Analytical Characterization of the Spectral Performance of Matrix Converters

Transcription

1 1 IEEE 7th International Power Electronics and Motion Control Conference - ECCE Asia June -5, 1, Harbin, China Analytical Characterization of the Spectral Performance of Matrix Converters Bingsen Wang and Emad Sherif Department of Electrical and Computer Engineering Michigan State University 1 Engineering Building East Lansing, MI s: bingsen, sherifem@egr.msu.edu Abstract This paper proposes an analytical method to obtain accurate spectrum of the switching functions of matrix converter. The challenges associated with the spectral analysis of matrix converter waveforms are two fold. On one hand, the modulation signal contains both the input and output frequencies. Unlike the injected harmonics in the modulation functions, the input frequency and the output frequency may not be related to each other by an integer ratio. On the other hand, it is very common that the switching frequency or the carrier frequency is not rational multiple of either the input frequency or the output frequency. These aforementioned challenges make it an almost impossible task to obtain accurate spectra of matrix converter waveforms through commonly resorted numerical methods such as fast Fourier transform (FFT). The proposed solution is an extension of the double Fourier series expansion for the pulse width modulated waveforms. I. INTRODUCTION Matrix converters continues to attract significant research attention over the last several decades [1]-[15]. Matrix converter is a direct AC to AC frequency converter which employs an array of controlled bidirectional switches as the main power elements to provide a variable AC load with unrestricted frequency [1]. Since the matrix converter does not have decoupling, consequently, the input current and output voltage are made directly from the output current and input voltage. Furthermore, the matrix converter does not require any large energy storage elements. It is well known that the matrix converter has several desirable features that include the following: 1) sinusoidal input and output current waveforms; ) generation of load voltage with variable magnitude and frequency; ) a simple and compact power circuit; 4) controlled input power factor for any load; and 5) inherent regeneration capability. The modulation of matrix converters has been one of the research areas that have attracted significant research effort. The high-frequency modulation of the matrix converter was first proposed by Alesina and Venturini, the maximum voltage transfer ratio q, which is the ratio between the magnitude of the output voltage and the magnitude of the input voltage, was limited to.5 [16]. An optimal Alesina-Venturini (OAV) method was proposed in [17]. The voltage transfer ratio q was subsequently increased to.866 by utilization of the third harmonic injection technique. The space vector modulation (SVM) approach was proposed in [18] to control only the output voltages. This strategy allows an immediate comprehension of the modulation process without the need for a fictitious dc link, and avoiding the addition of the thirdharmonic components [19]. One of the most widely utilized strategies for controlling the AC output of power electronic converters is the technique known as pulse width modulation (PWM). The PWM controller allows the use of complicated space vectors and sectorbased calculations which simplifies operation and increases reliability. The aim of PWM scheme is to create a switched pulse train that will be used to control the switching of the semiconductor devices in converters. These trains of switched pulses normally contain unwanted harmonic components in addition to the desired low-frequency component, which is typically the fundamental component. It is quite complex to accurately characterize the spectrum of PWM waveforms that are typically present in many converters. It is often resorted to a numerical approach such as fast Fourier transform (FFT) analysis for obtaining a quick result. This numerical approach does not demand significant mathematical effort. However, the obtained result from FFT is typically inaccurate unless the ratio between the fundamental frequency and switching frequency is integer and the interval used for FFT analysis has been appropriate chosen. In general case, the analytical method of determining the harmonic components of PWM such as double Fourier integral analysis has to be invoked []. This analytical solution ensures the correct harmonics magnitude when various PWM strategies are compared against each other. This paper presents a new analytical approach to the spectral analysis of matrix converters. This analytical solution is called triple Fourier integral analysis, which is an extension to the double Fourier integral analysis approach. The rest of the paper is organized as follows. In section II, the double Fourier integral analysis is reviewed for completeness. Section III presents the proposed triple Fourier integral analysis. Section IV demonstrates the application of the triple Fourier integral analysis. Section V presents the numerical verification followed by a summary in Section VI /11/$6.@1 IEEE

