Harmonic Spectrum of Output Voltage for Space Vector Pulse Width Modulated Ultra Sparse Matrix Converter

energies

Article Harmonic Spectrum of Output Voltage for Space Vector Pulse Width Modulated Ultra Sparse Matrix Converter

Tingna Shi * ID , Lingling Wu, Yan Yan and Changliang Xia ID

School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China; [email protected] (L.W.); [email protected] (Y.Y.); [email protected] (C.X.) * Correspondence: [email protected]; Tel.: +86-22-27402325

Received: 19 January 2018; Accepted: 2 February 2018; Published: 8 February 2018

Abstract: In a matrix converter, the frequencies of output voltage harmonics are related to the frequencies of input, output, and carrier signals, which are independent of each other. This nature may cause an inaccurate harmonic spectrum when using conventional analytical methods, such as fast Fourier transform (FFT) and the double Fourier analysis. Based on triple Fourier series, this paper proposes a method to pinpoint harmonic components of output voltages of an ultra sparse matrix converter (USMC) under space vector pulse width modulation (SVPWM) strategy. Amplitudes and frequencies of harmonic components are determined precisely for the ﬁrst time, and the distribution pattern of harmonics can be observed directly from the analytical results. The conclusions drawn in this paper may contribute to the analysis of harmonic characteristics and serve as a reference for the harmonic suppression of USMC. Besides, the proposed method is also applicable to other types of matrix converters.

Keywords: ultra sparse matrix converter (USMC); space vector pulse width modulations (SVPWM); triple fourier series; harmonic analysis

1. Introduction The ultra-sparse matrix converter (USMC) evolved from the conventional matrix converter (CMC) and inherited many merits from CMC including compact structure without intermediate storage, sinusoidal input and output current, adjustable input power factor, etc. Meanwhile, USMC possesses superiorities over CMC, such as simpler commutation technology, a decrease in number of switches, etc. Currently, USMC is considered as a promising type of matrix converter [1–4]. As an AC–AC power converter, the primary task of USMC is to obtain a sinusoidal output waveform with high quality. However, due to the characteristics of the power device and the modulation methods, the output voltages of USMC inevitably contain harmonic components, thus negatively impacting the practical application. Therefore, it is very important to analyze the harmonic components in the output waveform of USMC, which not only lays the foundation for the suppression of the harmonics but also improves the theory concerning matrix converter harmonic analysis. Up to now, there are some studies on the harmonic analysis of power electronics converters. Among them, fast Fourier transform (FFT) analysis is the most commonly used one [5–8]. It is simple and easy to implement. However, FFT involves sampling and window etc. in the processing, which will incur spectrum leakage, aliasing, and other issues. Therefore, when the frequencies of input, output and carrier signals are special, the results of FFT analysis are not accurate. Another commonly used method is calculating the analytical formula with Fourier series. In 1975, S. Bayes and B.M. Bird ﬁrst applied the double Fourier analysis method to power electronic converters which were originally used in the communication system [9]. Since then, the studies on harmonic analytic solution have

Energies 2018, 11, 390; doi:10.3390/en11020390 www.mdpi.com/journal/energies Energies 2018, 11, 390 2 of 13 mostly concentrated on the double Fourier analysis. For example, the harmonic analytical formulas of output voltage of two-level inverter are derived in [10] by using double Fourier analysis and then a comparison about the high-frequency harmonics is carried out when using saw-tooth carrier and triangular carrier in modulation process. In [11], Jacobi matrix is used to ﬁnd the closed solution of the double Fourier series; and the characteristics of output voltage of the inverter under different space vector pulse width modulation (SVPWM) are compared. In [12] the output voltage harmonics of inverters with different topologies under various modulation methods are analyzed in detail by double Fourier analysis. Like CMC, the output voltage of USMC is related to the frequencies of input, output and carrier signals, which are independent of each other. If using double Fourier series analysis, only two of these frequencies are considered at most, resulting in inaccurate spectral analytic results. In view of this problem, some scholars have tried to use triple Fourier series to analyze the harmonic of output waveform for CMC under “AV method” [13,14]. However, due to some differences in the topology and control performance between USMC and CMC, a mature harmonic analysis theory for USMC has yet to emerge. Compared with “AV method”, SVPWM not only increases the maximum voltage transmission ratio from 0.5 to 0.866, but also achieves digitization easily. As a result, SVPWM is currently widely applied [15–17]. Therefore, for USMC, the harmonic analysis of output voltage under SVPWM is of important signiﬁcance. In this paper, based on the characteristics of SVPWM and triple Fourier series, an output voltage harmonic calculation method for USMC is proposed, which can obtain an accurate harmonic spectrum and identify the relations between the frequencies of harmonic components and the frequencies of input, output and carrier signals. In addition, the harmonic distribution pattern and the main harmonic components of output voltage of USMC are analyzed and summarized according to the analytical results.

2. Triple Fourier Series It is assumed that the three-variable function g(x, y, z) is periodic in x, y, and z directions with the period of 2π, x, y and z being independent of each other. The triple Fourier series expression of g(x, y, z) can be deﬁned by the following:

+∞ +∞ +∞ j(kx+py+qz) g(x, y, z) = ∑ ∑ ∑ Fkpqe (1) k=−∞ p=−∞ q=−∞ where k, p, q are indices on the carrier frequency, output frequency and input frequency respectively; Fkpq is the triple Fourier coefﬁcient and its expression is

ZZZ 2π 1 −j(kx+py+qz) Fkpq = g(x, y, z)e dxdydz (2) 8π3 0

Let x = 2πf ct, y = 2πf outt + φ0, z = 2πf int + φi then Equation (1) can be regarded as the output voltage expression of any phase of USMC.; f c, f out and f in are respectively carrier frequency, output voltage frequency and input voltage frequency of USMC; φi is the input power factor angle; φ0 is the output voltage initial phase. In most applications, USMC is required to operate with unit power factor, so φi = 0. When φo is not zero, the integration limits of Equation (2) should be adjusted. Since the integrand g(x, y, z) is periodic in x, y, and z directions with the period of 2π, the integration results would not be affected by the speciﬁc values of the integral limits as long as the periodic variation of x, y and z over 2π intervals. Therefore, whether φ0 is zero does not have any effects on the calculation results. To simplify calculation, it is assumed that φ0 = 0. Energies 2018, 11, 390 3 of 13

