A Novel 3D Space Vector PWM Control Method for 3D Magnetic Property Measurement Apparatus

Transcription

1 389 Journal of International Conference on Electrical Machines and Systems A Novel 3D Space Vector PWM Control Method for 3D Magnetic Property Measurement Apparatus Changgeng Zhang* Qingxin Yang** and Yongjian Li* Abstract This paper presents a novel three-dimension space vector pulse width modulation (3D-SVPWM) control method for the 3D magnetic property measurement apparatus. Arbitrary 3D magnetic flux density can be obtained by carrier-based PWM technique independently in each axis and vector summation of three axes. The proposed 3D-SVPWM which directly obtains 3D flux density is more efficient and has lower harmonic flux root mean square(rms) than carrier-based PWM technique. Besides, core losses in motor supplied by PWM inverter are greatly different from that supplied by ideal sinusoidal power supply. Furthermore, the magnetic material measurement supplied by PWM inverter is closer to the real engineering situation. Firstly, the tetrahedron-based and cube-based space state vectors are proposed and analyzed. Secondly, the SVPWM method is extended from 2D vector space to 3D vector space. Thirdly, the optimization switching pulse sequence for 3D- SVPWM is researched. At last, the simulation proves that the three-level 3D-SVPWM drive for 3D magnetic property measurement is efficient. Keywords: Space Vector PWM, Magnetic Measurement, Tetrahedron Triangulation, Core Loss 1. Introduction The three dimension (3D) rotational magnetic flux brings more core losses in electrical machines than alternating flux and 2D rotational flux. Alternating core losses exists in magnetic yoke of power transformer, while 2D rotational flux exists in the cores of most electrical machine and T- joins of three-phase power transformer[1].the 3D magnetic flux rotation is observed in the transverse flux permanent magnet machines such as claw-pole machine and permanent-magnetic spherical motor[2]-[3]. The core losses of electrical soft magnetic material need to be directly measured by 3D magnetic property measurement apparatus. This paper designed a 3D space vector PWM control method to obtain the arbitrary desired magnetic flux for 3D magnetic property measurement apparatus. 3D-SVPWM is also beneficial for driving permanent-magnetic spherical motor. The aim of magnetic measurement drive is to acquire the desired 3D flux density in the sample. In the 2D situation, the core losses of the magnetic sample under circular or elliptic rotational magnetization are usually measured, as the circular or elliptic rotational magnetization is easy to get[4]. But these simple cases do not arise in practice for * Province-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability, Hebei University of Technology, Tianjin, China (chang_geng@qq.com) ** Tianjin Key Laboratory of AEEET Tianjin Polytechnic University Tianjin, China the field of transformer cores. Quasi-rhombic rotational magnetization is more similar for the real one in the power transformer and electrical machine cores[5]. Furthermore, the pattern of the flux density is more complicated in the 3D situation. The traditional method driving the magnetic measurement device is to amplify the signal with the A- class power amplifier which is expensive and relative low in rated power[6]. Moreover, 3D magnetic measurement apparatus is driven by at least three power amplifiers. So robust flexible electrical drive of 3D magnetic property measurement apparatus is needed to reproduce the desired patterns of the flux density[7]. Since the PWM technique would produce richer highorder harmonics in exciting current than power amplifier, more optimized methods must be considered to reduce the current ripple[8,9]. In fact, the main benefit of SVM is the explicit identification of pulse placement as an additional degree of freedom that can be exploited to achieve harmonic performance gains[10]. Based on the space vector theory, the output current ripple RMS in five-phase drives was analyzed[11]. Iron losses in PWM inverter are larger than those in ideal sinusoidal power source and the iron losses model in PWM waveform is more complex[12,13]. And the measurement result of magnetic material supplied by PWM inverter is closer to the real engineering situation. In this paper, the traditional SVPWM method in 2D situation is extended to the 3D situation. The optimization

