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

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

Hao Li * , Meng Zhao, Hao Yan and Xingwu Yang College of Electrical Power Engineering, Shanghai University of Electric Power, Shanghai 200090, China; [email protected] (M.Z.); [email protected] (H.Y.); [email protected] (X.Y.) * Correspondence: [email protected]; Tel.: +86-1811-750-1989

Received: 5 October 2019; Accepted: 18 October 2019; Published: 22 October 2019

Abstract: 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.

Keywords: condition monitoring; IGBT; switching time; sparse reconstruction

1. 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 [3–5]. 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

Electronics 2019, 8, 1203; doi:10.3390/electronics8101203 www.mdpi.com/journal/electronics Electronics 2019, 8, 1203 2 of 14 thermal activation are the main causes of chip-related failure. Previous studies [4–6] show that about 60% of IGBT failures are caused by overheating or other thermal-related problems, such as thermal resistance increased due to damage in IGBT module, and the load fluctuations. The IGBT junction temperature is a key parameter to directly reflect the IGBT safety margin, health states and operation performance, but it is difficult to be measured due to the encapsulated structure of the semiconductor device [7]. The existing IGBT junction temperature estimation methods can be mainly classified into a model-based method [8,9] and a thermo-sensitive electrical parameter (TSEP) method [7,10,11]. The model-based methods estimate IGBT junction temperature as the response of an equivalent thermal RC network to power losses, which require a high measurement accuracy of instantaneous power loss [11] and correct identification of the thermal RC network [12]. Practically, the thermal RC network is nonlinear especially at high temperatures and depends on ageing effects [13]. For the TSEP-based methods, the IGBT is taken as a temperature-sensitive sensor to estimate the junction temperature by using external observable electrical parameters [7], which are closely related to the junction temperature. The common TSEPs can be classified into static and dynamic TSEPs depending on time characteristics [10]. The dynamic TSEPs are extracted during IGBT switching transitions, such as turn-on delay time, turn-off delay time, and turn-on time [7,10,11], which show a promising and feasible way to indirectly reflect the IGBT junction temperature variation. As a high-speed power electronic device, the health state of an IGBT module is closely related to its dynamic switching characteristics. Switch-time degradation is a known problem of IGBT, which can signify several failure mechanisms [2], such as partial failure of a multichip device, increased leakage current, solder-fatigue, bond wire liftoff, and gate driver circuit degradation. In [13], the bond wire failures in IGBT module are detected by utilizing the changes of turn-on behavior [13]. Incipient degradation, e.g., IGBT latch-up [14], and die-attach aging [15] can be detected by using the IGBT turn-off time variation. Meanwhile, the IGBT switching time, as a dynamic TSEP, has good linearity and high sensitivity typically around 2 ns/◦C[7,10] for junction temperature measurement. The authors in [11] propose an online detection method of IGBT turn-off delay time for IGBT junction temperature estimation by utilizing the inherent emitter stray inductance in a high-power IGBT module. Alternatively, the authors in [16] develop a real-time junction temperature measurement method using the on-chip internal gate resistor as the sensitive TSEP and compares it with the UCE-method. The switching time is a good health indicator and a reliable TSEP for IGBT, which is relatively 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 [17–20]. 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 Electronics 20192019,, 88,, xx FORFOR PEERPEER REVIEWREVIEW 33 ofof 1515

ElectronicsInIn previousprevious2019, 8, 1203 studystudy [21],[21], aa compressedcompressed sensingsensing (CS)(CS) methodmethod isis proposedproposed forfor IGBTIGBT switchingswitching3 timetime of 14 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, sampling rate. To To further improve improve the the efficiency efficiency and reduce the computational time time for for online online CM, CM, an under-sampling reconstruction method of IGBT high-speed switching signal is presented in this an under-sampling reconstruction method of IGBT hi high-speedgh-speed switching signal is presented in this paper. First, the physical mechanism and signal characteristics of IGBT switching transitions are paper. First, First, the the physical physical mechanism mechanism and and signal characteristics of of IGBT switching transitions are analyzed. Then, by utilizing the sparse characteristics of IGBT switching signal in the wavelet analyzed. Then,Then, byby utilizing utilizing the the sparse sparse characteristics characteristics of IGBT of IGBT switching switching signal signal in the waveletin the wavelet domain, domain, the wavelet basis is used for sparse representation and the StOMP algorithm is adopted to domain,the wavelet the basiswavelet is used basis for is sparseused for representation sparse representation and the StOMP and the algorithm StOMP isalgorithm adopted tois adopted recover the to recover the IGBT switching transition signal from under-sampling data. Finally, the effectiveness of recoverIGBT switching the IGBT transition switching signal transition from under-samplingsignal from under-sampling data. Finally, data. the e ffFinally,ectiveness the ofeffectiveness the proposed of the proposed method is verified on the double-pulse test and the PWM converter test. themethod proposed is verified method on theis verified double-pulse on the testdouble-pulse and the PWM test and converter the PWM test. converter test.

