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Spread Spectrum Modulation by Using Asymmetric-Carrier Random PWM Laszlo Mathe, Member, IEEE, Florin Lungeanu, Dezso Sera, Member, IEEE, Peter Omand Rasmussen, Member, IEEE, and John K. Pedersen, Senior Member, IEEE
Abstract—This paper presents a new fixed-carrier-frequency random pulsewidth modulation method, where a new type of carrier wave is proposed for modulation. Based on simulations and experimental measurements, it is shown that the spread effect of the discrete components from the motor current spectra and acoustic spectra is very effective and is independent from the modulation index. The flat motor current spectrum generates an acoustical noise close to the white noise, which improves the acoustical performance of the drive. The new carrier wave is easy to implement digitally, without employing any external circuits. The modulation method can be used in both open- and closed-loop motor control applications. Fig. 1.
Index Terms—AC motor drives, acoustic noise, modulation, pulsewidth modulation (PWM) power converters.
N OMENCLATURE AC-RPWM CR CVx EMI FCF-RPWM m.i. PR PWM RCF-PWM RPP-PWM RPWM SPL
Asymmetric-carrier random pulsewidth modulation (PWM) (RPWM). Compare register. Compare value for the PWM unit. Electromagnetic interference. Fixed-carrier-frequency RPWM. Modulation index. Period register. Pulsewidth modulation. Random-carrier-frequency PWM. Random-pulse-position PWM. Random PWM. Sound pressure level. I. I NTRODUCTION
T
O REACH controllability, high efficiency, and dynamic performance in electrical drives, power electronic converters based on on/off control of power switches are employed (Fig. 1). To control the power switches from the converter, several modulation methods were proposed from which the PWM method is the most used technique. A drawback of this Manuscript received December 14, 2010; revised June 8, 2011 and November 3, 2011; accepted November 15, 2011. Date of publication December 22, 2011; date of current version April 27, 2012. L. Mathe, D. Sera, P. O. Rasmussen, and J. K. Pedersen are with Aalborg University, 9220 Aalborg, Denmark (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). F. Lungeanu is with the R&D Office, Danfoss Power Electronics A/S, 100027 Beijing, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2011.2179272
Topology of a standard full-bridge voltage-source inverter.
method is that it gives rise to discrete frequency components in the current spectrum which lead to EMI [1]–[4] and acoustic noise in the drive [5]–[7]. A cost-effective strategy to distribute the discrete components from the current spectrum of the motor is RPWM. Several concepts of RPWM strategies can be found in the literature [8]. The RPWM strategies from a switching frequency point of view can be classified into two main categories: 1) RCF-PWM; 2) FCF-RPWM. The spread of the discrete components from the motor current spectrum using RCF-PWM is more effective compared to the FCF-RPWM method [9]–[11]. In most of the practical applications, the control algorithm is synchronized with the switching; therefore, the variable switching frequency affects the performance of closed-loop applications [12], [13]. Usually, FCF-RPWM methods are based on the fact that the redistribution of zero vectors does not have an effect on the fundamental component but changes the high frequency content of the motor current spectra [14]–[16]. In [17], several FCF-RPWM methods were analyzed and compared. The authors concluded that the methods where positions of pulses were randomized (RPPPWM) have good performance only at low fundamental amplitude. An RCF-PWM method where the slope of the carrier wave varies randomly can be found in [18] and [19]. In [20], the authors introduce a limitation on variable-slope modulation method to realize a new FCF-RPWM method. In this method, the increasing and decreasing slopes of a triangular carrier wave were chosen such that the modulation period constant is maintained. A drawback of the variable-slope modulation technique is that external circuits are needed to generate PWM pulses. This paper proposes a new FCF-RPWM method called AC-RPWM. The advantage of the new AC-RPWM method compared to
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Fig. 2. Modulation used for a three-phase voltage-source converter. (a) Space vector diagram in the α−β plane. (b) Switching function during one period of the carrier wave.
