CDROM Proceedings
ISIE 2010 2010 IEEE International Symposium on Industrial Electronics
Palace Hotel Bari Bari, Italy 04 - 07 July, 2010 Sponsored by The Institute of Electrical and Electronics Engineers (IEEE) IEEE Industrial Electronics Society (IES) Co-sponsored by IEEE Control Systems Society (CSS) Society of Instrument and Control Engineers (SICE-Japan) Politecnico di Bari, Italy
© 2010 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. IEEE Catalog Number: CFP10ISI-CDR ISBN: 978-1-4244-6391-6
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Time Average Approach for the Calculation of Quasi-Subharmonics of PWM Technique in Ultra High Speed AC Motor Supply Peter Stumpf, Zoltán Varga, Peter Bartal, Rafael K. Járdán, István Nagy Faculty of Electrical Negineering, Budapest University of Technology and Economics H-1111 Budapest, Goldmann Gy. tér 3. Hungary, e-mail:
[email protected] The paper is concerned with a novel method for the calculation of the so called quasi-subharmonics generated by PWM controlled VSC supplying ultrahigh speed induction machine. It is generally believed that the impact of subharmonics is negligible in most modern drives. Our finding is that on the contrary, it can cause serious malfunction and breakdown in special cases like ultrahigh speed induction machines (USIM) applied in systems developed for renewable and waste energy recovery. The time average method is described and used for the determination of quasi-subharmonics. The results are verified by simulations and laboratory tests. Index Terms—Harmonic analysis, Frequency conversion, Pulse width modulated power converters, Induction machines
I.
INTRODUCTION
II.
COMMENTS ON PWM TECHNIQUE
Large number of publications was presented in the PWM techniques [2, 3, …, 16]. Following a number of unexpected failures of the USIM and that our laboratory tests revealed the presence of quasi-subharmonics with extraordinary levels, we have started focusing our attention to a comparatively neglected scope of the PWM technique, the generated quasisubharmonics. The maximum switching frequency of the converters that are achieved by IGBT converters and available in the market is typically below 20 kHz that is considerably lower than it would be necessary to get flux and current waveforms with low THD at the high stator frequencies of USIMs, which are well above 1 kHz. Asynchronous PWM can generate quasi-subharmonics producing considerable heating losses in the machine. Following several breakdowns of USIM, we have come to the conclusion that they most likely have been caused by low order quasi-subharmonics. Another paper [3] discusses the increased rotor losses caused by quasi-subharmonics. Here the generation of quasi-subharmonics is articulated. In order to have a deeper insight into the generation of subharmonics one of the most common form of naturally sampled PWM applying a triangular carrier signal vtri with amplitude Vtri , frequency fc to compare against the sinusoidal reference waveform vsine = Vtri sinωr t with frequency fr is studied (Fig.1.). The modulation produces the PWM signal vPWM at the output terminal of the converter measured to the centre of the DC supply voltage.
The basic features of the PWM converter-fed ultra high speed induction machines, USIMs (up to 100 krpm or over) are the necessarily high fundamental supply frequency and the limited switching frequency resulting in low and noninteger mf=fc/fr frequency ratio (here fc corresponds to the carrier frequency, fr is the reference or fundamental frequency) that leads to stator voltage and current harmonic spectra far more unfavorable as compared to those obtained at usual fundamental frequencies. Particular problem is that at low and noninteger mf values the converter is prone to generate so called quasi-subharmonics. The term quasi-subharmonic will be introduced and explained. Taking into account some special parameters of the USIM, namely the low leakage reactances and the low stator and rotor resistances, we can conclude that both the higher and the quasi-subharmonic current components can considerably be higher than in usual machines. Furthermore the low rotor resistance results in low rated and breakdown slip. The higher harmonic current components can be attenuated by applying low power series reactors between the converter and the machine but this solution is inefficient against the quasi-subharmonics as their frequency can be low. A possible method based on the application of current source inverters suggested for solving the problems resulting from the subharmonics is published in [3]. As subharmonics appear only at noninteger mf values, a simple and efficient approach would be avoiding them by Fig. 1. Naturally Sampled PWM applying synchronized carrier and reference frequencies at Applying the Fourier series approach for the output voltage least below a given level, eg. mf=21. Though the technique is of the converter v , the individual quasi-subharmonics with PWM well known, it is not applied by manufacturers of the very small amplitudes and low frequencies are obtained [4]. converters we have studied.