2. Under-Sampling Under-Sampling Reconstruction Method for the IGBT Switching Signal

2.1. The The Physical Physical Mechanism Mechanism behind the IGBT Switching Transitions IGBTIGBT is is aa compositecompositecomposite full-controlledfull-controlledfull-controlled powerpower semiconductorsemiconductorsemiconductor device.device. TheThe equivalent equivalent circuitcircuit of of an an IGBTIGBT is is illustrated illustrated ininin FigureFigure1 1.1.. CCgege,,, CCgcgcgc,, and,andand CCcecece areareare thethe the distributeddistributed distributed capacitancescapacitances capacitances amongamong among thethe the gate-emitter,gate-emitter, gate-emitter, thethe gate-collector,gate-collector, andand and thethe the collector-emitter,collector-emitter, collector-emitter, respectively.respectively. respectively. RRGG Gisis isthethe the gategate gate resistance,resistance, resistance, andand and Rdd Randandd and Rw areRarew thearethe thedriftdrift drift resistancesresistances resistances ofof N ofN Nandand and PP Pregions,regions, regions, respectively,respectively, respectively, andand and inductanceinductance inductance LsLs representsrepresentsrepresents thethe parasiticparasitic switchingswitching looploop inductance.inductance. TakingTaking thethethe IGBTIGBTIGBT collector–emitter collector–emittercollector–emitter voltage voltagevoltageV VCECECE andand the collector currentcurrent IIICCC intointo account, account, their their waveforms waveforms during during switching switching tran transitionstransitionssitions (i.e.,(i.e., turn-onturn-on andand turn-off) turn-oturn-off)ﬀ) are are shown shown in in Figure2 2.. TheThe physicalphysical mechanismmechanism behindbehind IGBTIGBT switchingswitching transitionstransitions isis describeddescribed as as follows: follows:

Figure 1. EquivalentEquivalent circuit circuit of IGBT.

VCECE Tonon IC 90% Vdc IC 90% Vdc IICC 90% Vdcdc

Vdcdc IICC

Tofof f f V VCE(sat)CE(sat) V 10% V VCE(sat)CE(sat) 10% Vdcdc 10% Vdcdc 0 0 tt0 tt1 tt2 tt3 tt4 tt5 tt6 tt7 tt 0 1 2 3 4 5 6 7 Figure 2. Waveforms of VCE and I during IGBT switching transitions. Figure 2. Waveforms of VCECE andand IICC duringduring IGBTIGBT switchingswitching transitions.transitions.

1. IGBTIGBT Turn-offTurn-off TransitionTransition

Electronics 2019, 8, 1203 4 of 14

1. IGBT Turn-oﬀ Transition

At the beginning time of turn-off transition, i.e., the turn-off voltage VGE is applied to the gate at t0, the Miller capacitor Cgc starts to charge, and the VCE rises slowly. The IC remains almost unchanged at the time period from t0 to t1 because the gate-emitter voltage VGE is still larger than the threshold voltage VTH. When Cgc reaches its steady state (VGE reaches VTH), VCE starts to rise rapidly at t1, and the change rate is determined by the carrier removal rate. The rapid rise of VCE causes IC to slowly fall at first. When VCE reaches the DC voltage Vdc at time t2, IC begins to fall rapidly. Due to the parasitic inductance in the circuit, an overshoot of VCE is induced. At t3, VCE maintains at Vdc, and IC has a small tail current. The change rate of VCE during IGBT turn-off transition is [17]:

dV 1 Vg(+) Vg( ) CE = ( − − ), (1) dt τG 1 + (CO/gmτG) where τG = RGCgc, CO is the charge extraction capacitance, Vg(+) and Vg(-) are the on-state and oﬀ-state gate-emitter voltages, respectively, and gm is the transconductance. The change rate of IC in the t2-t3 phase is [18]: dIc gmVg VT IL/gm = − − − . (2) t d Rg Cgc + Cge + gmLs 2. IGBT Turn-on Transition At the beginning of IGBT turn-on transient under inductive load conditions, the gate voltage begins to rise at t4, and the gate current starts to charge Cgc and Cge. Once the gate-emitter voltage VGE is greater than the threshold voltage VT (at t5), the MOS channel starts to conduct (hole injection) and IC rises rapidly. The rising speed of IC is determined by RG and the behavior of MOS channel, which is expressed as: dI gm(Vg+ VT) c = − . (3) t d Rg Cgc + Cge + gmLs