other FCF-RPWM methods is that it has good performance for both low and high m.i. values (m.i.-normalized output voltage amplitude [15]). A digital implementation requiring no external circuits is also proposed in this paper. Finally, the error introduced by the motor current sampling in the top and bottom of the carrier wave is analyzed. II. FCF-RPWM In a conventional construction of an inverter (Fig. 1), six active vectors and two zero sequence vectors can be generated. The six active voltage vectors represented in the α−β plane [Fig. 2(a)] form a hexagon, where each active vector points to the corner of the hexagon. When one of the six active vectors is generated, the load takes energy from the dc link, forming a circuit from the load impedances as shown in Fig. 2(a) and (b). The generation of the zero sequence vector is done by connecting the three legs of the load to the plus (V111 ) or minus (V000 ) of the dc bus. The position of a voltage vector Us in the α−β plane is defined by the ratio between the applied time lengths for the two adjoined active vectors. The zero sequence vectors are responsible for reducing the amplitude of the resultant voltage vector Us . During a modulation period where a triangular carrier wave is used, two resultant voltage vectors, with arbitrary amplitude and position in the α−β plane, can be generated. The first voltage vector is generated during the first part of the modulation period (rising slope, noted Trising ), and the second vector is generated during the second part of the modulation period (falling slope, noted Tfalling ). The fundamental requirement of FCF-RPWM is to have constant update frequency and to generate the same voltage vector in the α−β plane as it was before randomization. Due to the fact that the distribution of zero vectors (V111 , V000 ) does not influence the fundamental frequency component [15], the timelength distribution of the zero vectors can vary randomly. The
Fig. 3. Repositioning of the pulse in one leg. (a) (Continuous lines) Pulses generated by symmetrical modulation and (broken line) changed position of the pulse in leg c. (b) Vector diagram of the generated voltage vectors: The top figure represents the first part of the modulation; the bottom figure represents the second part of the modulation.
random variation of the zero sequence vectors gives a random position for the active region [the shadowed area in Fig. 2(b)] during the first and the second part of the modulation period. Using symmetrical modulation (the reference voltage vector is updated once per modulation period), the same resultant voltage vector is generated in the first and the second part of the modulation period. Fig. 3 shows the effect of repositioning of the pulse for one phase of the inverter. The pulses using symmetrical modulation for legs named qa , qb , and qc are represented by continuous lines, while the repositioning of the pulse for leg qc is represented by the dashed line. The average phase voltage during the modulation period in leg qc is the same, independently of the pulse position. The generated voltage vectors Us , Us1 , and Us2 differ in amplitude and position in the α−β plane, which means that the fundamental component is
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Fig. 4. PWM carrier waveforms: Symmetrical carrier wave in the first modulation period and asymmetric carrier wave in the second modulation period.
distorted by the individual randomized pulse position in one leg. The distorted fundamental component affects the stability of the drive, introducing current and torque ripple. Usually, the three phases of the motor are in delta or star connection; therefore, the current through one phase depends on the current in the other two phases. As a consequence, the PWM pulses have to be synchronized to have controllability of the motor current in each individual phase of the motor. This proves that the position of a pulse cannot be changed in an arbitrary manner for each leg. In the case of RPP-PWM, the pulse position of each leg is modified. However, the time spent for generating the active vectors is maintained the same. III. D ESCRIPTION OF THE P ROPOSED M ODULATION M ETHOD The PWM unit (used for motor control) of a commercial microcontroller usually consists of an up–down counter, three CRs, and a PR. An up-down counter is used for generating the carrier wave for the PWM unit. In traditional modulation methods like space vector modulation (SVM), the time required for up-counting mode (Trising ) is equal to the time required for down-counting mode (Tfalling ), as it is shown in the first modulation period in Fig. 4. By