Expending Equation (1) and it can be expressed in the following form:

 +∞  ∑ [Ak00 cos(kx) + Bk00 cos(kx)]+  =   k 1   +∞  A  ( ) + ( )+  g(x, y, z) = 000 +  ∑ A0p0 cos py B0p0 cos py  2  p=1     +∞   ∑ A00q cos(qz) + B00q cos(qz)  q=1  +∞ +∞  ∑ ∑ [A cos(kx + py) + B sin(kx + py)] +  kp0 kp0   k=1 p = −∞     6=   p 0     +∞ +∞  (3)  A cos(py + qz) + B sin(py + qz) +   ∑ ∑ 0pq 0pq  + p=1 = −   q ∞     q 6= 0     +∞ +∞   ∑ ∑ [Ak q cos(kx + qz) + Bk q sin(kx + qz)]   0 0   k=1 q = −∞    q 6= 0 " # +∞ +∞ +∞ Akpq cos(kx + py + qz)+ + ∑ ∑ ∑ ( + + ) k=1 p = −∞ q = −∞ Bkpq sin kx py qz p 6= 0 q 6= 0

In Equation (3) harmonic components are divided into four parts: the ﬁrst part is the DC component (k = 0, p = 0, q = 0); the second part is the fundamental voltage (k = 0, p = 1, q = 0) and harmonic components only associated with the carrier frequency (when k = 1, 2, 3, ... , p = 0, q = 0), the input frequency (when k = 0, p = 0, q = 1, 2, 3, ... ) or the output frequency (k = 0, p = 2, 3 ... , q = 0) respectively; the third part is the harmonic components related to any two of the three frequencies(input, output, and carrier frequencies), and their frequencies are the sum of the integral multiple of the two frequencies; the fourth part is the harmonic components relevant to the three frequencies, and their frequencies are sums of the integral multiples of the three frequencies. According to the Fourier analysis theory, the frequency of any harmonic is kf c ± pf out ± qf in and the harmonic amplitude Ukpq satisﬁes the following relationship with Fkpq:

Ukpq = 2 Fkpq (4)

3. Harmonic Calculation of Output Voltage of USMC Under SVPWM Strategy

3.1. SVPWM The topology of USMC is shown in Figure1, which is divided into two stages, the rectiﬁer stage and the inverter stage. There are three phases in rectiﬁer stage, each of which consists of one unidirectional IGBT and four diodes. The inverter stage has the same structure as the traditional two-level voltage source inverter. Energies 2018, 11, x FOR PEER REVIEW 3 of 13

+∞  [AkxBkxkk00 cos( )+ 00 cos( )]+ k=1 A +∞ 000  gxyz( , , )= + [ A00pp cos( py )+ B 00 cos( py )]+ 2 p=1 +∞ [cos(z)+cos(z)]AqBq  00qq 00 q=1

+∞ +∞ ++ [cos()+sin()]AkxpyBkxpykp00 kp + kp==−∞1 (3) p≠0 +∞ +∞ ++ + [A0pq cos()+sin()]py qz B0pq py qz + pq=1 =−∞ q≠0 +∞ +∞ ++ [cos()+sin()]AkxqzBkxqzkq00 kq kq==−∞1 q≠0 +∞ +∞ +∞ ()++ Akxpyqzkpq cos + +    ()++ kp==−∞=−∞1 q Bkxpyqzkpq sin pq≠≠00

In Equation (3) harmonic components are divided into four parts: the first part is the DC component (k = 0, p = 0, q = 0); the second part is the fundamental voltage (k = 0, p = 1, q = 0) and harmonic components only associated with the carrier frequency (when k = 1, 2, 3, …, p = 0, q = 0), the input frequency(when k = 0, p = 0, q = 1, 2, 3, …) or the output frequency(k = 0, p = 2, 3 …, q = 0) respectively; the third part is the harmonic components related to any two of the three frequencies(input, output, and carrier frequencies), and their frequencies are the sum of the integral multiple of the two frequencies; the fourth part is the harmonic components relevant to the three frequencies, and their frequencies are sums of the integral multiples of the three frequencies. According to the Fourier analysis theory, the frequency of any harmonic is kfc ± pfout ± qfin and the harmonic amplitude Ukpq satisfies the following relationship with Fkpq:

Ukpq=2 F kpq (4)

3. Harmonic Calculation of Output Voltage of USMC Under SVPWM Strategy

3.1. SVPWM The topology of USMC is shown in Figure 1, which is divided into two stages, the rectifier stage and the inverter stage. There are three phases in rectifier stage, each of which consists of one unidirectional IGBT and four diodes. The inverter stage has the same structure as the traditional two-levelEnergies 2018 voltage, 11, 390 source inverter. 4 of 13

Energies 2018, 11, x FOR PEER REVIEW 4 of 13

The SVPWM strategy of rectifier stage is shown in Figure 2a, in which the six effective vectors I1–I6 divide the space into six sectors. Taking I1(a, c) as an example, “a” represents that Sa and Sc are

