A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
A New Physics Based SPICE Sub-circuit Model for Insulated Gate Bipolar
Transistors (IGBTs)
Rui Chibante¹, Armando Araújo², Adriano Carvalho²
¹Instituto Superior de Engenharia do Porto (ISEP)
Rua Dr. António Bernardino de Almeida, 431 - 4200-072 Porto
Porto - Portugal
Tel +351 22 834 05 00 - ext. 1725 / Fax +351 22 832 11 59
rmc@isep.ipp.pt
²Faculdade de Engenharia da Universidade do Porto (FEUP)
Rua Dr. Roberto Frias, s/n 4200-465 Porto
Porto, PORTUGAL
Tel +351 22 508 14 00 / Fax +351 22 508 14 40
asa@fe.up.pt / asc@fe.up.pt
Acknowledgments - The authors would like to thank PRODEP (Programa de Desenvolvimento
Educativo para Portugal) for the support to this work.
Keywords - «Device modelling», «Simulation», «Power semiconductor devices».
Abstract - A physics based, Non-Punch-Through, Insulated Gate Bipolar Transistor (NPT-IGBT)
model is presented, as well as its porting into available circuit simulator SPICE. The developed model
results in a system of ODEs, from which time/space hole/electron distribution is obtained, and is based
on solution of ambipolar diffusion equation (ADE) trough a variational formulation, with posterior
implementation using one-dimensional simplex finite elements. Other parts of the device are modeled
using standard methods. Thus, this new hybrid model combines either advantages of numerical
methods or mathematical, through modeling charge carrier behavior with high accuracy even
maintaining low execution times. Implementation of the model in a general circuit simulator is made
by means of an electrical analogy with the resulting system of ODEs.
1 Introduction
Modeling charge carrier distribution, in low-doped zones, shown in all bipolar power
semiconductor devices, is known as the most important issue for accurate description of dynamic
behavior of these devices. Knowledge of hole/electron concentration in that region is crucial but it is
still a challenge for model designers [1, 2].
In what IGBT models is concerned, some important attempts include Hefner's physics-based
model [3, 4] that points out for non-quasi-static effects by introducing a so-called redistribution
current. As this current assumes a linear carrier distribution over n- region, model exhibits some
problems at describing switching behavior at high blocking voltages [5]. Although it is adequate for
most circuit simulations model is inadequate for predicting dynamic carrier distribution [6].
Goebel [7] and Metzner et al. [2] developed hybrid models where carrier distribution is calculated
by a numerical routine that is linked to circuit simulator. With this approach it is possible to achieve
accurate results (typical of numerical models) but model implementation becomes a hard task [5, 6].
Due to cumbersome implementation, models like those have been incorporated in powerful, but
also very expensive ones, simulation programs (like SABER). This is why there is a great demand for
accurate device models running on inexpensive simulators, like those of standard SPICE family [8].
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ISBN : 90-75815-07-7
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
So, in recent years several important SPICE models have been reported in literature, with an
interesting trade-off between accuracy and computation time.
Kraus et al. [8] have proposed a physics based semi-empirical model that uses an analytical
solution of ADE. By series expansion of ADE, Leturcq et al. [9] applied a methodology, based on
Fourier transform, where carrier distribution is obtained through a series of RC networks, making an
easy implementation in a general circuit simulator. Recently, Busatto et al. [5] experienced the
lumped-charge technique (introduced by C. L. Ma et al. [10]) for IGBTs modeling, including some
effects such as depletion capacitances, advanced mobility model, and separately handling of
hole/electron fluxes for a better fitting to device physics.
An alternative approach introduced by Araújo [11] is based on ADE solution trough a variational
formulation and simplex finite elements. This approach has already been shown to be successful for
PIN diodes and BJTs. One important advantage of this modeling approach is its easy implementation
into general circuit simulators by means of an electrical analogy with the resulting system of ODEs.
ADE implementation is made with a set of current controlled RC nets which solution is analog to the
ODE system (section 2).
