Nanoseconds Switching Time Monitoring of Insulated Gate Bipolar Transistor Module by Under-Sampling Reconstruction of High-Speed Switching Transitions Signal

An insulated gate bipolar transistor (IGBT) is one of the most reliable critical components in power electronics systems (PESs). The switching time during IGBT turn-on/off transitions is a good health status indicator for IGBT. However, online monitoring of IGBT switching time is still difficult in practice due to the requirement of extremely high sampling rate for nanoseconds time resolution. The compressed sensing (CS) method shows a potential to overcome the technical difficult by reducing the sampling rate. To further improve the efficiency and reduce the computational time for IGBT online condition monitoring (CM), an under-sampling reconstruction method of an IGBT high-speed switching signal is presented in this paper. First, the physical mechanism and signal characteristics of IGBT switching transitions are analyzed. Then, by utilizing the sparse characteristics of IGBT switching signal in the wavelet domain, the wavelet basis is used for sparse representation. The stagewise orthogonal matching pursuit (StOMP) algorithm is proposed to enhance the convergence speed for switching signal reconstruction. Experiments are performed on not only a double-pulse test rig but also a real Pulse-Width Modulation (PWM) converter. Results show that the IGBT high-speed switching transitions signal can be accurately recovered with a reduced sampling rate and the nanoseconds switching time change can be monitored for IGBT CM.


Introduction
The power electronics system (PES) plays an increasingly important role in all parts of modern power system including source, network, load, and storage fields [1]. As one of the most popular power electronic devices, insulated gate bipolar transistors (IGBTs), a fast switching semiconductor device with easy driving, low on-state voltage and fast switching speed, has been extensively applied in PES ranging from middle to high power areas. The industrial survey [2] shows that the power semiconductor devices ranked the most fragile components in PES with a failure rate of about 34%. To improve the operational reliability of PES, it is a strong demand to apply a timely and effective condition monitoring (CM) technique for IGBT, especially in the safety-critical and mission-critical applications such as aerospace, transportation, electric vehicle, and renewable energy.
The progressive failure mechanism of IGBT involves a combination of electrical, thermal, mechanical, and environmental factors [3][4][5]. During the life cycle of an IGBT module, the bonding wires and the solder layers are prone to aging and fatigue due to the thermo-mechanical fatigue stress experienced by the packaging materials, e.g., mismatch in the coefficients of thermal expansion. The electrical overstress (EOS) (overvoltage and/or overcurrent), electrostatic discharge (ESD), and the sensitive and accessible directly from the terminals of power converter. The typical switching time of IGBT is about hundreds of nanoseconds and the value varies with load current, junction temperature, and other factors [17][18][19][20]. However, the change of IGBT switching time is very small [4,5] (range from several to tens of nanoseconds) when the health status of the IGBT module changes. Thus, a high-resolution of the IGBT switching time is required for the purpose of CM. If the analog-to-digital conversion (ADC) is directly performed on the IGBT high-speed switching signal, according to the Nyquist sampling theorem, the sampling rate needs to be extremely high (up to a GS/s level) to meet the nanosecond time detection accuracy. In practice, the high-speed A/D converters with high sampling rate and auxiliary circuit are expensive. The relationship between the price of ADC chip and sampling rate is not linear, and the price is very expensive at a sampling rate up to a GS/s level. Moreover, due to the high sampling rate, the massive measurement data are difficult to be handled (including data transmission and storage) for continuous condition monitoring of the IGBT. Considering the several kHz switching frequency in a real power converter, a massive amount of switching transitions exists in a fundamental period; thus, the amount of measurement data at a high sampling rate is quite huge. For a multilevel power converter with tens or even hundreds of IGBTs, such as the neutral point clamped (NPC) thee-level, cascaded H bridge and modular multilevel converter (MMC) converters, the situation would be more severe. In addition, the nonignorable computational time for processing the massive CM data is not conducive to online monitoring.
In previous study [21], a compressed sensing (CS) method is proposed for IGBT switching time online monitoring, which shows a potential to break though the limitation of extremely high sampling rate. To further improve the efficiency and reduce the computational time for online CM, an under-sampling reconstruction method of IGBT high-speed switching signal is presented in this paper. First, the physical mechanism and signal characteristics of IGBT switching transitions are analyzed. Then, by utilizing the sparse characteristics of IGBT switching signal in the wavelet domain, the wavelet basis is used for sparse representation and the StOMP algorithm is adopted to recover the IGBT switching transition signal from under-sampling data. Finally, the effectiveness of the proposed method is verified on the double-pulse test and the PWM converter test.

