Discretization of trapezoidal power loss

The working mode of an accelerator power supply is unique; it works in the frequency-doubling chopper mode and outputs a DC or pulsed current. Figure 1 shows a typical current waveform of a quadrupole magnet pulsed power supply. The pulsed current waveform (black) can be approximated as a trapezoidal waveform (orange). The device power loss is a trapezoidal wave, which is divided into four stages: rising, flat top, falling, and flat bottom. t₁, t₂, t₁, and t₃ are the times for each stage. The rates of increase and decrease are equivalent.

Fig. 1Full-screen View

Power loss profile and discretization results: (a) Output current; (b) Power loss; (c) One-level square-wave loss; (d) k-level square-wave losses

As observed in Fig. 1, the trapezoidal power loss is divided into multiple square-wave pulses, where P_loss is the power loss of the IGBT module including conduction loss and switching loss; P₁, P₂, …, P_k are the discrete square-wave power pulses; and k is the classification level. When the divided pulse is narrower, the energy of the divided pulse is closer to the energy of the trapezoidal power loss. A one-level discrete pulse with k = 1 is expressed as: $P_{\text{1}}=P_{\min }+\left(P_{\max }-P_{\min }\right)\cdot \left(a+b\right)=P_{\text{ave}},$ (1) where P_max and P_min denote the flat-top and flat-bottom values of the power loss, respectively, P₁ denotes the amplitude of the square-wave loss, P_ave denotes the average power loss, f₀ denotes the frequency of the periodic current and the power loss, and a and b are the ratios of t₁ and t₂ to 1/f₀, respectively.

When the trapezoidal loss profile is divided into k parts, the amplitude of each discrete pulse in the rising section can be obtained using: $\begin{array}{c}{P}_i=\frac{1}{\Delta t}{\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}\left(\frac{P_{\max }-P_{\min }}{a/f_0}t+P_{\min }\right)\text{d}t}\\ =P_{\min }+\frac{P_{\max }-P_{\min }}{a/f_0}\cdot \frac{\Delta t}{2}\cdot \left(2i\text{-}1\right),\end{array}$ (2) where Δt represents the pulse interarrival time equal to 1/f₀k and P_i represents the amplitude of the ith pulse. The amplitude of each discrete pulse in the falling section is expressed as: $\begin{array}{c}{P}_i=\frac{1}{\Delta t}{\displaystyle {\int }_{(i-1)\Delta t}^{i\Delta t}\left(-\frac{P_{\max }-P_{\min }}{a/f_0}\left[t-\left(2a+b\right)\frac{1}{f_0}\right]+P_{\min }\right)\text{d}t}\\ =\frac{P_{\max }-P_{\min }}{a}\cdot \left(2a+b\right)+P_{\min }+\frac{P_{\max }-P_{\min }}{a/f_0}\cdot \frac{\Delta t}{2}\cdot \left(1-2i\right).\end{array}$ (3) Therefore, when the partition level of the trapezoidal power loss is k, the power amplitude is: $P_i=\{\begin{array}{ll}{P}_{\min }+C_1\cdot \frac{\Delta t}{2}\cdot \left(2i\text{-}1\right)\hfill & 1\le i\le \left[a\cdot k\right]\hfill \\ P_{\max }\hfill & \left[a\cdot k\right]<i\le \left[\left(a+b\right)\cdot k\right]\hfill \\ P_{\min }+C_1\cdot \frac{\Delta t}{2}\cdot \left(1-2i\right)+C_2\hfill & \left[\left(a+b\right)\cdot k\right]<i\le \left[\left(2a+b\right)\cdot k\right]\hfill \\ P_{\min }\hfill & \left[\left(2a+b\right)\cdot k\right]<i\le k\hfill \end{array},$ (4) where C₁ and C₂ are constants for simplicity of expression, written as: C₁=(P_max-P_min)f₀/a, C₂=C₁(2a+b)/f₀.

Junction temperature calculation

The IGBT module has a sandwich structure, including a chip layer, solder layer, copper layer, and ceramic layer, among others. The power loss is transferred to the heatsink through various layers of materials and is brought out through the cooling water. Thermal resistance and heat capacity are used to describe heat conduction according to the similarity between thermal and electrical engineering. The thermal impedance network of the IGBT module generally comprises multiple independent resistor-capacitor (RC) units and converts the power loss to a temperature difference. Specifically, the thermal impedance network constitutes the thermal impedance from junction to case Z_thjc, the thermal impedance from case to heatsink Z_thch, and the thermal impedance from heatsink to ambient Z_thha.

