kV IGBT module are presented for hard and soft switching. A loss model of the ... undesirable in propulsion applications since the mechanical design of the ...
Switching Frequency Limit for Soft-Switching MF Transformer System for AC-fed Traction Tommy Kjellqvist
Staffan Norrga
¨ Stefan Ostlund
Royal Institute of Technology Sweden
Royal Institute of Technology Sweden
Royal Institute of Technology Sweden
Catenary
Abstract— The migration to line side power conversion topologies comprising a medium frequency transformer in AC-fed propulsion systems may reduce size and weight of the conversion system. The switching frequency is a crucial factor affecting the viability of such a conversion system. An increased switching frequency of a mutually commutated conversion system comprising a medium frequency transformer employing soft switching is verified by single pulse measurements. A test bench to characterize standard IGBT modules under soft switching has been build and loss measurements on a 3.3 kV IGBT module are presented for hard and soft switching. A loss model of the investigated topology is developed and the maximum switching frequency for a converter equipped with the investigated module is estimated for hard and soft switching.
L line
MF transformer n:1 To propulsion and auxillary inverters
15 kV 16 2/3 Hz
Fig. 1. vtr
T
C
D
Investigated converter topology
Z
T
C
D
Z
T
I. I NTRODUCTION In present AC-fed propulsion systems the line side power conversion is generally made up of a step-down transformer followed by a four-quadrant voltage source converter (VSC). By this arrangement the single-phase AC voltage from the catenary is converted into a DC voltage that is fed to the propulsion and auxiliary power inverters. The transformer tends to be a heavy and bulky component which is highly undesirable in propulsion applications since the mechanical design of the vehicle is strongly influenced by the weight and size of the transformer. To avoid these disadvantages several solutions have been proposed that utilize transformers operating at frequencies well above the line frequency [1]–[5]. References [1], [2] describe solutions where the conventional system is altered in the sense that a cycloconverter is connected between the transformer and the supply. In this case the VSC applies a medium frequency voltage to the secondary winding of the transformer. This voltage is converted to a suitable PWM voltage by the cycloconverter that in turn is applied to a passive line filter and thereby the power flow of the system can be controlled. In [5] a concept was presented employing capacitive snubbers in the VSC and a new commutation algorithm to achieve reduced switching losses. The basis for the commutation algorithm is that the two converters are alternately modulated, thus mutually setting up the conditions for soft commutation. With this new concept the switching losses are predicted to be reduced. The converter topology is illustrated in Figure 1. Today, the operating frequency of the VSC is generally in the range of several hundred hertz up to one kilo hertz. The weight savings of a medium frequency transformer converter system operating at this frequency is not enough to motivate the cost and complexity of the line side cycloconverter. By introducing soft commutation of the VSC, the transformer frequency may be raised to several kilo-hertz. Raising the frequency also allows a reduction of the line side filter.
t
itr
t vac
t
Fig. 2. side.
Commutation sequence during power flow from the DC to the AC
This paper present a study on the maximum inverter frequency if a conventional VSC for traction is fitted with capacitive snubbers and operated according to [5]. II. S OFT-S WITCHING MF S YSTEM FOR AC -F ED T RACTION The investigated inverter topology, illustrated in Figure 1, comprise a conventional VSC, fitted with capacitive snubbers, connected via a medium frequency transformer to a cycloconverter. By proper modulation of the to converters, soft switching is achieved for all semiconductors in the circuit. The voltage source inverter applies a medium frequency voltage to the low voltage side of the transformer. The high voltage side is connected to the catenary via a cycloconverter. If the VSC and the cycloconverter is properly modulated, all switchings occur at zero voltage or zero current. The commutation sequence during power flow from the DC to the AC side is illustrated in Figure 2, and is as follows. Initally, the current flow is through the IGBTs (T). The IGBTs are then turned off and the current will commutate to the snubber capacitors (C). As the snubber capacitors recharge, the diodes become forward biased and current is commutated
(A)
(B)
AUX
(C)
AUX
I
AUX
I DUT
I DUT
VCE
DUT
VCE
IC
IC
(D)
IC
(E)
AUX
(F)
AUX
I
AUX
I DUT
VCE
VCE
I DUT
DUT
VCE
IC
VCE
IC
IC
