a behavioural thyristor model was built. After the DC analysis, eq is made equal to g_var and the equation OCR OutputSchottky and fast recovery diodes. On the basis of this diode model. 'liie solution being stored, g__var no longer has any significance.forward and reverse recovery) have been experimentally verified on The solution value _g_var is then passed to a state variable g_state.are fully described. The dynamic features (junction capacitance,

g_ var : eq = Olanguage. The diode model and its automatic parameter exnactionThey are implemented in MAST, the Saber simulator modelling eq = g_var- exp(-k.g_var)them into the physical constants of the underlying model equations.

equations to be solved are :directly available databook parameters and automatically convertDC and transient analysis. During the DC analysis, the relevant'liie power diode and thyristor models presented here usevariable that allows the equation expression to be changed betweencomponents.the conductance g from the external parameter k. eq is a valcharacteristics, and finally to choose the most appropriate

The template provided in figure fl uses this method to extractof device features on the design, to identify the required devicethat reduces calculation time.level, such a model family would greatly help to study the infiuenceequations section, it is transformed into a simple artificial equationexplores concepts with different devices and topologies. At thistransient analysis. As this equation cannot be removed hom thethe beginning of a circuit design, the application engineer typicallybecomes redundant but is still to be solved at each time step ofcharacterization was the motivation of the present work. Indeed. atis recorded in a state variable. The conversion equation thenThe absence of advanced generic models with simplealgorithms. Once the DC analysis has been perfomied, the solutioncommercial components.conversion equation is solved during the DC analysis by the Saberthe basis for building libraries of characterized models forex¤·action in the equations section of a MAST template [6]. Theadvanced models are mainly used by simulator manufacturers as

The first method consists of performing the parameterremains a nontrivial issue for the end-user. For this reason, thesimulation.for parameter exaaction are described in related papers, thissolve such equations for parameter extraction at the beginning of aestimated from measurable extemal parameters. Even if proceduresApplied to this example, two general methods are now presented toconcentration, mobility, etc) are not readily available but must beClearly, the intemal constant g can not be found analytically.required physical input constants (carrier life time, doping

intemal device physics. Their characterization is difficult since the(1)g—¤¤<r><—/<-g>= 0

The second category comprises more advanced models based oninclude intemal effects such as diode reverse recovery. parameter k through the following conversion equation :these models are insufficient for accurate simulations as they do not the conductance g of a resistor model is related to an externalsemiconductor devices as switches. Although fast and easy to use, way to calculate the intcmal constants. Let us consider such a casemodels. The first consists of simple models that basically represent is not always possible. In some cases, solving equations is the only

provide two categories of power semiconductor device expressed as explicit functions of the external parameters. But thisIRCUIT simulators, such as PSpice or Saber. generally Parameter extraction is easy if all internal constants can be

I. Introduction II. Solving Equations during DC Analysis

compared with measurements. A new thyristor model featuring reverse recovery is also presented.and robustness. It includes junction capacitance, forward and reverse recovery phenomena. For all these effects, simulation results are

Relevant features found in existing models arc combined to get an efficient power diode model in terms of accuracy, calculation speedthis conversion with the MAST modelling language, techniques are proposed to solve equations at the beginning of a simulation.available databook parameters and automatically convert them into the physical constants of the underlying model equations. To achievedoping concentration or mobility. Unfortunately, these are generally not accessible to the user. The models described here use readily

Abstract - Most power semiconductor models available in circuit simulators require physical input parameters, such as carrier lifetime,

Geneva, Switzerland.CERN'CN'95'16

CERNI UWIIIIIIIIIIIIIIllllllllllllllllllllllll European Laboratory far Particle Physics,Alan COURTAY CERN LIBRARIES, GENEVA

Automatic Parameter Extraction

MAST Power Diode and Thyristor Models Including

Brighton. UK, September 1995 S6/K 1 '?Presented at the SABER User Group Meeting CN 95/16

1/_ L r" W J ` A ‘ QA } kx rg [\+ { ·\, *{\ »

