PSPICE Implementation of a ew Electro-hermal Model For High Power Diodes F.Profumo, * S.Facelli, * B.Passerini, **S. Giordano Dipartimento di Ingegneria Elettrica Politecnico di orino, C.so Duca degli Aruzzi 4, 9, orino, Italy. el. 39--56477 Fax 39--564799 e-mail: profumo@polito.it *Green Power Semiconductors ia XX Settemre /5 6 Genova, Italy el: 39--5854 Fax: 39--5834 e-mail: info@gpsemi.it **Dipartimento di Ingegneria Biofisica ed Elettronica, Università di Genova, ia Opera Pia -a, 645 Genova, Italy el: 39--353 899 Fax: 39--353-777 e-mail: giorda@die.unige.it Astract-- In this paper, a new PSPICE implemented electro thermal model of high power diodes is descried. he model is ased on an approximated solution of the minority carriers diffusion equation. In the first part of the paper, the model is riefly descried pointing out the assumptions on which it is ased and descriing the implementation approach. hen, the model is validated y means of comparison with experimental measurements and with the results otained from other two reference models. I. IRODUCIO In the field of power electronics, electro-thermal models are essential tools for predicting devices long term reliaility, designing the optimum cooling system and accurately choosing the circuit protections. he rigorous approach to the power device electro-thermal simulation requires generally too much effort and it is not a viale choice for circuit simulations. In fact, the complete solution of the prolem would require to account for non-linearities and 3D effects in oth the electrical and the thermal models that are extremely demanding oth in terms of computational resources and model definition complexity. However, it is generally possile to reduce the complexity of the prolem y taking some simplifying assumptions and y limiting the types of studies to which the model is devoted. A common approach consists in using a standard circuit simulator (like SPICE, SABER, etc. ) to implement the model of the circuit to e analysed. his generally gives a good understanding of the gloal circuit functioning without eing extremely demanding in terms of computational resources. Unfortunately, the device models that come with these simulators are generally not adequate when the attention of the design engineer must focus on the component ehaviour. he method that has een widely adopted to overcome this inconvenience, has een to use the same circuit simulator to uild more sophisticated circuit models in the form of either su-circuit or suroutines depending on the type of simulator. hese models can then e used within the complete circuit model in those critical positions where the device ehaviour needs to e studied. he topic of device modelling for circuit simulation has een addressed y many authors in a quite large numer of papers. In [] Kraus and Mattausch make an exaustive analysis of the status and trends of this suject. he concepts used to implement device models in circuit simulators, from the most accurate and time consuming, can e concisely summarised in: - numerical solution (direct discretisation of device equations); - approximated solution; - transformation; - lumped models; - functional models; he model presented in this paper elongs to the second category of the aove list. he aim of this new power diode model implementation has een to reduce the required model parameters. Less parameters, in fact, simplify the model tuning for a given device and reduce the need of experimental tests and of data which are often difficult to otain. In order to compare the presented approach with other existing ones, two diode models elonging to the third and the fourth category have een also implemented. he two models chosen as reference for this work have een the ones presented y Lauritzen in [] and [3] (lumped model) and y Strollo in [4] and [5] (transformation). In order to compare the three modelling approaches, the results otained from oth the reference models and the presented model have een compared with experimental measurements. he comparison have een carried on for the case of surge current transient and high di/dt reverse recovery. II. MODELLIG APPROACH he modelling approach is ased on a simplified circuital representation of the diode equations that govern the forward conduction, reverse recovery and reverse locking states. In Fig. a schematic representation of the PI diode structure during the forward conduction reverse recovery transition is reported. As it can e noted, the intrinsic ( - ) diode ase, is flooded y a carriers plasma which average concentration is much higher than the ase doping concentration (high injection level). Part of the diode -783-644-X//$. (C)