2 Changgeng Zhang Qingxin Yang and Yongjian Li 390 of switching pulse sequence for 3D-SVPWM is analyzed. A novel 3D-SVPWM drive is designed for 3D-measurement apparatus. 2. Three Dimension SVPWM FOR 3D MAGNETIC MEASUREMENT 2.1 The 3-level 3D-SVPWM for 3D magnetic measurement The magnetic pole is three-axis orthogonal structure, and the measurement sample is located in the center of the apparatus[14]. And the magnetic measuring coils are wound on the square sample with one layer in order to minimize the linkage flux. When all of the three axes are excited by currents, the magnetic flux density of structure is plotted in Fig.1 simulated by finite element analysis software Ansys.Maxwell. Fig. 1. Three dimension magnetic measurement apparatus Fig. 2. 3D-SVPWM inverter for three magnetic measurement apparatus The 3D-SVPWM topology for 3-level inverter is singlephase full-bridges to make up each phase leg of the main inverter, as shown in Fig.2. Each full-bridge can switch between +V dc, 0, -V dc, in which the switching signal (1010) and (0101) is corresponding to a short circuit on one phase output. Fig. 3. Voltage vectors for 3D-SVPWM inverter Due to the special orthogonal excitation structure, the space vector on different axis is independent on each other. All the 3 3 ( = 27) states can be considered to form stationary vectors which are composed of a cube of side length V dc in x-y-z space as shown in Fig The Expected Magnetic Flux Density and Voltage for 3D Magnetic Measurement Three dimensional rotational flux exists in T-joints of power transformer and the claw-pole machines. Different patterns of 3D rotating magnetic flux density are observed in T-joint of power transformer and special motor. 3D Quasi-rhombic rotational flux density exists in the real power transformer and electrical machine cores. It is a new challenge for the drive of 3D measurement apparatus to obtain various patterns of flux density. Fig.4a shows the exactly sphere B(t) loci. The changing induction frequency in the z-axis is 3Hz much slower than that in x-y plane. As the induction frequency in z-axis is growing bigger, the sphere shape is blurred as shown in Fig.(4c, 4e). In real situation, the flux density rotates as quasi-rhombi shape in x-y planes. Fig.4g shows the exactly rotational rhombi shape in x-y plane and sinusoidal periodic changing in z-axis. According to the Faraday Theorem, dφ E = N (1) dt the exciting voltage to generate the desired flux density can be computed. The voltage vectors shown in Fig. (4b, 4d, 4f) remain sinusoidal periodic changing. The voltage in Fig.4h changes irregularly in 3D space. The traditional method needs three high-bandwidth power amplifiers to acquire the three channel voltage vector dependently. And then, magnetic flux density in each axis is obtained independently and summed up to get the

3 391 A Study on The Noise and Vibration Analysis of 200kW PMSM for Electric Propulsion Ship expected vector flux density. This paper designs the drive controlled by the 3D-SVPWM algorithm to directly provide 3D space voltage vector and 3D space flux density. the 3D state space D-SVPWM based on Tetrahedron The vector state space is divided into several tetrahedrons as shown in Fig.3 using the complex mesh generation algorithm Delaunay triangulation which maximizes the minimum angle of all the angles of the Tetrahedron in the triangulation. They tend to avoid skinny Tetrahedron, which is beneficial to reduce the current ripple. When the target voltage V * entered into one tetrahedron, the subinterval switching vector state are the four vertexs of the tetrahedron. (c) (d) (e) (f) (g) (h) Fig. 4. Different types of magnetic flux density B(t) and the corresponding exciting space voltage vector U(t) B x, B y 50Hz sinusoidal periodic change and B z 3Hz sinusoidal periodic changes, (c) B x, B y 50Hz sinusoidal periodic change and B z 20Hz sinusoidal periodic changes, (d) B x, B y B z, 50Hz sinusoidal periodic change, (e) B x, B y 50Hz exactly rhombic periodic change and B z 5Hz sinusoidal periodic changes, (d)(f)(g) are the corresponding exciting voltage vector for (c)(d)(e). 3. Principal of 3D-SVPWM As the general theory of SVPWM, an arbitrary target output voltage vector V 0 * can be formed by the summation of a number of these space vectors within one switching period T/2. There are several different methods to divide Fig. 5. The expected Voltage located in the standard tetrahedron The expected Voltage located in the standard cube Firstly, both the arbitrary tetrahedron X can be transformed into the standard form ˆX by linear transformation. The original four vertex are denoted as (A x, A y, A z ), (B x, B y, B z ), (C x, C y, C z ), (D x, D y, D z ), and the standard tetrahedron vertex are (1 0 0), (0 1 0), (0 0 1) and (0 0 0) as shown in Fig.5a. X = FXˆ + d (2) where F is transformation matrix and d is displacement vector. Ax Dx Bx Dx Cx Dx F = Ay Dy By Dy Cy Dy (3) Az Dz Bz Dz Cz D z