The aim of this paper is to present the practice of this approach for IGBTs with non-punch-through
structures. In order to complete the IGBT model it has to be supplemented with a few sub-circuits,
modeling other regions of the device: emitter, junctions, space charge and MOS regions. These subcircuits, based on classical approaches, are presented in section 3. According to this hybrid approach it
is possible to model charge carrier distribution, with high accuracy, even maintaining low execution
times. Implementation and simulated SPICE results are presented in sections 4 and 5 respectively.
2 FEM ADE solution
Assuming a high-level injection condition for low-doped base zone we have n( x, t ) p( x, t ) . With
this condition charge carrier distribution is given by the well-known ambipolar diffusion equation
(ADE):
∂p
p
∂2 p
=− +D 2
∂t
τ
∂x
(1)
with:
D=2
D p Dn
D p + Dn
.
(2)
The general boundary condition associated with (1) is:
∂p
1 I n D p − I p Dn
=
∂x 2qA
Dn D p
(3)
where In/Ip are the electron/hole currents at the border of the n- region. For an NPT-IGBT these
currents are defined in terms of total current IT as follows:
∂p
∂x
∂p
∂x
=
Ip
IT
− l
2qD p qD
(4)
=
In
IT
− r .
2qD p qD
(5)
xl
xr
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
I pl is a recombination term, modeled with "h" parameter theory. I nr is channel current from MOS
part of the device. Bipolar part of a one-dimensional NPT-IGBT model indicating the most important
variables is presented in Figure 1.
Fig. 1: Bipolar part of the NPT-IGBT.
In Araújo [11] it is shown that ADE can be solved by a variational formulation with posterior
solution by finite element method. This approach uses the following functional associated with (1) and
(3):
1 ∂p
2 ∂x
2
Π=
Ω
p2
p ∂p
+
+
dΩ −
2 Dτ D ∂t
Γ
Jp
Jn
−
p dΓ .
2qDn 2qD p
(6)
Its minimization results in a system of ordinary differential equations like:
∂p
M
+ G [ p ] + L = [0] .
∂t
(7)
Carrying out minimization of (6) with a finite element formulation (FEM) using linear, onedimensional, shape functions and r finite elements, matrices associated with (7) are:
Aele
6D
M=
2
1
1
4
1
1
4
1
1
2
2
−2
−2
4
Ae
2le
G=
(8)
−2
−2
4
−2
+
Aele
6 Dτ
−2
1
1
4
1
1
2
4
1
1
2
(9)
2
!
"
#
L=
$
'
'
J
− n A1 0
2qD p 2qDn
)
$
'
Jp
(
(
%
+
#
Jp
%
*
-
0
'
J
− n Ar .
2qD p 2qDn
&
(
(
&
)
(10)
*
,
Ae and le are, respectively, area and width of each finite element.
As the resulting FEM matrices are symmetric, it enables an electrical sub-circuit equivalent to a
set of current controlled variable RC nets as can be seen in figure 2:
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
Fig. 2: Simplex element electrical circuit equivalent.
with:
Aele
Al
; Ci = C j = e e
6D
2D
.
6 Dτ le
2 Dτ
Rij =
; Ri = R j =
Ae le
6 Dτ Ae − Aele2
Cij = −
(11)
Each RC net is associated with one part of the domain (the low doped zone) corresponding to
FEM formulation for one element. Voltages in each node of the net are an image of hole/electron
concentration. To solve ADE it is just enough to get:
1. A RC sub-circuit that emulates one element. The number of RC nets (sub-circuits) in series
equals the number of elements necessary for partition of the domain.
2. A sub-circuit for calculus of element width le that is emulated as a current that controls RC
net values.
Note that:
1. Width of each element, le, is a sub-circuit parameter, so the elements can be lowered where
it is known that concentration changes fast and enlarged where concentration changes
slowly.
2. Device properties, D and , in each element can also be sub-circuit parameters that enable
solutions for heterogeneous materials.
3. It is easy to increase accuracy by adding more elements to the solution (it means to add
more RC nets in series).
3 Circuit solvers for remaining zones
3.1 Emitter model
The contribution of carrier concentration for the total current is well described by the theory of "h"
parameters [7, 11, 12], for high doped emitters, assuming a high injection level in the carrier storage
region:
I nl = qhp Ap02 .