injected current, due to recomination in the emitter regions, does not reach the - region (J n and Jp). hus, the currents that effectively contriutes to the carriers plasma formation are just Jp (at the P - - junction) and Jn (at the - - - - junction). When the reverse recovery phase egins (diode current ecomes negative) the carrier concentration in the P - ase starts to decrease. At a certain point, the carrier concentration at the P - - junction ecomes zero. his is the time instant at which the diode starts to lock reverse voltage. As the reverse recovery phase proceeds the plasma detaches from the junction and the space charge region starts to uild up. he reverse recovery phase terminates when all the excess carriers have een removed from the diode ase. P Jn 3 Excess carriers concentration Jp -- Jn Jp X XW Base doping Profile currents are generally expressed in terms of the carriers concentration at the junctions y means of Eq. 5 and 6: J n h c (5) x J p h c (6) x W where h and h [A cm 4 ] are constant parameter ( -4 <h ;h <5-4 ). It can e shown that Eq., under the simplifying assumptions already mentioned, can e circuitally represented y means of the R-C network of Fig.. oltages, currents and the parameters R, C, G and dx have the following meaning: - the voltage at the nodes,,..., is equivalent to the excess carriers concentration in the diode ase; - the current in the resistors Ri represents dc/dx; - dx is the space discretisation step equal to W/ where is the numer of the circuit elementary RC cells; - Ri*dx can e chosen equal to. dx; - Gi*dx is given y dx L where L is the amipolar diffusion length defined as: D τ ; a Fig.: Diode ase during forward conduction and reverse recovery. he carrier diffusion and storage within the ase of the PI diode, under some simplifying assumptions (monodimensionality, high injection levels, quasi-neutrality) can e descried y means of the well known diffusion equation: d c d c c Da () dt dx τ where: c [cm -3 ] is the free carriers concentration in the ase region, τ [s] is the high injection levels carriers lifetime and D a [cm /s] is the amipolar diffusion constant given y: D D D n p a () Dn Dp where D n and D p are the electron and holes diffusion constants. he oundary conditions associated with Eq. are usually given in terms of injected current J t [A/cm ] and emitters recomination currents J n and J p as in Eq. 3 and 4. dc q Da J (3) n dx x d c q D a J (4) p dx x W /(Da*q)**/() where is the electron and holes moility ratio µn/µp and J n, J p are the emitter recomination currents. hese Jn*/(Da*q) n R*dx C*dx - C*dx is dx/d a n G*dx C*dx R*dx G*dx Ci*dx Ri*dx ni Gi*dx R*dx C*dx G*dx R*dx Fig. : Circuital representation of the carrier diffusion and storage within the ase of the PI diode. It must e pointed out that when the carrier concentration at x ecomes zero (final stage of the reverse recovery) the carries plasma detaches from the P - - junction (see Fig. ). Starting form this instant, the oundary conditions Eq.(3) and Eq.(4) are no longer valid. hus, the circuit of Fig. which intrinsecally implements these oundary conditions y means of the current sources at oth ends should e modified in order to account for moving oundaries as the space charge region uilds up. his prolem can e solved as explained in [6] y introducing an auxiliary variale u which has the following properties: - u(x)c(x)-kj r (where k is a constant parameter and J r is the reverse recovery current density) within the carriers storage region; - d u outside the carriers storage region. dx he carrier concentration c outside the storage region is forced to zero. By using this simplified approach it is possile to maintain the same oundary conditions for the n Jp*/(Da*q) n /(Da*q)**/( -783-644-X//$. (C)