The leakage current I, and the series resistance Ron are directly OCR Outputgt8 = lim (4) gives sufficient accuracy and makes parameter extraction easier.

simulation, one exponential law to describe the junction voltageequadon (2), g can be approximated by gn with n sufficiently large :curve is still governed by the series resistance. For power circuitAs the sequence limit when n tends to infinity is the solution ofoperation, the diode current remains constant, and above V0,. , themodel characteristics near-zero. In the reverse-biased region ofgin = flat) l3)different parameters L and Vb. This only has a small effect on thesequence {gn} is defined by :several ideal diodes in parallel using the same relationship (8) withthe crossing point between the curve y=f(x) and the line y=x. Thelevel recombination and emitter recombination effects by adding

In the graphical illustration of figure 2, the solution g appears asSome models ([2] and [4]) distinguish high·level injection, low

(2)8 = f lg) .

itself : (8)I - I exp -ld .1*I vb Iform where the unknown variable g is expressed as a function ofsolution. To build this sequence, the equation has to be written in a junction voltage Vj by the exponential law :of calculating the first elements of a sequence that converges to the resistance Rim in series with an ideal diode that represents theequation is solved in the parameters section. This method consists dynamic effects are going to be added. It consists of a contact

The second method avoids this inconvenience since the The diode static submodel is the basis to which all diodesophisticated encrypted models.(5g.l), it becomes more problematic, if not impossible, for

IH. Diode Static Modelparameter as a val variable. If this task is easy for a resistancethe main template and adapted in order to accept the convertednot be called from the netlist section. It has to be incorporated into

Fig. 2. Example of converging sequenceA template requiring the converted parameter as an input data canvariable cannot be passed as an argument to a lower level template. a¤ e= z· a> g¤ er

This is the main disadvantage of the method. The value of a valconverted parameter during both DC and transient analysis.transient analysis. Only the val variable g_val is equal to theundefmed during DC analysis and g_var is equal to zero duringwhich is the simplest cquadon possible. Note that g_state is

g_va: : eq = O

eq = g_vat

to be solved reduces to g_var=0:

y=xY=f(X)Eq. 1. Example ofparamclcr conversion usinz the equations section

template can directly be called from the netlist section.i(p->m)+=(v(p)—v(m))*g__valprovided solution is a parameter variable, so the resistance

# The resistance template has 10 be rewrittensolve the conversion equation for any value of k. Note that the

g_var : eq=0 The template given in figure 3 uses the described method toequations { Then condition (5) applied to f;(x) is valid for k > exp(l) only.

tog — -¥ - file)eq = g_var

k > exp(l). equation (6) has to be written as follows :g_val = g_s1a1eUnder this form, condition (5) is only true for k < exp(l). Forelse {

(6)tt = ¤<t>(—l<-a) = Ale).eq = g_var - exp(—l;*g_var)

previously given in example :g_vaI = g_varsatisfying (5). This is the case with the conversion equationJ (dc_d0main) {the equation then has to be written into another form (2)

values {condition (5) may not be verified for all their possible values. lf so,

when (dc_d0ne) {g_s1‘ale = g_vaI} As the function f(x) depends on the external parameters,val nu eq has to belong to this neighbourhood.slate nu g_sm1e neighbourhood of g. To ensure the sequence convergence, gg then

of x, but it is critical if condition (5) is only valid in theval nu g_valelement go can be arbitrary if this condition is verified for any valuevar nu g_varAppendix A explains why (5) is necessary. The choice of the first

dx ,,8number k (5)l<< l .electricalpm

element templaze parameter_c0nversi0n p m = k following condition on the function f(x) is satisfied :It is important to note that this method only works if the

Ez. 4. Diode DC Model

switch. The subcircuit Ri, // L now acts independently of the OCR Outputzero. After t> ts, the ideal diode can be regarded as an open

Von/,, ls Vd ideal diode then gets blocked since its current has decreased tothrough the controlled source. At time t= ts, Id reaches — llrrm. The

IAnegative voltage across L commands a constant reverse current Irrmdetermines the diode current, As Li linearly decreases, the constant

For t < ts, the ideal diode conducts and the external circuit

provides the typical diode reverse current waveform of tigure 6.Turned off with an inductive load, the circuit model (tig.5)\na¤

ROI!