on-state, while Sb is off-state. In each sector, there is always one phase remaining the maximum current amplitude in theFigure Figurerectifier 1. 1. Typology Typologystage. In of of orderUltra Ultra Sparse Sparseto obtain matrix matrix the converter. converter. maximum voltage utilization, the switching device corresponding to maximum current amplitude is kept open, and the other two switchesThe SVPWMare switched strategy on alternately. of rectiﬁer stageThe duty is shown cycles in of Figure the two2a, adjacent in which effective the six effective vectors required vectors Ito1– Isynthesize6 divide the the space reference into six input sectors. current Taking vectorI1(a, Iref c) can as anbe example,obtained by “a” the represents following that equation: Sa and Sc are on-state, while Sb is off-state. In each sector, there is always one phase remaining the maximum current   πππ−−() −  amplitude in the rectiﬁer stage. In order sin to obtain zk the+1 maximumin  voltage utilization, the switching   336  device corresponding to maximum currentdm = amplitude is kept open, and the other two switches are  −−π switched on alternately. The duty cycles of thecos twozk adjacent (in 1) effective vectors required to synthesize  3 the reference input current vector Iref can be obtained by the following equation: (5) ππ  sin zk+1−−()  π π inπ sin[ 36−z−3 6 + 3 (kin−1)] ddnm= = π  cos[zπ− (kin−1)]  coszk−− (3 1) (5) sinz+ π − π (ink −1)  [6 3 3 in ]  dn = π cos[z− 3 (kin−1)] where kin is the sector number of the reference input current vector. whereWhenkin is theIref is sector in the number first sector, of the the reference expression input of current average vector. DC voltage udc in a carrier cycle is as follows:When Iref is in the first sector, the expression of average DC voltage udc in a carrier cycle is as follows:

3U3Uimim udc ududu= d=muab ++=dnuac = (6)(6) dcm ab n ac 2cos2 cosz z √ √ wherewhereu uabab andand uuacac are input line voltage, uab == 3Uimimcos(zcos(z + + ππ/6),/6), uuacac = = 33UUimimcos(zcos(z − −π/6);π/6); UimU isim theis theamplitude amplitude of input of input phase phase voltage. voltage.

FigureFigure 2.2. TheThe spacespace vectorvector modulationmodulation strategystrategy ofof UltraUltra SparseSparse matrixmatrix converterconverter ((aa)) rectiﬁerrectifier stagestage (b(b)) inverter inverter stage. stage.

Similarly, when Iref is in other sectors, the values of udc and the duty cycles da, db and dc of the Similarly, when Iref is in other sectors, the values of udc and the duty cycles da, db and dc of the switches Sa, Sb and Sc can be obtained as shown in Table 1. switches Sa,Sb and Sc can be obtained as shown in Table1.

Table 1. The expression of da, db, and dc when Iref in diffirent vectors.

z kin da db dc udc [−π/6,π/6) 1 1 dm dn 3Uim/2cosz [π/6,π/2) 2 dm dn 1 −3Uim/2cos(z + 2π/3) [π/2,5π/6) 3 dn 1 dm 3Uim/2cos(z − 2π/3) [5π/6,7π/6) 4 1 dm dn −3Uim/2cosz [7π/6,3π/2) 5 dm dn 1 3Uim/2cos(z + 2π/3) [3π/2,11π/6) 6 dn 1 dm −3Uim/2cos(z − 2π/3)

Energies 2018, 11, 390 5 of 13

Table 1. The expression of da, db, and dc when Iref in difﬁrent vectors.

z kin da db dc udc

[−π/6,π/6) 1 1 dm dn 3Uim/2cosz [π/6,π/2) 2 dm dn 1 −3Uim/2cos(z + 2π/3) [π/2,5π/6) 3 dn 1 dm 3Uim/2cos(z − 2π/3) [5π/6,7π/6) 4 1 dm dn −3Uim/2cosz [7π/6,3π/2) 5 dm dn 1 3Uim/2cos(z + 2π/3) [3π/2,11π/6) 6 dn 1 dm −3Uim/2cos(z − 2π/3)

Figure2b shows the SVPWM strategy of inverter stage, in which the six effective vectors V1–V6 divide the space vector into six sectors. Taking V1(1, 0, 0) as an example, “1” means that SpA is on-state in the upper bridge arm of phase A; and the second and third bits “0” represent that SNB and SNC are turned on in the lower bridge arm of phase B and phase C respectively. The duty cycles of the two adjacent effective vectors and zero vectors required to synthesize the reference output voltage vector Vref can be obtained by the following equation:  √ = 3Uref π − + π ( − )  d1 u sin 3 y 3 kout 1  √ dc 3Uref π (7) d2 = sin y − (kout − 1)  udc 3   d0 = 1 − d1 − d2 where kout is the sector number of the reference output voltage vector, Uref is the amplitude of output phase voltage. The duty cycles of SPµ (µ = A, B, C) are deﬁned as dPµ. When Vref is located in different sectors, the values of dPµ are shown in Table2. Since the conducting state of upper and lower bridge arm switches in the same phase of A, B and C are complementary, the duty cycles of SPµ and SNµ satisfy the following relationship: dNµ = 1 − dPµ.

Table 2. The expression of dPµ when Vref is in different vectors.

y kout dPA dPB dPC

[0,π/3) 1 (1 + d1 + d2)/2 (1 − d1 + d2)/2 (1 − d1 − d2)/2 [π/3,2π/3) 2 (1 + d1 − d2)/2 (1 + d1 + d2)/2 (1 − d1 − d2)/2 [2π/3,π) 3 (1 − d1 − d2)/2 (1 + d1 + d2)/2 (1 − d1 + d2)/2 [π,4π/3) 4 (1 − d1 − d2)/2 (1 + d1 − d2)/2 (1 + d1 + d2)/2 [4π/3,5π/3) 5 (1 − d1 + d2)/2 (1 − d1 − d2)/2 (1 + d1 + d2)/2 [5π/3,2π) 6 (1 + d1 + d2)/2 (1 − d1 − d2)/2 (1 + d1 − d2)/2

3.2. Harmonic Calculation of Output Voltage Since the three-phase output voltages of USMC are symmetrical, the following calculation and analysis take Phase A as an example. It can be seen from Equation (4) that the amplitude corresponding to each harmonic can be determined by the value of Fourier coefficients. Considering the SVPWM process, the integration limit can be redefined without altering the analytical solution, requiring only cyclic variation of x, y and z over 2π intervals. So the expression of FA_kpq in Equation (2) can be defined as follows: 11π π π 1 Z 6 Z 2 Z F = u e−j(kx+py+qz)dxdydz (8) A_kpq π3 −π A 8 6 0 −π

In the operation process of USMC, uA is actually a pulse wave. Therefore, to solve Equation (8), it is necessary to determine the trip point of the output voltage pulse and the value of uA. The following is the process to solve various trip points of output voltage pulse based on the analysis of SVPWM of USMC. Energies 2018, 11, 390 6 of 13

When both the reference input current vector Iref and output voltage vector Vref are in the ﬁrst sector, the conduction state of each switch in one carrier period and instantaneous value of A-phase output voltage can be determined based on the switching sequence of the rectiﬁcation stage and the inverterEnergies 2018 stage,, 11, x as FOR shown PEER inREVIEW Figure 3. 6 of 13

Figure 3. Modulation process when both Iref and Vref are in the first sector. Figure 3. Modulation process when both Iref and Vref are in the ﬁrst sector.