whole reverse recovery transient and directly otain the position of the space charge moving oundary x sc. III. PSPICE MODEL IMPLEMEAIO he circuit of Fig. can e conveniently implemented starting from the network node potential equations (see Eq.7) and representing them in PSPICE y means of ABM locks. ni ni R dx d dt ni ( I ni C dx G dx ni he otained elementary cells have the form represented in Fig.3, 4, 5. he first and the last cell implement Eq.7 plus the oundary conditions (Eq.3 and 4) and the recomination in the emitter regions (Eq. 5 and 6). n Da PWR n {h/(da*q)} I )/((%I)m)*(%I3)/((%I3))} ni Da n PWR u {dx} {Da/(dx**)} n {/tau} Fig. 3: First elementary cell (P side). ni- {Da/(dx**)} {/tau}. Fig. 4: Middle elementary cell. {h/(da*q)} n- {dx} {Da/(dx**)} v {/tau}. v. v ni ) (7) n n he circuit of Fig. implemented with the functional locks of Fig.3, 4 and 5 together with the sustitution of u(x) allow to otain the excess carrier concentration at nodes n...n 8. he voltage drop across the carrier storage region can e expressed in terms of carrier concentration and current density y means of: W J t dx q ( µ µ ) (c n p where [cm -3 ] is the doping concentration. In Eq. 8 the contriution of the demer voltage has een neglected. Eq. 8 can e directly used for the calculation of y sustituting the integral with the summatory of Eq. 9 or approximated y means of an averaged function of the carriers concentration at x and xw [5]: J t q ( µ µ n p ) dx i ci where is the numer of nodes (eight in our case). In the transformation of the integral (8) in the summatory (9), the simplifying assumption of constant moility has een made. In order to take into account moility degeneration due to carrier carries scattering the series resistance R [5] can e added to the diode model. In order to complete the calculation of the voltage across the diode terminals, other two contriution must e considered. he first is the junction voltage which, according to the Boltzmann approximation can e expressed as: i ) (8) (9) k c cw j ln( ) () q n where: k [e/k] is the Boltzman constant, [K] is the junction temperature, c [cm -3 ] and c W [cm -3 ] are the carriers concentrations calculated for x and xw respectively. Finally, the contriution of the voltage across the space charge region can e computed y using the arupt depletion layer approximation [7] y means of: sc q Jp x ε q vl sc () where ε [F/cm] is the asolute dielectric constant of silicon, v l is the limit velocity for carriers (holes) and X sc [cm] is the width of the space charge region. {(%I)/((%I))*/((%I3))} n Fig. 5: Last elementary cell ( side). For the presented simulation results a total of eight nodes have een used. In order to allow the potential of nodes n i (representing the carrier concentration) to e numerically compatile with the PSPICE implementation, their value has een normalised y multiplying for the electron charge q. I. RASIE HERMAL IMPEDACE MODEL he transient thermal impedance of a power semiconductor device can e generally approximated y means of a distriuted RC network equivalent. he network is otained y connecting elementary RC cells in series. he parameters R i [ C/W] and C i [J/ C] are otained as follows: C Γ C [J C - ] () i i thi i -783-644-X//$. (C)
where Γ i [kg m -3 ] is the specific weight, C thi [j C - kg - ] is the specific heat and i [m 3 ] is the volume of the element; R i xi x dx σ x i S ( x) i eff_i [ C/W] (3) where σ i [W/ C m - ] is the thermal conductivity, x i [m] is the element thickness and S eff_i [m ] is the effective section of the heat-path. means of : µ n Ploss 35 3 Rth Rth Rthi Fig. 7: Simplified paralleled RC network. j Rth c kmn Cth Cth Cthi Cth j µ p 48 3 It is of common use to simplify this representa_ tion y considering only the effect of the dominant time constants. By doing this, the Z th curve can e expressed as finite series of exponential terms like in Eq.(3). his allows to reduce the model to the R- C network of Fig.7. In some cases R thi and τ i are otainale directly from the device datasheet. hus, the Z th equivalent depicted in Fig. 3 it is the most convenient to use for circuital simulation. In the presented model the temperature j has een used for the calculation of the electron and holes moility of Eq.