Influence of Power Cycling Test Methodology on the Applicability of the Linear Damage Accumulation Rule for the Lifetime Estimation in Power Devices

The lifetime of power semiconductor devices, operating under a given mission profile and subjected to power cycling stress, is conventionally estimated under the assumption of linear damage accumulation rule, that is the application of the Miner's rule. To this purpose, lifetime models must be properly defined allowing to take into account for the relevant parameters of power cycling stress. This work shows how to estimate a cumulative distribution function in the case of an arbitrary temperature swing profile, starting from the statistical distribution at constant power cycling conditions. It is found that the accuracy of the linear damage accumulation rule is related to the experimental methodology adopted for power cycling tests. A detailed experimental activity is carried out on packaged insulated gate bipolar transistor (IGBT) devices, providing useful guidelines for the definition of lifetime models to be adopted in the Miner's rule.

, [7]. The wire bonds degradation is related to the different CTEs between Si and Al. The stress is then localized at the interface between materials and cracks are likely to be formed and to propagate in the aluminum wire [8]. In the case of aluminum reconstruction, thermo-mechanical stresses result in tensile stress at the metallization layer [9]. This type of process can exceed the elastic limits of the aluminum layer, leading to the formation of extrusion granules. In addition, the compressive stress, which is established at the aluminum/mold compound interface, induces a plastic behavior of the aluminum. As a result, an increase of series resistance can be observed [8]. The mismatch between the copper tab and the silicon die, which is a typical interface in discrete power devices, leads to the degradation of the solder joint with possible cohesion fracture or creep fatigue [10]. This type of degradation affects the thermal impedance of the component, since the heat dissipation takes place through the copper tab.
The above-mentioned degradation mechanisms are mainly triggered by the temperature cycling, but they are also affected by the average temperature and the heating time. Several models have been reported in literature, allowing to account for these parameters [11], [12], [13], [14], [15], [16], [17].
In general, there is a strong request for an accurate prediction of the lifetime in power electronics, in order to satisfy the reduction of development and testing time [18]. In a consolidated approach, the analysis of the reliability of a generic power system begins with the study of the mission profile [19], [20], [21]. Based on the electric and thermal models of the system, the mission profile is translated in a temperature profile in power semiconductor devices. The rainflow algorithm can be adopted to evaluate the number and the amplitude of temperature cycles [22]. Lifetime models are used to predict number of cycles to failure as a function of relevant parameters: temperature swing ΔT j , minimum temperature T j,min , heating time t on , and current density per wire [5], [10], [11], [12], [14], [23]. The number of cycles to failure can be defined either as the average number or as the number leading to a given probability of failure (PoF). Hence, based on the considered lifetime model, Miner's rule is adopted to predict the lifetime consumption for a given temperature profile, under the assumption of linear damage accumulation (LDA). The applicability of LDA is a fundamental point, which have been considered in literature. In [21], LDA rule was validated considering the superimposition of different temperature profiles, having different heating times. Moreover, the analysis in [21] was carried out at different values of PoF. In [24], the application of combined power cycling stresses led to an underestimation in the lifetime prediction, which was explained assuming a dual degradation mechanism, resulting in a prediction error. In [25], under the assumption of a single degradation mechanism, combined power cycling stresses verified the applicability of LDA rule. In [26], combined experimental tests at different ΔT j values did not verify the linearity of Miner's rule, particularly in the case of a combined stress with significantly different ΔT j values (varying between 110 and 70°C ). In [27] and [28], the impact of combined vibrating and thermal cycling stresses was analyzed. An overestimation of the lifetime was found by applying Miner's rule. This inconsistency was ascribed to a change of the thermo-mechanical response due to the interaction between different types of stress [27] or to an additional stress phenomenon due to random vibrations during the test being temperature dependent [28]. In [29], a nonlinear cumulative damage model was proposed for ceramic column grid array electronic package subjected to a combination of thermal cycling and vibration. Also in [30], combined thermal cycling and vibration stress, under the assumption of a single failure mechanism, i.e., solder fatigue, led to an overestimation of lifetime with Miner's rule, because of dynamic effects combined with thermal stress. In [31], a nonlinearity in the accumulation of damage to solders in combined thermal cycling and vibration stress was found. In this case, the prediction error, despite the hypothesis of a single degradation mechanism and no interaction between the two stresses, was ascribed to the formation of intermetallic material at the interfaces or to the increase of voids size, amplifying the degradation process.