to the diodes (D). Due to the capacitive snubbers, the IGBTs will turn off at zero voltage. Since the power flow is reversed, the cycloconverter may be naturally commutated. By shortcircuiting the the primary terminals of the transformer the full VSC voltage appears across the leakage inductance of the transformer and the transformer current is reduced linearly to zero. Due to the leakage inductance, the diode is turned off at reduced current derivative. There is now no power flow in between the AC and DC sides (Z). By commutating the second cycloconverter leg reverse polarity is applied across the leakage inductance and current is increased in the opposite direction. Thus, the IGBTs are passively turned on (T). The power flow is reversed and is now again from the DC to the AC side. By alternating the duration of the various states, it is possible to produce a three-level PWM output. To ensure soft switching of both VSC and cycloconverter all states must occur in the specified order, resulting in a short deviation from the desired output voltage during the commutation of the VSC. If power flow from the DC to the AC side is desired, the first cycloconverter leg is commutated directly after the commutation of the VSC as illustrated in Figure 2. If power flow in the opposite direction is desired, the commutation of the first cycloconverter leg is delayed until the desired pulse width is achieved. The VSC is then commutated right after the commutation of the second cycloconverter leg, reducing the power flow in the undesired direction to a minimum. Zero AC voltage may be applied from the primary side by the cycloconverter as illustrated in Figure 2, or it may be applied from the secondary side by the VSC. If zero voltage is applied from the secondary side, line current will always flow through the transformer windings, as well as through the VSC. However, the mean flux density in the transformer is reduced since it is only magnetized during the voltage pulses. III. C HARACTERIZATION OF IGBT M ODULES Most commercially available IGBTs are optimized for hard switching and, hence, very little is reported on soft switching
vCE / 1800 V, iC / 400 A, p / 360 kW
Fig. 3. Operating principle of the test bench. A: Charging of the DC link. B: Current ramp up. C: Zero voltage IGBT turn-off. D: Current freewheeling. E: Current ramp down. F: Discharging of the DC link.
vCE
iC
1 0.8 0.6 0.4
p
0.2 0 −0.2 0
Fig. 4.
5
10 t [µs]
15
20
Experimental result for snubbered turn-off with Cs = 470 nF.
of these devices. During zero voltage turn-off an elevated tail current has been reported in literature [6] as well as an elevated forward voltage drop during passive turn-on [7], [8]. The passive turn-on forward voltage drop is due to conductivity modulation lag present in bipolar devices and is heavily dependent on the impressed current derivative. The elevated tail current is the dominating loss mechanism during zero voltage switching and is clearly visible in Figure 4. A. Test Bench To characterize the modules, a test bench has been built comprising two inverter legs connected to an inductive load. The device under test (DUT) is an Eupec FF200R33KF2C 3.3kV 200A IGBT module containing a full converter leg with both IGBTs and anti parallel diodes. To permit zero voltage switching (ZVS) of the device under test one converter leg is fitted with capacitive snubbers. The modules are mounted on an aluminium block, heated by power resistors. In that way the junction temperature of the device under test can be controlled.
B. Forward voltage drop The forward voltages across the IGBT and the diode are approximated by linear equations:
400
VF Current, [A]
The test sequence is illustrated in Figure 3. The test is initiated by charging of the DC-link capacitor to the desired test voltage (A). During charging the IGBT under test is in on state and the voltage across the snubber capacitors are forced to a defined voltage. By turning on the opposite IGBT in the auxiliary leg the DC link voltage is applied across the load inductor (B). The circuit is kept in this state until the desired test current has been reached. The IGBT in the snubbered leg is then turned off, forcing the load current through the snubber capacitors (C). As the snubber circuit is recharged, the diode in the snubbered leg is forward biased and current is commutated to the diode (D). The test is terminated by turning off the auxiliary switch, feeding the inductor energy back to the capacitor bank (E). Since there is no way to reverse the load current, the DC link must be discharged before another test sequence may be initiated (F). During a test sequence, both voltage and current through the device under test are measured. Energy loss during switching is found by integration of the instantaneous product of measured voltage and current. Currents are measured by Rogowski-coils and voltages are measured relative to ground by single ended voltage probes. Rogowski-coils have the advantage of a relative high bandwidth and only a small inductive loading on the circuit. Since the coils can be opened the transducers can easily be mounted in the main circuit. In the following, an IGBT module is characterized. The purpose is to illustrate to what extent the switching frequency can be increased by using soft commutation.