(12)
That relates electron current I nl to carrier concentration at left border of the n- region (p0). Emitter
zone is seen as a recombination surface that models the recombination process of electrons that
penetrate p+ region due to limited emitter injection efficiency.
3.2 MOSFET model
3.2.1 MOS current
MOS part of the device is well represented with standard MOS models. For DC characteristics, a
basic expression for channel current is:
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
I mos = K p
V2
Vgs − Vth Vds − ds
2
(
)
(13)
in linear region and:
I mos =
(
K p Vgs − Vth
)
2
(14)
2
in saturation region.
This model can be improved taking in account several important phenomena such as:
1. Trans-conductance reduction in linear region.
2. Mobility reduction due to transverse electric field for high gate voltages.
3. Avalanche breakdown for high drain voltages.
These effects are modeled respectively by:
1. An empirical parameter K plin .
2. An empirical parameter that represents the reduction of trans-conductance.
3. An avalanche multiplication factor M.
Thus, equations (13) and (14) are rewritten as:
I mos = K plin
and:
I mos =
(V
)
− Vth Vds −
gs
K psat (Vgs − Vth )
2
K plin Vds2
2 K psat
×
1 + θ Vgs − Vth
2
×
M
(
M
1 + θ (Vgs − Vth )
)
(
Vds < Vgs − Vth
Vds > (Vgs − Vth )
K psat
)K
(15)
plin
K psat
K plin
(16)
where [13]:
4 −1
M = 1−
Vds
Vbr
(17)
Vbr = 5,34 × 1013 kv N D−0,75 .
(18)
3.2.2 MOS capacitances
Transient behavior is ruled by capacitances between device terminals as illustrated in figure 3:
Fig. 3: MOSFET part of IGBT (MOS notations).
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
Well-known nonlinear Miller capacitance is the most important one in order to describe switching
behavior of MOS part. It is comprehended of a series combination of gate-drain oxide capacitance
(Cox) and gate-drain depletion capacitance (Cgdj) resulting in the following expression:
C gd =
Cox
W' C
1 + sc ox
ε si Agd
(19)
with:
2ε siVgd
Wsc' =
qN D
.
(20)
W’sc is depletion width formed under the gate, ND is base doping concentration and Agd is MOS
region area.
Drain-source capacitance (Cds) is defined in the same way as Cgdj :
Cds =
ε si Ads
(21)
Wsc
with:
2ε siVds
.
qN D
Wsc =
(22)
Wsc is total depletion width developed under p+ body and Ads is the respective area. Note that Agd +
Ads represents total device area (A). As depletion zone widths support different voltages [(20) and (22)
] and remembering that Vgd = Vds - Vgs it can be written:
Wsc' = 0
Wsc'
=
−
Wsc2
2ε siVgs
qN D
Wsc <
2ε siVgs
Wsc >
2ε siVgs
qN D
.
(23)
qN D
Gate-source capacitance is normally extracted from capacitance curves and a constant value may
be used producing generally good results [9, 14].
3.3 Voltage drops
As the global model behaves like a current controlled voltage source, it is necessary to evaluate
voltage drops over the several regions of the IGBT. Thus, neglecting the contribution of the highdoped zones (emitter and collector) the total voltage drop (forward bias) across the device is composed
by the following terms:
VIGBT = V p + n − + VΩ + Vsc .
(24)
3.3.1 Junction voltage
The p+n- junction voltage drop can be calculated according to Boltzmann approximation:
V p +n−
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p2
= VT ln 02 .
ni
(25)
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
3.3.2 Voltage over storage region
Voltage drop across the lightly doped storage region is described integrating electrical field:
xr
VΩ = Edx .