4 Changgeng Zhang Qingxin Yang and Yongjian Li 392 D d (4) x = Dy D z Secondly, after the target voltage V * and its enclosure tetrahedron are transformed to the standard form as shown in Fig.5a, all the vertex vector states are computed ˆ * 1 ( * V = F V d ) (5) Consequently, the interval time of each switch vector state is * * * TP 1 = 1 Vx Vy Vz * TP2 = Vx (6) * TP3 = Vy * TP4 = Vz Besides, the displacement of the four state vectors is arbitrary, and the optimal displacement scheme to reduce the ripple current is discussed in the next section D-SVPWM based on Cube The procedure of 3D-SVPWM based on Cube is similar to that based on Tetrahedron. Following the natural perspective, the space is divided into 9 smaller cube of length side V dc. When the target voltage V * entered into one cube, the subinterval switching vector state are the eight vertex of the cube. And then the arbitrary cube is transformed into the standard cube which vertex are (0 0 0), (0 1 0), (1 1 0), (1 0 0), (0 0 1), (0 1 1), (1 1 1), (1 0 1) as shown in Fig.5b. The transformed target voltage ˆ * V is denoted as ( xyz ˆ, ˆ, ˆ), and the time of each state vector is computed as following P1 = (1 x)(1 y)(1 z) P2 = (1 xy ) (1 z) T ˆˆˆ P3 = xyz P4 = x(1 y)(1 z) (7) P5 = (1 x)(1 yz ) T ˆ ˆˆ P6 = (1 x) yz T ˆˆ ˆ P7 = xy(1 z) P8 = x(1 yz ) The method using the cube triangulation is easy to be programmed, but more switching times in T/2 period bring more switching losses. The proposed algorithm is compared with carrier-based PWM for 3D magnetic measurement apparatus. The most straightforward modulation strategy is naturally sampled PWM, which compares a low-frequency target reference waveform for three-axis against a high-frequency saw tooth carrier waveform. The circuit topology is three-level PWM full bridge inverter as shown in Fig.3. The expected voltages of three-axis are independent of each other and controlled independently, because the exciting axis are orthogonal to each other and the mutual inductance of the different axis windings is zero in theory. 3.3 Zero Space Vector Placement Modulation Strategies Note that the balance of the half carrier period is made up of any combination of the vertex space vectors P 0... P n. This freedom of choice allows the placement of the space vectors to be varied anywhere within the half carrier period, which is the basis of most of the various space vector modulation alternatives. The space vector state on any axis must contain redundant zero space vector placement. Furthermore, any tetrahedron (or cube) in vector space contains one or more vector point on the axis, which is transformed to P 0 in standard form. The 3D-SVPWM implementation centers the active space vectors in each half carrier period, and places the redundant zero space vector in the head and rail of each carrier period. This creates a space vector sequence for standard tetrahedron triangulation: P1 P2 P3 P4 P1 P4 P3 P2 P1 T1 T2 T3 T4 T1 T4 T3 T2 T (8) Optimization of the Switched Pulse As noted, the current ripple analysis is based on the harmonic distortion factor (HDF) approach in and utilizes the complex space vector approach [8, 9, 10], which is already used in the evaluation of the continuous and discrete PWM techniques. The harmonic flux represents the time integral of the harmonic voltage vector. This can be done by performing the following calculation over the first half of the switching period (only the first half of the switching period is necessary for analysis because of symmetry) t * λ () t = ( v v ) dt (9) 0 where v* is the expected voltage vector, v is the generated voltage vector by 3D-SVPWM, λ(t) is the harmonic flux. 4.1 Comparative analysis on triangulation Schemes The trajectories of the harmonic flux generated by numerical computation are plotted in the vector space. Figs.6 shows the harmonic flux trajectory over a sub-cycle

5 393 A Study on The Noise and Vibration Analysis of 200kW PMSM for Electric Propulsion Ship ΔT/2 for the applied vectors in the center of tetrahedron and in the center of the cube. /2 2 2 Ts 2 path rms λ 0 path Ts λ = dt (10) Figs.7 shows the harmonic flux RMS comparison between different schemes, carrier-based, tetrahedron-based and cube-based. Fig. 7. Harmonic Flux RMS along the line from the vertex to the center among different 3D-SVPWM Schemes (c) Fig. 6. Harmonic flux trajectory of center point over the first half of the switching period by the: Carrier-Based Tetrahedron-Based (c) Cube-Based 1) All the harmonic flux trajectories start at origin of the vector space and go back to the origin. 2) All the trajectories are the combinations of lines and the number of line segments are the number of the vertex, four lines for the tetrahedron and eight lines for the cube. 3) The harmonic flux trajectories is not a function of the zero space vector displacement, and therefore, is with the same characteristic for different zero space vector displacement schemes. The squared value of the harmonic flux over the switching period can be calculated as following Fig. 8. Regular tetrahedron used for optimizing the switched pulse sequence, the shadow area is the cross section of regular tetrahedron and x-o-z plane Harmonic flux RMS along different path applied by the same switching sequence P1 P2 P3 P4 where Path 1: line from P1 to center, Path 2: line from P2 to center, Path 3: line from P3 to center