VCE
200
0
VF,0 V CE,0
0
Fig. 5.
3 Voltage, [V]
6
IGBT and diode forward voltage drop.
If sinusoidal current is assumed the mean commutation loss can be derived. Z 1 π hEi = E(Iˆ sin θ)dθ (5) π 0 Assuming second order approximation of the energy loss according to (4) yields hEi =
a ˆ2 2b ˆ I + I + c. 2 π
(6)
vCE (iC ) = VCE,0 + rCE iC
(1)
IV. C ONVERTER L OSS M ODEL
vF (iF ) = VF,0 + rF iF
(2)
Both on state losses and switching losses are derived for the VSC operation in a medium frequency transformer conversion system as well as in a conventional low frequency transformer conversion system.
The constants in (1) and (2) are found by least square fitting to measured data or from data sheets. That is, Z In 2 ((V0 + ri) − v(i)) di. (3) min
A. The Conventional Inverter On State Losses
0
Typical forward voltage as a function of current and corresponding linear approximations are shown in Figure 5. C. Switching Losses During test, voltage and current has been measured and the instantaneous power has been calculated and integrated to derive the energy loss. For hard switching, turn-on losses in the IGBT and reverse recovery losses in the diode has been measured. IGBT losses has been measured for both hard and snubbered turn-off. The test has been carried out for several values of current and temperature. In Figure 6, 7, and 8 typical hard switching waveforms are illustrated. The switching losses are are approximated by a second order function E(i) = ai2 + bi + c (4) where a, b, and c are found by least square fitting to experimental data. Figure 9, 10, and 11 show the estimated polynomials as well as experimental data at elevated temperatures.
Calculating the on state losses for an inverter employing sinusoidal pulse width modulation (PWM) is a complex task and involves the computation of Bessel functions and a detailed analysis is presented in [9]. However, if a high pulse number is assumed significant simplifications can be applied. In the following, similar results are derived under this assumption. Due to symmetry in the modulation scheme, only one switch and one rectifier element has to be considered. In the following pure sinusoidal voltage and current are assumed. v(θ) = M Vd F (θ) F (θ) = sin (θ + φ) i(θ) = Iˆ sin θ
(7) (8) (9)
As an inverter leg is modulated, current will alternate between the upper and lower switch and the current will flow through the diode or the IGBT depending on the sign of the load current. Since the current is sinusoidal, one single diode or IGBT will only conduct during one half of the fundamental
500 °
T =125 C
°
T =25 C
j
j
400
2 / [mJ]
300
on
iC 1
E
vCE / 1800 [V], iC / 200 [A], pon / 360 [kW]
3
200
pon vCE
100
0 0
1
2
3 t / [µs]
4
5
6
0
0
100
200 I / [A]
300
400
C
Fig. 6. Typical IGBT hard turn-on waveforms. Vd = 1800V, IC = 200A, Tj = 25◦ C.
Fig. 9.
Measured IGBT hard turn-on losses.
Cs=0 nF
Tj=80 C
iC
Cs=220 nF Cs=470 nF
1
Poff
300
200
100 0 0 0
1
2
3 t / [µs]
4
5
1
0
6
Fig. 7. Typical IGBT hard turn-off. Vd = 1800V, IC = 200A, Tj = 80◦ C.
Fig. 10. C.
100
200 IC [A]
300
400
Measured IGBT hard and snubbered turn-off losses at Tj = 95◦
300
VR
°
Tj=125 C
Tj=80°C
°
Tj=80 C
250
Prr 0
200 Err / [mJ]
iR / 200 [A], vR / 1800 [V], prr / 360 [kW]
Cs=100 nF
400
vCE Eoff [mJ]
vCE / 1800 [V], iC / 200 [A], poff / 360 [kW]
500 °
T =25°C j
150
−1 100
iR 50
−2 1
2
3
4 5 t / [µs]
6
7
8
9 0
0
50
100
150
200 I / [A]
250
300
350
400
F
Fig. 8. Typical hard diode reverse recovery. UR = 1800V, IF = 200A, Tj = 80◦ C.
Fig. 11.
Measured diode hard reverse recovery losses.