2 II. DOUBLE FOURIER ANALYSIS The Double Fourier Integral Analysis is utilized as an analytical approach to identify the harmonic components of carrier-based PWM. This analytical approach was originally proposed for communication system by Bennet [1] and Black [], later by Bowes and Bird [] for power converters. The sine-triangle naturally sampled modulation is considered for this analysis. The analysis process assumes the existence of two time variables x(t) and y(t) to denote the high-frequency carrier signal and low-frequency modulation function, respectively. y x x(t) = ω c t θ c y(t) = ω m t θ m (1) ω c and ω m are the angular frequencies of carrier signal and modulation function, respectively. θ c and θ m are the phase angles of carrier signal and modulation function, respectively. Without loss of generality, the carrier angle θ c and modulation function angle θ m are assumed to be zero, i.e. θ c = ; θ m = If the triangular carrier is assumed, the carrier signal c(x) can be mathematically described by () c(x) = 1 arccos(cos x) () π Furthermore, the modulation function m(y) is defined by m(y) = 1 M cos(y) The switching function f(x, y) is then determined by the comparison of modulation function against the carrier signal, as described by (4) f(x, y) = Φ [m(y) c(x)] (5) the Φ( ) is the signum function, i.e { 1 if v > Φ(v) = otherwise It can be observed from (4) that the switching function f(x, y) takes the values of 1 or for any combination of x and y. Since both x and y are functions of time t as shown in (1), the switching function f(x, y) is a function of time t as well. In addition, the function f(x, y) is periodic in x and y with the period of π, i.e. (6) f(x π, y) = f(x, y π) = f(x, y) (7) The Fourier expansion can be conducted in two steps. First, the function is expanded in x-direction while y is kept constant. If the complex form is utilized, then the following will hold. [ f(x, y) = Fm (y)e jmx] (8) m= Fig. 1. Illustration of the unit cell with the shaded area indicating the region f(x, y) takes the unity value. F m (y) = 1 π π f(x, y)e jmx dx (9) It is observed that F m (y) is periodic since f(x, y) is periodic in y-direction. π F m (y π) = 1 π = 1 π = F m (y) π f(x, y π)e jmx dx f(x, y)e jmx dx Hence, F m (y) can be further expanded. F m (y) = F mn = 1 π π (1) [ Fmn e jny] (11) F m (y)e jny dy (1) Combination of the Equations (8) and (11) yields the double Fourier series. f(x, y) = m= F mn e j(mxny) (1) F mn can be obtained by substituting (1) into (9). F mn = 1 4π π f(x, y)e j(mxny) dxdy (14) It is worth noting the integral in (14) is carried out in the unity cell as shown in Fig. 1. Furthermore, the unit cell replicates in both x and y directions due to the periodicity of c(x) and m(y) as defined in () and (4), respectively. 679

3 The real form expansion is given by the following f(x, y) = A [A n cos(ny) B n sin(ny)] n=1 [A m cos(mx) B m sin(mx)] n A mn = 1 π B mn = 1 π [A mn cos(mx ny) B mn sin(mx ny)] π π f(x, y) cos(mx ny) dx dy f(x, y) sin(mx ny) dx dy (15) The complex coefficient F mn is related to A mn and B mn as the following. F mn = A mn jb mn (16) On the right-hand side of (15), the first term represents the dc component of the switching function f(x, y). The second term defines the modulation function harmonics or baseband harmonics while the third term defines the carrier harmonics. The last term defines the side-band harmonics around the carrier harmonics []. III. TRIPLE FOURIER ANALYSIS Let a triple-variable function f(x, y, z) be periodic in both x, y, and z directions. It is further assumed that x, y and z are angular variables and the period in all directions is π, i.e. f(x, y, z) = f(xπ, y, z) = f(x, yπ, z) = f(x, y, zπ) (17) With reference to the double Fourier expansion, a triple Fourier expansion is possible. If a function f(x, y, z) is periodic in x, y, z directions with the period of π. The triple Fourier expansion can be obtained by. f(x, y, z) = F kmn = 1 8π k= m= F kmn e j(kxmynz) (18) π f(x, y, z)e j(kxmynz) dxdydz (19) The real form expansion similar to (15) is carried out as the following. f(x, y, z) = A }{{} k=1 n=1 DC Offset [A k cos(kx) B k sin(kx)] [A m cos(my) B m sin(my)] [A n cos(nz) B n sin(nz)] }{{} k=1 n=1 k=1 n=1 Base Bands [A km cos(kx my) B km sin(kx my)] [A mn cos(my nz) B mn sin(my nz)] [A kn cos(kx nz) B n sin(kx nz)] } {{ } Inter-Harmonics Between Two Independent Frequencies ] k=1 m=, m n [ Akmn cos(kx my nz) B kmn sin(kx my nz) } {{ } Inter-Harmonics Between Three Independent Frequencies () the Fourier coefficients A kmn and B kmn are determined by the following triple integral. A kmn = 1 π 4π f(x, y, z) cos(kx my nz)dx dy dz B kmn = 1 π 4π f(x, y, z) sin(kx my nz)dx dy dz (1) The real coefficients A kmn and B kmn and complex coefficient F kmn by F kmn = A kmn jb kmn () IV. APPLICATION OF TRIPLE FOURIER ANALYSIS TO PWM WAVEFORMS The modulation of ac/ac converter typically involves three different frequencies: two low-frequencies that are related the input and output frequencies, respectively, and one carrier frequency. With proper formulation of the switching function, the spectrum of the modulated PWM waveforms can be accurately characterized by the analytical approach specified in (). The critical step in obtaining the analytical result is to carry out the integrals in (19) or (1). The procedure is demonstrated using the modulation function proposed in [16] as an illustrative example. Due to space limitation, only a single switching function will be analyzed. 68