It should be noted that this paper is aimed at symmetrical regularly sampled SVPWM. It should be noted that this paper is aimed at symmetrical regularly sampled SVPWM. The carrier The carrier function c(x) used in this paper can be expressed as: function c(x) used in this paper can be expressed as: 1 cx( )=⋅1 arccos(cos x ) , x ∈−[] π, π (9) c(x) = π· arccos(cos x) , x ∈ [−π, π] (9) π Combine the duty cycle of each switch with Equation (9), and α1, α2 and α3 in Figure 3 can be Combine the duty cycle of each switch with Equation (9), and α , α and α in Figure3 can be determined as follows: 1 2 3 determined as follows:  α1 = ==−dNAdn = (1 − dPA)dnπ  α1NAndd(1 d PAn ) dπ α = d π (10) 2α =nd π (10)  =2n( + )π α3 =+d()n dPAdm α3nPAmdddπ When the reference vectors of input current and output voltage are located in other sectors, the valuesWhen of α1 ,theα2 referenceand α3 can vectors be analyzed of input in current a similar and way. ou Ittput is foundvoltage that are the located expressions in other of sectors,α1, α2 and the 1 2 3 1 2 αvalues3 are the of sameα , α whenand αI refcanand be Vanalyzedref are in in different a similar sectors, way. exceptIt is found that that the speciﬁc the expressions values of ofdPA α, ,d αm andand 3 ref ref PA m dαn areare changedthe same with whenkin Ior kandout. V are in different sectors, except that the specific values of d , d and dBasedn are changed on the above with analysis,kin or kout. Equation (8) can be further simpliﬁed as follows: Based on the above analysis, Equation (8) can be further simplified as follows: FA_kpq ==+FA1 + FA2 (11) FA_kpq FF A1 A2 (11) where where 6 Z z Z y 1 f f −j(py+qz) FA1 = D1 · e dydz (12) π3 ∑ ∑ 6 8 1 zr zyffyr −+j()py qz F =⋅kin=1,3,5 kout=1 Dyzedd A13 zy 1 (12) 8π kk=1,3,5 =1 rr in6 out Z z Z y 1 f f −j(py+qz) FA2 = D2 · e dydz (13) 3 ∑ ∑ 6 zy −+() 8π 1 zr ffyr j py qz F =⋅kin=2,4,6 kout=1 Dyzedd A23 zy 2 (13) 8π kk= 2,4,6 =1 rr where in out − − D = R α3 u e−jkxdx + R α1 u e−jkxdx + R α1 u e−jkxdx where 1 −π A1 −α3 A2 −α1 A3 π (14) +R α3 u e−jkxdx + R u e−jkxdx α A4--ααα311−−−α A5 Du1 =edededjjjkx xu++3 kx xu kx x 1A1−−− A2 A3 π αα31 α π (14) ++3 −−jjkx kx uxuxA4 ed A5 ed αα13

EnergiesEnergies 20182018, 11, 11x FOR, 390 PEER REVIEW 7 of7 13 of 13

---α321 jjjkxkxkx α α − − − Duxuxux2A1A2A3R α3 eded+ed−jkx R α2 −jkx R α1 −jkx D = πu e dx + α32u e dαx + u e dx 2 −π A1 −α3 A2 −α2 A3 α α α R α1 123 − jjjkxkxkx R α2 − R α3 − + eded+edu uxuxuxe jkxdx + u e jkxdx + u e j kxdx (15) −α1A4A4A5A6 α1 A5 α2 A6 (15) α112 α α R π −jkx + uπA7e jdkxx α3 ed uxA7 α3 The integration limits zr and zf in Equations (12) and (13), and the values of uAλ (λ = 1–7) in EquationsThe integration (14) and (15)limits are zr related and zf toin the Equations sector number (12) and of reference(13), and inputthe values current of as u shownAλ (λ = in1– Table7) in3 .; Equationsyr and y f(14in) Equationsand (15) are (12) related and (13) to theare relatedsector number to the sector of reference number input of reference current output as shown voltage, in Tableas shown 3.; yr and in Table yf in4 .Equations (12) and (13) are related to the sector number of reference output voltage, as shown in Table 4. Table 3. Value of uAλ, zr and zf when kin changes from 1 to 6.

Table 3. Value of uAλ, zr and zf when kin changes from 1 to 6. kin zr zf uA1 uA2 uA3 uA4 uA5 uA6 uA7 kin zr zf uA1 uA2 uA3 uA4 uA5 uA6 uA7 1 −π/6 π/6 ub ua uc ua ub -- b a c a b 2 1 π−/6π/6 ππ/6/2 uuc uu a u ub u ucu u-b -u a uc 3 2 π/2π/6 5ππ/2/6 uc uau b ub ua uc ubub uca uc-- π π 43 5 π/2/6 75π/6/6 uac ubu b ua uc ub uauc u-c -u b ua 5 7π/6 3π/2 ua uc ub uc ua -- 4 5π/6 7π/6 ua ub uc ua uc ub ua 6 3π/2 11π/6 ub uc ua ub ua uc ub 5 7π/6 3π/2 ua uc ub uc ua - - 6 3π/2 11π/6 ub uc ua ub ua uc ub Table 4. Value of yr and yf when kout change from 1 to 6.