Hg. 6. Diode current tum-off waveform

-lrrrn

. where Von corresponds to a current of l Amp through the diode.-tm/10

1nB *_—_·. `(l+ I

(9)V

v_ = é. . . . barrier potential V., is also easily extracted 1

dir/dtprovided by the diode I-V characteristics (tig;1). The junction

¢ Id lfo iEg. 3. Solving the conversion equation in the parameters section

by the DC submodel of the previous section.the diode works as a generator under zero-bias. It was thus replaced

[rp m :1/gseries with a constant voltage source VJ. is not satisfactory since

#Resistance template directly called from rietlist section I 0¤g1nal model [1], composed of a bmary TCSISIHHCC-ROB / Roff incurrent source (ftg.5). Note that the DC representation in thesimply consists of a resistance, an inductance and a controlled

gzgg submodel proposed by [l] is extremely easy to implement since itg9=_]n(g8)/k _ Based on a macro-modelling approach, the reverse recoveryg8=-In(g7)/kg7=-ln(g6)/k

Fig. 5. Diode reverse recoverv modelg6= `ln(g5)/kg5=-In(g4)/kg4=-ln(g3)/kg3=-ln(g2)/lcg2=-ln(g1)/kg1=-In(g0)/kg0=0_ 5 Vi- I 25 1. RL

i=K.VL

8=89g9=exp(—lc*g8)g8=expl-k*g7)g7=e¤t>l-i¢*g6lg6=exp(-k*g5)

Raig5=exp(-l~*g4lg4=exp(·lc*g3)g3=exp(-lc*g2)

A°°°°2 =¢·¤p - °“g

it should be an open circuit.g;=€xp;current can flow through it. It then works as a short circuit whereasg0=UDuring this time, the diode remains conducting and a reverseUr (l<<€1P(l l) {doped region during conduction, takes some time to be removed.Pammelefs {diode is turned off rapidly. The excess charge, stored in its lightly—”“”*b€’ 8·80·81-82·83»84»85-86»87»88-S9power dissipation. The phenomenon occurs when a forward biasedpower circuit designers as it introduces overvoltages and high”W”b€' k

Reverse recovery is the most important diode dynamic effect forelectrical PJ"element template parameter_c0nversi0n p m =k

IV. Diode Reverse Recovery

excess charge in the diode lightly-doped region. OCR Output

constants and variables such as transit time, carrier lifetime orcomparison between measured and simulated tum—off waveforms.

currents of the circuit model [l] can thus be related to physicalby an optimized geometric circuit design, Figure 8 shows the

correspondences between both models. The internal constants andcurrent slopes, the parasitic inductance Lp was reduced to 130 nl-I

developed by Lauritzen and Ma [2]. Table l gives thetested in the chopper circuit of igure 7. In order to get steep

shown in Appendix B to be strictly equivalent to the micro—modelTo validate the model experimentally, a fast recovery diode was

different blocking conditions (fig.9) Moreover, this model isstill gives good agreement with measurement when simulated in

= I" parameters have been calculated for a given tum-off waveform, it17 ( )1nlO 1 1 dl ` -;- - ———- % 1,,,,,iQ” (2 mioiidzia wide range of operating conditions. Whereas its internalaccuracy, but the reverse recovery circuit model remains valid over

Macro-models are often considered to have a narrow range of—-——— - [“ -L-] -— = O i6 tr.,) -—~1.w— Q-V <>1 1 d1 " . 1¥(2 ollThomson fast recovery diode ESM 244— lOOand Qrr are the following :

Eg. 8. Simulated (a) and measured (b) tum-off waveforms of theto calculate RL and K. The geometric relationships linking lrrm, trr

(b)parameters, but if all three are provided, Irrm and trr only are useddiode template is implemented to handle any combinadon of two

100nscurrent tum-off waveform, the third one being redundant. The ~60 .. ·80...3-zotwo parameters out of Irrm, trr and Qrr are needed to define the

Note that once the tum-off conditions Ifo and dlf/dt are given, only.4 ..·...· +.·l ·-~- >{ ...--- 4 ..-.-... ; ........ L ...... .{ -15-40L ...... i..recovery is removed from the model (as well as forward recovery).

