Simplified Fourier Series Based Transistor Open-Circuit Fault Location Method in Voltage-Source Inverter Fed Induction Motor

Transistors in three-phase voltage-source inverter often suffer from open-circuit failures due to the lifting of bonding wires caused by thermic cycling, resulting in performance degradation with ripple torque and current harmonics. Current-spectral-analysis based methods are widely applied to failure diagnosis; however, high calculation consumption and complex implementation limit their application in some real-time occasion. In this paper, a simplified Fourier series method is proposed by the product between reconstructed phase currents and reference signals. Meanwhile, a novel normalized method for DC and fundamental components of simplified Fourier series are proposed to locate twenty-one transistor open-circuit faults. Numerical results show that the proposed Fourier series method coincides with that of Fast Fourier Transform. Experimental results and the comparison with previous methods show high efficiency and merits of its application to transistor open-circuit fault location in the voltage-source inverter.


I. INTRODUCTION
Three-phase VSIs are widely used in industrial applications due to their superior performance. Their health condition makes a vital contribution to the reliability of motor drives. It is reported that the failure rate of power semiconductor takes up a large scale of the failure in motor drives, followed by capacitor and gate drives [1]. Consequently, lots of researches had been done on the fault diagnosis in three-phase VSIs to reach convenient maintenance and fault-tolerant in recent decades [2]- [6]. Generally, failures in three-phase VSIs can be divided into two categories: short-circuit and open-circuit. Shortcircuit will cause overcurrent and has great damage to VSIs immediately [7]. Some positive protection is often taken, such as fault isolation by hardware design, as well as shut down the drive system immediately. The open-circuit fault is less destructive and causes system performance degraded by generating torque ripple and harmonics. The healthy components continue suffering overcurrent and overvoltage, which is extremely easy to cause secondary failure or even destroy the system if no positive actions are taken.
It should be noticed that the fault diagnosis consists of fault detection and fault location. Fault detection is applied to monitor system healthy conditions and provide safety strategies, such as shutting down the system. Fault location focuses on the position of fault after detection, which makes much sense to maintenance or switch to tolerant strategies.
The reported fault diagnosis methods in three-phase VSIs include two kinds: signal of component-based and signal of system-based. The former one uses the internal signal as diagnostic features by failure mechanism in the physical and electrical model of power component, such as collectorto-emitter voltage, the drain-to-source voltage, and gate-tosource voltage during the IGBT turn-on transient are used for healthy condition [8], [9]. Fast-detection, fair robustness to system disturbance and portability can be achieved by these methods. However, extra electrical circuit equipped with every power components will increase the cost and system volume, which limit their applications. System signalbased fault diagnosis methods are the most widely researched for feasible design and effective performance in recent years. They can be divided into two categories: voltage-based and current-based.
Voltage-based methods are proposed in [10]- [13], they may be further classified by the type of measured voltagespole [10], phase [12], line or neutral voltage [13]. Fast diagnosis and fair robustness are the most advantages of these methods; however, extra sensors or electrical circuits are required, resulting in high cost. Recently, [14] proposes a voltage-based method by replacing the measured voltage with observed voltage. This method improves calculation, and no voltage sensor is required.
Current-based methods attract more attention because the diagnostic signals can be shared with the feedback current sensors. The diagnostic methods can be easily inserted into the controller as an independent subroutine. They can be further classified by currents-in αβ-axis, dq-axis, abc-axis. Park vector [15] and its modified approaches [16], [17] are the most classical diagnostic methods in αβ-axis. These approaches are not suitable for integration into the drive controller because they require complex pattern recognition algorithms and poor robustness. Reference [18] proposes to divide three-phase currents into six stages and diagnosis in dq-axis, current in d-axis is used for fault detection and the current vector rotating angle is used to locate the faulty switches. Current-based diagnostic methods in abc-axis are the most widely researched, normalized DC current and its modified method are proposed in [19], [20]. They have some drawbacks when implemented in a closed-loop scheme and poor robustness to transients. Reference [21] proposes load currents are applied to realize fault diagnosis with fuzzy classifier, [22] proposes to extract the fault features by the errors between the reference currents and the measured currents, [23], [24] proposes to use three-phase currents to describe the symmetry both in healthy and faulty conditions to realized fault detection and location, all these methods show fast diagnosis, fair robustness and acceptable tuning efforts. Besides, features extracted by signal processing and identified by pattern recognition, are also very popular as off-line techniques, such as discrete wavelet transform with support vector machine [25], fuzzy system [26], etc. These methods need a longer detection time and a complex implementation, compared with real-time methods, however, they have fair portability.
Lots of spectral analysis are used to locate the faulty transistor by analyzing the spectral distribution of phase currents [27], [28]. It should be noticed that these methods need extra signal processing hardware due to the complex calculation, which limits their application. In [19], the Fourier Series of currents are estimated by the motor rotating angle [19]; however, the estimation will be biased after fault occurrence, what is more, these methods are only available to a single open-circuit fault.
To overcome the complex calculation of traditional current spectral analysis methods. A simplified Fourier Series method is proposed, novel normalized DC and fundamental components are proposed for fault location for both single and multiple open-circuit faults. Two contributions are made in this paper, listed as follows.
1) A simplified Fourier series method is proposed by the product between reconstructed CLs and reference signals, and the computational complexity is O(log 2 L). 2) Novel normalized method for DC and fundamental components by simplified Fourier series are proposed to locate twenty-one transistor open-circuit, experimental results show high efficiency and merits. The structure of this paper is as follows: Section II elaborates the proposed simplified Fourier series theory, including continuous and discrete systems, the concept of CDR. Section III concentrates the novel normalized method for DC and fundamental components and transistor open-circuit location. Section IV gives out experimental results, including the comparison of proposed simplified Fourier series method with FFT, fault location results and the comparison with previous spectral-analysis based methods. A conclusion is made in Section V.