4 components in the modulated switching function. The Fourier coefficient is determined to be π π πm(y,z) F mn = 1 8π e j(mynz) dxdydz π π πm(y,z) 1 for m = n = M for m = ±1, n = = M for m =, n = ±1 for m, n (6) Fig.. Illustration of the unit cube. Within the region enclosed by the two curved surfaces, the switching function f(x, y, z) takes the unity value. Without loss of generality, the sine-triangle naturally sampled modulation is considered for this analysis. The carrier c(x) is given by (). The modulation function m(y, z) is assumed the following m(y, z) = 1 M cos y M cos z () M is the modulation index, which is also the attainable voltage transfer ratio. The phase angles y and z are further defined as functions of time t by y(t) = (ω i ω o )t z(t) = (ω i ω o )t (4) ω i and ω o of angular input frequency and output frequency, respectively. The switching function f(x, y, z) is then determined by the comparison of the modulation function given in () and the carrier signal given in (), i.e. f(x, y, z) = Φ [m(y, z) c(x)] (5) Since f(x, y, z) is periodic in x, y, z-directions, the Fourier series of f(x, y, z) will be determined by (). Moreover, Fourier coefficients are given by (1). It is worth noting that the triple integral in (1) will be conducted in the unit cube as illustrated in Fig., which is similar to the two-dimensional unit cell shown in Fig. 1 for the case of double Fourier integral analysis. The evaluation of the Fourier coefficients given the particular carrier signal in () and the modulation function in () can be carried out for the two difference cases of k = and k. Case#1: when k = Since x is the phase angle of the carrier signal and the index k is for the carrier frequency harmonics, the Fourier coefficients for the case of k = represent low-frequency It is worth noting that F mn = when m, n can be interpreted as the none existence of the interaction between the two low-frequency components in the modulation. Furthermore, if the modulation function contains higher-order harmonics, such as the triplen harmonics utilized in various third-harmonic injection methods, then these harmonics are expected to show in (6). Case#: when k The Fourier coefficients for the case of k mainly represent the harmonic components of the frequencies that are at or close to the switching frequency and its multiples in the modulated switching function. F kmn = 1 π π πm(y,z) 8π π π πm(y,z) ) ] π = sin [( k mn kπ e j(kxmynz) dxdydz J n ( kπ M ) J m ( kπ M ) (7) J n ( ) and J m ( ) are the nth- and mth-order Bessel functions of the first kind, respectively. It can be observed that F k = when k = l for l = 1,... Hence, the magnitude of the harmonic component at the triplen times switching frequency is zero. V. NUMERICAL VERIFICATION In order to verify the results presented in Sections III and IV, the numerical spectrum has been obtained through FFT. Fig. (a) illustrates the PWM switching function for one throw of the matrix converter. The switching function is generated on at the switching frequency or carrier frequency of 1 Hz. a reference frequencies of 1 Hz and 11 Hz respectively, and the voltage transfer ratio of.4. The particular choice of various frequencies allows for an accurate result from numerical FFT. The spectra that are obtained from FFT and from the proposed analytical approach are overlaid in Fig. (b), which clearly demonstrates the agreement between the analytical spectrum and the numerical spectrum. Hence, the analytical result is accurate. 681

5 Switching & Modulation Functions Spectra of Switching Function Switching Function Modulation Function t (sec) (a) Numerical Spectrum Analytical Spectrum Frequency (khz) Fig.. Illustration of (a) The time domain pulse width modulated waveform; (b) the analytical and numerical (FFT) spectra of the PWM waveform. (b) VI. CONCLUSIONS In this paper, an analytical approach for characterizing the spectrum of PWM waveform that is modulated by two independent low-frequency signals. The proposed approach extends the double Fourier analysis for ac/ac converters has been proposed. The spectrum of the PWM waveform that is obtained by the analytical approach shows very good agreement with the one obtained by the numerical method such as FFT. Once the spectrum of the switching function has been accurately characterized, the various quantities, such as the synthesized terminal voltages and currents in an ac/ac converter can be characterized accordingly. This proposed approach is not only applicable to conventional direct matrix converters. It is equally applicable to the indirect matrix converter as well. Moreover, upon the accurate knowledge of spectral contents of switching functions, the normally assumed approximations, such as the sinusoidal input voltage and output current, in analysis of matrix may be relaxed. Therefore, the characterization of the harmonic interaction the coupled input and output quantities across the matrix converter becomes feasible. REFERENCES [1] P. W. Wheeler, J. Rodriguez, J. C. Clare, L. Empringham, and A. Weinstein, Matrix converters: A technology review, IEEE Trans. on Industrial Electronics, vol. 49, no., pp , April. [] J. Rodriguez, M. Rivera, J. W. Kolar, and P. W. Wheeler, A review of control and modulation methods for matrix converters, IEEE Transactions on Industrial Electronics, vol. 59, no. 1, pp. 58-7, 1. [] S. M. Ahmed, A. Iqbal, and H. 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