B. Capacitively Snubbered Inverter On State Losses When using a line side cycloconverter, zero voltage can be produced either on the primary or secondary side. When zero voltage is generated on the primary side, the current through the transformer and VSC is zero, reducing conduction losses in the VSC and transformer. However, when zero voltage is generated on the secondary side, the mean magnetic flux in the transformer is reduced. In the following, both cases are considered. First, losses during non zero power flow are considered. During non zero power flow, current though the IGBTs or the diodes depend on the direction of the instantaneous power flow. One single rectifier or switch will only conduct during every second voltage pulse. Thus, Z δ(θ) 1 π−φ dθ (16) i(θ) · VCE (i(θ)) PI = π 0 2 Z 1 π δ(θ) PD = i(θ) · VF (i(θ)) dθ (17) π π−φ 2 where
i(θ) = Iˆ sin(θ)
(18)
δ(θ) = M · |sin(θ + φ)|.
(19)
and
400 350 300 Losses [W]
period. During this half period, the duty ratios of the IGBT and the diode are, 1 δI (θ) = (1 + M F (θ)) (10) 2 1 δD (θ) = (1 − M F (θ)) . (11) 2 The average power dissipation can be found from, Z π 1 vCE (i(θ)) i(θ)δI (θ)dθ (12) PI = 2π 0 Z π 1 PD = vF (i(θ)) i(θ)δD (θ)dθ (13) 2π 0 thus, M 1 ˆ CE,0 + cos φ IV PI = 2π 8 1 M + + cos φ Iˆ2 rCE (14) 8 3π M 1 ˆ F,0 − cos φ IV PD = 2π 8 1 M + − cos φ Iˆ2 rF . (15) 8 3π The results in (14) and (15) are consistent with the simplified expressions presented in [9].
IGBT zero voltage
250 200
Diode zero voltage
150 100 IGBT pulse Diode pulse
50 0 0 Braking
1.0472
2.0944 φ [rad]
3.1416 Accelerating
Fig. 12. Conduction loss distribution in the VSC. Modulation index M = 1 and phase current I = 200 A. Losses corresponds to a single IGBT and diode module respectively.
If zero voltage is applied by the VSC, conduction losses during the zero voltage intervals must be considered. During zero voltage an IGBT and a diode participate in the current conduction and the load is evenly distributed over time among all four switches. The losses in a single switch then is, Z 1 π 1 − δ(θ) PI,D = dθ. (22) i(θ)v(i(θ)) π 0 4 Solving this equation for the IGBT and the diode respectively yields, π VCE,0 Iˆ 2−M PI = − φ cos φ + sin φ 4π 2 rCE Iˆ2 π 2M 2 (23) − 1 + cos φ + 4π 2 3 π VF,0 Iˆ PD = 2−M − φ cos φ + sin φ 4π 2 rF Iˆ2 π 2M (24) − 1 + cos2 φ . + 4π 2 3 Note that the sum of (20) and (23) is equal to (14) while the sum of (21) and (24) is equal to (15). Figure 12 illustrates the conduction losses during different operating modes where φ near zero corresponds to regenerative braking (net power flow from the DC side to the AC side) while φ close to π corresponds to motoring (net power flow from the AC side to the DC side). C. Switching Losses
Solving equation (16) and (17) result in: ˆ VCE,0 IM PI = ((π − φ) cos φ + sin φ) 4π Iˆ2 M rCE 1 + cos2 φ + 2 cos φ + 6π ˆ VF,0 IM (sin φ − φ cos φ) PD = 4π Iˆ2 M rF 1 + cos2 φ − 2 cos φ + 6π
(20)
For a VSC operating in a conventional converter system comprising a low frequency transformer, a power device only conduct current during one half of the fundamental period due to the sign of the load current. Therefore, the power loss due to switching of one power device is one half of the mean energy loss times the switching frequency.