changing the ratio between Trising and Tfalling , the resultant voltage vectors generated in the first and the second part of the modulation period will be similar in position and amplitude, with the difference being that the resultant voltage vector is created with different modulation frequency. Generating the same voltage vectors after and before randomization fulfills the second requirement for FCF-RPWM described in the previous paragraph. To fulfill the first requirement for FCF-RPWM, the distribution of the time length between the first and the second part of the modulation (Trising and Tfalling ) is done such that the modulation period Tmod constant is maintained. The following equation has to be fulfilled to maintain constant modulation period: Tmod = Trising + Tfalling = constant =
1 fsw
(1)
where Tmod is the modulation period, Trising is the time length where the counter is counting up (the first part of the modulation period), Tfalling is the time length where the counter counts
down (the second part of the modulation period), and fsw is the switching frequency. The second modulation period in Fig. 4 represents the proposed asymmetrical carrier waveform, where the modulation period (Tmod ) is maintained constant, but the distributions of the time length between Trising and Tfalling are not equal. Having a constant time base (Tclk ) for the PWM module counter, the slope of the carrier wave cannot be changed. As it is shown in Fig. 4, for digital implementation of the asymmetrical carrier waveform, two different values are used for the PR, creating an asymmetrical carrier wave for modulation. From a mathematical point of view, after the duty cycles are calculated conformably [21], the compare values can be calculated for the first and second parts of the modulation period by using PRrising =
Trising Tclk
PRfalling =
Tfalling Tclk
CV1 = dzv0 · PRrising CV1 = dzv0 · PRfalling CV2 = CV1 + dav1 · PRrising CV2 = CV1 + dav1 · PRfalling CV3 = CV2 + dav2 · PRrising CV3 = CV2 + dav2 · PRfalling
(2)
where PRrising and PRfalling are the PR values for the PWM timer in the first and second parts of the modulation period, dx is the duty cycle for the active and zero sequence voltage vectors, and CVx represents the compare values in the CRs of the PWM module of the microcontroller. By choosing a random time length for Trising or Tfalling in every modulation period, the time length for the active vector regions (highlighted in Fig. 4) will vary randomly. In other words, the voltage vector generated in the Trising period is generated at different switching frequency than the voltage vector generated in the Tfalling period. From this point of view, the AC-RPWM method can be interpreted as an RCFPWM, where the update of the new voltage vector is done with constant frequency. The spread effect of discrete components from the motor current spectra using this method is effective even at high m.i., where the time spent for generation of the active vectors is longer than the time spent for generation of zero vectors. For low m.i., a very good spread effect of the discrete components can be reached by redistributing the time length of the zero sequence vectors (using the RPP-PWM strategy). As it was presented in the introduction, the redistribution of the time length between the zero voltage vectors affects the current ripple, without having any influence on the fundamental component [14]. Redistribution of the zero vectors modifies the position of the active vector regions (highlighted in Fig. 5) in the rising and falling modulation periods. In the first modulation period in Fig. 5, redistribution of the zero vectors modifies the position of the active vector; both active regions are centered (Tzv0 = Tzv1 ). In the second modulation period,
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sequence vectors’ time length within a modulation period. In this block, PRrising and PRfalling are also calculated, and the duty ratios are converted into compare values for the PWM module, based on (2). It can be seen from the block diagram in Fig. 6 that the main advantage of AC-RPWM is that it is very straightforward to include into an existing closed- or open-loop control algorithm, without the need for changing the control structure or adding hardware components. As a disadvantage, it can be mentioned that the PWM module has to be updated twice during the modulation period (double update is needed).
Fig. 5. First modulation period of SVM (Tzv0 = Tzv1 ). Redistribution of the zero vectors (Tzv0 = Tzv1 ). TABLE I M EASURED SPL OF D IFFERENT M ODULATION S TRATEGIES AND F UNDAMENTAL F REQUENCY
Fig. 6. loops.