The initial fall of the collector voltage VCE appears due to the back electromotive force (EMF) across the stray inductance LS. The VCE falling speed is determined by LS and the change rate of IC. The load current communicates from the freewheeling diode to IGBT. When the IC reaches the load current IL (at t6), the freewheeling diode starts to turn oﬀ. Due to the reverse recovery current of the freewheeling diode, an overcurrent of IC will occur [18]. The stored charge within a diode is extracted, and the diode current falls below zero [19]. The collector voltage VCE may rise a little, like the concave shown in Figure2. Then, VCE begins to drop rapidly to the saturation on-state voltage VCE (sat). The rate of VCE change at this stage is deduced as: " # dV 1 Vg+ VT IL/gm dI CE = − − − + C . (4) dt Cgc Rg dt

As shown in Figure2, the IGBT switching time Tsw (including the turn-on time Ton and the turn-off time Toff) are defined as the time durations of VCE 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 Toff. Moreover, with the increasing junction temperature, the carrier recombination becomes slower, leading to the increasing CO and Toff. Equation (4) shows that an increase of current would cause a slower rate of VCE change, which will result in an increase of Ton. 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 Electronics 2019, 8, 1203 5 of 14 other healthy chips [2]. Other degradations such as solder fatigue and latch-up may also prolong the switching time of IGBT [4]. The Toff value is an early indicator to latch-up failure caused by damage in the die-attach layer [14]. On the other hand, the Ton can be also used a precursor of IGBT gate degradation [20].

2.2. Compressed Sensing of the IGBT Switching Signal Switching time has been veriﬁed adequately to monitor the operational condition of the IGBT module [3–5], [10–15]. 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 VCE 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 Figure2, the VCE during IGBT switching transitions is not sparse in the time domain. However, if there exists a transform basis such that the VCE switching signal is sparse in that transform basis, a low sampling rate can be performed by designing a suitable measurement matrix, and then the VCE 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]: y = Φ f + e = Φ Ψ θ + e, (5) · · · 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 switching signal from · CE 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 l0 norm problem, which can be expressed as: ( θˆ= arg min θ k k0 , (6) s.t. Φ Ψ θ = y · · where ||θ||0 is the l0 norm, and Equation (6) is a complete NP-hard (non-deterministic polynomial) problem [21]. For a sparse signal, the l1 norm is the convex function closest to the l0 norm [24], and the minimum l1 norm and the minimum l0 norm have equivalent solutions. Therefore, the l1 linear Electronics 2019, 8, 1203 6 of 14 Electronics 2019, 8, x FOR PEER REVIEW 6 of 15 programmingthe original V algorithmCE switching can besignal used tocan solve be therecovered global optimalfrom the solution under-sampling of the sparse data vector byθ , andinverse the originaltransformation.VCE switching signal can be recovered from the under-sampling data by inverse transformation.

2.3. Sparse Reconstruction Algorithm Figure3 3 shows shows thethe schematicschematic ofof thethe proposedproposed CSCS basedbased sparsesparse reconstructionreconstruction methodmethod forfor IGBTIGBT switching time monitoring with a reduced sampling rate. First, an N× M measurement matrix switching time monitoring with a reduced sampling rate. First, an N ×M measurement matrix is isconstructed constructed for forrandom random unde under-samplingr-sampling of the of theIGBT IGBT high-speed high-speed switching switching signal signal VCE; VinCE this; in way, this way,the high-dimensional the high-dimensional data dataare areprojected projected onto onto the the low-dimensional low-dimensional space. space. The The common common used measurement matrices include Gaussian matrix, partialpartial Hadamard matrix, and a random Bernoulli matrix [24], [24], which meet meet both both the the RIP RIP and and irrelevance irrelevance criteria. criteria. Then, Then, the the sparse sparse basis basis is constructed is constructed to tosparsely sparsely represent represent the the high-dimensional high-dimensional VVCECE switchingswitching signal. signal. Finally, Finally, the the original VVCECE switching signal is recovered byby usingusing thethe reconstructionreconstruction algorithmalgorithm toto extractextract thethe IGBTIGBT switchingswitching time.time.