(21)
fsw hEi (25) 2 Due to the constraints put on the modulation scheme to achieve natural commutation of the line side cycloconverter, Psw =
TABLE I
6
Inverter modules DC link voltage Modulation index Nominal current Average power, IGBT Average power, diode
EUPEC Ud M In Pmax,I Pmax,D
Maximum switching frequency [kHz]
S UMMARY OF I NVERTER O PERATING C ONDITIONS
FF200R33KF2C 1800 V 0.7 200 A 500 W 250 W
all four IGBTs in both inverter legs must be commutated each cycle. Therefore, the total power loss due to commutations is doubled compared to the conventional inverter. Psw = fsw hEi
(26)
V. L OSS C OMPARISON BETWEEN S OFT AND H ARD S WITCHING To compare losses at hard and soft switching, an inverter equipped with the investigated IGBT-modules has been studied under different operating conditions. Some of the inverter operating conditions are summarized in Table I. The loading capability of an inverter is determined by the maximum temperature rise, but also by power cycling and thermal cycling of the component. As the component is thermally cycled different thermal expansion coefficients cause mechanical stress in the device. Power cycling cause junction temperature swings and cause bond-wire lift off. Thermal cycling may cause delamination due to thermal expansion incompatibility between baseplate and substrate. In traction, power cycling and thermal cycling are often the limiting factors and measures are taken by the manufacturers to constantly improve packaging and bonding technologies [10], [11]. However, thermal cycling of the component must be considered when determining the inverter ratings. To determine the life time the inverter must be modelled thermally in the time domain. Using this model the thermal loading can be determined for a specific load cycle. Lifetime prediction models [12]–[14] are used to estimate lifetime from cycling capability data from accelerated testing and thermal loading determined from simulations. Since this procedure require extensive knowledge on the application of the vehicle and the packaging and cooling of the module, this approach cannot be applied here. For example the thermal impedance is heavily dependent on the packaging and varies for different current ratings. Instead reasonable maximum losses are assumed in Table I. These losses corresponds to a junction to heatsink temperature rise of approximately 20 K for a full size converter and the losses are 1.7% of the transfered power. The influence of these figures on the relative difference between hard and soft switching has been found to be limited. Figure 13 compare the maximum switching frequency for different operating conditions at nominal current. The lowest curve represent maximum switching frequency at nominal current when the VSC is operated in a conventional system comprising a VSC connected to the line through a line
5
Primary ZV 4
Soft
MF
3 2
Hard MF
1
0 0 Braking
Hard LF
0.5
1
1.5 φ [rad]
2
2.5
3 Accelerating
Fig. 13. Maximum switching frequency concidering maximum allowed junction temperature for different snubber configurations and operating modes.
frequency transformer. During braking, corresponding to φ close to zero, frequency is limited by the losses in the IGBTs. However, at acceleration, corresponding to φ close to π, the frequency is limited by the losses in the diodes. For the second curve, a line side conversion system is introduced and the current is commutated at reduced di/dt from the diode to the IGBT that is passively turned on, thus the IGBT turnon losses and diode recovery losses are heavily reduced. However, IGBT turn-off losses are doubled since every IGBT need to be commutated in each cycle. Therefore, maximum switching frequency at acceleration is significantly increased, while switching frequency at braking is only slightly increased. Maximum switching frequency is limited by the IGBT for all load angles. When the IGBT is snubbered, IGBT turn-off losses are halved and thus the maximum switching frequency is doubled. The losses can be decreased further by applying zero voltage from the primary inverter, thus increasing the switching frequency by approximately 25% at the investigated modulation index. Copper losses in the transformer windings are also reduced but when applying zero voltage from the primary, volt-second area applied to the transformer is increased and thereby the mean flux density. Therefore, core losses in the transformer are increased. In Figure 14 and 15 switching frequency is held constant and the maximum inverter current is derived for the conventional conversion system comprising a line frequency transformer and the medium frequency transformer conversion system respectively. The maximum switching frequency is increased by more than four times in a wide load current range compared to the conventional inverter. The size of the snubber is limited by its physical size and the voltage reduction due to losst volt-second area during commutation. The dependency of the mean turn off losses on snubber size at nominal current is illustrated in Figure 16. Volt-second area is lost during commutation of both the VSC and the cycloconverter. VI. C ONCLUSIONS By replacing the heavy and bulky line frequency transformer in an electric rail vehicle by a medium frequency transformer,
300 250 500 Hz
200
Irms [A]
1000 Hz
150
2000 Hz 100
3000 Hz
4000 Hz
50 0 0 Braking
1.0472
2.0944
3.1416 Accelerating
φ [rad]
Fig. 14. Peak inverter current at acceleration and braking at different switching frequencies for conventional operation.