If not, the internal constants cannot be calculated, and reverse-—- -----· A-10

2 dz(15)

OE Q Vdt · T

(14)lm < rn.dl' ..1 5

40 L..

y ( J).;_.; Q"' < 2 dz..- ..........

1* dl J 10wt 2 ld

reverse current geometry (fig.6), must be respected :

<A>template. In particular. the following conditions, imposed by the (a)(V)These parameters are initially checked to be valid by the

reverse recovery input parameters.

$|3¤ SOL: $4].4u $|3.6¤ (1)Slléu 54Muguaranteed to verify condition (5) for any acceptable values of theis J 4adescribed in section H. Note that the corresponding function f(K) is

solved in the parameters section, using the second methodfixed, RL can be directly deduced from (ll). Equation (l2) is

..».. .... r....]..A/ ..... 1.is no ...... 1 ...... J.probe measuring dIr/dt for the controlled current source. L beingnegligible compared to Vj. With K>> l, the inductance acts as a 40% —Z0 ,. -Z0i·6D ..is arbitrarily set equal to 10 pH. This very low value makes VL

*v•·f··w······—t··A degree of liberty exists between L and RL, so the inductance rl tl_.LQ’£

D.] Z0L dt RL

d IL Lg K +

1 -1

-K=_.m(_;) _};-cxpl;.L|| =f(K) (12)1041 uz "

I5, $0

(A) (V}

Dbdc

(11)—l L; ,"-]m{EL) RL 1nlO dtHg. 7. Chopper circuit for measuring diode switching transients

(1 l) and K is obtained by solving equation (l2).related to these external parameters. The time constant is given byfor speci5ed Ifo and dlr/dt conditions, L/ RL and K have been s F >—<source). As most power diode databooks provide Irrm, trr and Qrr LL Power MOSFETL/RL and K (the coefHcient of the voltage controlled current

In fact, the model only depends on two intrinsic parameters20 v | sour

200 pH

out(10)R _ - Id(1)=I,,,,,.e.><p—£{%]0.65 rz

a time constant L / RL :external circuit, and imposes an exponential current decrease with

Lv·I30 u.H

forward current, the voltage overshoot peak value Vg, and itsinjected carriers. As this concentration is established by the Ez. 10. Simulated (al and measured (la) forward recovery waveforms OCR Outputconducnvity rapidly increases due to the increased concenuation of

(b)initial low conductivity in the lightly-doped region. Thistransient, a voltage overshoot develops across the diode because ofswitches from its nonconducting to its conducting state. During this j j j sonar

The forward recovery phenomenon occurs when the diode

JUL. VdV. Diode Forward Recovery

. . jig

L, ,RK 2. 1 . V 1= 1. 1+— 14 = ¢

M = T — c = +—- 1 +1 + (1) KL I (' (I) `(O) RRK " RK RL L1 LL 1l L L

. qm(1)= KL 1+;1L(1)1 = KL 1+-RK 1 RLK1 - L(a) (A)

variablu in the circuit model of iigure 5 (cf. Appendix B)Equivalent expresions for IAuritzen`s micro-model parameters and

edi xm om ${9311 nb mia wl. wth 992u sm. wl- n1.5 .1 -20

TABLE I

0 J -11

in different blocking conditions 2/dir/dt:Eg. 9. Simulated (a) and measured (b) reverse recoverv cuncms

(b)

..1 ...- ---3- ···--· (··

...... , ....... _ ....... , ...... 1. W] 1] .._... E ...... L ...... Q .....is` 0

.;.4 ...`..... mum ··¢zng s_, ,..... ; ...... 4.