Table 4. Value of yr and yf when kout change from 1 to 6. kout 1 2 3 4 5 6

kyrout 1 0 2 π/33 2π/3 4 π 5 4π/36 5π/3 yyf r 0π /3π/3 2π /32π/3 π π 4π/34π/3 5π5π/3/3 2π yf π/3 2π/3 π 4π/3 5π/3 2π

Based on Equations (11) and (15), the analytical solution of FA_kpq can be obtained by further calculationBased on in Equations MATLAB; (11) then and the (15), amplitude the analytical of each harmonicsolution of can FA be_kpq got. can be obtained by further calculation in MATLAB; then the amplitude of each harmonic can be got. 4. Spectrum Analysis of Harmonic Characteristics of Output Voltage of USMC 4. Spectrum Analysis of Harmonic Characteristics of Output Voltage of USMC When the voltage transmission ratio m is 0.5 (m is the ratio of Uref/Uim), the input frequency f inWhenis 50 Hz,the thevoltage output transmission frequency ratiof out ism 70is 0.5 Hz(m and is the the ratio carrier of frequencyUref/Uim), thef c isinput 5 kHz, frequency the analytical fin is 50 spectrumsHz, the output of output frequency phase voltage fout is u 70A andHz line and voltage the carrieruAB can frequency be obtained fc is by 5 thekHz, calculation the analytical formula spectrumsin Section of3 .output The spectrums phase voltage are shown uA and in Figure line 4 voltage, where uABh Acanand behAB obtainedare normalized by the calculationvalues based formulaon the in fundamental Section 3. The amplitude spectrums of phaseare shown voltage in Figure and line 4, voltagewhere h respectively.A and hAB are It normalized can be seen value thats the basedmain on harmonics the fundamental are concentrated amplitude around of phase the fundamentalvoltage and line (k = 0)voltage and 1 respectively. to 4 times of carrierIt can be frequency. seen thatEnlarge the main part harmonics of Figure4 are and concentrated mark the main around harmonic the fundamental frequency, as (k shown = 0) and in 1 Figure to 4 times5. of carrier frequency. Enlarge part of Figure 4 and mark the main harmonic frequency, as shown in Figure 5. 100 100 80 80

60 B 60

A h 40 4h 0 20 20

00 5 10 15 20 25 00 5 10 15 20 25 Frequency(kHz) Frequency(kHz) (a) (b)

FigureFigure 4. 4.TheoreticalTheoretical spectral spectral of output of output voltage voltage (m (=m 0.5=, 0.5,fc = f5c =kHz 5 kHz,, fin = f50in =Hz 50, fout Hz, = f7out0 Hz)= 70 (a Hz)) spectr(a) spectrumum of uA ( ofb)u spectrA (b)um spectrum of uAB. of uAB.

Energies 2018, 11, 390 8 of 13 Energies 2018, 11, x FOR PEER REVIEW 8 of 13

100 100 0.07 80 80 0.07

60 0.15 B 60

A h 40 h 40 20 0.21 20 00 0.1 0.2 0.3 0.4 0.5 00 0.1 0.2 0.3 0.4 0.5 100 100 80 80

A 60 4.7 5.3 60

h A 40 4.4 5.7 h 40 20 3.84.1 5.96.2 20 3.5 4.85 5.15 6.5 0 0 3.5 4.0 4.5 5.05.5 6.0 6.5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 100 100 10.0

80 80 A

60 B 60

h A 40 9.7 9.9310.07 10.3 h 40 9.9310.07 20 9.77 10.23 20 9.77 10.33 9.86 10.1410.37 9.63 9.86 10.1410.37 09.63 0 9.6 9.8 10.0 10.2 10.4 9.6 9.8 10.0 10.2 10.4 100 100

80 80 B

A 60 60

A h 4014.63 h 40 15.3 14.63 15.37 20 14.77 15.23 20 14.77 15.23 14.7 15.37 0 0 14.6 14.8 15.0 15.2 15.4 14.6 14.8 15.0 15.2 15.4 100 100

80 80 B

A 60 60

A h 40 20.0 h 40 19.63 20.37 19.63 20.37 20 19.77 20.23 20 19.77 20.23 0 0 19.6 19.8 20.0 20.2 20.4 19.6 19.8 20.0 20.2 20.4 Frequency(kHz) Frequency(kHz) (a) (b)

FFigureigure 5 5.. TheThe partial partial enlargement enlargement of of Figure Figure 44 (aa) )spectral spectral of of uuAA (b(b) spectral) spectral of of uABuAB. .

FromFrom Figure Figure 5,5 ,it it can can be be seen seen that that the the main main components components of of low low-frequency-frequency harmonics harmonics in in output output phase voltage are 3fin and 3fout. The amplitude of 3fin is higher, which is up to 0.51 when m = 0.5. phase voltage are 3f in and 3f out. The amplitude of 3f in is higher, which is up to 0.51 when m = 0.5. TheseThese two two components components belong belong to to the the common common-mode-mode voltage voltage,, so so the theyy are are not not contained contained in in line line voltagevoltage spectrum. spectrum. WithWith respect respect to high-frequency high-frequency harmonics, harmonics, they they cluster cluster around around an integer an integer multiple multiple of the of carrier the carrierfrequency frequency and are and symmetrical are symmetrical about about the carrierthe carrier frequency. frequency. Harmonics Harmonics with with high high amplitudes amplitudes in inFigure Figure5 are5 are listed listed in Tablein Table5. The 5. frequencyThe frequency expression expression of each of harmoniceach harmon in theic tablein the shows table theshows relation the relationbetween between the frequency the frequency of main of harmonics main harmonics and the and frequencies the frequencies of input, of input, output output and carrier and carrier signals. signalsComparing. Comparing the spectrum the spectrum of phase of voltage phase with voltage line withvoltage, line it voltage, is found it that is exceptfound thethatharmonics except the of harmonics of 3fin and 3fout, there are other harmonic components which also belong to the common 3f in and 3f out, there are other harmonic components which also belong to the common mode voltage, mode voltage, as marked gray in Table 5. The amplitude of 2fc is the highest among all the harmonic as marked gray in Table5. The amplitude of 2 f c is the highest among all the harmonic components, components, up to 0.93 under the condition of m = 0.5, followed by the component of fc ± 6fin with up to 0.93 under the condition of m = 0.5, followed by the component of f c ± 6f in with the amplitude theof amplitude 0.53. According of 0.53. to literatureAccording [ 18to– 20literature], high-frequency [18–20], high common-mode-frequency common voltages-mode are the voltages main cause are theof main bearing cause damage, of bearing winding damage, fault, winding electromagnetic fault, electromagnetic interference. Theinterference. above analysis The above can accurately analysis can accurately describe the amplitude of these common mode voltages, thus providing reference for the suppression of common mode voltage.