This work aims at extending a recent conference paper [32], in which the cumulative distribution function (CDF) was built up in the case of non-constant power cycling tests by means of Miner's rule and the results were compared with experiments. Different from [32], this work investigates how the power cycling test methodology affects the applicability of the linear damage accumulation rule. More specifically, power cycling tests are performed by following two different approaches: 1) "non-controlled ΔT j " approach, in which a constant heating current is used to achieve the desired ΔT j ; 2) "active control of ΔT j " approach, in which the heating time is modulated in order to keep the ΔT j value close to the desired value for the entire experiment. Therefore, this work investigates the influence of accelerated testing methodology on the lifetime estimation. Finally, compared to [32], the experimental activity is extended to different discrete power devices and with a larger records of test conditions. The remainder of this article is organized as follows. Section II describes in detail the experimental setup adopted for power cycling tests, focusing on the different methodologies considered in the article. Section III reports the experimental results in the case of constant ΔT j stress and in the case of an arbitrary profile of ΔT j . In the latter case, experimental CDFs are compared with the predictions arising from the Miner's rule application. Finally, in Section IV, the main conclusions are summarized.

A. Experimental Setup
Power cycling experiments are performed by adopting a custom designed board, whose schematic representation is reported in Fig. 1. Four devices are stressed simultaneously under the same value of I dc current, considering a multiplexing approach. Electronic switches S 0 -S 3 are adopted to divert the current between the four devices under test (DUTs). A CompactRio board is used to control electronic switches S 0 -S 3 , to acquire the voltage drop V ce and to communicate with the PC. Fig. 2 illustrates the typical current and voltage waveforms in one DUT. During the heating time (ON), a large current I dc flows in the device, while during the cooling-down phase (OFF) a reference current I ref of 50 mA is injected in the device in order to sense V ce . As reported in Fig. 2(b), the V ce,off profile is used to Fig. 3. Picture of the setup. The test circuit is placed on a liquid-cooled thermal plate, whose temperature is fixed by means of a temperature controller [17]. estimate the junction temperature (T j ) profile. To this purpose, the T j vs. V ce characteristic at I ref = 50 mA is first derived by means of a source measure unit and with the device placed at controlled temperature in an oven. Therefore, a temperature sensitive electrical parameter (TSEP) method is used to derive the temperature profile [33], [34]. The inset of Fig. 2(b) shows the result of the TSEP methodology. The profile starts with a maximum junction temperature T j,max , arising from the heating phase, and approaches to a minimum value T j,min at the end of the cooling-down phase. The temperature cycling is defined as ΔT j = T j,max -T j,min . According to Fig. 2, the experimental activity has been carried out considering a periodicity of 2.5 s on four samples, hence leading to t on = 0.625 s and t off = 1.875 s, being t on and t off the duration of ON and OFF phases, respectively. According to [33] and [34], the selected value of t on allows for an accurate temperature estimation when a TSEP method is considered.
As reported in Fig. 1, a bypass transistor can be adopted to sustain the I dc current. When a DUT fails, the corresponding switch must be permanently opened and the switch S 4 is enabled in order to maintain the same t on and t off values on the remaining DUTs.
A picture of the experimental setup is reported in Fig. 3. The custom board, including the DUTs, is mounted on a liquidcooled thermal plate. A thermal controller, Julabo Presto A40, is then adopted to control the temperature of the thermal plate. DUTs are placed on the back side of the custom board and are in contact with the thermal plate. The average junction temperature can be estimated as follows: where P av is the average power dissipated in the DUTs, R th,jh is the thermal resistance between the junction of the DUT and the thermal plate, and T ref is the temperature of the thermal plate, being fixed by the temperature controller. Hence, for each experiment the T ref is properly tuned in order to achieve the desired T j,av . It is worth noting that the R th,jh value of a DUT is also influenced by adjacent devices (mutual heating effects). Hence, when a DUT fails and is then permanently turned-off, small changes of T j are observed in adjacent DUTs. This effect is compensated by properly modifying the T ref value. However, a transient effect may be visible in T j and V ce profiles. DUTs used for experiments are commercial insulated gate bipolar transistor (IGBT) with TO-247 package, with a maximum pulsed current of 120 A, a rated voltage of 650 V, and a maximum junction temperature of 175°C. The gate of DUTs is biased with a constant dc voltage of 15 V.