Conventional high-frequency Tsw monitoring method

Analog Sampling with f signal f high-speed ADC (high-frequency) Tsw

Requirement of high sampling rate and massive data transmission&storage Tsw extraction

IGBT high-speed switching signal

The sparse reconstruction method for Tsw monitoring

Random Sparse θ ˆ Analog y StOMP f sampling with transform signal f algorithm low-speed ADC Data transmission (Wavelet basis) Toff & storage (low-frequency) Recovered signal CS computational platform

Figure 3. The propose sparse reconstruction meth methodod of IGBT switching signal for CM.CM.

As shown in Figure2, the VCE switching signal has a multi-resolution feature, which can be 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 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. 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. The VCE transient state corresponds to the behavior of IGBT turn-on and/or turn-off transitions. As As analyzed in Section 2.1, the characteristics of the V switching transition signal are related to the analyzed in Section 2.1, the characteristics of the VCECE switching transition signal are related to the distribution parametersparameters of of IGBT IGBT module. module. To To accurately accurately identify identify the nanosecondsthe nanoseconds IGBT IGBT switching switching time, it is required to sparsely represent the switching transient characteristics of V signal, i.e., V signal time, it is required to sparsely represent the switching transient characteristicsCE of VCE signal,CE i.e., VCE issignal transformed is transformed and projected and projected onto an onto orthogonal an orthogonal transform transform base. The base. wavelet The transformwavelet transform consists ofconsists different of different spatial resolution, spatial resolution, frequency frequency and directional and directional characteristics, characteristics, which defineswhich defines a well-sparse a well- representationsparse representation of signal of transientssignal transients and singularities. and singularities. Previous Previous study in study [24] showed in [24] showed that the waveletthat the basiswavelet could basis sparsely could representsparsely therepresent IGBT switching the IGBT transientswitching signal transient more signal effectively more than effectively the Fourier than basis. the Similarly, this paper uses the Mallat algorithm to construct an N N Haar wavelet× basis to sparsely Fourier basis. Similarly, this paper uses the Mallat algorithm to construct× an N N Haar wavelet basis represent the V signal [25]. to sparsely representCE the VCE signal [25]. The reconstruction algorithm is vital for IGBT switching time monitoring, which recovers the high-dimensional signal from the under-sampling data.data. As As analyzed in Section 2.2,2.2, the essence of sparse reconstruction of V signal is a sparse decomposition process of compressing measurement sparse reconstruction of VCE signal is a sparse decomposition process of compressing measurement data on redundant dictionaries [[23].23]. For the orthog orthogonalonal 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 signal can be reconstructed with a high probability, as applied in previous so that the original VCE signal can be reconstructed with a high probability, as applied in previous study [[25].25]. However, However, the the OMP algorithm will in introducetroduce new computation overheads during the Schmidt orthogonalization, and only one atom is sele selectedcted in each iteration to to update update the the support support set. set. If the amount of data is large, thethe iterationiteration eefficifficiencyency may be low or even the reconstruction may be failed. Considering Considering the the IGBT IGBT switching switching frequency frequency in in a a real real power converter is generally from a few kHz to several tens of kHz, there areare numerousnumerous switchingswitching transitions in a fundamentalfundamental period with massive CM data. The computation time of the reconstruction algorithm for processing the massive

Electronics 2019, 8, x FOR PEER REVIEW 7 of 15

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 Electronics 2019, 8, 1203 7 of 14 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 massive CM data. The computation time of the reconstruction algorithm for processing the massive atoms by maximizing the residual inner product and updating the residual by minimizing the data isresidual important norm. for However, online monitoring. unlike the OMP Different algorithm, from usingin which the only OMP one algorithm optimal solution in [24], theelement Stagewise is Orthogonalselected Matching per iteration, Pursuit the StOMP (StOMP) algorithm algorithm forms isa matched used to improvefilter by setting the e ffia thresholdciency and and reduce selects the computationall atoms time with for a residualVCE switching inner product signal greater reconstruction than the threshold [25]. The St flowchart in each iteration, of the StOMP thus reducing algorithm to reconstructthe matching the V pursuitCE switching times and signal speeding is presented up the convergence in Figure4 of. the algorithm [25].

Initialization：Set index set Λ0=∅, the residual r =y, and iteration counter t=1.

ϕ > Step 1：Calculate ur =| < , j | and select the set of indexs λ corresponding to u larger than the threshold St by a matched filter.