The Physical Mechanism behind the IGBT Switching Transitions
IGBT is a composite full-controlled power semiconductor device. The equivalent circuit of an IGBT is illustrated in Figure 1. C ge , C gc, and C ce are the distributed capacitances among the gate-emitter, the gate-collector, and the collector-emitter, respectively. R G is the gate resistance, and R d and R w are the drift resistances of N and P regions, respectively, and inductance Ls represents the parasitic switching loop inductance. Taking the IGBT collector-emitter voltage V CE and the collector current I C into account, their waveforms during switching transitions (i.e., turn-on and turn-off) are shown in Figure 2. The physical mechanism behind IGBT switching transitions is described as follows: In previous study [21], a compressed sensing (CS) method is proposed for IGBT switching time online monitoring, which shows a potential to break though the limitation of extremely high sampling rate. To further improve the efficiency and reduce the computational time for online CM, an under-sampling reconstruction method of IGBT high-speed switching signal is presented in this paper. First, the physical mechanism and signal characteristics of IGBT switching transitions are analyzed. Then, by utilizing the sparse characteristics of IGBT switching signal in the wavelet domain, the wavelet basis is used for sparse representation and the StOMP algorithm is adopted to recover the IGBT switching transition signal from under-sampling data. Finally, the effectiveness of the proposed method is verified on the double-pulse test and the PWM converter test.

The Physical Mechanism behind the IGBT Switching Transitions
IGBT is a composite full-controlled power semiconductor device. The equivalent circuit of an IGBT is illustrated in Figure 1. Cge, Cgc, and Cce are the distributed capacitances among the gate-emitter, the gate-collector, and the collector-emitter, respectively. RG is the gate resistance, and Rd and Rw are the drift resistances of N and P regions, respectively, and inductance Ls represents the parasitic switching loop inductance. Taking the IGBT collector-emitter voltage VCE and the collector current IC into account, their waveforms during switching transitions (i.e., turn-on and turn-off) are shown in Figure 2. The physical mechanism behind IGBT switching transitions is described as follows:   In previous study [21], a compressed sensing (CS) method is proposed for IGBT switching time online monitoring, which shows a potential to break though the limitation of extremely high sampling rate. To further improve the efficiency and reduce the computational time for online CM, an under-sampling reconstruction method of IGBT high-speed switching signal is presented in this paper. First, the physical mechanism and signal characteristics of IGBT switching transitions are analyzed. Then, by utilizing the sparse characteristics of IGBT switching signal in the wavelet domain, the wavelet basis is used for sparse representation and the StOMP algorithm is adopted to recover the IGBT switching transition signal from under-sampling data. Finally, the effectiveness of the proposed method is verified on the double-pulse test and the PWM converter test.

The Physical Mechanism behind the IGBT Switching Transitions
IGBT is a composite full-controlled power semiconductor device. The equivalent circuit of an IGBT is illustrated in Figure 1. Cge, Cgc, and Cce are the distributed capacitances among the gate-emitter, the gate-collector, and the collector-emitter, respectively. RG is the gate resistance, and Rd and Rw are the drift resistances of N and P regions, respectively, and inductance Ls represents the parasitic switching loop inductance. Taking the IGBT collector-emitter voltage VCE and the collector current IC into account, their waveforms during switching transitions (i.e., turn-on and turn-off) are shown in Figure 2. The physical mechanism behind IGBT switching transitions is described as follows:

IGBT Turn-off Transition
At the beginning time of turn-off transition, i.e., the turn-off voltage V GE is applied to the gate at t 0 , the Miller capacitor C gc starts to charge, and the V CE rises slowly. The I C remains almost unchanged at the time period from t 0 to t 1 because the gate-emitter voltage V GE is still larger than the threshold voltage V TH . When C gc reaches its steady state (V GE reaches V TH ), V CE starts to rise rapidly at t 1 , and the change rate is determined by the carrier removal rate. The rapid rise of V CE causes I C to slowly fall at first. When V CE reaches the DC voltage V dc at time t 2 , I C begins to fall rapidly. Due to the parasitic inductance in the circuit, an overshoot of V CE is induced. At t 3 , V CE maintains at V dc , and I C has a small tail current. The change rate of V CE during IGBT turn-off transition is [17]: where τ G = R G C gc , C O is the charge extraction capacitance, V g(+) and V g(-) are the on-state and off-state gate-emitter voltages, respectively, and g m is the transconductance. The change rate of I C in the t 2 -t 3 phase is [18]:

IGBT Turn-on Transition
At the beginning of IGBT turn-on transient under inductive load conditions, the gate voltage begins to rise at t 4 , and the gate current starts to charge C gc and C ge . Once the gate-emitter voltage V GE is greater than the threshold voltage V T (at t 5 ), the MOS channel starts to conduct (hole injection) and I C rises rapidly. The rising speed of I C is determined by R G and the behavior of MOS channel, which is expressed as: The initial fall of the collector voltage V CE appears due to the back electromotive force (EMF) across the stray inductance L S . The V CE falling speed is determined by L S and the change rate of I C . The load current communicates from the freewheeling diode to IGBT. When the I C reaches the load current I L (at t 6 ), the freewheeling diode starts to turn off. Due to the reverse recovery current of the freewheeling diode, an overcurrent of I C will occur [18]. The stored charge within a diode is extracted, and the diode current falls below zero [19]. The collector voltage V CE may rise a little, like the concave shown in Figure 2. Then, V CE begins to drop rapidly to the saturation on-state voltage V CE (sat) . The rate of V CE change at this stage is deduced as: As shown in Figure 2, the IGBT switching time T sw (including the turn-on time T on and the turn-off time T off ) are defined as the time durations of V CE decreasing from 90% to 10%, and increasing from 10% to 90% of its initial value, respectively. The IGBT switching time is related to current, junction temperature, and gate parameters. According to Equation (1), the increase of current will increase the carrier concentration, leading to a faster IGBT turn-off transition and thus a decrease of T off . Moreover, with the increasing junction temperature, the carrier recombination becomes slower, leading to the increasing C O and T off . Equation (4) shows that an increase of current would cause a slower rate of V CE change, which will result in an increase of T on .
From the perspective of CM, the IGBT switching time is a reliable condition parameter that is related well to the health state of IGBT. For example, partial failure of a multichip device would change the switching time for a given current due to the changes in current density and stored charge of other healthy chips [2]. Other degradations such as solder fatigue and latch-up may also prolong the switching time of IGBT [4]. The T off value is an early indicator to latch-up failure caused by damage in the die-attach layer [14]. On the other hand, the T on can be also used a precursor of IGBT gate degradation [20].