The state-of-health of DUTs, subjected to power cycling stress, is estimated by monitoring the junction-to-case thermal impedance (Z th,jc ) and the V ce voltage at the end of the ON phase (V ce,on ). The failure event is determined by an increase of Z th,jc by 20% or V ce,on by 5% [2]. For each power cycling condition, this work aims at analyzing the statistics of failure events. For this reason, 12 samples are tested under the same conditions.
As far as power cycling tests are concerned, different methodologies can be adopted to obtain the desired junction temperature cycling [35], [36], [37], [38]. In this work, the considered approaches are: "non-controlled ΔT j " and "active control of ΔT j ." Details are reported in Section II-B.

B. Methodologies Adopted for Power Cycling Tests
The power cycling tests, as discussed in Section II-A, allows achieving the desired ΔT j by properly selecting the heating current I dc and the heating time t on . This type of calibration is performed at the beginning of the experiment. In the case of "non-controlled ΔT j " approach, I dc and t on are kept constant for the entire experiment.
As reported in Fig. 4 (blue curves), the actual ΔT j value is not constant over the device lifetime. In fact, in the wear-out region, the device is subjected to a degradation (increase) of the thermal impedance Z th,jc or voltage drop V ce,on . As a consequence, the internal temperature of the device increases. The ΔT j value is a crucial parameter in determining the lifetime of power components [39]. Its increase causes a faster degradation, leading to a lower number of cycles to failure. 2) Active Control of ΔT j : In order to keep the ΔT j value actually constant, or within a limited range of variation, the "active control of ΔT j " is adopted. As shown in Fig. 4 (red curves), the increase of temperature (due to the wear-out phenomenon) is compensated by reducing the heating time of the DUT. More specifically, a hysteretic control is considered in this work. The hysteresis thresholds are ±1°C with respect to the reference ΔT j value. Hence, every time the ΔT j value is out of the thresholds, the t on time is reduced, or eventually increased. In this work, the change of t on is limited to 30% with respect to the initial value. It is worth noting that the heating time is also a parameter affecting the lifetime of power components, even if to a lesser degree with respect to ΔT j . Hence, it is important to avoid significant changes of t on during the test.
This method is applied to all four devices mounted on the test board. As a consequence, the sum of all t on times could be different (lower) with respect to the periodicity of the control signals. In this case, the bypass transistor (see Fig. 1) is activated, for a limited time, in order to warranty a constant periodicity of power cycling tests.
3) Non-Constant Cumulative Stress: Regardless of the previously discussed methods being adopted to control ΔT j , this work aims at evaluating the case of non-constant power cycling tests, in which the ΔT j is changed on purpose during the test. This is achieved by modifying the I dc value during the power cycling test.
Non-constant power cycling tests are designed by considering the combination of two specific ΔT j values: 120 and 140°C. The detailed list of tests is reported in Table I. It can be observed that non-constant power cycling tests foresee a different order of ΔT j stresses and a different switching point (expressed as a number of cycles).