The conversion from the power loss to the thermal profile is depicted in Fig. 2. For two power dissipation pulses, P₁ and P₂, the junction temperature fluctuation was calculated using ^[18]: $\{\begin{array}{l}\Delta T_{\text{1}}=P_1\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)\\ \Delta T_{\text{2}}=P_1\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{2\Delta t}{{\tau }_{\text{th}}}}\right)+\left(P_2-P_1\right)\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)\\ \begin{array}{cc}& =\end{array}\Delta T_{\text{1}}\cdot e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}+P_2\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)\end{array},$ (5) where P₁ and ΔT₁ are the power loss and temperature at Δt, respectively; thermal resistance R_th and time constant τ_th constitute an RC combination of the heat transfer; P₂ and ΔT₂ are the power loss and temperature at 2Δt, respectively. The junction temperature change at moment iΔt is calculated in the same way as for the two power pulses. Thus, for i power pulses, the junction temperature fluctuation is calculated using: $\{\begin{array}{l}\Delta T_{i-\text{1}}=P_{i-1}\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)\\ \Delta T_i=\Delta T_{\text{i-1}}\cdot e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}+P_i\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)\end{array},$ (6) where P_i-1 and ΔT_i-1 denote the power loss and temperature at previous times, respectively; P_i and ΔT_i are the power loss and temperature, respectively, at the subsequent times.

Fig. 2Full-screen View

Junction temperature curves obtained for (a) two power pulses and (b) multiple power pulses

According to Fig. 2, the temperature reaches a minimum at the kth pulse. Substituting Eq. (4) into Eq. (6), the minimum temperature fluctuation is obtained as: $\begin{array}{l}\Delta T_{\min }=R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right){\left[\begin{array}{l}{P}_k\\ P_{k-1}\\ ⋮\\ P_1\end{array}\right]}^T\left[\begin{array}{l}1\\ e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\\ ⋮\\ e^{-\frac{\left(k-1\right)\Delta t}{{\tau }_{\text{th}}}}\end{array}\right].\\ \begin{array}{cc}& \end{array}={\displaystyle \sum _{i=1}^k{P}_i\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)}\cdot e^{-\frac{\left(k-i\right)\Delta t}{{\tau }_{\text{th}}}}\end{array}$ (7) The temperature reaches a maximum at a pulse time of [(a+b)k]+1. $\Delta T_{\text{max}}={\displaystyle \sum _{i=1}^{[\left(a+b\right)k+1]}{P}_i\cdot R_{\text{th}}\cdot \left(1-e^{-\frac{\Delta t}{{\tau }_{\text{th}}}}\right)}\cdot e^{-\frac{\left[\left(a+b\right)k-i+1\right]\Delta t}{{\tau }_{\text{th}}}}.$ (8) The maximum junction temperature can be expressed as: $T_{\text{j}\max }=\Delta T_{\text{thjc}}+\Delta T_{\text{thch}}+\Delta T_{\text{thha}}+T_0,$ (9) where T₀ denotes the ambient temperature; ΔT_thjc, ΔT_thch, and ΔT_thha are the temperature differences from junction to case, case to heatsink, and heatsink to ambient, respectively. Accordingly, the maximum and minimum junction temperatures were obtained.

The calculation error of the junction temperature decreases with an increase in the divided level k, but a sufficiently accurate division will lead to computational complexity. The timescale of the thermal impedance is above 1 ms ^[16]; therefore, the pulsed interval Δt = 1 ms is considered as the appropriate value in this study. In addition, k^*= 1000/f₀.

The relative error of estimated temperature is given by: $\epsilon =\frac{T_{\text{j}\max }\left(k^{*}\right)-T_{\text{j}\max }\left(k\right)}{T_{\text{j}\max }\left(k^{*}\right)}=1-\frac{T_{\text{jm}ax}\left(k\right)}{T_{\text{jm}ax}\left(k^{*}\right)}.$ (10)

Accelerated power cycling test setup

The lifetime model of the IGBT module can be divided into physical and empirical models. The aging process of a device is regarded as a physical or chemical process in the physical lifetime model. The damage model obtained using the plastic and creep strains of materials must be combined with FE analysis. The empirical lifetime model adopts a statistical principle to establish a model between the number of cycles to failure and external conditions. Accelerated power cycling is an important test for establishing a lifetime model. Most reliability evaluations have been performed with theoretical estimations and are separate from power cycling tests. As described herein, an AC power cycling test under realistic electrical conditions was performed, and the test results with 600-A 1,200-V IGBT modules were used to develop simple corrected lifetime model parameters.