Step 2：Augment the index set Λt=Λt-1∪{λ} ,

and sensing matrix At=[At-1, Aλ].

Iteration：： θθˆˆ t=t+1 Step 3 Solve =argmin|| y -A t || 2 by least square method to get a new sparse coefficient estimate.

Step 4：Calculate the new approximation of the data and renew the residual ryA=−θˆ

If t > K No Yes

Step 5：Stop iteration and recover the VCE signal by the estimated sparse coefficients ：fˆ =Ψ⋅ θ ˆ

FigureFigure 4. Flowchart 4. Flowchart of theof the StOMP StOMP algorithm algorithm for for reconstructionreconstruction of of VVCECE signalsignal..

The3. Experimental basic steps ofVerification 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.3.1. However, Double-Pulse unlike Tests the OMP algorithm, in which only one optimal solution element is selected per iteration, theTo StOMPverify the algorithm feasibility formsand effectiveness a matched of filter the proposed by setting sparse a threshold reconstruction and selects method all for atoms IGBT with a residualswitching inner time product monitoring, greater the than IGBT the double-pulse threshold Sttestin is each implemented iteration, first. thus The reducing schematic the diagram matching pursuitof timesthe H-bridge and speeding based double-pulse up the convergence test rig is show of then in algorithm Figure 5. [The25]. adjustable DC power supply is used to regulate the DC voltage (0–1000 V). The DSP controller (TI, F28335) is used for IGBT switching 3. Experimentalcontrol. A heating Verification 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 3.1. Double-Pulsean accuracy Testsof 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 Toinductance verify the isfeasibility 2.2 mH. 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 Figure5. 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. Electronics 2019, 8, 1203 8 of 14 Electronics 2019, 8, x FOR PEER REVIEW 8 of 15

△T1 △T3

VDC △T2

T1 D1 T3 Adjustable DC power supply DC capacitor AB (0~1000V/5 Load inductance C kW) IGBT DUT G T2 T4

E Thermal couple 400W PTC heater Heat △T △T △T +PID Temperature controller 1 2 3 sink （resolution +/–1.0°C） VGE control pulse

FigureFigure 5. Schematic5. Schematic of of the the IGBTIGBT double-pulse test test rig rig..

The IGBTThe IGBT can can be be controlled controlled at at anyany preset voltage, voltage, current, current, and and temperature temperature by using by the using test therig. test Taking the IGBT T2 as an example of DUT, the IGBT T1 and IGBT T4 are controlled turn off during the rig. Taking the IGBT T2 as an example of DUT, the IGBT T1 and IGBT T4 are controlled turn off double-pulse test, while the IGBT T3 is controlled turn-on. During the time period of ΔT1, the DC during the double-pulse test, while the IGBT T3 is controlled turn-on. During the time period of capacitor starts to charge the load inductor through T3 and T2. The magnitude of the collector current ∆T1, the DC capacitor starts to charge the load inductor through T3 and T2. The magnitude of the is determined by the pulse width of ΔT1. The collector current increases with the ΔT1. During ΔT2, T2 collectoris turned current off, isand determined the load current by the continues pulse widthto flow ofthrough∆T1. Thethe anti-parallel collector current diode D1 increases and the IGBT with the ∆T1. DuringT3. If the∆ Tload2,T inductance2 is turned is o fflarge, and enough, the load the current load current continues changes to flowslowly through and remains the anti-parallel stable. diodeTherefore, D1 and the the IGBTIGBT T3.T2 can If thebe controlled load inductance turned-on is largeand turn-off enough, under the loaddifferent current load changescurrents slowlyby and remainscontrolling stable. the pulse Therefore, width of the ΔT IGBT1. Figure T2 6can shows be controlledthe VCE waveform turned-on of T2 measured and turn-o byff aunder differential different load currentsvoltage probe by controlling with a sampling the pulse rate widthof 1 GS/s, of ∆whenT1. Figure the load6 shows current the is 100VCE A andwaveform 200 A, respectively. of T2 measured by a diIt ffcanerential be seen voltage that the probe VCE withwaveform a sampling during rateIGBT of turn-on 1 GS/s, transition when the has load an currentobviousis concave 100 A and characteristic in the decreasing process, which agrees well with the theoretical analysis in Section 2.1. 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 SectionElectronics 2.1 2019. , 8, x FOR PEER REVIEW 9 of 15

(a)

(b)

Figure 6. Experimental waveforms of VCE in double-pulse test: (a) IC = 100 A; (b) IC = 200 A. Figure 6. Experimental waveforms of VCE in double-pulse test: (a) IC = 100 A; (b) IC = 200 A.