Compressed Sensing of the IGBT Switching Signal
Switching time has been verified adequately to monitor the operational condition of the IGBT module [3][4][5], [10][11][12][13][14][15]. However, it requires a high-resolution and high accurate measurement of the small nanoseconds change during the fast switching transient. In practice, the conventional data acquisition (DAQ) with an extremely high sampling rate is very expensive with a heavy burden of data transmission, storage and computation. Compressed sensing is an emerging technique that allows sampling of analog signals at sub-Nyquist rates (i.e., the Shannon/Nyquist sampling theorem can be broken through) while avoiding the traditional issue of aliasing, and allowing for effective compression of signals during the sampling process [21].
According to the CS theory, a high-frequency V CE switching signal can be represented by a small amount of sampling points when expressed in terms of a proper basis. The basis in which a compressible signal is sparse is called a "sparsifying basis". Sparse means that the signal has only a few non-zero elements. For an N-dimensional signal, if it has only K non-zero elements, it is K-sparse. As shown in Figure 2, the V CE during IGBT switching transitions is not sparse in the time domain. However, if there exists a transform basis such that the V CE switching signal is sparse in that transform basis, a low sampling rate can be performed by designing a suitable measurement matrix, and then the V CE switching signal can be recovered from a small number of sampling points.
The mathematical model for the sparse reconstruction method of IGBT high-speed switching signals is [22]: where f is the original N × 1 dimension high-speed switching signal, y is the M × 1 dimension (M << N) sampling signal at a low sampling rate, e is a Gaussian white noise with dimension of N × 1, and Φ is a M × N dimension measurement matrix that is used to perform dimension-reduced projection of signal f to achieve compressed sampling. The premise of Equation (5) is that there exists an N × N-dimensional orthogonal transform base Ψ that can sparsely represent f, i.e., most components of the θ vector in f = Ψ·θ are zero. According to the theory of CS, the keys to recover the V CE switching signal from under-sampling data are: (1) the design of the measurement matrix Φ to obtain the under-sampling data y; (2) the selection of the orthogonal transform base Ψ to obtain the sparse representation θ of the switching signal; (3) and the utilization of the under-sampling data y to estimate the sparse vector θ and recover the original switching signal (f = Ψ·θ).
Since the dimension of the measurement signal y is much smaller than the dimension of the original signal f (M << N), the inverse problem of solving Equation (5) is actually an ill-posed problem, i.e., the original signal f cannot be directly estimated by M measurement values. According to the CS theory [23], in order to reconstruct the signal from a small amount of data, the sensing matrix A = Φ·Ψ needs to satisfy the Restricted Isometric Property (RIP) criteria, and the sparse basis Ψ must be irrelevant to the measurement matrix Φ. Considering the sparse property of vector θ (K non-zero terms), solving Equation (5) can be further transformed into a minimizing l 0 norm problem, which can be expressed as: where ||θ|| 0 is the l 0 norm, and Equation (6) is a complete NP-hard (non-deterministic polynomial) problem [21]. For a sparse signal, the l 1 norm is the convex function closest to the l 0 norm [24], and the minimum l 1 norm and the minimum l 0 norm have equivalent solutions. Therefore, the l 1 linear programming algorithm can be used to solve the global optimal solution of the sparse vector θ, and the original V CE switching signal can be recovered from the under-sampling data by inverse transformation. Figure 3 shows the schematic of the proposed CS based sparse reconstruction method for IGBT switching time monitoring with a reduced sampling rate. First, an N × M measurement matrix is constructed for random under-sampling of the IGBT high-speed switching signal V CE ; in this way, the high-dimensional data are projected onto the low-dimensional space. The common used measurement matrices include Gaussian matrix, partial Hadamard matrix, and a random Bernoulli matrix [24], which meet both the RIP and irrelevance criteria. Then, the sparse basis is constructed to sparsely represent the high-dimensional V CE switching signal. Finally, the original V CE switching signal is recovered by using the reconstruction algorithm to extract the IGBT switching time.  Figure 3 shows the schematic of the proposed CS based sparse reconstruction method for IGBT switching time monitoring with a reduced sampling rate. First, an N × M measurement matrix is constructed for random under-sampling of the IGBT high-speed switching signal VCE; in this way, the high-dimensional data are projected onto the low-dimensional space. The common used measurement matrices include Gaussian matrix, partial Hadamard matrix, and a random Bernoulli matrix [24], which meet both the RIP and irrelevance criteria. Then, the sparse basis is constructed to sparsely represent the high-dimensional VCE switching signal. Finally, the original VCE switching signal is recovered by using the reconstruction algorithm to extract the IGBT switching time.

Conventional high-frequency T sw monitoring method
Sampling with high-speed ADC  As shown in Figure 2, the VCE switching signal has a multi-resolution feature, which can be decomposed into on/off steady state and on/off transient state. In the steady state, VCE is either the IGBT saturation voltage or the DC voltage Vdc, and the signal can be characterized as a DC component. The VCE transient state corresponds to the behavior of IGBT turn-on and/or turn-off transitions. As analyzed in Section 2.1, the characteristics of the VCE switching transition signal are related to the distribution parameters of IGBT module. To accurately identify the nanoseconds IGBT switching time, it is required to sparsely represent the switching transient characteristics of VCE signal, i.e., VCE signal is transformed and projected onto an orthogonal transform base. The wavelet transform consists of different spatial resolution, frequency and directional characteristics, which defines a wellsparse representation of signal transients and singularities. Previous study in [24] showed that the wavelet basis could sparsely represent the IGBT switching transient signal more effectively than the Fourier basis. Similarly, this paper uses the Mallat algorithm to construct an N × N Haar wavelet basis to sparsely represent the VCE signal [25].