Block diagram of a motor control using AC-RPWM in open and closed
the active vectors are repositioned (Tzv0 = Tzv1 ). From the point of view of the current ripple, the optimal position for active vectors is in the middle like in the SVM. In Table I, the acoustic performances of four different modulation methods are presented, at an average switching frequency of 4 kHz. The SPL was measured with a Bruel and Kjaer sound level meter type 2230 using A-weighting. The increased current ripple causes louder acoustic noise in the motor (Table I) but transforms the whistling noise into a white noise. IV. D IGITAL I MPLEMENTATION OF AC-RPWM In Fig. 6, a general block scheme of a typical motor control structure in open or closed loop is shown. The outcome of the control block is always a calculated reference voltage vector in the α−β plane. This voltage vector [Us from Fig. 2(a)] is decomposed into two adjacent active voltage vectors, and the compare values for the PWM module are usually calculated using the SVM block. The AC-RPWM block is an additional block which makes the randomization of the active and zero
V. S IMULATION R ESULTS To compare the spectra of the motor current obtained by using different modulation techniques, a number of simulations were carried out in the power electronic simulation software PLECS 3.1 [22]. In order to make the comparison between different modulation methods with maximum randomization freedom, the nonlinearities like dead time and minimum pulse filter have been neglected in the simulation. Fig. 7 shows the current spectra produced by four modulation methods: SVM, RPP-PWM, AC-RPWM, and the combination of AC-RPWM with RPP-PWM. The simulations were made for three different values of the m.i.: at low speed with m.i. = 0.1, at medium speed with m.i. = 0.5, and at high speed with m.i. = 0.9. At low speed, the RPP-PWM method has good performance; the discrete frequency components disappear from the spectrum around the switching frequency. However, the discrete components are present at double the switching frequency. In the case of AC-RPWM, the discrete components are located around the switching frequency. The combination of the two methods nearly entirely eliminates the discrete components from the current spectrum. At medium speed, the time spent for generation of zero sequence vectors is less; the possibility for repositioning the active vectors is reduced. This results in the appearance of the discrete components in the case of RPP-PWM. At double the switching frequency, the amplitude of the discrete components is considerably reduced when AC-RPWM is used. The combination of the two methods (RPP–AC-RPWM) gives again the best performance. At high speed, the time spent for generation of zero sequence vectors is minimal; there is not a big difference between the spectra obtained by using SVM or RPP-PWM. For high speed, the performance of AC-RPWM is better than that of RPP-PWM. In conclusion, the combination of the two modulation methods, RPP-PWM and AC-RPWM, has the best performance of the considered techniques in all speed ranges. In Table II, the total harmonic distortion (THD) of the simulated motor current is presented for the three different m.i. values. It should be noticed that, for high m.i., the THD of the RPP-PWM method is lower but the discrete components are still present in the spectrum (the randomization effect is small); the THDs of RCF-PWM and AC-RPWM are almost the same in this case.
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Fig. 7. Simulation results of the motor current spectrum. SVM with m.i. values of (a) 0.1, (e) 0.5, and (i) 0.9. RPP-PWM with m.i. values of (b) 0.1, (f) 0.5, and (j) 0.9. AC-RPWM with m.i. values of (c) 0.1, (g) 0.5, and (k) 0.9. RPP–AC-RPWM with m.i. values of (d) 0.1, (h) 0.5, and (l) 0.9. TABLE II THD OF THE M OTOR C URRENT
Fig. 8.
VI. E XPERIMENTAL R ESULTS To show the acoustic performance and the effect of the hardware limitations (dead time and minimum pulsewidth) of the proposed modulation method, an experimental setup was built, as shown in Fig. 8. The setup consists of a 2.2-kW asynchronous motor, a 2.2-kW Danfoss VLT AutomationDrive FC 302, and a motor control running on a Texas Instruments 320F28335 floating-point microcontroller. The motor current and vibrations on the motor shell were measured with a Bruel and Kjaer pulse multianalyzer type 3560. In this work, vibration measurements have been used to analyze the performance of the proposed modulation method. It is well known that the vibration spectrum is similar to that of the radiated acoustic noise [23]. For randomization purposes, the built-in pseudorandom number generator function from “C” programming language was used. Figs. 9 and 10 show the measured spectra of the phase current and the vibrations on the shell of the asynchronous motor for m.i. values of 0.1, 0.5, and 0.9. The measured current spectrum shows similar results as the simulation from the previous section despite the fact that, for the experimental tests, limitations to ensure the minimum pulsewidth were introduced.
Block diagram of the experimental setup.
The spectrum of the motor shell vibration (Fig. 10) shows that the discrete components nearly disappeared when the ACRPWM technique was used, regardless of the m.i. The small peaks from the spectrum were difficult to be distinguished by ear, when the AC-RPWM method was used. The current ripple is minimal in the case of SVM; by using AC-RPWM, this ripple was increased, which leads to higher SPL. By eliminating the discrete components from the vibration spectra, the acoustic noise in the case of AC-RPWM becomes close to white noise. VII. C URRENT S AMPLING The current sampling error can cause torque oscillation which deteriorates the performance of the control in the case of closed-loop applications [24]–[27]. Usually, the acquisition of the motor currents is done on the top and/or the bottom of the triangular carrier wave [25], [28]. In the case of SVM, this means that the sampling of the motor currents is done in the middle of the time length spent for zero sequence vector generation. To quantify the current sampling error, the ideal case was simulated without nonlinearities (like dead time, minimum pulse filter, and saturation). The motor from the schematic shown in Fig. 1 was replaced with an R−L load.