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. For comparison, a 100 V vertical shift is inserted intentionally between the recovered and reconstructed VCE signals. From the experimental results, the difference between the reconstructed VCE signal and the original VCE 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 Figure 7b and 8b, the concave and overshoot behaviors of the VCE signal during IGBT turn-on and turn-off transitions are well characterized by the proposed sparse reconstruction method.

Electronics 2019, 8, 1203 9 of 14

Figures7a and8a show the under-sampling points during IGBT turn-on and turn-o ff transitions, respectively. From these under-sampling points, the VCE switching transitions signals are recovered by the proposed sparse reconstruction method, as shown in Figures7b and8b, 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. For comparison, a 100 V vertical shift is inserted intentionally between the recovered and reconstructed VCE signals. From the experimental results, the difference between the reconstructed VCE signal and the original VCE 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 Figures7b and8b, the concave and overshoot behaviors of the VCE signal during IGBT turn-on and turn-off transitions Electronics 2019, 8, x FOR PEER REVIEW 10 of 15 areElectronics well characterized 2019, 8, x FOR PEER by the REVIEW proposed sparse reconstruction method. 10 of 15

(a) 1000 (a) 1000

0 0 Amplitude/V Amplitude/V

-1000 -1000 1200 1200 VCE signal measured at sampling rate of 1GS/s 1000 VCE signal measured at sampling rate of 1GS/s 1000 Recovered V signal from down-sampled points Recovered V CE signal from down-sampled points 800 (b)CE 800 (b) 600 600

Amplitude/V

400

Amplitude/V 400 200 III 200 I II III 0 I II 0 -200 -2000.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 t/us t/us

Figure 7. Reconstruction of VCE signal during IGBT turn-on transition: (a) random under-sampling FigureFigure 7. 7.Reconstruction Reconstruction of ofV VCECE signalsignal during IGBT turn-on turn-on transition: transition: (a ()a random) random under-sampling under-sampling points (100 MS/s); (b) comparison of the recovered VCE signal and original VCE signal. pointspoints (100 (100 MS MS/s);/s); (b ()b comparison) comparison of of the the recoveredrecovered VCE signalsignal and and original original VCEV CEsignal.signal.

1000 (a) 1000 (a) 500 500

0 0

-500

Amplitude/V -500 Amplitude/V -1000 -1000 1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2 1200 VCE signal measured at sampling rate of 1GS/s 1200 VCE signal measured at sampling rate of 1GS/s 1000 Recovered VCE signal from down-sampled points 1000 Recovered VCE signal from down-sampled points 800 800 600 600

Amplitude/V 400

(b)

Amplitude/V 400 (b) 200 200 0 0 -200 -200 1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6t/us 1.8 2.0 2.2 t/us

FigureFigure 8. 8.Reconstruction Reconstruction of ofV VCECE signalsignal duringduring IGBT turn-off turn-oﬀ transition:transition: (a ()a )random random under-sampling under-sampling Figure 8. Reconstruction of VCE signal during IGBT turn-off transition: (a) random under-sampling pointspoints (100 (100 MS MS/s);/s); (b ()b comparison) comparison of of the therecovered recovered VVCE signalsignal and and original original VCEV CEsignal.signal. points (100 MS/s); (b) comparison of the recovered VCE signal and original VCE signal.

To quantify the percent error between the original VCE signal f and the recovered VCE signal fr, 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]: the reconstruction error Ce is defined as [27]: ff- r 2 ff- r ⋅ Ce =2 ⋅ 100% . (7) Ce =f 100% . (7) f 2 2 Different random measurement matrices including Gauss matrix, Bernoulli matrix, and a partial Different random measurement matrices including Gauss matrix, Bernoulli matrix, and a partial Hadamard matrix are used to the VCE switching signal and the statistical results of the reconstruction Hadamard matrix are used to the VCE switching signal and the statistical results of the reconstruction error are listed in Table 1. The VCE reconstruction error during IGBT turn-off is less than 1%. The VCE error are listed in Table 1. The VCE reconstruction error during IGBT turn-off is less than 1%. The VCE turn-on reconstruction errors using the random Gauss matrix and the random Bernoulli matrix are turn-on reconstruction errors using the random Gauss matrix and the random Bernoulli matrix are less than 1.5%. The VCE turn-on reconstruction error is slightly larger than of the turn-off—that is, less than 1.5%. The VCE turn-on reconstruction error is slightly larger than of the turn-off—that is,