The sparse reconstruction method for T sw monitoring
The reconstruction algorithm is vital for IGBT switching time monitoring, which recovers the high-dimensional signal from the under-sampling data. As analyzed in Section 2.2, the essence of sparse reconstruction of VCE signal is a sparse decomposition process of compressing measurement data on redundant dictionaries [23]. For the orthogonal matching pursuit algorithm (OMP), the local optimal solution is selected based on greedy iteration to gradually approximate the original signal, so that the original VCE signal can be reconstructed with a high probability, as applied in previous study [25]. However, the OMP algorithm will introduce new computation overheads during the Schmidt orthogonalization, and only one atom is selected in each iteration to update the support set. If the amount of data is large, the iteration efficiency may be low or even the reconstruction may be failed. Considering the IGBT switching frequency in a real power converter is generally from a few kHz to several tens of kHz, there are numerous switching transitions in a fundamental period with massive CM data. The computation time of the reconstruction algorithm for processing the massive As shown in Figure 2, the V CE switching signal has a multi-resolution feature, which can be decomposed into on/off steady state and on/off transient state. In the steady state, V CE is either the IGBT saturation voltage or the DC voltage V dc , and the signal can be characterized as a DC component. The V CE transient state corresponds to the behavior of IGBT turn-on and/or turn-off transitions. As analyzed in Section 2.1, the characteristics of the V CE switching transition signal are related to the distribution parameters of IGBT module. To accurately identify the nanoseconds IGBT switching time, it is required to sparsely represent the switching transient characteristics of V CE signal, i.e., V CE signal is transformed and projected onto an orthogonal transform base. The wavelet transform consists of different spatial resolution, frequency and directional characteristics, which defines a well-sparse representation of signal transients and singularities. Previous study in [24] showed that the wavelet basis could sparsely represent the IGBT switching transient signal more effectively than the Fourier basis. Similarly, this paper uses the Mallat algorithm to construct an N × N Haar wavelet basis to sparsely represent the V CE signal [25].
The reconstruction algorithm is vital for IGBT switching time monitoring, which recovers the high-dimensional signal from the under-sampling data. As analyzed in Section 2.2, the essence of sparse reconstruction of V CE signal is a sparse decomposition process of compressing measurement data on redundant dictionaries [23]. For the orthogonal matching pursuit algorithm (OMP), the local optimal solution is selected based on greedy iteration to gradually approximate the original signal, so that the original V CE signal can be reconstructed with a high probability, as applied in previous study [25]. However, the OMP algorithm will introduce new computation overheads during the Schmidt orthogonalization, and only one atom is selected in each iteration to update the support set. If the amount of data is large, the iteration efficiency may be low or even the reconstruction may be failed. Considering the IGBT switching frequency in a real power converter is generally from a few kHz to several tens of kHz, there are numerous switching transitions in a fundamental period with massive CM data. The computation time of the reconstruction algorithm for processing the massive data is important for online monitoring. Different from using the OMP algorithm in [24], the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm is used to improve the efficiency and reduce the computation time for V CE switching signal reconstruction [25]. The flowchart of the StOMP algorithm to reconstruct the V CE switching signal is presented in Figure 4. data is important for online monitoring. Different from using the OMP algorithm in [24], the Stagewise Orthogonal Matching Pursuit (StOMP) algorithm is used to improve the efficiency and reduce the computation time for VCE switching signal reconstruction [25]. The flowchart of the StOMP algorithm to reconstruct the VCE switching signal is presented in Figure 4. The basic steps of the StOMP algorithm are similar to the OMP algorithm, such as matching atoms by maximizing the residual inner product and updating the residual by minimizing the residual norm. However, unlike the OMP algorithm, in which only one optimal solution element is selected per iteration, the StOMP algorithm forms a matched filter by setting a threshold and selects all atoms with a residual inner product greater than the threshold St in each iteration, thus reducing the matching pursuit times and speeding up the convergence of the algorithm [25].
Step 1：Calculate and select the set of indexs λ corresponding to u larger than the threshold St by a matched filter.
Step 3：Solve by least square method to get a new sparse coefficient estimate.
Step 4：Calculate the new approximation of the data and renew the residual Step 5：Stop iteration and recover the V CE signal by the estimated sparse coefficients ： ˆf

Double-Pulse Tests
To verify the feasibility and effectiveness of the proposed sparse reconstruction method for IGBT switching time monitoring, the IGBT double-pulse test is implemented first. The schematic diagram of the H-bridge based double-pulse test rig is shown in Figure 5. The adjustable DC power supply is used to regulate the DC voltage (0-1000 V). The DSP controller (TI, F28335) is used for IGBT switching control. A heating system consisting of a PTC heater and a PID temperature controller is applied to regulate temperature of the IGBT. The IGBT case temperature is measured by a thermal couple with an accuracy of 1 °C. The parameters of the IGBT double-pulse test rig are in accordance with [24]. The device under test (DUT) is an Infineon IGBT module [26] FF450R17ME4 (1.7 kV/450 A) and the load inductance is 2.2 mH. The basic steps of the StOMP algorithm are similar to the OMP algorithm, such as matching atoms by maximizing the residual inner product and updating the residual by minimizing the residual norm. However, unlike the OMP algorithm, in which only one optimal solution element is selected per iteration, the StOMP algorithm forms a matched filter by setting a threshold and selects all atoms with a residual inner product greater than the threshold St in each iteration, thus reducing the matching pursuit times and speeding up the convergence of the algorithm [25].