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Fig. 9. Measured phase currents of the motor. (a) SVM at 0.1 m.i. (b) SVM at 0.5 m.i. (c) SVM at 0.9 m.i. (d) RPP–AC-RPWM at 0.1 m.i. (e) RPP–AC-RPWM at 0.5 m.i. (f) RPP–AC-RPWM at 0.9 m.i.
Fig. 10. Measured vibrations on the motor shell. (a) SVM at 0.1 m.i. (b) SVM at 0.5 m.i. (c) SVM at 0.9 m.i. (d) RPP–AC-RPWM at 0.1 m.i. (e) RPP–AC-RPWM at 0.5 m.i. (f) RPP–AC-RPWM at 0.9 m.i.
The impedance of the simplified R−L circuit was set to be similar with the impedance of the motor. Using a symmetrical regular sampled SVM method, a balanced three-phase sinusoidal voltage can be generated. The R−L circuit will act as a first-order low-pass filter, creating a current with the same fundamental frequency as the motor phase voltage. To extract the fundamental current component (purely sinusoidal without ripple) from the measured current through the inductance, a resonant filter was used. The advantages of the resonant filter are that its phase shift is zero at its resonant frequency and that it has high attenuation outside of the resonant frequency. Subtracting the fundamental current value from the measured current value gives the current measurement error when the sampling is made on the top and on the bottom of the carrier wave. Fig. 11 shows the simulation results of the macroscopic
and microscopic scales of the filtered current and the measurement error. As it can be concluded from Fig. 11, the sampling error is high when the reference voltage vector is in the middle of a sector, and it is low when the reference voltage vector has the same position as an active vector. This means that, when the reference voltage vector is close to an active voltage vector, the sampling in the top and bottom of the triangular carrier has minimal error. At zero crossing, the slope of the current is maximum, which results in maximum error. Fig. 11(c) shows that, when zero vectors are generated, the reference signal is always between the values of the current sampled in the first and second parts of the modulation period. This error is mainly caused by the regular sampling. The maximum error for this case is approximately ±0.05 A. For this example, where 10 A
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Fig. 11. Simulation results of the current using SVM with 0.5 m.i. (a) One fundamental period of the current. (b) Sampling error during the fundamental period. (c) Zoom of the current plot at zero crossing. (d) Zoom of the current plot around the maximum value of the fundamental term. (e) and (f) Zoom of the error plot.
is the peak nominal current, 15 A can be considered as a maximum measurable current. Considering the relatively high current measurement error (1%), even a resolution as low as 8 b is enough for analog-to-digital (AD) conversion. By using the RPP-PWM method and by sampling the motor current on the top and bottom of the triangular carrier, the current will not be sampled in the middle of the time spent for generation of the zero sequence vectors, which further increases the sampling error. This error can be reduced by using the same position for the active vector region in the first and the second half of the modulation period (single update for the voltage vector). However, by introducing limitations on the possible position for the active region, the spread effect of the discrete components from the motor current spectra is reduced. Fig. 12 shows the inductor current in macroscopic time scale when the RPP-PWM and AC-RPWM techniques are used. The inductor current is sampled in both cases in the middle and in the top of the triangular carrier; the sampled value is subtracted
from the filtered current, which gives the sampling error. As it was expected, the sampling error increases in the case of random modulation. In the case of AC-RPWM, the error was slightly smaller than that in the case of RPP-PWM because the active vector region is placed in the middle of the half modulation period. Taking into consideration the error made by sampling on the top and bottom of the carrier, for the random modulation, a resolution of 6 b is enough for AD conversion. In the case of applications where this sampling error can cause problems, limitations like maximizing the time length Trising or Tfalling to, for example, 80% of the time length of the modulation period can be introduced. However, this kind of limitations reduces the effectiveness of the randomization (the discrete components are going to be more dominant). The usage of a low-pass filter and a high sampling frequency for current measurement could be a solution [26]. However, low-pass filters affect the phase of the filtered signal which can introduce even higher error. Using a notch filter (like the previously presented resonant filter) which
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Fig. 12. Macroscopic time scale of (top) the inductor current and (bottom) the sampling error at 25-Hz fundamental frequency; RPP-PWM in the left column and AC-RPWM in the right column.