Electronics 2019, 8, 1203 10 of 14

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]:

f fr 2 Ce = k − k 100%. (7) f · k k2 Different random measurement matrices including Gauss matrix, Bernoulli matrix, and a partial Hadamard matrix are used to the VCE switching signal and the statistical results of the reconstruction error are listed in Table1. The VCE reconstruction error during IGBT turn-off is less than 1%. The VCE turn-on reconstruction errors using the random Gauss matrix and the random Bernoulli matrix are less than 1.5%. The VCE turn-on reconstruction error is slightly larger than of the turn-off—that is, because the characteristics of VCE waveform during IGBT turn-on are more complex [17,18]. As shown in Figure7, the VCE waveform has three transient stages during IGBT turn-on process. In the first stage, VCE is decreased by the voltage drop of stray inductance during the fast rising of IC. In the second stage, the freewheeling diode sustains a reverse voltage and VCE rises a little and then drops quickly as the depletion layer shrinks toward the drift region. In the third stage, VCE 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 VCE signal by the StOMP algorithm is 0.32 ms when compared with 2.89 ms by the OMP algorithm.

Table 1. The reconstruction error of IGBT switching signal with diﬀerent measurement matrices.

Random Measurement Reconstruction Error C e Matrix ¬ Gauss matrix 1.35% 0.76% Bernoulli matrix 1.22% 0.85% Partial Hadamard matrix 2.01% 0.77%

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. Figure9 shows the VCE 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. Electronics 2019, 8, x FOR PEER REVIEW 11 of 15 because the characteristics of VCE waveform during IGBT turn-on are more complex [17,18]. As shown in Figure 7, the VCE waveform has three transient stages during IGBT turn-on process. In the first stage, VCE is decreased by the voltage drop of stray inductance during the fast rising of IC. In the second stage, the freewheeling diode sustains a reverse voltage and VCE rises a little and then drops quickly as the depletion layer shrinks toward the drift region. In the third stage, VCE 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 VCE signal by the StOMP algorithm is 0.32 ms when compared with 2.89 ms by the OMP algorithm.

Table 1. The reconstruction error of IGBT switching signal with different measurement matrices.

Random Measurement Matrix Reconstruction Error Ce Gauss matrix 1.35% 0.76% Bernoulli matrix 1.22% 0.85% Partial Hadamard matrix 2.01% 0.77%

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 VCE 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 Electronics 2019, 8, 1203 11 of 14 time based IGBT junction temperature estimation.

1200 38℃ 1000 48℃ 60℃ 800 69℃ (a) ℃ ) 79℃ ≈1.54≈≈≈ 600 (V 88℃ ce

V 400 97℃ 110℃ 200 120℃ 130℃ 0

-200

1200 38℃ 1000 48℃ (b) 60℃ 800 69℃ ≈1.52≈≈≈℃ 79℃

） 600 88℃ V 97℃ （ 400

CE 110℃ V 200 120℃ 130℃ 0

-200 1000 1500 2000 2500 3000 3500 4000 t(ns) Figure 9. IGBTIGBT turn-off turn-oﬀ transitions under different diﬀerent case temperatures: (a) direct sampling at sampling rate of 1GS1GS/s;/s; ((b)) sparsesparse reconstruction.reconstruction.

3.2. PWM Converter Test As analyzed in Section 2.12.1,, the IGBT switching timetime varies with the instantaneous current (since the carriercarrier concentration concentration speed speed is diisff differenterent [16 ]).[16]). The The proposed proposed sparse sparse reconstruction reconstruction method method is further is 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 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 turn-off 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 [13–15]. The potential applications of the proposed method for IGBT junction temperature measurement will be further studied. Electronics 2019, 8, x FOR PEER REVIEW 12 of 15 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 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. Electronics 2019, 8, 1203 12 of 14 The comparative results are exhibited in Figure 12.

T D VCE 1 1 T3 D3 T5 D5 + A B - Vdc IC C C

T2 D2 T4 D4 T6 D6

(a)

IGBT module

Gate Driver

DC supply

DSP controller

Three-phase PWM converter

Load inductor

(b)

FigureFigure 10. 10. TheThe PWM PWM converter converter test test for for IGBT IGBT switching switching time time monitoring: monitoring: (a) schematic (a) schematic diagram; diagram; (b) Electronicspicture(b) picture2019 of, 8 test, x of FOR testrig. PEER rig. REVIEW 13 of 15

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 turn- off 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 [13–15]. The potential applications of the proposed method for IGBT junction temperature measurement will be further studied.