Figure 3 presents the configuration of the accelerated power cycling test setup based on a quadrupole magnet pulsed power supply. Accordingly, six constituent parts can be seen: voltage source, DC-link capacitor, IGBT modules, filter inductor, filter capacitor, and quadrupole magnet. The quadrupole magnet power supply operates in double frequency chopper mode, adopts a full bridge topology, and comprises two half-bridge modules (Infineon FF600R12ME4). When the power supply outputs a pulsed current, V1 and V4 switch on and off, and VD2 and VD3 are used for freewheeling.

Fig. 3Full-screen View

(Color online) Configuration of the power cycling test setup

The parameters of the AC power cycling test are listed in Table 1. Module-1 and module-2 in the pulsed power supply produce the same power loss and junction temperature fluctuation and endure the same electrical-thermal stress. The maximum estimated junction temperature T_{jmax_est} before the compensation was approximately 101 ℃, and the minimum estimated junction temperature T_{jmin_est} was approximately 57 ℃. T_{jmax_IR} measured using an IR camera was approximately 100.29 ℃, while T_{jmin_IR} was approximately 57.54 ℃.

According to existing research ^[17], the on-state saturation voltage V_CE between the collector and emitter of the IGBT module is selected as a characteristic parameter reflecting the aging state, and an increase in V_CE of 5% is considered as the judgment criterion of failure. V_CE of the IGBT is measured offline by the circuit shown in Fig. 3. The driving signals of V1 and V4 were connected with a constant 15 V and the tubes were conductive; therefore, the circuit became a current path.

The experiment was accomplished as follows: the initial on-state saturation voltage change rate ΔV_CE was set to 0, the initial on-state saturation voltage V_CE1 was measured, and then ΔV_CE was determined as greater or less than 5%. If ΔV_CE was greater than 5%, the power-cycling test was terminated. Otherwise, 25,000 cycles of the power cycling test were performed, the present on-state saturation voltage V_CE2 was measured, and the new voltage change rate ΔV_CE was calculated. This process was repeated until the failure criterion was reached, which was increased by 5%. The voltage measurement points were led out through the wires.

Test results

Figure 4(a) depicts the relative variation in V_CE for the two IGBT modules during the power cycling test, where the red arrows indicate the failure points. In this test, there was a slight increase in V_CE before 7×10⁵ cycles. From 9×10⁵ to 2.5×10⁶ cycles, the value of V_CE was stable. Subsequently, it suddenly increased. After approximately 2.55×10⁶ cycles, the saturation voltage of module-1 increased by more than 5% from the initial value. After approximately 2.68×10⁶ cycles, the saturation voltage of module-2 increased by 5%. Degradation of the IGBT modules was realized, and the power cycling test was completed. Therefore, the number of cycles until failure were N₁ = 2.5×10⁶ and N₂ = 2×10⁶.

Fig. 4Full-screen View

Power cycling test results and N_f-ΔT_j curve. (a) Results of accelerated power cycling tests; (b) Fitting curves obtained from power cycling tests

The power cycling capability of power semiconductor modules is mostly modeled by the Coffin-Manson law, in which the number of cycles to failure (N_f) is assumed to be proportional to ∆T_j^{n [19]} (∆T_j represents the junction temperature fluctuation per power cycle), expressed as: $N_{\text{f}}=\alpha {\left(\Delta T_{\text{j}}\right)}^n,$ (11) where α and n are the fitting parameters. Equation (11) is rewritten in logarithmic form as follows: $\ln N_{\text{f}}=\ln \alpha +n\ln \left(\Delta T_{\text{j}}\right).$ (12) This appears as a straight line when plotting log(N_f) over log(∆T_j). As shown in Fig. 4(b), the blue curve represents the IGBT4 1200V industrial model PC curve at T_vjmax=125 ℃. log(N_f) and log(∆T_j) can be approximated by a linear function with the slope of −6.9062. Combined with the slope, the test results were fitted to obtain a simple corrected lifetime model for quantitatively evaluating the lifetime of an IGBT module. The values of α and n are α = e^40.886 = 5.7091×10¹⁷ and n = −6.9062, respectively. The corrected Coffin-Manson model is written as: $N_{\text{f}}=5.7091\times {10}^{17}{\left(\Delta T_{\text{j}}\right)}^{-6.9062}.$ (13)

Small bubbles were observed inside the failed modules after the experiments. The appearance of bubbles impacts IGBT modules, and can cause unstable performance. If the module continues to operate, more serious failures will occur, including bond wire deformation, solder layer delamination, and core melting. Owing to the different thermal expansion coefficients of the materials in each layer inside the IGBT module, the materials in each layer were aged under the impact of accumulated thermal stress.