$1 Wl .... .-4 ----·-- L- ··---· 1 ----—- <]--—\---:-

1 ‘‘‘ J. .... L. ’°] "t ···‘‘· 2 ······ T ‘·‘··‘ 2 ·‘‘·‘‘ J] :

xs, 20

IA} (V)

D&F¤rvrard

above equations are easily adapted to the reverse recovery circuit.Using the equivalence between models [1] and [2] (table 1), the

(19)(1)<0 : V,,_(z)·= 0.dqm

l+iId(1)+—(A)

(a)

TT1 -() D 1 R0W (1) *‘ HQQ

W % % 7% m % ln (1} DT R0*° FTcw) , l.,<r>

-xs-+ -—···—- + -·—·—-·>— /*0) W (f) = TL.10+ ··‘· " ‘’‘‘ ' `'`` "" ```-’` ‘? ‘‘‘‘‘‘ ?\>`~i `‘‘‘ 77 ‘‘`‘‘

(18). (O20 - V,,.(f)=lG(!)··lQ(f).s+" ``-_- T dqm

forward recovery submodel consists of the following equations :a more satisfactory forward recovery than in [2]. The improvedmodel, it uses the same equadons for reverse recovery, but presents

\0+ ‘‘‘‘·········· \···\····· al. in their power diode model [3]. Derived from Laurit2en'scircuit. The analydcal description of Vm was proposed by Batard et

UT ···‘··· T ······ \ '········ P dependent voltage source V, in series with the reverse recoveryThe voltage overshoot at turn·on can be modelled by a

dif/dt.

mrresponding time tr, mainly depend on the rate of current riseKsvm.@v:w Cmm fur Uffmm Tum-u

H2. 14. Thvristcr model OCR Outputoff-state under positive voltage, Rog blocks the current that wouldRm, / Rog controlled by a digital gate. When the thyristor is in the(fig.l4). This scr_logic module is basically a binary resistanceresistance with a module that realizes the thyristor static propertiesthe diode reverse recovery circuit by replacing the contactalso includes reverse recovery. The model is directly derived from

The new thyristor model presented here has a digital gate and vt I zi t Rlproviding more accuracy in the main circuit response. i=K.VLthyristors, the simuladon complexity is greatly increased withoutcomponent. If the simulated circuit has a large number ofThe analog gate obliges the user to add a triggering circuit for eachfollowing the same micro-modeling approach as for their diode [2].

A thyristor model with reverse recovery is proposed by [5]phenomenon like the power diode.

Refipulse. When tuming off, the thyristor presents a reverse recovery _ Gm ¤.......I / | r¢rr_logrc

ROI!be regarded as diodes that turn—on with a gate terminal currentnetworks due to their high power-handling capabilities. They can

AnodeThyristors (or SCRs) are widely used in power switching

VH. Derived Thyristor Model Critical Turn-on tq, Vbo

Turn·on Transient

Junction Capacitance Ccomplexity makes the model very convenient for the user.

Reverse Recovery lfo, dir/dt, Irrm, trr, Qrrcan then be added one by one. This gradual approach intocomputational time. When more insight is needed, dynamic effects I-V Characteristics Ron, Von, Is, init_statethe DC submodel may provide sufficient accuracy with lowfor his simulation. For example, in the first approach of a design,

Features Parametersparameters, the user is able to specify the features to be retainedwith different degrees of accuracy. By assigning zero to some input

Ttiyristor input parametersBecause of its modular structure, the diode model can be used

TABLE H]

Eg. 13. Complex: power diode modelthyristor Graetz bridges.that this capacitance helps convergence in the simulation ofeasily replaced by the diode junction capacitance submodel. Notesimplicity, the junction capacitance is a constant C but it can beprovided V.], is positive at the begnning of the simulation. Forparameter init_state allows the thyristor initial state to be specified,

vt. I g 1. Rt. The thyristor input parameters are listed in table 3. Thei=K.V1.