II. SIMPLIFIED FOURIER SERIES IN VSI FED IM
The structure of three-phase VSI fed IM is showed in Fig. 1, U d is the input voltage of the drive system, C 1 , C 2 are two symmetrical capacitors to eliminate high frequently harmonics of the supply voltage. T1, T2, T3, T4, T5, T6 are six power transistors, they are the core components of the drive system. D1, D2, D3, D4, D5, D6 are six stream diodes to avoid the impact caused by inductive rotating load. Three-phase currents (i a , i b , i c ), measured speed (ω), reference current in d-axis (i d * ) and reference speed are the inputs of control systems, pulse-width modulation signals are generated to switch their operation states between 'on' and 'off', alternately. As a result, DC to AC energy conversion is achieved. In healthy conditions, three-phase currents are sinusoidal time series; their frequencies and amplitudes are the same, only with 2/3π phase difference between every two among them. In faulty conditions with open-loop control strategy, the phase currents of undamaged legs still keep the same as healthy condition, while the phase current of the damaged leg is distorted. However, in faulty condition with closedloop control strategy, phase currents of undamaged legs will be spread by the phase current of the damaged leg for the feedback strategy. This paper focuses on transistor opencircuit fault location in VSI fed IM with FOC.

A. FOURIER SERIES IN CONTINUOUS SYSTEM
Three-phase output currents are periodic time series with period T , T is given as following Equation (1), Considering measurement error and system noise, reference speed is used in Equation (1).
Three-phase currents can be represented by Fourier Series corresponding to a sum of harmonically related time series. The frequencies of these Fourier Series are integer multiples of the fundamental frequency (ω 0 = 2π/T ). These periodic Fourier Series are of the form, A m,h are the amplitude of related exponential time series with frequency of hω 0 . The integration of Equation (2) in a period is given, can be given as follows, Equation (2) is multiplied by a reference signal, whose frequency is zω 0 , phase angle is the same as original signal, there is, The integration of Equation (5) in a period is given, The orthogonality of trigonometric functions has a characteristic showed as Equation (7), Substituting Equation (7) in (6) and simplifying yields, (8) A m,z can be calculated as Equation (8),

B. FOURIER SERIES IN DISCRETE SYSTEM
In discrete system, three-phase currents in a period are composed of L samples, named CLs, their length is given as following.
In k instant, t is defined as the ending position of CL, Three phase CLs are given, DC component can be easily calculated. However, the realtime calculation of AC component is challenging because ϕ a,b,c (k) are uncertain variables. In every sampling instant, ϕ a = ϕ b = ϕ c , what's more, in any two sampling instants during a period, there is ϕ m (p) = ϕ m (g), where p, g ∈ [t, t + 1, · · · , k].
In order to eliminate the difference between ϕ a , ϕ b , ϕ c and the difference in any two sampling instants during a period ϕ m (p), ϕ m (g), a CDR algorithm is proposed.