Energies 2018, 11, 390 9 of 13

Energies 2018, 11, x FOR PEER REVIEW 9 of 13 describe the amplitude of these common mode voltages, thus providing reference for the suppression of common mode voltage.Table 5. The main harmonic components in high-frequency bands.

Table 5. The mainf (kHz) harmonic components inf high-frequencyhA (%) bands. 3.5/6.5 fc ± 30fin 9.69

f (kHz)3.8/6.2 fc ±f 24fin 12.74 hA (%) 3.5/6.54.1/5.9 f c f±c ±30 18f infin 17.669.69 3.8/6.2 f ± 24f 12.74 4.4/5.6 c fc ± 12infin 27.19 4.1/5.9 f c ± 18f in 17.66 4.7/5.3 fc ± 6fin 53.29 4.4/5.6 f c ± 12f in 27.19 c in 4.7/5.34.85/5.15 f c f± ±6 3f inf 9.7053.29 9.63/9.77/10.23/10.374.85/5.15 2fcc ±± fout3f in± 6fin 15.919.70 9.63/9.77/10.23/10.379.7/10.3 2f c ±2ffoutc ± ±6fin6 f in 28.0815.91 9.7/10.3 2f c ± 6f in 28.08 9.86/10.14 2fc ± 2fout 11.31 9.86/10.14 2f c ± 2f out 11.31 9.93/10.17 2fc ± fout 28.89 9.93/10.17 2f c ± f out 28.89 10.010.0 2f2cfc 93.3293.32 14.63/14.77/15.23/15.3714.63/14.77/15.23/15.37 3f c3±fc ±f out fout± ± 66ffinin 22.9822.98 14.7/15.3 3f ± 6f 22.50 14.7/15.3 3c fc ± 6infin 22.50 19.63//19.77/20.23/20.37 4f c ± f out ± 6f in 18.21 19.63//19.77/20.23/20.37 4fc ± fout ± 6fin 18.21 20.0 4f c 31.13 20.0 4fc 31.13

According to Equations (11) to (15), the amplitudeamplitude of each harmonic is related to the voltage transmission ratioratio mm.. When Whenm mincreases increases from from 0.1 0.1 to 0.8,to 0.8, the relationshipthe relationship between between amplitudes amplitudes of main of harmonicsmain harmonics and m andcan bem obtained,can be obtained, as shown as in shown Figure 6in. The Figure harmonics 6. The inharmonics Figure6a arein Figure components 6a are exceptcomponents for the except common for the mode common voltage mode in the voltage output in phasethe output voltage. phase It canvoltage. be seen It can that be except seen that the harmonicexcept the of harmonic 2f c ± 2f outof most2fc ± harmonics’2fout most harmonics’ amplitudes amplitudes decrease when decreasem increase. when m The increase. harmonics The inharmonics Figure6b in are Figure common 6b are mode common components. mode components. When m ≤When0.5, them ≤ amplitudes0.5, the amplitudes of 2f c, f ofc ± 2f6c,f finc ±, 36ffin, and3fin and 3f c ±3fc 6±f in6f,in are, are much much higher higher than than those those of of other other harmonics. harmonics. Besides,Besides, theirtheir amplitudesamplitudes decreasedecrease rapidly withwith the the increase increase of mof. Whenm. Whenm > 0.5,m > the0.5, amplitudes the amplitudes of harmonics of harmonics in Figure in6b Figure are gradually 6b are approaching,gradually approaching, but still decrease but still with decrease the increase with the of increasem. of m.

Figure 6. TheThe relationship relationship between between main main harmonic harmonic co componentsmponents and and m min inoutput output phase phase voltage voltage (a) (componentsa) components other other than than common common mode mode voltage voltage (b) (commonb) common mode mode components. components.

From the above analysis, it can be seen that the amplitudes of high-frequency common mode From the above analysis, it can be seen that the amplitudes of high-frequency common mode components are much higher than those of other harmonics, especially when m is low. Besides, the components are much higher than those of other harmonics, especially when m is low. Besides, the harmonics’ amplitudes are affected by the value of m. Most harmonics’ amplitudes decrease gradually harmonics’ amplitudes are affected by the value of m. Most harmonics’ amplitudes decrease gradually with the increase of m, but a few increase. with the increase of m, but a few increase.