must be introduced to interface the high frequency transformer to the line. To make the medium frequency concept competitive to the conventional system, the transformer weight must be reduced enough to make up for the cost of the cycloconverter. Standard 3.3 kV IGBT-modules, used in traction today, has been characterised under hard and soft switching conditions. For the investigated IGBT, turn-off losses can be reduced to at least half by introducing capacitive snubbers. The characterized modules are standard modules and are not optimized for soft switching. Modules optimized for soft switching exists, but are not readily available for traction applications today. A loss model for the voltage source inverter has been developed, and the losses of the VSC operated in soft switching and hard switching are compared. It has been found that the switching frequency may be increased by more than four times compared with hard switching. R EFERENCES
300 250 1000 Hz 2000 Hz
Irms [A]
200
4000 Hz
150
8000 Hz
100
16000 Hz
50 0 0
1.0472
2.0944
3.1416
φ [rad]
Braking
Accelerating
Fig. 15. Peak inverter current at acceleration and braking at different switching frequencies for soft switced medium frequency operation. 120 100
off [mJ]
80 60 40 20 0 0
100
200
300
400
500
Cs [nF]
Fig. 16.
Mean turn-off losses at rated current I=200 A.
weight and volume of the line side conversion system may be reduced significantly. However, a primary side power converter
¨ [1] S. Ostlund, “A primary switched converter system for traction applications,” Ph.D. dissertation, KTH, Stockholm, Sweden, 1991. ¨ [2] P. Kjaer, S. Norrga, and S. Ostlund, “A primary-switched line-side converter using zero-voltage switching,” IEEE Trans. Ind. Applicat., vol. 37, pp. 1824–1831, Nov/Dec. 2001. [3] L. Heinemann, “An actively cooled high power, high frequency transformer with high insulation capability,” in Applied Power Electronics Conference and Exposition, 2002. APEC 2002. Seventeenth Annual IEEE, vol. 1, 2002, pp. 352–357. [4] A. Rufer, N. Schibli, C. Chabert, and C. Zimmermann, “Configurable front-end converters for multicurrent locomotives operated on 16 2/3 hz ac and 3 kv dc systems,” IEEE Trans. Power Electron., vol. 18, pp. 1186–1193, Sept. 2003. [5] S. Norrga, “A soft-switched bi-directional isolated ac/dc converter for ac-fed railway propulsion applications,” in International Conference on Power Electronics, Machines and Drives, 2002, June 2002, pp. 433–438. [6] M. Trivedi and K. Shenai, “Modeling the turn-off of igbt’s in hard- and soft-switching applications,” in Electron Devices, IEEE Transactions on, vol. 44, no. 5, May 1997, pp. 887–893. [7] T. Reimann and J. Petzoldt, “The dynamic behaviour of power transistors at impressed di/dt in zvs applications,” in Power Electronics and Applications, EPE ’95. 6th European Conference on, vol. 1, Sept. 1995, pp. 571–576. [8] I. Widjaja, A. Kurnia, K. Shenia, and D. M. D. Divan, “Switching dynamics of igbt’s in soft switching converters,” in Electron Devices, IEEE Transactions on, vol. 42, no. 3, Mar. 1995, pp. 445–454. [9] L. K. Mestha and P. D. Evans, “Analysis of on-state losses in pwm inverters,” Electric Power Applications, IEE Proceedings B, vol. 136, no. 4, pp. 189–195, July 1989. [10] T. Schuetze, H. Berg, and M. Hierholzer, “Further improvements in the reliability of igbt modules,” Conference Record - IAS Annual Meeting (IEEE Industry Applications Society), vol. 2, pp. 1022 – 1025, 1998. [11] T. Schuetze, H. Berg, and O. Schilling, “The new 6.5kv module: a reliable device for medium voltage applications,” in Power Conversion and Intelligent Motion (PCIM), Aug. 2001. [12] M. Ciappa, P. Malberti, W. Fichtner, P. Cova, L. Cattani, and F. Fantini, “Lifetime extrapolation for igbt modules under realistic operation conditions,” Microelectronics and Reliability, vol. 39, no. 6-7, pp. 1131 – 1136, 1999. [13] M. Ciappa and W. Fichtner, “Lifetime prediction of igbt modules for traction applications,” Annual Proceedings - Reliability Physics (Symposium), pp. 210 – 216, 2000. [14] M. Ciappa, F. Carbognani, and W. Fichtner, “Lifetime modeling of thermomechanics-related failure mechanisms in high power igbt modules for traction applications,” IEEE International Symposium on Power Semiconductor Devices and ICs (ISPSD), pp. 295 – 298, 2003.