Double-Pulse Tests
To verify the feasibility and effectiveness of the proposed sparse reconstruction method for IGBT switching time monitoring, the IGBT double-pulse test is implemented first. The schematic diagram of the H-bridge based double-pulse test rig is shown in Figure 5. The adjustable DC power supply is used to regulate the DC voltage (0-1000 V). The DSP controller (TI, F28335) is used for IGBT switching control. A heating system consisting of a PTC heater and a PID temperature controller is applied to regulate temperature of the IGBT. The IGBT case temperature is measured by a thermal couple with an accuracy of 1 • C. The parameters of the IGBT double-pulse test rig are in accordance with [24]. The device under test (DUT) is an Infineon IGBT module [26] FF450R17ME4 (1.7 kV/450 A) and the load inductance is 2.2 mH.  Figure 5. Schematic of the IGBT double-pulse test rig.
The IGBT can be controlled at any preset voltage, current, and temperature by using the test rig. Taking the IGBT T2 as an example of DUT, the IGBT T1 and IGBT T4 are controlled turn off during the double-pulse test, while the IGBT T3 is controlled turn-on. During the time period of ΔT1, the DC capacitor starts to charge the load inductor through T3 and T2. The magnitude of the collector current is determined by the pulse width of ΔT1. The collector current increases with the ΔT1. During ΔT2, T2 is turned off, and the load current continues to flow through the anti-parallel diode D1 and the IGBT T3. If the load inductance is large enough, the load current changes slowly and remains stable. Therefore, the IGBT T2 can be controlled turned-on and turn-off under different load currents by controlling the pulse width of ΔT1. Figure 6 shows the VCE waveform of T2 measured by a differential voltage probe with a sampling rate of 1 GS/s, when the load current is 100 A and 200 A, respectively. It can be seen that the VCE waveform during IGBT turn-on transition has an obvious concave characteristic in the decreasing process, which agrees well with the theoretical analysis in Section 2.1. The IGBT can be controlled at any preset voltage, current, and temperature by using the test rig. Taking the IGBT T 2 as an example of DUT, the IGBT T 1 and IGBT T 4 are controlled turn off during the double-pulse test, while the IGBT T 3 is controlled turn-on. During the time period of ∆T 1 , the DC capacitor starts to charge the load inductor through T 3 and T 2 . The magnitude of the collector current is determined by the pulse width of ∆T 1 . The collector current increases with the ∆T 1 . During ∆T 2 , T 2 is turned off, and the load current continues to flow through the anti-parallel diode D1 and the IGBT T3. If the load inductance is large enough, the load current changes slowly and remains stable. Therefore, the IGBT T 2 can be controlled turned-on and turn-off under different load currents by controlling the pulse width of ∆T 1. Figure 6 shows the V CE waveform of T 2 measured by a differential voltage probe with a sampling rate of 1 GS/s, when the load current is 100 A and 200 A, respectively. It can be seen that the V CE waveform during IGBT turn-on transition has an obvious concave characteristic in the decreasing process, which agrees well with the theoretical analysis in Section 2.1.  Figures 7a and 8a show the under-sampling points during IGBT turn-on and turn-off transitions, respectively. From these under-sampling points, the VCE switching transitions signals are recovered by the proposed sparse reconstruction method, as shown in Figures 7b and 8b, respectively. The compression ratio of sampling rate is 10. To evaluate the performance of signal reconstruction, the recovered VCE transient signals are compared with the original VCE signal at a sampling rate of 1 GS/s.  