EnergiesEnergies 20182018, 11, 11, x, 390FOR PEER REVIEW 1010 of of 13 13

5. Comparative Analysis with FFT 5. Comparative Analysis with FFT The spectrums of the experimental waveforms cannot be obtained based on the triple Fourier series Theby the spectrums existing ofmeasurement the experimental techniques waveforms and an cannotalysis tools. be obtained FFT is basedusually on used the tripleto deal Fourier with theseries experimental by the existing waveforms. measurement Since it techniques is very sensit andive analysis to thetools. periodicity FFT is and usually frequency used to resolution deal with theof theexperimental waveforms, waveforms. when the frequencies Since it is very of input, sensitive output to the and periodicity carrier signals and frequency are all integers, resolution and of the the timewaveforms, interval over when which the frequencies FFT is conducted of input, as outputan integer and multiple carrier signals of the period are all integers,of these three and thesignals, time theinterval results over obtained which FFTby the is conductedFFT analysis as anare integer accurate. multiple However, of the when period the of thesevalues three of these signals, three the frequenciesresults obtained are special by the (for FFT example, analysis arethe accurate. multipliciti However,es of the when three the frequencies’ values of these reciprocals three frequencies are large orare the special three (forfrequencies example, include the multiplicities non-integer), of the it threeis difficult frequencies’ to ensure reciprocals that the are time large is oran theinteger three multiplefrequencies of the include carrier non-integer), cycle, input itand is difﬁcult output tovoltage ensure cycle. that theUnder time this is ancondition, integer multiplethe harmonic of the spectrumscarrier cycle, obtained input andby FFT output are not voltage accu cycle.rate. The Under following this condition, are the examples. the harmonic spectrums obtained by FFTThe arespectrums not accurate. obtained The following by FFT areon thethe examples. experimental waveforms of output voltage are comparedThe spectrumswith those obtained analytical by results FFT on based the experimental on the method waveforms proposed of in output this paper. voltage Figure are compared 7 shows thewith USMC those experimental analytical results system. based When on thethe methodamplitude proposed of input in phase this paper. voltage Figure is 427 V shows and the the voltage USMC transmissionexperimental ratio system. m is When0.5, fin the= 50 amplitude Hz, fout = of70 inputHz, fc phase= 5 kHz. voltage The time is 42 domain V and the waveforms voltage transmission of output voltageratio m andis 0.5, theirf in =FFT 50 Hz,resultsf out are= 70 shown Hz, f c in= 5Figure kHz. The8, where time domainFigure 8a waveforms denotes the of outputwaveform voltage of phaseand their voltage FFT u resultsA and its are FFT shown results in Figure and Figure8, where 8b Figuredenotes8a the denotes waveform the waveform of line voltage of phase uAB voltageand its FFTuA andresults. its FFTThe results amplitudes and Figure of all8 bthe denotes harmonic the waveformcomponents of linein Figure voltage 8 areuAB theand nominal its FFT results.value basedThe amplitudeson the fundamental of all the amplitude. harmonic componentsThe FFT result ins Figure in Figure8 are 8 thecuts nominal out a section value of based the output on the voltagefundamental waveform amplitude. for seven The output FFT results voltage in Figureperiods8 cuts with out a frequency a section of resolution the output of voltage 10 Hz. waveformFrom the magnitudesfor seven output of fin, voltagefout, and periodsfc, the time with corresponding a frequency resolution to seven ofoutput 10 Hz. voltage From periods the magnitudes is an integer of f in , multiplef out, and off c ,the the carrier, time corresponding input and output to seven period output so the voltage FFT periodsresults are is an accurate. integer multipleThe main of harmonic the carrier, componentsinput and output in Figure period 8 are so the listed FFT in results Table are 6 accurate.and compared The main with harmonic the results components obtained infrom Figure the8 analyticalare listed incalculation. Table6 and It comparedcan be seen with that the the results frequencies obtained of from primary the analytical harmonics calculation. in Figure It can8 are be consistentseen that thewith frequencies the analytical of primary results harmonicsbut the amplitud in Figurees 8of are them consistent are slightly with thedifferent. analytical This results is due but to thethe existence amplitudes of the of them switching are slightly dead-time, different. the conduction This is due tovoltage the existence drop of ofthe the switching switching device dead-time, and thethe ripple conduction of the voltageinput voltage drop of at the line switching side. The device largest and error the between ripple of the the measured input voltage spectrum at line and side. theThe theoretical largest error results between is only the 3.06%. measured spectrum and the theoretical results is only 3.06%.

FigureFigure 7. 7. ExperimentalExperimental system system of of ultra-sparse ultra-sparse matrix matrix converter converter (USMC). (USMC).

Energies 2018, 11, 390 11 of 13 Energies 2018, 11, x FOR PEER REVIEW 11 of 13 (50v/div) A u A h (50v/div) AB u AB h

FigureFigure 8.8. TimeTime domaindomainwaveforms waveforms ofofoutput outputvoltage voltageand and their their FFT FFT results results ( a(a) )u uAAand and its its FFT FFT ( b(b))u uABAB andand itsits FFT.FFT.

TableTable 6.6. TheThe mainmain harmonicharmonic componentscomponents inin FigureFigure8 .8. FFT Results Analytical Results FFT Results Analytical Results f (kHz) hA (%) f (kHz) hA (%) f (kHz)0.15 h49.54A (%) f 0.15(kHz) h51.51A (%) 0.150.21 49.5420.73 0.21 0.15 21.51 51.51 0.213.5/6.5 20.739.56 3.5/6.5 0.21 9.69 21.51 3.5/6.5 9.56 3.5/6.5 9.69 3.8/6.23.8/6.2 11.69 3.8/6.2 3.8/6.2 12.74 12.74 4.1/5.94.1/5.9 18.18 4.1/5.9 4.1/5.9 17.66 17.66 4.4/5.64.4/5.6 25.72 4.4/5.6 4.4/5.6 27.19 27.19 4.7/5.34.7/5.3 51.07 4.7/5.3 4.7/5.3 53.29 53.29 4.85/5.15 10.17 4.85/5.15 9.70 9.63/9.77/10.23/10.774.85/5.15 14.5010.17 9.63/9.77/10.23/10.774.85/5.15 9.7015.90 9.63/9.77/10.23/10.779.7/10.3 30.4014.50 9.63/9.77/10.23/10.77 9.7/10.3 15.90 28.06 9.86/10.149.7/10.3 12.1430.40 9.86/10.149.7/10.3 28.06 11.31 9.93/10.079.86/10.14 27.4512.14 9.86/10.14 9.93/10.07 11.31 28.89 10.0 90.26 10.0 93.32 9.93/10.07 27.45 9.93/10.07 28.89 14.63/14.77/15.23/15.37 23.07 14.63/14.77/15.23/15.37 22.98 14.7/15.310.0 24.3690.26 14.7/15.310.0 93.32 22.50 19.63/19.77/20.23/20.3714.63/14.77/15.23/15.37 19.0723.07 14.63/14.77/15.23/15.3719.63/19.77/20.23/20.37 22.9818.21 20.0014.7/15.3 28.2024.36 14.7/15.3 20.00 22.50 31.13 19.63/19.77/20.23/20.37 19.07 19.63/19.77/20.23/20.37 18.21