Lifetime analysis under two operating conditions

Figure 6 displays a flowchart of the lifetime estimation, including the aforementioned details. It is divided into two stages: one comprises power loss discretization and junction temperature computation of a square-wave pulse, while the other involves an accelerated power cycling test and result processing. The junction temperature is the core factor in the proposed lifetime prediction scheme, as verified in the previous section. For applications involving a trapezoidal pulsed current, a more practical set of parameters was selected to estimate the lifetime.

Fig. 6Full-screen View

Flowchart of lifetime estimation

In practice, an accelerator pulsed power supply outputs various pulsed currents with different parameters. Two typical working conditions were selected from multiple operating pulse conditions: small and large pulsed period and current amplitude. The parameters of the two current waveforms are detailed in Table 3, including the rising time, falling time, flat-top time, flat-bottom time, turning time, flat-bottom current, and flat-top current. The pulsed periods were 7 and 17 s, and the corresponding values of the flat-top current were 300 and 500 A.

The junction temperature under two typical waveforms was calculated at k^*, where the maximum temperatures T_jmax were 74 and 113.8 ℃, and the temperature fluctuations ΔT_j were determined as 31.7 and 54.4 ℃. Figure 7 depicts the periodic current and junction temperature profiles. The junction temperature increased with an increase in current value. The maximum and minimum junction temperatures were realized at the end of the flat-bottom section and flat-top section, respectively. The blue circles in the figure represent the maximum and minimum junction temperatures. The thermal stress generated by current waveform-1 is lower than that generated by current waveform-2.

Fig. 7Full-screen View

Periodic junction temperature profiles with two different current profiles for (a) waveform-1 and (b) waveform-2

Based on the junction temperature results, the number of cycles to failure (N_f) of the two current conditions, calculated using Eq. (13), were N_f1 = 2.4544×10⁷ and N_f2 = 5.8165×10⁵. The empirical lifetime models also include the Coffin-Manson-Arrhenius and Bayerer models. The Bayerer model was adopted here as a reference because it considers more factors. As a result, the number of cycles to failure (N_f) obtained using different lifetime models are listed in Table 4. It can be observed that the prediction results of each model decrease with an increase in junction temperature fluctuation. The lifetime prediction results of the Coffin-Manson-Arrhenius model were several orders of magnitude larger than those of the other models. By contrast, the prediction results of the Coffin-Manson model based on parameter fitting in Sect. 3 were closer to the Bayerer model and better than the parameters of the Coffin-Manson model in the literature [22].

Assuming that the module is always in a working state, the lifetimes of the IGBT modules under two different current waveforms were calculated as 5.448 and 0.3176 years, respectively. $\begin{array}{c}\frac{N_{\text{f}1}\times 7}{60\times 60\times 24\times 365}=5.448\text{ years},\frac{N_{\text{f2}}\times 17}{60\times 60\times 24\times 365}\\ =0.3176\text{ years}\text{.}\end{array}$ (14)

Influence of electrical parameters on IGBT module reliability

The electrical parameters affecting power device losses are current, voltage, and switching frequency. The parameters of the pulsed current include rising time, falling time, flat-top time, flat-bottom time, flat-top current, and flat-bottom current. Using the aforementioned pulsed current waveform-1 as a reference, the effects of the flat-top current, switching frequency, front-stage voltage, flat-top time, and flat-bottom time on the junction temperature fluctuation and lifetime were investigated, as shown in Fig. 8. When the value of the flat-top current increases by 100 A, the junction temperature fluctuation of the module increases by more than 10 ℃, and the number of failures also decreases rapidly as 1.2561×10⁹, 2.4544×10⁷, 1.3543×10⁶, and 1.2899×10⁵. The logarithmic function for the number of failures is provided in Fig. 8 (a). When the switching frequency increases by 5 kHz, the junction temperature fluctuation of the module increases by 4.5 ℃, and the number to failure decreases, but they are all of the order of 10⁷.