Figure 11. Experimental waveforms of VCE andand IICC inin the the PWM PWM converter converter test. test.

500 T (Direct sampling at 1GS/s) 450 on Toff (Direct sampling at 1GS/s) 400 Ton (Under-sampling reconstruction) 350 Toff (Under-sampling reconstruction) 300

250 Time/ns 200

150

100

50 0 5 10 15 20 25 30 I /A C

Figure 12. Ton and Toff with different IC.

4. 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.

Author Contributions: Conceptualization, H.L.; methodology, H.L.; validation, M.Z. and H.Y.; investigation, X.Y.; writing—original draft preparation, H.L.

Funding: This work was funded in part by the National Natural Science Foundation of China (NSFC) under Grant No. 51907116, and Shanghai Science and Technology Innovation Action under Grant No. 19DZ1205402 and Shanghai Green Energy Grid Connected Technology Engineering Research Center: No. 13DZ2251900.

Electronics 2019, 8, x FOR PEER REVIEW 13 of 15

Electronics 2019, 8, 1203 13 of 14 Figure 11. Experimental waveforms of VCE and IC in the PWM converter test.

500 T (Direct sampling at 1GS/s) 450 on Toff (Direct sampling at 1GS/s) 400 Ton (Under-sampling reconstruction) 350 Toff (Under-sampling reconstruction) 300

250 Time/ns 200

150

100

50 0 5 10 15 20 25 30 I /A C

Figure 12.12. Ton andand TToffoﬀ withwith different diﬀerent I ICC. .

4.4. Conclusions Conclusions ToTo overcome the technical difficulty diﬃculty of the ex extremelytremely high sampling rate rate for for IGBT IGBT nanoseconds nanoseconds switchingswitching time time online online monitoring monitoring,, this this paper paper proposes proposes a a CS CS base basedd under-sampling reconstruction monitoringmonitoring method. method. By By analyzing analyzing the the physical physical me mechanismchanism behind behind the the IGBT IGBT switching switching transitions transitions,, the V switching signal is sparsely represented by the wavelet basis, and the StOMP algorithm the VCECE switching signal is sparsely represented by the wavelet basis, and the StOMP algorithm is is proposed to recover the high-speed V signal for under-sampling data. Finally, experimental proposed to recover the high-speed VCECE signal for under-sampling data. Finally, experimental verificationveriﬁcation is conducted conducted on not not only only the the double-pul double-pulsese test, test, but but also also a a real real PWM PWM converter. converter. The The results results showshow that: that: (1)(1) The The characteristics characteristics of of VCE VCE signal signal during during sw switchingitching transitions transitions can can be be recovered recovered from from the the under-samplingunder-sampling data data with with a a reconstruction reconstruction error error less than 2%. The The StOMP StOMP algorithm algorithm shows shows a a faster faster convergenceconvergence speed speed than than the the traditional traditional OMP algorithm for sparse reconstruction. (2)(2) The The under-sampling under-sampling reconstruction reconstruction method method can can monitor monitor the the nanoseconds nanoseconds change change of of IGBT IGBT switchingswitching timetime variationvariation with with a largelya largely reduced reduced sampling sampling rate. Atrate. a compressedAt a compressed ratio 10 ratio of the 10 sampling of the samplingrate, the detection rate, the detection accuracy ofaccuracy less than of 2less ns canthan be 2 ns achieved. can be achieved.

AuthorAuthor Contributions: Conceptualization, H.L.;H.L.; methodology, methodology, H.L.; H.L.; validation, validation, M.Z. M.Z. and and H.Y.; H.Y.; investigation, investigation, X.Y.; X.Y.;writing—original writing—original draft draft preparation, preparation, H.L. H.L.

Funding:Funding: ThisThis work work was was funded funded in in part part by by the the National Natural Natural Science Science Foundation Foundation of of China China (NSFC) (NSFC) under under Grant No. 51907116, and Shanghai Science and Technology Innovation Action under Grant No. 19DZ1205402 and GrantShanghai No. Green51907116, Energy and Grid Shanghai Connected Science Technology and Technolo Engineeringgy Innovation Research Action Center: under No. Grant 13DZ2251900. No. 19DZ1205402 and Shanghai Green Energy Grid Connected Technology Engineering Research Center: No. 13DZ2251900. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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