6 Changgeng Zhang Qingxin Yang and Yongjian Li 394 harmonic flux RMS for all three schemes. The scheme of tetrahedron triangulation brings up smallest current ripple among the three schemes. Tetrahedron-based 3D-SVPWM has the least mean of the harmonic flux RMS, and carrier-based 3D- SVPWM has the largest mean and standard deviation. Fig. 9. Harmonic flux RMS on shadowed cross section applied by the Seq 1. The best switching sequence having the smallest harmonic flux RMS among the four schemes on the shadowed cross section Table 1. The Performance of Different Schemes Schemes Mean Std. Carrier-Based e e-02 Tetrahedron- Based e e-02 Cube-Based e e-02 Table 2. Different Schemes for Tetrahedron Triangulation 3D-SVPWM Schemes T 1 T 2 T 3 T 4 Seq 1 P 1 P 2 P 3 P 4 Seq 2 P 2 P 1 P 3 P 4 Seq 3 P 2 P 3 P 4 P 1 Seq 4 P 3 P 4 P 1 P 2 The mean and standard deviation of all the points in standard geometry are compared in the Table.\ref{tab:seq} for all the schemes. The following important facts are emphasized The line for benchmark test is selected from the origin to the center. The center of standard tetrahedron is ( ) and the center of the cube is ( ). The center of the geometry has the worst feature on 4.2 Optimization of the Switched Pulse for Tetrahedron Schemes Tetrahedron-based 3D-SVPWM is better than other schemes and is further optimized to select the sequence of the switched pulse. Regular tetrahedron is to be selected to compare different switch sequence, as regular tetrahedron shown in Fig.8 is axis symmetric to narrow the discussed point set. Fig.9 shows the simulation with the same switching sequence P1 P2 P3 P4 (11) which is applied on the points along different path. Fig.10a gives the harmonic flux RMS on the shadowed section, Fig.10b computes the optimal scheme among the four schemes shown in Tab.2. The following important facts are summarized. The expected voltage in Path 1 with the largest switching time vector at the first switching place brings up the smallest harmonic flux RMS, while the expected voltage in Path 4 with the largest switching time vector at the last switching place brings up the largest harmonic flux RMS. The point in the area closest to the point P 1 have the optimized sequence Seq1 started at P 1. Similarly, the optimized Seq2 and Seq3 are closest to the point P 2 which is the first state vector in Seq2 and Seq3. Furthermore the optimized Seq2 is second closest to the point P1 which is the second state vector in Seq2, the optimized Seq3 is second closest to the point P3 which is the second state vector in Seq3. In sum, the optimized state vector sequence is selected as the descend sequence of each vector state's switching time. 5. Simulation Results The expected vector flux density is shown in Fig.4c where B x, B y 50Hz sinusoidal periodic change and B z 20Hz sinusoidal periodic changes. And the corresponding vector voltage is shown in Fig.4d. Tetrahedron-based 3D-SVPWM and the optimized vector voltage schemes are simulated by Simulink. Fig.11 and Fig.12 show the voltage and current waveform controlled by 3D-SVPWM.