C. CDR AND SIMPLIFIED FOURIER SERIES
There are two ZCSs in every CL, one is up-to-down ZCS, the other one is down-to-up ZCS. The positions of down-toup ZCS in CLs I m are marked as H m . In, all samples before H m are removed back to the last sample i m (k) to formÎ m . The CDR process of phase-a when T4 fails is showed as subplot (a) in Fig. 2, and the reconstructed result is showed  Here, assuming that the phase angle of ZCS in every instant is approximated to ϕ m , then ϕ a is showed in Fig. 2, the reconstructed CLs are nearly the same with ϕ m ≈ 0, shown in Fig. 3.   3) UpdateÎ m . Current harmonics and system noise will cause fluctuation near ZCS, which has a negative influence on the calculation of H m . Here, a ZCS calculation algorithm is given.
Three-phase currents are firstly filtered, as following, Then, H m = k − N 2 . The proposed ZCS calculation algorithm has a short delay-time with (16) These features focus on the characteristics without considering the interaction. However, in the closed-loop system, transistor open-circuit fault on the faulty leg will propagate to healthy legs. Hence, a novel normalized method for DC and AC components is proposed by taken into the interaction. AC components are divided by the maximum amplitude of fundamental components; DC components are divided by the maximum absolute value of the sum of CLs, shown as follows.
Substituting Equation (17) in (19) and simplifying yields, normalized AC components can be online calculated by (20), Table 1 gives the FFT and proposed normalized Fourier Series of currents under 30% IM RLT at 1000r/min, these data are from experimental board and off-line calculated by Matlab. The motor parameters are listed in Table 2. The first part in Table 1 gives (D m , A m,z (z = 1, 2, 3, 4, 5)) calculated by Equation (18). The second part in Table 1 are normalized Fourier series |D m |, |Â m,z |(z = 1, 2, 3, 4, 5) calculated by Equation (19). Compared with the FFT, features are much more recognizable in the proposed normalized Fourier Series.
1) The normalized fundamental components of healthy legs are larger than that of faulty legs. 2) The normalized DC components are not equal to zero in fault legs. The operation principle of VSI is applied to explain the mentioned features. Fig. 5 shows the negative and positive current flows in a faulty leg with lower transistor T k+1 open-circuit, T k is turned on and off alternatively. When i m > 0, the current flows from DC-link to IM in two cases, one case is through T k directly if it is turned on, another case is through V k+1 during the dead-time interval that is applied to   Considering system noise and measurement errors, two symmetrical boundaries near predefined constant are set as Fig. 6, where the predefined constant is v, the distance between boundaries is 2ξ . If a variable is inside two boundaries, shown as the light green zone, it is considered equal to v. Otherwise, if a variable is larger than the upper boundary,  it is considered larger than the v, if a variable is smaller than the lower boundary, it is considered smaller than the v, the criterion is given as, Based on the analysis above, a fault location table to locate twenty-one single and multiple open-circuit faults are proposed as Table 3. Where × means a does not care condition, D * m , A * m are filtered values ofD m ,Â m , given as, Absolute values of A * m,1 are applied to locate faulty legs, sign, and values of D * m are applied to locate the position of the faulty transistor. The flowchart of the proposed fault location algorithm is given as Fig. 7, which includes three steps: 1) CDR to eliminate ϕ m (k) 2) Normalized DC and fundamental components calculation, equation(17)-(22) 3) Look up Table 3 C. TUNING EFFORT An important property of diagnosis algorithm is low tuning effort. The proposed fault location method needs four parameters, N 1 , N 2 , ξ 0 and ξ 1 . N 1 , N 2 are two constants applied to ZCS calculation, N 1 is a filter constant, it makes sense when the currents contain high harmonics. N 2 is applied to search for the position of ZCS. It plays an important role in the proposed algorithm. In fact, the number of positive samples on the right of ZCS is smaller than L/2. Hence, the upper limit of N 2 is L/2, and N 2 is suggested to set as large enough to improve the accuracy of CDR. ξ 0 , ξ 1 are applied to measure the equivalent relationship with 0, 1, respectively. The value ranges of these four parameters are shown in Table 4, where large efficient value ranges show low tuning effort.