(9) y kmp (4) where kmn and kmp > are the temperature exponents of the lattice moilities and in the Boltzman approximation of the junctions voltage Eq.().. MODEL ALIDAIO he electro-thermal model presented in the previous sections has een validated comparing the simulation of a reverse recovery and a surge current transients with measurement performed on two sample devices. he sample devices are oth 3mm, 5 Hockey Puk diodes ut, for the surge current test, a standard recovery diode type has een used while, the reverse recovery test, has een performed on a fast recovery one. In oth cases the used values of τ i and R thi of the thermal impedance model are the ones reported in ale I. ABLE I ALUES OF R HI AD τ I OF HE SAMPLE DEICE Z H-JC CURE. R thi [ C/W] τ i [s] st Exp. erm.5.649 nd Exp. erm.5.69 3 rd Exp. erm.8.5 4 th Exp. erm.65.489 he surge current test has een performed y applying to the Device Under est (D.U..) a sinusoidal current pulse with a duration of ms and an amplitude of 75 A. he resulting current and voltage waveforms have een recorded and compared with the simulated ones. As can e seen from Fig.8 the thermal effect on the diode forward characteristic is quite evident and characterised y the typical eight-shaped waveform. he model parameters have een adjusted in order to otain the measured forward voltage drop at the peak current. It can e noted that, for lower current, the model tend to over-estimate the forward voltage drop. his can e compensated y introducing the carrier-carrier scattering effect on the moilities. Forward current [A] 8 7 6 5 4 3 Simulated voltage Measured oltage 3 Forward voltage drop [] Fig.8: Forward voltage drop model validation. he reverse recovery tests have een performed at high di/dt (aout A/µs) in order to stress the diode dynamic ehaviour. he current waveform (see Fig. 9) consists of a sinusoidal pulse of aout 8 A amplitude and 4.4 µs period. he comparison etween simulated and measured waveforms evidences a discrepancy in the first part of the forward current pulse that can e justified with a difference in the forward recovery ehaviour. he asence of oscillation in the ending part of the reverse recovery is due to the asence, in the simulated circuit, of the parasitic elements. -783-644-X//$. (C)
current [A] 8 6 4 - -4 Measured current -6 4 6 time [us] Simulated current Fig.9: Reverse recovery model validation. I. COMPARISO WIH HE REFERECE MODELS A. Lumped model Lauritzen and Ma proposed a simple power diode model, ased on an extension of the charge-control diode model, using the lumped charge approach of Linvill [8]. his approach yields a very simple model for reverse recovery [] and can e easily enriched including forward recovery and emitter recomination [3]. Moreover, with little efforts we have generalised the model taking into account some electrothermal interaction as well. he model is gloally descried y the equations: qe qm i M M dqm qm qe qm dt τ M ve K B qe ISτ e ; q MR Mi vm qmr M M ve i E ISEτ e v ve vm RSi i ie im (L) (L) (3L) (4L) (5L) (6L) (7L) Equation -3L take into account the reverse recovery ehaviour and are descried in []; the Equations 4L and 5L descrie the voltage drop in the conductivity-modulated ase region and the emitter recomination [3]. Equation 8L is the Laplace transformation of the relationship which relates the junction temperature to the dissipated power in the diode. he model has the advantage of simplicity and simulation speed; moreover all the equations reported are valid over all the regions of operation and no conditional statements are needed to implement the whole model. B. Model ased on the Laplace transform method Strollo et al. presented different versions of a model ased on Laplace transform solution for the time-dependent amipolar diffusion equation [4,5]. he starting point of the model here presented and generalised with electro-thermal interaction is a continued-fraction expression in the Laplace domain of the carrier distriution in the ase region. By truncating the continued-fraction expansion it is possile to derive a lumped RC representation of the ase region of the diode [4]. he equations