(26)
xl
Assuming a uniform doping level and quasi-neutrality (n = p + ND) over the n- zone, and neglecting
diffusion current, we have:
xr
1
J
dx .
q x p (µ n + µ p ) + µn N D
VΩ
(27)
l
Equation (27) can be seen as a voltage drop across a conductivity modulated resistance:
VΩ = IT RD
(28)
with:
xr
1
dx
.
qA x p ( µ n + µ p ) + µ n N D
RD =
(29)
l
Applying FEM formulation, an approximation can be made without a significant loss of
accuracy[11], in order to avoid the logarithms resultant from integration of (29). This approximation
consists in taking the mean value of p in each finite element, so it is assumed that the concentration is
constant between two consecutive nodes. This yields:
r
le
RD
e =1
qAe pav ( µ n + µ p ) + µ n N D
(30)
with:
pav =
pe + pe+1
.
2
(31)
3.3.3 Depletion voltage
Voltage drop over the space charge region is calculated by integrating Poisson equation. For a
uniformly doped base the classical expression [15] is the following:
Vsc =
qN D
Wsc (Wsc + 2Wvb )
2ε si
(32)
where
Wvb =
2ε siVbi
qN D
(33)
is a function of the external parameter junction in-built voltage Vbi.
I
Under high reverse current conditions ND in previous equations must be replaced by N D + qAVp l if
it is intended to have in account the large amount of holes that flows through the depletion region.
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
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4 Implementation in SPICE
Figure 4 shows a complete IGBT model obtained linking all sub-circuits with associated boundary
conditions. Notice, at the top, FEM solution for time/space hole/electron distribution in low-doped
IGBT base. Spatial distribution is calculated using nine simplex elements (nine current controlled RC
nets). Controlling parameter le is determined by the circuit at middle left and boundary conditions, at
borders of IGBT base, with circuit in center. Circuit at middle right emulates voltage drops in
junctions, space charge and ohmic zone along the base. Circuits at bottom left emulate MOSFET part
of IGBT: relation of currents (IT = Ip + Imos) and variable capacitances. Circuit at bottom right calculate
total voltage drop that depends on anode current, so the IGBT is seen as current controlled voltage
source.
15
23
26
16
30
32
9
27
31
33
17
64
14
44
5
7
10
12
22
6
8
11
13
18
24
25
34
35
19
85
86
90
91
28
87
88
93
62
94
52
1
4
3
37
21
2
92
89
95
V(29)
VGATE
29
V(40) VANODO
40
I(VIPR) IPR
56
53
20
59
38
55
I(VIMOS) IMOS
57
I(V5) ITOTAL
41
Fig. 4: SPICE FEM based IGBT model.
A detailed description of some sub-circuits, namely those ones related with FEM formulation and
variable capacitances are available in [11].
5 SPICE IGBT model results
In order to model validation some experimental data from [4] were used as reference. The test
circuit is composed with a resistor-inductor load and a resistive gate drive. Figure 5 shows simulated
static characteristics for various gate voltages.
10
20 V
ANODE CURRENT (A)
9
12 V
9V
8
7
8V
6
5
4
7V
3
2
Vgs = 6 V
1
0
0
1
2
3
4
5
6
7
8
9
10
ANODE VOLTAGE (V)
Fig. 5: Simulated static IGBT characteristics.
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
Notice that model implements with accuracy:
•
•
•
Diode voltage offset due to anode-epitaxial layer p-n junction.
Low resistance in on-state (due to conductivity modulation).
MOSFET channel current saturation.
Figures 6, 7 and 8 show simulated gate voltage, anode current and anode voltage for Vcc = 300V,
load inductor of 80 H, load resistance of 30 and various gate resistances 1 - 1 K , 2 - 2 K and 3 3K .
12
ANODE CURRENT (A)
GATE VOLTAGE (V)
20
15
10
1
5
2
9
6
1
2
3
3
3
0
0
5.00U
15.0U
25.0U
35.0U
5.00U
45.0U
15.0U
25.0U
35.0U
45.0U
TIME (s)
TIME (s)
Fig. 6: IGBT gate voltage.
Fig. 7: IGBT anode current.
ANODE VOLTAGE (V)
450
350
250
1
2
150
3
50
5.00U
15.0U
25.0U
35.0U
45.0U
TIME (s)
Fig. 8: IGBT anode voltage.