The study of the lifetime under non-constant cumulative stress requires the knowledge of the statistics of failure events occurring under constant stress conditions. The cumulative distribution function (CDF) gives the probability that a device will fail within a given number of cycles N (in order words the percentage of population expected to be failed as a function of N). The Weibull statistics is widely adopted to describe thermal/power cycling phenomena in power semiconductor devices [1]. The CDF is expressed as follows: where α is the shape parameter and β is the scale parameter. By considering the reverse function of (2), it is possible to estimate the expected number of cycles to failure at different probabilities of failure (PoFs): 10%, 25%, 50%, and 75%. In order words, this is the number of cycles at which a given percentage of a population will have failed. This kind of estimation is performed for both 120 and 140°C constant stresses. The Miner's rule is usually adopted in order to estimate the lifetime consumption (LC) of devices subjected to non-constant thermal/power cycling stresses [40], [41]. Under the assumption of linear damage accumulation, the LC at a given y percentage of PoF is given by the following: where n i is the number of cycles for the ith type of stress and N i,y is the expected lifetime, in terms of number of cycles, at the y percentage of PoF for the ith type of stress. Hence, N i,y is estimated by reversing (2) for ΔT j = 120°C and ΔT j = 140°C. LC represents the fraction of life consumed by the application of a non-constant power cycling stress. When LC y reaches the value of 1, the failure occurs and the lifetime can be determined. Being a statistical event, this lifetime represents the number of cycles with y percentage of probability of failure. The data extrapolated with the Miner's rule can be then compared with experimental results arising from the application of non-constant cumulative stress.

A. Power Cycling Tests Under Constant ΔT j Stress
In this section, the experimental results of power cycling tests are reported, by considering constant ΔT j values: 120 and 140°C. In both cases, "non-controlled ΔT j " and "active control of ΔT j " approaches are considered for the sake of comparison. V ce,on and ΔT j profiles are reported in Fig. 5 for the nominal ΔT j = 120°C. The adopted heating current is 63.5 A with a and ΔT j = 140°C. Results arising from both techniques, "active control of ΔT j " and "non-controlled ΔT j ," are reported. Experimental data are fitted assuming a Weibull distribution. Prediction bounds (99%) are also included in the plots.  Table I). Lifetime consumption (LC) is calculated according to (3), by considering the expected number of cycles estimated in Fig. 6 for a PoF of 10%. The prediction interval arises from the prediction bound of Fig. 6. maximum variation of ±0.5 A, with T j,min = 25°C. In the case of "active control of ΔT j ," the temperature swing is kept constant to the nominal value of 120°C, within the hysteresis threshold of 1°C [see Fig. 5(a)]. The V ce,on profile, reported in Fig. 5(b), is initially flat, while it sharply increases close to the end of life of the components. In all 12 experiments the failure is determined by an increase of V ce,on by 5%, while Z th,jc is almost unchanged (not shown here). On the other hand,  Table I) Fig. 5(b) is responsible of modifications in the number of cycles to failure. In the case of ΔT j = 140°C, an heating current of 68.5 A is adopted, with T j,min = 25°C. V ce,on and ΔT j profiles (not reported here) are analogous to those reported in Fig. 5.
The experimental number of cycles to failure can be adopted to build the CDF plot. By means of the Bernard formula [42], the experimental CDF is expressed as follows: where N k is the number of cycles to failure of the kth experiment (with experiments sorted in ascending order according to the number of cycles to failure) and N tot = 12 is the total number of experiments. Results are reported in Fig. 6 for both ΔT j = 120°C and ΔT j = 140°C and for both "non-controlled ΔT j " and "active control of ΔT j " approaches. Aiming at linearizing the dependence between CDF and N, the expression of (2) can be written as follows: Hence, in the case of Weibull distribution, a linear fitting can be adopted to estimate both α and β parameters. Lines at specific PoFs are reported in Fig. 6 and are labeled B10, B25, B50, and Fig. 9. Experimental non-constant ΔT j stresses for Test 1. In (a) the temperature cycling profile is obtained by actively controlling the heating time. In (b) a constant heating current is adopted, leading an increase of temperature close to the end of life. Experimental CDFs, for both ΔT j profiles, are reported in (c) and compared with those calculated according to the Miner's rule (see Fig. 8).
B75. In general, the adoption of "active control of ΔT j " approach leads to a larger number of cycles to failure for a given PoF with respect to the "non-controlled ΔT j " approach. In the latter case, according to [43], a positive feedback relationship between the wire bonds degradation and ΔT j leads to lower lifetimes.