7 395 A Study on The Noise and Vibration Analysis of 200kW PMSM for Electric Propulsion Ship Yet, it can be seen that the most pronounced sideband harmonics around the 33th harmonic order shown in Fig.13. In sum, the simulation results prove the algorithm of 3D- SVPWM is efficient. been proposed to implement the 3D-SVPWM. Although, magnetic flux density in each axis are obtained independently by carrier-based PWM technique and are summed up to get the expected vector flux density. Extensive analyses in this paper have shown that tetrahedron-based 3D-SVPWM brings up the smaller current ripple than carrier-based and cube-based methods. The sequence of the switched pulse of tetrahedron-based 3D-SVPWM is optimized which brings up the smallest harmonic flux RMS. This method is easy to be extended from three-level to multi-level in order to get lower flux density THD in the sample. Acknowledgements Fig. 10. Simulation recorded voltage waveform of three axis by 3 D-SVPWM with f c /f 0 = 33. Fig. 11. Simulation recorded current waveform of three axis by 3D-SVPWM with f c /f 0 = 33. Fig.12. Simulation recorded high frequency current spectrum of X axis with f c /f 0 = Conclusion In this paper, the technique of space vector PWM has been extended from 2D to 3D. Then, 3D-SVPWM technique is applied to 3D magnetic measurement to obtain arbitrary 3D rotational magnetic flux density with low THD in the measurement sample. The tetrahedron-based and cube-based schemes have This work was supported by National Natural Science Foundation of China, under Grant and References [1] J. Sievert, "The measurement of magnetic properties of electrical sheet steel -survey on methods and situation of standards," Journal of Magnetism and Magnetic Materials, vol , no. 6, pp , [2] Y. G. Guo, J. G. Zhu, Z. W. Lin, J.J. Zhong, "Measurement and Modeling of Core Losses of Soft Magnetic Composites Under 3D Magnetic Excitations in Rotating Motors," IEEE Transactions on Magnetics, vol. 41, no. 10, pp , [3] C. L. Xia, P. Song, H. F. Li, B. Li and T. N. Shi, "Research on Torque Calculation Method of Permanenet-magnet Spherical Motor Based on the Finite-Element Method," IEEE Transactions on Magnetics, vol. 45, no. 4, pp , [4] M. Enokizono, T. Suzuki, and J. D. Sievert, "Measurement of Iron Loss Using Rotational Magnetic Loss Measurement Apparatus," IEEE Translation Journal on Magnetics in Japan, vol. 6, no. 6, pp , [5] H. Pfutzner, E. Mulasalihovic, H. Yamaguchi, D. Sabic, G. Shilyashki, and F. Hobauer, "Rotational Magnetization in Transformer Cores - A Review," IEEE Transactions on Magnetics, vol. 27, no.11, pp , [6] Y. J. Li, J. G. Zhu, Q. X. Yang, "Study on Rotational Hysteresis and Core Loss Under Three-Dimensional Magnetization," IEEE Transactions on Magnetics, vol. 47, no.10, pp , [7] A. J. Moses and N. Tukun, "Investigation of Power Loss in Wound Toroidal Cores under PWM Excitation," IEEE Transactions on Magnetics, vol. 33, no. 5, pp , [8] D. Dujic, M. jones, and E. Levi, "Analysis of output current ripple RMS in multiphase drives using space

8 Changgeng Zhang Qingxin Yang and Yongjian Li 396 vector approach," IEEE Trans. Power Electron., vol. 24, no. 8, pp , [9] D. Dujic, M. Jones, E. Levi, J. Prieto, and F. Barrero, "Switching ripple characteristics of space vector PWM schemes for five-phase two-level voltage source inverters Part 1: Flux harmonic distortion factors," IEEE Trans. Ind. Electron., vol. 58, no. 7, pp , [10] H. W. Van der Broeck, H. Skudelny, and G. Stanke, "Analysis and realization of a pulse width modulator based on voltage space vectors," IEEE Transactions on Industry Applications, vol. 27, no. 1, pp , [11] J. Prieto, M. Jones, F. Barrero, E. Levi, S. Toral, "Comparative Analysis of Discontinuous andcontinuous PWM Techniques in VSI-Fed Five- Phase Induction Motor," IEEE Transactions on Industrial electronics, vol. 58, no.12, pp , [12] A. Boglietti, P. Ferraris, M. Lazzari, and F. Profumo, "Effects of modulation index on the iron losses in soft magnetic materials supplied by PWM inverter," IEEE Transactions on Magnetics., vol. 29, no. 6, pp , [13] A. Boglietti, A. Cavagnino, D. M. Ionel, M. Popescu, D. A. Staton, S. Vaschetto, "A General Model to Predict the Iron Losses in PWM Inverter-Fed Induction Motors," IEEE Transactions on Industry Applications, vol. 46, no. 5, pp , [14] Y. J. Li, Q. X. Yang, J. G. Zhu, Z. G. Zhao, X. J. Liu and C. G. Zhang, "Design and Analysis of a Novel 3-D Magnetization Structure for Laminated Silicon Steel," IEEE Trans. Magn., vol. 50, no. 2, p , include electromagnetic field computation and contactless power transfer. Dr. Yang is the President of China Electrotechnical Society. Yongjian Li received B.E, M.E and Ph.D. degree in electrical engineering from Hebei University of Technology(HEBUT), Tianjin, China. He is currently a professor in HEBUT. His research interests include measurement magnetic properties, modeling of magnetic materials and power electronics. Changgeng Zhang He received B.E degree in Electrical Engineering from Tianjin Polytechnic University and M.S. degree in Computing Mathematics from Nankai University. He is Ph.D. student in electrical engineering in Hebei University of Technology. His research interests are magnetic measurement and modeling of magnetic property. Qingxin Yang received the B.E., M.E. and Ph.D. degrees from Hebei University of Technology, Tianjin, China, in 1983, 1986, and 1997, respectively. He is the president of Tianjin Polytechnic University, Tianjin. His research interests