Notice that model predicts all dynamic IGBT characteristics, namely:
•
•
•
Slowly decaying current at turn-off (tailing phenomenon characteristic of IGBTs due to
bipolar transistor part).
Control of anode voltage rise through gate resistance.
Gate controlled turn-off delay time.
Simulation time for a complete turn-on/turn-off cycle is about 2 seconds in a AMD Athlom XP
1800 processor running at 1.5 GHz with 512Kb of RAM.
6 Conclusions and perspectives for future work
This paper presents a new hybrid NPT-IGBT model based on a finite element formulation (FEM)
that accurately predicts time/space hole/electron distribution. Knowledge of this time/space charge
carrier distribution in IGBT low-doped base, supplemented with usual models for emitter, junctions,
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A New Physics Based SPICE Subcircuit Model for Insulated Gate Bipolar Transistors (IGBTs)
ADRIANO CARVALHO Adriano
space charge and MOS regions, enables fast and efficient implementation of a complete IGBT model
in SPICE like circuit simulators. Refinement of model characteristics, such as, non-equally spaced
elements, non-homogeneous doping and diffusivities and number and type of elements is a simple task
and left as a choice for the user.
In near future authors intend to:
•
•
•
Validate the model with own experimental results and take a closer look to the parameter
extraction problem (considered as new critical issue in power semiconductor models [1, 8,
13].
Extend the model to a punch-through structure.
Link a FEM based thermal model enabling implementation of electro-thermal modeling of
the IGBT.
References
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Simulation. IEEE Trans. Power Electron., 13(3): p. 452-465, 1998.
[2]. D. Metzner, T. Vogler and D. Schröder. A Modular Concept for the Circuit Simulation of Bipolar Power
Semiconductors. IEEE Trans. Power Electron., 9(5): p. 506-513, 1994.
[3]. A.R. Hefner. An Improved Understanding for the Transient Operation of the Power Insulated Gate Bipolar
Transistor (IGBT). IEEE Trans. Power Electron., 5(4): p. 459-468, 1990.
[4]. A.R. Hefner and D.M. Diebolt. An Experimentally Verified IGBT Model Implementation in the Saber
Circuit Simulator. IEEE Trans. Power Electron., 9(5): p. 532-542, 1994.
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[6]. K. Sheng, B.W. Williams and S.J. Finney. A Review of IGBT Models. IEEE Trans. Power Electron., 15(6):
p. 1250-1266, 2000.
[7]. H. Goebel. A Unified Method for Modeling Semiconductor Power Devices. IEEE Trans. Power Electron.,
9(5): p. 497-505, 1994.
[8]. R. Kraus, P. Turkes and J. Sigg. Physics-Based Models of Power Semiconductor Devices for the Circuit
Simulator SPICE. in 29th Annual IEEE Power Electronics Specialists Conference: IEEE 1998.
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[10]. C.L. Ma, P.O. Lauritzen and J. Sigg. Modeling of Power Diodes with the Lumped-Charge Modeling
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[11]. A. Araújo. Modelação de Semicondutores Bipolares - Formulação de um Novo Método para Simulação em
Circuitos Electrónicos de Potência, in Faculdade de Engenharia da Universidade do Porto: Porto,
Portugal,1998.
[12]. M.O. Berraies. Modéles de Composants Semiconducteurs pour la Simulation des Circuits en Électronique
de Puissance, in Université Paul Sabatier de Toulouse: Toulouse, France,1998.
[13]. C.M. Tan and K. Tseng. Using Power Diode Models for Circuit Simulations - A Comprehensive Review.
IEEE Trans. Industrial Electron., 46(3): p. 637-645, 1999.
[14]. K. Sheng, S.J. Finney and B.W. Williams. A New Analytical IGBT Model with Improved Electrical
Characteristics. IEEE Trans. Power Electron., 14(1): p. 98-107, 1999.
[15]. P.M. Igic, et al. Investigation of the Power Dissipation during IGBT Turn-off using a New Physics-Based
IGBT Compact Model. Microelectronics Reliability, 42(7): p. 1045-1052, 2002.
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