CDFs, estimated in the case of "active control of ΔT j " approach, exhibit a similar shape parameter α, while the change of the stress level leads to a modification of the scale parameter β. On the other hand, the adoption of "non-controlled ΔT j " approach leads to a statistic in which the shape parameter α is significantly reduced in the case of ΔT j = 140°C. It is possible that during the degradation phase the non-controlled increase of temperature can cause some early failures, hence modifying the α parameter of the distribution.

B. Power Cycling Tests Under Non-Constant ΔT j Stress
In the case of non-constant ΔT j stress, Miner's rule is adopted for the lifetime estimation. The case of Test 1 (as illustrated in Table I) is reported in Fig. 7. Non-constant stress is defined as: 6000 cycles at ΔT j = 140°C and the remaining cycles at ΔT j = 120°C, with T j,min = 25°C. According to (3), the lifetime consumption is calculated by considering the expected number of cycles at ΔT j = 120°C and ΔT j = 140°C. These values can be directly derived from Fig. 6(a) and (b) (active  control of ΔT j ) for the given PoF (10%). However, the CDFs of Fig. 6 are defined within given prediction bounds with a level of certainty of 99%. Consequently, the LC profile is also known in a prediction interval, as reported in Fig. 7. The lifetime is then calculated as the number of cycles leading to LC = 1.
Overall, a lifetime interval can be estimated, arising from the limited statistics in the experimental activity. For the sake of comparison, the analysis of Test 1 is then carried out by considering lifetime models derived with both "non-controlled ΔT j " and "active control of ΔT j " approaches and for the probability of failure ranging from 10% to 75%. The application of Miner's rule for both cases is reported in Fig. 8. The lifetime consumption is estimated in Fig. 8(a) and (c) at different PoF. By using these pairs of values, i.e., the number of cycles to failure and the PoF, a CDF can be predicted according to Miner's rule [see Fig. 8(b) and (d)]. Although both predicted CDFs are included in the range of constant stresses (ΔT j = 120°C and ΔT j = 140°C), the adoption of lifetime models  Table I). In all considered cases, the experimental CDF is within the prediction bound of the Miner's rule prediction. calibrated with a "non-controlled ΔT j " approach leads to higher probability of failure (under non-constant stress).
Experimental non-constant ΔT j stresses are reported in Fig. 9 in the case of Test 1. In Fig. 9(a), ΔT j profiles were obtained by actively controlling t on and hence exactly matching the conditions of Test 1 (see Table I). This is the most appropriate profile to consider, since the only available lifetime models are those for ΔT j = 120°C and ΔT j = 140°C. For the sake of comparison, in Fig. 9(b) ΔT j profiles were generated by only controlling the heating current, hence an uncontrolled temperature increase close to the end of life is observed. In Fig. 9(c), by considering the CDF calculated on the basis of models calibrated with the "active control of ΔT j " methodology, the application of the Miner's rule leads to a lifetime prediction being in a very good agreement with the experimental CDF deriving from the tests of Fig. 9(a). As reported in Table II, the experimental number of cycles to failure is always included in the prediction interval (associated to the Miner's rule estimation) for the full range of PoFs. In the case of "non-controlled ΔT j " approach, the application of Miner's rule leads to a lifetime prediction which is accurate in the case of large PoFs, while at low PoF the experimental results differ from the calculated values (they are even outside of the prediction intervals). Considering the non-constant stress profile of Fig. 9(b) (in which the stress methodology is analogous to the one adopted for the calibration of lifetime models) the difference between the Miner's prediction and the experimental CDF decreases but it is still relevant in the case of PoF close to 10%. The error around PoF = 10% can be explained by considering CDFs at constant stress reported in Fig. 6. More specifically, in Fig. 6(d) the number of cycles to failure for ΔT j = 140°C is very low in the case of PoF = 10%. As discussed in Section III-A, this is probably due to the positive feedback relationship between the wire bonds degradation and ΔT j , possibly leading to the premature failure of samples in which the thermo-mechanical stress is not kept constant. As a result, the application of the (3) in the case of combined 140°C/120°C stress leads to an underestimation of the lifetime with respect to the experimental value at PoF = 10%.