When f in = 50 Hz, f out20.00= 70.5 Hz, f c = 528.20 kHz, m = 0.5, the20.00 output voltage harmonic31.13 spectrums are shown in Figure9. Under this condition, at least 141 periods of output waveform has to be intercepted to ensureWhen the fin accuracy= 50 Hz, offout FFT= 70.5 results. Hz, fc But= 5 thekHz, amount m = 0.5, of the data output corresponding voltage harmonic to 141 output spectrums periods are isshown too large in Figure to be processed. 9. Under Thethis FFTcondition, results inat Figureleast 1419 intercepted periods of a sectionoutput ofwaveform the output has voltage to be waveformintercepted for to seven ensure output the accuracy voltage periods of FFT withresult thes. frequencyBut the amount resolution of data of 10.07 corresponding Hz, it can be to seen 141 thatoutput there periods exist large is too errors large between to be processed. the FFT analysis The FFT and results the analytical in Figure results.9 intercepted However, a section the analytic of the calculationoutput voltage method waveform based onfor tripleseven Fourier output seriesvoltage in periods this paper with is notthe affectedfrequency by resolution the magnitude of 10.07 of Hz, it can be seen that there exist large errors between the FFT analysis and the analytical results. However, the analytic calculation method based on triple Fourier series in this paper is not affected

Energies 2018, 11, 390 12 of 13

Energies 2018, 11, x FOR PEER REVIEW 12 of 13 input, output and carrier frequencies. The analytical results can accurately determine the amplitudes determineand frequency the amplitudes of each harmonic and frequency component of each regardless harmonic of component the magnitude regardless of the of three the frequencies.magnitude ofIn the practical three frequencies applications,. In thepractical carrier applicati frequencyons, andthe carrier the output frequency frequency and the of USMCoutput frequency usually vary of USaccordingMC usually to application vary according requirements. to application Therefore, requirements the method. Therefore, proposed the inmethod this paper proposed has a in wider this

paperscope has of application a wider scope than of FFT. application than FFT.

)

0 v

0 5 0

( 0

(

u u t(5ms/div) t(5ms/div) 100 100 FFT results FFFTF理T 理re理su理lts 80 80

60 B 60

A h h 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 Frequency(kHz) Frequency(kHz) 100 100 Theoretical calculation results Theoretical calculation results 80 理理理理理理 80

60 B 60

A h h 40 40 理理理理理理 20 20 00 5 10 15 20 25 0 0 5 10 15 20 25 Frequency(kHz) Frequency(kHz) (a) (b)

Figure 9. Time domain waveforms of output voltage and their spectrum (m = 0.5, f in = 50 Hz, Figure 9. Time domain waveforms of output voltage and their spectrum (m = 0.5, fin = 50 Hz, fout = 70.5 f out = 70.5 Hz, f c = 5 kHz) (a) uA and its spectrum (b) uAB and its spectrum. Hz, fc = 5 kHz) (a) uA and its spectrum (b) uAB and its spectrum. 6. Conclusions 6. Conclusions In this paper, a theoretical calculation method based on three Fourier series is proposed for USMC underIn SVPWM. this paper, Taking a theoretical input, output, calculation and carrier method frequencies based oninto three account, Fourier amplitudes series is and proposed frequencies for USMCof harmonic under components SVPWM. Taking of output input, voltage output, are and determined carrier frequencies precisely for into the account, first time. amplitudes Including bothand frequenciesthe low-frequency of harmonic harmonics components and the high-frequencyof output voltage harmonic are determined components, precisely the relation for the between first time. the Includingfrequencies both of harmonic the low-freque componentsncy harmonics and the frequencies and the high of input,-frequency output harmonic and carrier components signals can, the be relationdetermined. between Through the frequenc spectrumies analysis of harmonic in this components paper, the following and the frequencies conclusions of can input, be drawn: output and carrier signals can be determined. Through spectrum analysis in this paper, the following conclus1. Theions frequencies can be drawn: of low-frequency harmonics are triple that of the input or output frequencies. The high-frequency harmonics are organized in sidebands and symmetrical about integer 1. The frequencies of low-frequency harmonics are triple that of the input or output frequencies. multiples of carrier frequency. The high-frequency harmonics are organized in sidebands and symmetrical about integer 2. The high-frequency common-mode components are kf + 6f , kf , etc., whose amplitudes are multiples of carrier frequency. c in c high. This feature is obvious, especially when the voltage transmission m is low. 2. The high-frequency common-mode components are kfc + 6fin, kfc, etc., whose amplitudes are 3. The harmonic amplitudes are affected by the value of m, and most of which decrease gradually high. This feature is obvious, especially when the voltage transmission m is low. with the increase of m, while a few decrease. 3. The harmonic amplitudes are affected by the value of m, and most of which decrease gradually with the increase of m, while a few decrease. Acknowledgments: The authors are grateful to the financial support from the National Natural Science Acknowledgments:Foundation of China The under authors Grant are 51777135 grateful and to the the Application financial support Foundation from and the Front National Technology Natural Research Science Program of Tianjin under Grant 15JCQNJC04300. Foundation of China under Grant 51777135 and the Application Foundation and Front Technology Research ProgramAuthor Contributions: of Tianjin underTingna Grant Shi,15JCQNJC04300. Lingling Wu, and Changliang Xia put forward the idea; Lingling Wu and Yan Yan performed the experiments and analyzed the results; all authors contributed to the writing of the paper. Author Contributions: Tingna Shi, Lingling Wu, and Changliang Xia put forward the idea; Lingling Wu and Conflicts of Interest: The authors declare no conflict of interest. Yan Yan performed the experiments and analyzed the results; all authors contributed to the writing of the paper. References Conflicts of Interest: The authors declare no conflict of interest. 1. Kolar, J.W.; Schafmeister, F.; Round, S.D.; Ertl, H. Novel three-phase AC–AC sparse matrix converters. ReferencesIEEE Trans. Power Electron. 2007, 22, 1649–1661. [CrossRef]

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