does not affect the phase of the current can cause problems by filtering out the low-frequency components in the measured current. In this case, the current regulators from the closed-loop control will not be able to compensate for these unwanted lowfrequency components. VIII. C ONCLUSION A new FCF-RPWM method called AC-RPWM has been presented and analyzed in this paper. The modulation method has fixed update frequency, which has the advantage of easy implementation and integration into an existing open- or closedloop motor control algorithm, without using any additional hardware. The simulations and the experimental measurements show that the AC-RPWM method effectively spreads the discrete components of the current and vibration spectra independent of the m.i. This flat motor current spectrum is the main advantage compared to PWM methods like SVM and discontinuous PWM (DPWM). For those applications where high demands for shaft-torque dynamics are needed, the current sampling has to be improved. However, the AC-RPWM method is well suited for applications like heating ventilation and air conditioning (HVAC), where the demand on shafttorque dynamics is moderate and the acoustic noise issue is more important. The proposed method can be used for those HVAC applications where the less efficient asynchronous motor is replaced by a permanent-magnet synchronous motor, which requires closed-loop control. R EFERENCES [1] S. Kaboli, J. Mahdavi, and A. Agah, “Application of random PWM technique for reducing the conducted electromagnetic emissions in ac-
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[13] A. M. Trzynadlowski, K. Borisov, Y. Li, and L. Qin, “A novel random PWM technique with low computational overhead and constant sampling frequency for high-volume, low-cost applications,” IEEE Trans. Power Electron., vol. 20, no. 1, pp. 116–122, Jan. 2005. [14] V. Blasko, “Analysis of a hybrid PWM based on modified space-vector and triangle-comparison methods,” IEEE Trans. Ind. Appl., vol. 33, no. 3, pp. 756–764, May/Jun. 1997. [15] G. D. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice. Hoboken, NJ: Wiley, 2003. [16] K.-S. Kim, Y.-G. Jung, and Y.-C. Lim, “A new hybrid random PWM scheme,” IEEE Trans. Power Electron., vol. 24, no. 1, pp. 192–200, Jan. 2009. [17] M. M. Bech, F. Blaabjerg, and J. K. Pedersen, “Random modulation techniques with fixed switching frequency for three-phase power converters,” IEEE Trans. Power Electron., vol. 15, no. 4, pp. 753–761, Jul. 2000. [18] C. M. Liaw and Y. M. Lin, “Random slope PWM inverter using existing system background noise: Analysis, design and implementation,” Proc. Inst. Elect. Eng.—Elect. Power Appl., vol. 147, no. 1, pp. 45–54, Jan. 2000. [19] J. C. Salmon, “A new “slope-modulated” PWM strategy for implementation in a single chip gate array,” in Conf. Rec. IEEE IAS Annu. Meeting, 1988, vol. 1, pp. 388–394. [20] L. P. Luis Graces and V. T. N’Guyen Phuoc, “Inverter control device,” U.S. Patent 5 552 980, Sep. 3, 1996. [21] J. Holtz, “Pulse width modulation for electronic power conversion,” Proc. IEEE, vol. 82, no. 8, pp. 1194–1214, Aug. 1994. [22] PLECS Piece-Wise Linear Electrical Circuit Simulator User Manual, Plexim GmbH, Zurich, Switzerland, 2008. [23] W. C. Lo, C. C. Chan, Z. Q. Zhu, L. Xu, D. Howe, and K. T. Chau, “Acoustic noise radiated by PWM-controlled induction machine drives,” IEEE Trans. Ind. Electron., vol. 47, no. 4, pp. 880–889, Aug. 2000. [24] D. Antic, J. B. Klaassens, and W. Deleroi, “Side effects in low-speed AC drives,” in Proc. 25th Annu. IEEE Power Electron. Spec. Conf., 1994, vol. 2, pp. 998–1002. [25] V. Blasko, V. Kaura, and W. Niewiadomski, “Sampling of discontinuous voltage and current signals in electrical drives: A system approach,” in Conf. Rec. 32nd IEEE IAS Annu. Meeting, 1997, vol. 1, pp. 682–689. [26] L. Accardo, M. Fioretto, G. Giannini, and P. Marino, “Sampling problems using mixed random modulation techniques (MRMT) for the reduction of magnetic noise in traction motors,” in Proc. SPEEDAM, 2008, pp. 1199–1204. [27] T. Lu, Z. M. Zhao, Y. C. Zhang, and L. Q. Yuan, “Research on influences of sampling errors on performances of three-level PWM rectifier,” in Proc. ICEMS, 2009, pp. 1–6. [28] Y.-C. Son, S.-H. Song, and S.-K. Sul, “Analysis and compensation of current sampling error in AC drive with discontinuous PWM,” in Proc. 14th Annu. APEC, 1999, vol. 2, pp. 795–799.