CKVJ) (zz)ru = ·.¤p-¤q -.1é%[%{%,,,]]given by equation (28).calculated from tq and the specified reverse current waveform, isthe correspondence between ir and qu,). This critical current,Vm I \decreased to the corresponding critical current ir; (table 1 showsequivalently when the current through the inductance L hastransition Ro,. to Roy; is uiggered when qm has decreased to qm., , or

RDR the relevant internal parameter to command thyristor tum·0ff : theindependent of these conditions. This critical charge turns out to bebut it can be related to a critical amount of stored charge qw that is

Ama:conditions (I0, dlr/dt, lrrm...), its range of validity is quite narrow,guarantee thyristor turn-off. As tq is measured for given blockingJunction Capacitance CI , V; , C; , V;time interval for which V.], must be maintained negative to

dir/dt, Vfp_ tfpForward Recovery from the lightly-doped region. The parameter tq is the minimumfor sufncient time to have most of its excess charge qm removedReverse Recovery Ifc, dlr/dt, Irrm, rrr, Qrr

The thyristor remains in the on—state until it gets reverse biasedI-V Characteristics Ron, Von, Is losses.

order to reproduce accurately turn-on transients and switchingvoltage Vu,. The turn-on time ty is the parameter to be adjusted in

ParametersFeaturesgate active. It will also happen if V.]; exceeds the break-over

The transition Rog to Rm, occurs when V.], is positive and theDiode input parameters as in the diode model.

OCR OutputTABLE II flow through the diode. In the on-state, Rm, is the contact resistance

Fig. 15. Diverzina seouenw

gi =RLK 1+-- .iR +i,_—i,_ L (Bll)g·5 /| z= e¤ z¤ \ ea sw X }()RLK OCR Output

(B10)i =R Ki +——£—+" L h{R ll J— iRLK RLKmodel (table 1). Transforming again equation (B7)linking r and qm to the parameters and variables of the circuit

Comparing equation (B9) with (B2) gives the relationships

(B9)id = K 1+--xé-+iLLigRLK) dty*f(><>>=><

(B8)id=K.VL+g_i+r`L

or equivalently,

which sequence {gn} diverges. xd ¤ l+lR’ +lL (B7)Figure 15 exhibits a function f(x) that does not satisfy (A4) and

Kirchhoff’s current law givesthe curve y=f(x) cuts the line y=x with a slope in the interval [-1,1].to be derived from the circuit model. Referring to figure 5.In other words, the described method to solve g=f(g) applies ifdemonstration, the micro-model equations (Bl) and (B2) are goingequivalent since they are ruled by the same equations. As a

Regarding reverse recovery, the models are also strictlygi l < lwhich is equivalent to (8).

The above inequality is guaranteed if we have

(A3) I ="f<g,+.>—f<e.>I < I gun —g. B6 ( ’l` T V: —-$4 -*7 (rm +1)i°°°p(2v; ior equivalently,

By substitution into (B3), we getgn; —2,.r| < mm -2. <A2>

(B5)qc =(·t+Tm).ldThe sequence convergence implies that

DC conditions, (B 1) and (B2) givecircuit model. (B3) will now be compared to equation (8). Under(AllSM = f(g,.)

Referring to figure 4, equation (B4) is clearly valid for theLet {gn} be a sequence defined as follows z

Rm, is the internal series resistance.in the mrameters section vi is the junction barrier potential,

Necessary condition 10 solve an equation Vj is the junction voltage,L` is the diffusion leakage current,APPENDIX AT is the carrier lifetime,

Tm is the transit time.

qm(t) is the charge in the 1ightly—doped region,where qs(t) is the junction charge varaible,

simulation of choppers, pole inverters and Graetz bridges.(B4)Vd(I)=Vi(l)+RO”.[d(l;r

by the end-user. They have been succesfully used at CERN in thepossible to make accurate models that can be easily characterized

However, the examples given in this paper show that it is (B3)q (z)=IQ.·r exp -1 {bench and parameter Etdng by means of a computer.only be processed by extensive measurements on an experimental

dz Tprocedure for parameter extraction. The model characterization can im =<¤>+¤<r> (B2)d i€equations that it is nearly impossible to establish a systematicinternal constants so closely tied and entangled in a set of complexadvanced models, such as the IGBT model by Hefner et al.[7], have