of the model used here are the followings: v J i IS e (S) τi q i (S) IKF L{ i }( ) { }( ) s L q s sτ tanh sτ (3S) τ τ τ q (4S) i i τ IE (5S) v vj vm (6S) ( ) vm ir epi // Rlim R mod (7S) R mod λq ( λ ) qm τ τ dqm qm i dt τ (8S) Equation 3S represents the solution otained y means of the Laplace transform and it can e written also as follows [4]: { i}( s) { q }( s) L L τ sτ 3 τ 5 sτ 7 τ 9 sτ... (9S) In a practical circuit implementation, only a finite numer of sections are used of the RC ladder corresponding to the previous relationship. he accuracy depends on the numer of the sections included in the model. R epi is the unmodulated ase region resistance while the resistor R lim represents the limit to the degree to which conductivity -783-644-X//$. (C)
modulation can occur; R mod takes into account the conductivity modulation of the ase and depend on q and q m. he implementation shows good convergence property and very fast simulation time. It can e oserved that if one uses a second order approximation of the continued fraction he otains a lumped charge model very similar to that of Lauritzen previously descried. In Fig. and the reference models simulation of the same surge current and reverse recovery transient descried efore are reported. Diode Current [A] 8 7 6 5 4 3 Measured current PSPICE Simulation Strollo PSPICE Simulation Lauritzen.5.5.5 3 Forward oltage Drop [] Fig.: Reference models surge current simulation. current [A] 8 6 4 - -4-6 4 6 time [us] current Strollo Lauritzen Fig.: Reference models reverse recovery simulation. III. COCLUSIOS In this paper a new PI diode model ased on an approximated solution of the minority carriers diffusion equation has een presented. he otained time dependent carriers distriution has een used to implement temperature-dependent su-models for the diode reverse recovery and forward voltage drop simulations From the comparison of the presented model simulations with experimental results and reference models simulations the following conclusions can e drawn: the presented approach shows a fairly similar ehaviour to the transformation ased models (slightly etter dynamic ehaviour with respect to the lumped models); y using this modelling approach the model parameters can e reduced, if the case, to only the ones needed for the physical device description. For the presented simulation results the model parameters were only: intrinsic ase width (W), active area (S), high injection level carriers lifetime (τ), emitter recomination coefficients (h), ase doping concentration ( d ), temperature exponents of the lattice moilities (kmn, kmp). It must e pointed out that during the parameters optimisation procedure, in order to fit experimental data, the aove mentioned physical parameters can loose part of their physical meaning y assuming values that are out of the typical range of validity. evertheless this does not appear to have negative influence on the model validity; the simulation time is slightly higher with respect the reference models. REFERECES [] R. Kraus, H. J. Mattausch, Status and rends of Power Semiconductors Models for Circuit Simulation IEEE ransactions On Power Electronics, vol. 3, no. 3, May 998, pp. 45-465. [] P.O.Lauritzen and C.L.Ma, A simple power diode model with reverse recovery, IEEE ransactions On Power Electronics, vol.6, no., April 99, pp.88-9. [3] C.L.Ma and P.O.Lauritzen, A simple power diode model with forward and reverse recovery IEEE ransactions On Power Electronics, vol.8, no.4, Octoer 993, pp.34-346. [4] A.G.M. Strollo, SPICE Modeling pf Power Pi Diode Using Asymptotic Waveform Evaluation, Conf. Rec. IEEE PESC 96, ol., pp.44-5. [5] A.G.M. Strollo, A new SPICE sucircuit model of powe Pi diode, IEEE ransactions On Power Electronics, vol.9, no.6, ovemer 994, pp.553-559. [6] F. Bertz, J. Pritchard, A. B. Crowley, Modelling PI Diode Switch-off with the Enthalpy Method, Solid State Electronics ol. 7 os. 8/9 pp. 769-774, 984. [7] Ph. Leturcq, M.O. Berraies, J.L. Massol, Implementation and alidation of a ew Diode Model for Circuit Simulation, Conf. Rec. IEEE- PESC 96, vol., pp. 35-43. [8] J.G.Linvill, Lumped models of transistors and diodes, IRE Proc., pp.4-5, June 958. -783-644-X//$. (C)