Figures 7a and 8a show the under-sampling points during IGBT turn-on and turn-off transitions, respectively. From these under-sampling points, the V CE switching transitions signals are recovered by the proposed sparse reconstruction method, as shown in Figures 7b and 8b, respectively. The compression ratio of sampling rate is 10. To evaluate the performance of signal reconstruction, the recovered V CE transient signals are compared with the original V CE signal at a sampling rate of 1 GS/s. For comparison, a 100 V vertical shift is inserted intentionally between the recovered and reconstructed V CE signals. From the experimental results, the difference between the reconstructed V CE signal and the original V CE signal is very small. The specific characteristics of the IGBT switching signal can be recovered well with a largely reduced sampling rate (10-fold). As shown in Figures 7b and 8b, the concave and overshoot behaviors of the V CE signal during IGBT turn-on and turn-off transitions are well characterized by the proposed sparse reconstruction method.  To quantify the percent error between the original VCE signal f and the recovered VCE signal fr, the reconstruction error Ce is defined as [27]:  To quantify the percent error between the original VCE signal f and the recovered VCE signal fr, the reconstruction error Ce is defined as [27]: To quantify the percent error between the original V CE signal f and the recovered V CE signal f r , the reconstruction error C e is defined as [27]: Different random measurement matrices including Gauss matrix, Bernoulli matrix, and a partial Hadamard matrix are used to the V CE switching signal and the statistical results of the reconstruction error are listed in Table 1. The V CE reconstruction error during IGBT turn-off is less than 1%. The V CE turn-on reconstruction errors using the random Gauss matrix and the random Bernoulli matrix are less than 1.5%. The V CE turn-on reconstruction error is slightly larger than of the turn-off-that is, because the characteristics of V CE waveform during IGBT turn-on are more complex [17,18]. As shown in Figure 7, the V CE waveform has three transient stages during IGBT turn-on process. In the first stage, V CE is decreased by the voltage drop of stray inductance during the fast rising of I C . In the second stage, the freewheeling diode sustains a reverse voltage and V CE rises a little and then drops quickly as the depletion layer shrinks toward the drift region. In the third stage, V CE decays slowly because the carrier diffusion velocity in the drift (base) region is less than the disappearance velocity of depletion layer. In addition, the StOMP algorithm used in this work and the OMP algorithm used in [23] are tested under the same conditions. The StOMP algorithm has a faster reconstruction speed than the OMP algorithm, and the computation time of the V CE signal by the StOMP algorithm is 0.32 ms when compared with 2.89 ms by the OMP algorithm. As described in Section 2.2, the temperature affects the recombination velocity of the carriers, thereby affecting the IGBT turn-off time. The IGBT case temperature is controlled by the PID temperature controller, and the switching times are monitored under thermal equilibrium conditions. Figure 9 shows the V CE turn-off signal in the temperatures range of 38-130 • C. It is found that the IGBT turn-off time increases with temperature. This is because the carrier in the space charge region is removed from the depletion layer at a slower rate as the junction temperature and the charge extraction capacitor increases. As a result, the IGBT turn-off speed becomes slower. The temperature-sensitive coefficient of the turn-off time calculated by the least squares fitting method under direct high-frequency sampling is about 1.54 ns/ • C. For the proposed sparse reconstruction method, the derived temperature sensitivity is about 1.52 ns/ • C. In the experiment, the measurement error of IGBT turn-off time is less than 2 ns at different temperatures. Such an accuracy is adequate for the switching time based IGBT junction temperature estimation. extraction capacitor increases. As a result, the IGBT turn-off speed becomes slower. The temperaturesensitive coefficient of the turn-off time calculated by the least squares fitting method under direct high-frequency sampling is about 1.54 ns/°C. For the proposed sparse reconstruction method, the derived temperature sensitivity is about 1.52 ns/°C. In the experiment, the measurement error of IGBT turn-off time is less than 2 ns at different temperatures. Such an accuracy is adequate for the switching time based IGBT junction temperature estimation.

