 publications.

Marcin Janicki received the Ph.D. degree in elec-tronics from the Technical University of Lodz(TUL), Lodz, Poland, in 1999.

Since 1999, he has been with TUL. His interestsinclude inverse heat transfer problems and VLSItechnology.

Tomasz Torzewicz was born in Poland. He receivedthe M.Sc. degree in electronics and telecommunica-tions from the Technical University of Lodz, Lodz,Poland, in 2010, where he is currently working to-ward the Ph.D. degree.

Małgorzata Napieralska (M’12) received the Ph.D.degree in microelectronics from the Institut Nationaldes Sciences Appliquées, Toulouse, France, in 1991.

She is currently with the Technical University ofLodz, Lodz, Poland. She is an author or co-author ofover 120 publications.

Grzegorz Jabłonski (M’12) was born in 1970. Hereceived the Ph.D. degree in electrical engineeringfrom the Technical University of Lodz (TUL), Lodz,Poland, in 1999.

He is currently an Assistant Professor with TUL.

Andrzej Napieralski (SM’91) received the D.Sc.degree in microelectronics from Université PaulSabatier, Toulouse, France, in 1989.

Since 1973, he has been with the TechnicalUniversity of Lodz, Lodz, Poland, where he is aProfessor.