Fig. 8Full-screen View

Effects of different parameters on junction temperature fluctuation ΔT_j and lifetime N_f. (a) Flat-top current; (b) Switching frequency; (c) Front stage voltage; (d) Flat-top time; (e) Flat-bottom time

The magnet is a resistive and inductive load. According to the voltage equations of resistive and inductive loads, the maximum output voltage is calculated as follows: $\begin{array}{c}{V}_{\text{o}\max }=L\frac{\text{d}I}{\text{d}t}+I_{\max }R=0.06\times 146+300\times 0.062\\ =27.36\text{ V,}\end{array}$ (15) where R and L are the resistance and inductance of the magnet, respectively; dI/dt is the change rate of the magnet current; and I_max is the maximum current, that is, the flat-top current. According to the power supply design requirements, the voltage of the front voltage source is greater than the maximum output voltage, V > V_omax; therefore, the front voltage is higher than 30 V. Front stage voltages of 30, 50, 70, and 90 V are discussed herein. With an increase in the previous voltage source, the junction temperature fluctuation increased by approximately 1-3 ℃, and the number of cycles to failure decreased, also of the order of 10⁷. The junction temperature fluctuation and lifetime increased slightly with increasing rising and falling times. Therefore, the flat-top current has a crucial influence, followed by the switching frequency, front-stage voltage, flat-bottom time, and flat-top time.

Reducing the switching frequency and flat-top current value can effectively improve the reliability of the IGBT module, but it can also affect other power supply performances. For example, reducing the switching frequency leads to an increase in the current ripple and filter element volume. Consequently, there should be some redundancy in designing the power supply. The lifetime and reliability of the power device increase with a reduced ratio of the maximum operating current to the rated current.

Influence of heatsink on IGBT module reliability

The function of the heatsink is to remove the heat generated by electronic devices. Thermal resistance is a significant parameter for heatsinks. The heatsink of an accelerator pulsed power supply system is a water-cooled heatsink. Regulating the flow rate of cooling water can change the thermal resistance. The performance of the heatsink directly impacts the reliability of the power supply. We analyzed three existing heatsinks. Figure 9(a) presents the obtained curves of the thermal resistance of different heatsinks with the cooling water flow. The water flow rate range of the power supply system was 3−7 L/min. Accordingly, the performances of the heatsinks under two water flow conditions were analyzed, including 4.3 and 6.5 L/min. The circle points in the figure represent the different thermal resistance values of the heat sink. The performance of heatsink-3 was better than that of heatsink-1.

Fig. 9Full-screen View

Effects of different parameters on junction temperature fluctuation ΔT_j and lifetime N_f. (a) Thermal resistance of heatsink vs. cooling water flow rate; (b) Fluctuation of ΔT_j and maximum T_jmax of junction temperature under different water flow rates and thermal resistances

Figure 9(b) displays the variation curves of the junction temperature fluctuation and maximum junction temperature with different heatsinks and water flow rates. The junction temperature fluctuation is approximately 31.4 ℃, but the maximum junction temperature decreases with reduced thermal resistance, indicating that the average junction temperature also decreases. Moreover, the thermal resistance of the heatsink is not affected by the device power loss or junction temperature fluctuation. Thermal resistance influences average junction temperature. The maximum and average junction temperatures decrease with improving heatsink performance.

Analysis and discussion

The developed method does not apply to nonperiodic or high-frequency temperature fluctuations. In particular, the thermal impedance parameters employed in this study were provided by a datasheet ^[23], and an FE simulation can be performed to compute more accurate thermal impedance parameters. In addition, owing to the limitation of the test conditions, the number of power cycling tests performed was low, and the test results could indicate the approximate range of the lifetime. More modules should be tested under the power cycling test, and more data are required for computing the error band with the Weibull distribution and improving the degree of confidence. Based on the lifetime model, the lifetime distribution of the device was simulated using Monte Carlo methods, which can improve the accuracy of the results ^[24].

This paper discussed the lifetime estimation of an IGBT module under a single-pulsed current condition. In practice, a power supply outputs multiple pulsed currents with different parameters. Lifetime prediction under multi-pulsed current conditions can be analyzed based on a single pulsed profile. For example, the operating conditions of the accelerator pulsed power supply were recorded in one year, including the pulsed current parameters and running time. The rainflow-counting algorithm and Miner’s rule were used for the fatigue evaluation. This study is expected to be useful for the design and maintenance of an accelerator pulsed power supply. In the design stage, lifetime evaluation is performed, and the results presented herein are useful for such designers. In the operation stage, the potential faulty devices can be replaced in time to reduce maintenance costs.

The fault statistics of power electronic systems show that in addition to semiconductor devices, the reliability of capacitors is crucial. Capacitance plays a role in decreasing the DC-link voltage ripple and balancing the transient power between the front and rear stages of power converters ^[25]. Capacitance is sensitive to electrical and thermal stresses. Therefore, the reliability of the capacitance needs to be studied further.