PWM Converter Test
As analyzed in Section 2.1, the IGBT switching time varies with the instantaneous current (since the carrier concentration speed is different [16]). The proposed sparse reconstruction method is

PWM Converter Test
As analyzed in Section 2.1, the IGBT switching time varies with the instantaneous current (since the carrier concentration speed is different [16]). The proposed sparse reconstruction method is further verified in a three-phase PWM converter. The schematic diagram and picture of the PWM converter test rig are shown in Figure 10. The tested IGBT module is Infineon IGBT module FF50R12RT4 (1200 V/50 A) (Neubiberg, Germany) [26]. The DC voltage is 600 V, and the load inductance is 8 mH/phase. The fundamental frequency of the PWM converter is 50 Hz with a peak current of 30 A, while the switching frequency of IGBT is 8 kHz. Figure 11 shows the experimental waveforms of V CE and I C . The IGBT switching time at different I C (0-30 A) are extracted by the direct high sampling rate method and the proposed under-sampling reconstruction method, respectively. The comparative results are exhibited in Figure 12.
It can be clearly seen that the IGBT switching time changes with the instantaneous current. With the increasing instantaneous current, the IGBT turn-off time (T off ) decreases while the IGBT turn-on time (T on ) increases. When the value of I C is very small, the IGBT turn-on speed is faster than the turn-off speed. T on is 84 ns and T off is 448 ns when I C = 0.63 A. When I C = 21.9 A, T on and T off tend to be the same value of ca. 140 ns. The extracted IGBT switching time by the proposed sparse reconstruction method is close to the direct high sampling rate method (the average error is less than 2 ns). Experimental results demonstrate that the under-sampling reconstruction method could reflect the IGBT nanoseconds switching time variation at a 10-fold reduction of sampling rate. For the purpose of fault prognosis, the change of switching time at a certain current and case temperature can be used to reflect the health status of IGBT [13][14][15]. The potential applications of the proposed method for IGBT junction temperature measurement will be further studied.
FF50R12RT4 (1200 V/50 A) (Neubiberg, Germany) [26]. The DC voltage is 600 V, and the load inductance is 8 mH/phase. The fundamental frequency of the PWM converter is 50 Hz with a peak current of 30 A, while the switching frequency of IGBT is 8 kHz. Figure 11 shows the experimental waveforms of VCE and IC. The IGBT switching time at different IC (0-30 A) are extracted by the direct high sampling rate method and the proposed under-sampling reconstruction method, respectively. The comparative results are exhibited in Figure 12. It can be clearly seen that the IGBT switching time changes with the instantaneous current. With the increasing instantaneous current, the IGBT turn-off time (Toff) decreases while the IGBT turn-on time (Ton) increases. When the value of IC is very small, the IGBT turn-on speed is faster than the turnoff speed. Ton is 84 ns and Toff is 448 ns when IC = 0.63 A. When IC = 21.9 A, Ton and Toff tend to be the same value of ca. 140 ns. The extracted IGBT switching time by the proposed sparse reconstruction method is close to the direct high sampling rate method (the average error is less than 2 ns). Experimental results demonstrate that the under-sampling reconstruction method could reflect the IGBT nanoseconds switching time variation at a 10-fold reduction of sampling rate. For the purpose of fault prognosis, the change of switching time at a certain current and case temperature can be used to reflect the health status of IGBT [13][14][15]. The potential applications of the proposed method for IGBT junction temperature measurement will be further studied.

Conclusions
To overcome the technical difficulty of the extremely high sampling rate for IGBT nanoseconds switching time online monitoring, this paper proposes a CS based under-sampling reconstruction monitoring method. By analyzing the physical mechanism behind the IGBT switching transitions, the VCE switching signal is sparsely represented by the wavelet basis, and the StOMP algorithm is proposed to recover the high-speed VCE signal for under-sampling data. Finally, experimental verification is conducted on not only the double-pulse test, but also a real PWM converter. The results show that: (1) The characteristics of VCE signal during switching transitions can be recovered from the under-sampling data with a reconstruction error less than 2%. The StOMP algorithm shows a faster convergence speed than the traditional OMP algorithm for sparse reconstruction.
(2) The under-sampling reconstruction method can monitor the nanoseconds change of IGBT switching time variation with a largely reduced sampling rate. At a compressed ratio 10 of the sampling rate, the detection accuracy of less than 2 ns can be achieved.

Conclusions
To overcome the technical difficulty of the extremely high sampling rate for IGBT nanoseconds switching time online monitoring, this paper proposes a CS based under-sampling reconstruction monitoring method. By analyzing the physical mechanism behind the IGBT switching transitions, the V CE switching signal is sparsely represented by the wavelet basis, and the StOMP algorithm is proposed to recover the high-speed V CE signal for under-sampling data. Finally, experimental verification is conducted on not only the double-pulse test, but also a real PWM converter. The results show that: (1) The characteristics of VCE signal during switching transitions can be recovered from the under-sampling data with a reconstruction error less than 2%. The StOMP algorithm shows a faster convergence speed than the traditional OMP algorithm for sparse reconstruction.
(2) The under-sampling reconstruction method can monitor the nanoseconds change of IGBT switching time variation with a largely reduced sampling rate. At a compressed ratio 10 of the sampling rate, the detection accuracy of less than 2 ns can be achieved.