According to the analysis reported in Fig. 9 and Table II, the way in which accelerated lifetime tests are performed can have an impact on the accuracy of the linear damage accumulation theory. On one hand, if lifetime models are calibrated by means of accelerated tests with "active control of ΔT j ," the thermo-mechanical stress can be considered constant, since ΔT j  is fixed at the nominal value. Consequently, Miner's rule gives an accurate prediction when the considered stress is a combination of the stresses at constant ΔT j . On the other hand, during the calibration of lifetime models based on a "non-controlled ΔT j " approach, power devices are subjected to a temperature cycling exceeding the nominal ΔT j value. Therefore, the effective ΔT j value to be considered for the lifetime modeling purpose should be higher. When applying Miner's rule for a given (non-constant) temperature profile, the adopted lifetime model is based on the nominal ΔT j value rather than the effective ΔT j value. Hence, some inaccuracies are introduced in the lifetime estimation.
The "active control of ΔT j " approach is extensively verified in all the test conditions reported in Table I and the results are illustrated in Fig. 10. The different conditions foresee the same ΔT j stresses (120 and 140°C), but with different orders and switching points. Therefore, Miner's rule predictions can be again estimated from the results of Fig. 6. As illustrated in Fig. 10 the experimental CDFs are always well aligned with the application of the Miner's rule. The maximum error, which is reported in Table III, is in the order of 10%, which typically falls in the prediction bound calculated for the lifetime estimation. Therefore, we can conclude that the application of Miner's rule allows accurately calculating the number of cycles to failure at any PoF.

C. Analysis of Degradation Mechanisms
Device under tests considered in this article are discrete IGBTs in TO-247 package. They are characterized by a typical lead-frame substrate and solderable pins as terminal contacts. Discrete devices are encapsulated in a transfer mold compound based on an epoxy resin [44].
In order to observe the presence of stress in the solder joint region, X-ray images are captured by means of an EasyTom tomograph. The fresh sample, reported in Fig. 11(a), shows some voids at the interface between the silicon die and the copper tab, which can be ascribed to the manufacturing process [21], [23], [25]. Similarly, devices subjected to power cycling (both constant and non-constant temperature cycling) exhibit some voids, but no signs of delamination can be found in Fig. 11(b), (c), or (d). It is worth noting that the solder joint has a significantly larger volume, with respect to wire bonds, with a consequent higher thermal time constant. For this reason, the solder joint fatigue typically occurs when considering a longer heating time than the value considered in this article (t on = 0.625 s) [8].
The packages of some samples have been opened for the inspection of wire bonds. As reported in Fig. 12, in all considered cases (both constant and non-constant temperature cycling, both "active control of ΔT j " and "non-controlled ΔT j ") the formation of a crack at the Al/Si interface is visible in the images acquired through a Leica MS5 microscope. Therefore, according to this study, the discussion about the applicability of the LDA theory is related to the wire bonds degradation mechanism. It is worth mentioning that Miner's rule can be considered only if a single failure mode occurs in the component [25].
The analysis of degradation mechanisms is in agreement with the electrical wear-out of the components. In fact, as reported in Fig. 5, the failure events are associated to an increase of V ce,on , while Z th,jc is basically unchanged [45].

IV. CONCLUSION
In this article, different methodologies are considered for the accelerated lifetime testing of TO-247 IGBT devices subjected to power cycling stress, aimed at understanding, from an academic point of view, differences in the lifetime estimation under non-constant stress. More specifically, the "active control of ΔT j " approach consists in dynamically modulating the heating time in order to keep a constant ΔT j . In this case, the analysis of an arbitrary temperature profile by means of Miner's rule, allows to predict the CDF in very good agreement with experimental tests (under non-constant stress). Hence, the applicability of the linear damage accumulation rule is confirmed under the hypothesis that lifetime models are calibrated with an "active control of ΔT j " approach. However, guidelines for the qualification of power devices, such as [46], typically do not allow for modifications of the heating time during the power cycling test. This approach is referred to as "non-controlled ΔT j " method in this article, since the application of a constant heating current leads to a change of the temperature swing during the test (due to the modification of self-heating effects). In this case, the application of the Miner's rule for an arbitrary mission profile can lead to less accurate results, but still in the prediction bounds if large probabilities of failure are considered.