IV. EXPERIMENTAL RESULTS
The following analyses are based entirely on the experimental results since they give an understandable presentation of the VOLUME 8, 2020 algorithm performance in the presence of nonideal properties, such as model uncertainty, measurement noise, dead-time effects, etc. FOC with SVPWM was the control algorithm in the experiment. Some indices were presented to evaluate the performance of the proposed fault location method, such as location time, effectiveness, etc. Four typical faulty operating conditions were investigated. All kinds of transistor opencircuit faults were performed by inhibiting their respective gate signals while keeping the bypass diode still connected. The experimental results are presented by signal FaultType.
The experimental validation of the proposed fault diagnosis method was implemented in a TMS320F2806 board. The experimental setup was shown in Fig. 8, consisting of a 2.2kW squirrel-cage IM with 380V rated voltage, 4.9A rated current, a power converter with a switching frequency of 20kHz and the dead time of 3.2µs, a control board, a magnetic power brake, and a constant current source. The parameters of IM were listed in Table 2. The thresholds N 1 , N 2 for CDR were set as 4, 50, respectively. The thresholds ξ 0 was set 0.2, ξ 1 was set 0.25.  at 1000rpm with 30% RLT, the left subfigures are the 3-D view of the reconstructed CLs during 0.4s, the color of bar represents the amplitude, the right subfigures are the side view of reconstructed CLs during 0.4s. The number of current samples during a period is 300, calculated by Equation (10), representing the length of CLs, the fault occurs at 0.28s. All phase differences are eliminated in three-phase currents in every instant. For ∀p, g(p = g), there is ϕ m (p) = ϕ m (g), showed in three subfigures respectively.There is ϕ a (k) = ϕ b (k) = ϕ c (k), showed among three subfigures, the phase angles of three-phase CLs nearly coincide. In every phase, reconstructed CL has two states, healthy states, and faulty states, shown as the side view subfigures. Fig. 11 presents the experimental results for a single fault in T6, under 30% IM RLT and 500r/min reference speed. Torque ripples and distorted currents occur after fault, showed as subfigure 1, 2. The normalized DC and fundamental components of the simplified Fourier series by the proposed method are presented in subfigure 3, 4, respectively. Subfigure 5 gives out the fault location result.
In healthy conditions, three-phase currents are sinusoidal, |Â m,1 | = 1, normalized DC components, and harmonics are nearly equal to zero. After T6 fails, DC component of leg-c, D c raises. Due toD a +D b +D c = 0, DC components will also exist in healthy legs. Meanwhile, the fundamental component is nearly equal to 1 in the healthy leg, while it is smaller than 1 in the faulty leg. In subfigure 4,Â a,1 ,Â b,1 are inside the boundaries made up by ξ 1 , whileÂ c,1 are outside ξ 1 , which indicates leg-c is faulty. In subfigure 3,D c is larger than ξ 0 , the DC component in phase current of leg-c is positive, which indicates the lower transistor is faulty. Combining the results of subfigure 3 and 4, the fault can be located to T6 by looking up Table 3, the location flag FaultType raises to 8, showed in subfigure 5. Fig. 12 presents the experimental results for multiple opencircuit faults, under 30% IM RLT and assuming a reference    Real-time calculated normalized DC, fundamental components are showed in subfigure 3, 4, respectively. In subfigure 4, only A * a,1 is inside ξ 1 , A * b,1 , A * c,1 are outside the boundaries made up by ξ 1 , |A * a,1 | = 1, |A * b,1 |, |A * c,1 | < 1, which indicates phase-a is healthy. In subfigure 3,   a reference speed of 1000r/min. Motor measured speed and three phase currents are showed in subfigure 1, 2, respectively. Real-time calculated normalized DC, fundamental components are showed in subfigure 3, 4, respectively. In subfigure 4, only A * a,1 is inside ξ 1 , A * b,1 , A * c,1 are outside the boundaries made up by ξ 1 , |A * a,1 | = 1, |A * b,1 |, |A * c,1 | < 1, which indicates phase-a is healthy. In subfigure 3, D * b > 0, D * c > 0 and |D * a | = |D * b | + |D * c |, which indicates the lower transistors of leg-b and leg-c are broken. As a result, fault is located to T4 and T6, showed in subfigure 5.

C. COMPARISON WITH PREVIOUS SPECTRAL ANALYSIS BASED VSI FAULT LOCATION METHODS
The performance of the proposed transistor open-circuit fault location method is compared with previous methods in calculation consume, efficiency, cost, implementation, and tuning effort. Reference [27] and the proposed method can locate both single and multiple open-circuit faults. References [19], [20] and the proposed method are both low-implemented. In [19], [20], fundamental components are calculated in αβ-axis by calculating the rotating angle, the Fourier series are calculated by FFT in [27], [28], the calculation consume is O(L log L). In the proposed method, the fundamental components are approximated by CDR. The calculation complex is O(log 2 L). The tuning effort is relatively lower because of the large efficient value ranges of thresholds.
Consequently, the proposed method shows advantages in efficiency, calculation consumption, and low implementation.

V. CONCLUSION
Signal spectral analysis based methods are widely used in fault diagnosis. However, the calculation consumes limits the application in the real-time system, such as transistor opencircuit fault diagnosis in VSIs fed IM. In this paper, a realtime, easy-implemented simplified Fourier series algorithm for low-frequency periodic signals by data reconstruction with the position of ZSC is proposed, the Fourier series can be calculated in every sampling instant by product between reconstructed signals and reference signals. Especially, a novel normalized method of DC and fundamental components of simplified Fourier series is proposed to VSIs transistor open-circuit fault location. Comparison results show that the proposed simplified Fourier series algorithm nearly coincides with FFT. Experimental results show the high efficiency of proposed methods, both single and multiple of transistor open-circuit fault can be located in VSIs fed IM.