Laszlo Mathe (S’07–M’10) received the B.Sc. degree in electrical engineering and the M.Sc. degree from the Technical University of Cluj-Napoca, ClujNapoca, Romania, in 2000 and 2002, respectively, and the Ph.D. degree in electrical engineering from the Department of Energy Technology, Aalborg University, Aalborg, Denmark, in 2010. Between 2002 and 2007, he was working for industry as a Control Development Engineer. He is currently an Assistant Professor with Aalborg University. His current research activities are in power electronics, specifically in modulation and motor control.
Florin Lungeanu received the M.Sc. degree in electrical engineering from the Lower Danube University of Galati, Galati, Romania, in 1996. He was an Assistant Professor with the Lower Danube University of Galati until 2000. He was a Guest Researcher with the Department of Energy Technology, Aalborg University, Aalborg, Denmark, until 2001. He moved to industry and worked as a Control Engineer for Danfoss Drives A/S and as a Power System Engineer for Vestas Wind Power Systems A/S. He is currently a Motor and Application Control Specialist with the R&D Office, Danfoss Power Electronics A/S, Beijing, China. His main research interests are in control for power electronics, motor and application control, and new pulsewidth modulation techniques applied to various state-of-the-art and novel power electronic topologies.
Dezso Sera (S’05–M’08) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technical University of Cluj-Napoca, Cluj-Napoca, Romania, in 2001 and 2002, respectively, and the M.Sc. and Ph.D. degrees from the Department of Energy Technology (DET), Aalborg University, Aalborg, Denmark, in 2005 and 2008, respectively. Since 2009, he has been the Coordinator of the Photovoltaic Systems and Microgrids Research Program at DET, Aalborg University, where he is currently an Associate Professor. His current research activities are in photovoltaic (PV) power systems in general, specifically in the modeling, characterization, diagnostics, and maximum power point tracking of PV systems, and in the grid integration of PV power.
Peter Omand Rasmussen (M’98) was born in Aarhus, Denmark, in 1971. He received the M.Sc. degree in electrical engineering and the Ph.D. degree from Aalborg University, Aalborg, Denmark, in 1995 and 2001, respectively. In 1998, he joined Aalborg University as an Assistant Professor, where he has been an Associate Professor since 2002. His research areas are within magnetic gears and design and control of switched reluctance and permanent-magnet machines.
John K. Pedersen (M’91–SM’00) was born in Holstebro, Denmark, on September 2, 1959. He received the B.Sc.E.E. degree from Aalborg University, Aalborg, Denmark. He was with the Department of Energy Technology (DET), Aalborg University, as a Teaching Assistant from 1983 to 1984 and as an Assistant Professor from 1984 to 1989, where he has been an Associate Professor since 1989. He is currently the Head of DET, Aalborg University. He is the author or coauthor of more than 170 publications, and he is involved in a number of research projects in collaboration with industry. His research areas are in power electronics, power converters, and electrical drive systems, including modeling, simulation, and design with focus on optimized efficiency. Mr. Pedersen was the recipient of the 1992 Angelos Award for his contribution in the control of induction machines. In 1998, he was the recipient of an IEEE T RANSACTIONS ON P OWER E LECTRONICS Prize Paper Award for best paper published in 1997. He was also the recipient of the ABB Prize Paper Award at the Optimization of Electrical and Electronic Equipment (OPTIM) 2002 Conference.