(B1)[dw: 4,0):4,,0)internal constant to be related to measurable quantities. Somemodel equations have to be rearranged in a way that allows every

recovery behaviours of Lauritzen’s diode model [2]However, the limits of this concept must be acknowledged. TheThe following set of equations describes the DC and reverseextraction goes beyond the scope of semiconductor modelling.

beginning of a simulation, the concept of automatic parameterand Micro~ModeI [2]Based on general methods for solving equations at the

Equivalence between Circuit Model [1]

APPENDIX B

as the corresponding var variable. OCR OutputTo avoid division by zero, the equation is solved by choosing l/R0

(L/P) = O(C15). m 2 with d = -T.rfp

T

V»·(’n») : Vrp(d b](c lp]; b (4 by -—+———+— -———+—- $3 Ro B R0 R0 5 Ro

solve the systemAt nme tfp, Vm attains its maximum value Vfp. We thus have to

following form Zunknown, R0. (C8) is solved in the equations section under theBy substituting (C9) into (C8), we now get one equation with one

,,, - 1+ - t + -—-— D ( 1 ] dl! R0(C5)VA!) = ————-g-——— 1 Tdz 1

(cis)A: £Q,+@.;€’Q -4g" Z fpg% VIP R0 R0_ ‘¤ ( Tm) (-1) dz 1 - 1+-r-11-exp-— +—--— I5 1 1 dl I R0

(C12)c=11+—”—zipW U) = (C4) (7;) T

dl!L®=Mw-%w Cll ( )dr b = ——

The voltage drop Vu, becomes (C10)with a = i-’i(Tm +1},;,):,;,

qe(z)==1-l+”—z—1l—e 2acdl .. Ll;(L)(xp(-€))] (C3) dt 1 ·c(C9)

<_.¤¤> ,+._..(°""°’ NK-l ; ,p g)I1 __ = fpUsing (B 1), q¤(t) at turn-on is defined as

Using (C7), B can be expressed as a function of R01Q] dt ·c 1dl - 1- qm([) ¤ +L

the boundary condition q,,,(O)=O ,,, dz T dg 1 ·-(Tm+t )z +——-— ..1+.; [ +__ T fp YP d[{P1 T “ f T] d[fR0=The charge q,,.(t) can then be found by solving equation (B2) with (cg)

dz 1at 1 1};,.7,,.

Ido)-`=—·:—.[dl I

turn—on (T,,_+¢>r +———— . ] . ’*’ LP * ‘*We assume that the diode current slope dlf/dt is constant at dt 1 2E5 ijdr, Ro p T »a1, R0

£L__.__(C7)v,_ = ......2Parameters R0 and D.

Automatic Extraction ofthe Forward Recovery

The system to be solved now becomesAPPENDIX C

DT ”` dl I, R0M dl 1 -(T +t)t +-

(C6)V = ((1)

expressions for Tm and qs in the circuit model.we obtain a simplified expression for V]By comparing (B12) with (B1), we obtain thc equivalent

q 1-+2L 1 .... 1+ Q. RL RLK LL 1 1 211<L1+ .-ik +1;- 1<L1 + _.L L()}(B12;1 {K RL1 {J R i, = LK

By using the polynomial expansion

IO

Electronics Specialists Conference, Cambridge, MA, June 1991.model implemented in the Saber circuit simulator", IEEE Power

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Drives, No. 399, pp. 447-152, October 1994.Conference Publication on Power Electronics and Variable-SpeedHigh Power Diode Model with both Forward and Reverse Reoovery", IEE

[3] C. BATARD. D.M .SMl’1'H, H. ZELAYA, CJ. GOODMAN - "New

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REFERENCES

measurement.

work from the technical point of view and gave me his support withI also wish to thank Frederick BORDRY who followed this

research, for having proposed it to me.I am grateful to Rudi ZURBUCI-EN, who initiated this

when writing this paper.

I would like to thank John EVANS for his useful discussions

ACKNOWLEDGEMENTS