978-1-4244-6391-6/10/$26.00 ©2010 IEEE
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Their effects are usually considered marginal, negligible. The conclusion is misleading, the effects resulting from the special voltage wave vPWM,DC called quasi-subharmonics prove to be very significant in our special case. The main source of the quasi-subharmonic is the small DC contributions of the carrier and sideband harmonics in each reference period while the effect of the real subharmonic is marginal and can practically be neglected. Introducing a new time average approach, quasisubharmonics with relatively low amplitudes compared to the fundamental and with low frequencies are obtained. The quasi-subharmonics are not directly visible from the Fourier series approach, but of course they can be calculated from the Fourier series components [20]. The period Ts of the subharmonic component is multiple of the reference (fundamental) period Tr=1 pu. The ratio of Vsub /fsub, which is fairly proportional to the quasi-subharmonic stator flux, can be significant and responsible for the very high additional rotor copper loss (comparible to the rated copper loss [3]) and the failures of USIM. The study was confined to one phase of the 3 phase VSC with peak output voltage of ± 1pu. The ratio of ma= Vsine /Vtri
was kept constant at ma=0.8 and the frequency of the sinusoidal reference signal was fr=1/Tr=1 pu.
Fig. 2.
Frequency spectrum of vPWM
It was assumed that both the triangular and the sinusoidal signals were perfect waveforms and the supply DC voltage was smooth, ripple free. Both vsine and vtri are zero at the beginning of subperiod and vtri starts with negative slope.The Fourier series of vPWM contains a single fundamental component with frequency fr and the groups of sideband harmonics around the carrier and multiple carrier harmonics (Fig.2.). The frequency of the sideband harmonics grouped around the multiples of the carrier frequency fc is: fsideb=±(m × mf ±n)·fr
a, mf=6.99; n=6
b, mf =7.01=8-0.99; n=8 Fig. 3.
vPWM,DC(Tr) and ΣvPWM,DC(Tr). Ts=99Ts’=100pu
Fig. 4.
vPWM,DC(Tr) and ΣvPWM,DC(Tr). Ts=98Ts’=50pu
Fig. 5.
vPWM,DC(Tr) and ΣvPWM,DC(Tr). Ts=97Ts’=100pu
a, mf=6.98; n=6
b, mf =7.02=8-0.98; n=8
a, mf=6.97; n=6
b, mf =7.03=8-0.97; n=8
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(1)
where mf = fc / fr , m=1,2,... and n=1,2,... It is assumed that mf is rational number but in general not integer: mf=integer ± fraction. One constraint is that when m is odd then n = 2, 4,... and when m is even then n = 1, 3,... The subharmonics are the lowest sideband harmonics intruding in to the frequency range below fr=1 pu (Note that Tr=1/fr=1pu.).. In such a case fsideb=fsub and the subharmonic period when m=1 is Ts’=1/fsub = integer fraction. III. TIME AVERAGE APPROACH FOR THE DETERMINATION OF SUBHARMONICS OF vPWM In addition to Ts’ we can always define an other integer quasi-subharmonic period Ts=pTs’ where p=1,2,3… The reason is that mf can be written as mf=N/D=Ns /Ds where N, D, Ns and Ds are integers. Ns/Ds is the simplest form, that is, any common factors in the ratio have been removed (e.g. mf=7.02=702/100=351/50). Taking into account that mf=fc/fr=Tr /Tc=Ns /Ds, that is, the Ts quasi-subharmonic period Ts=DsTr=NsTc=integer (Tr=1 pu)
(2)
we conclude that the number of reference period is Ds and the number of carrier period is Ns in period Ts. In general Ts’ is not always integer but Ts’=integer±fraction. Being both Ns and Ds integers, the voltage vPWM is in frequency-locked state. (When mf is irrational, we have quasi-periodic state but it is out of the scope of the current paper.) Voltage vPWM has no DC component in period Tr when mf is integer and when there is no phase shift between the reference and carrier wave. On the other hand, when mf=integer.fraction=int.fra then it does have a DC component vPWM,DC practically in each Tr period
TABLE I QUASI-SUBHARMONIC PERIODS BELONGING TO DIFFERENT FRACTIONS (FRA)
fra [pu] ±0.1 ±0.2 ±0.3 ±0.4 ±0.5
Ts [pu] 10 5 10 5 2
Ts’ [pu] 10 5 3.33 2.5 2
fra [pu] ±0.6 ±0.7 ±0.8 ±0.9
Ts [pu] 5 10 5 10
Ts’ [pu] 1/0.6 1/0.7 1.25 1/0.9
As examples a few results of the numerical calculation are shown in Fig.3, 4 and 5 for mf=7±0.01; 7±0.02 and 7±0.03. There two quasi-subharmonic periods Ts and Ts’ are given in the captions. Ts’ is not integer but p=Ts/Ts’ is integer. Note, that Ts can be two orders of magnitude longer than Ts’≈Tr, when e.g. the fraction in mf is 0.99 (Fig.3). As it is expected the time functions vPWM,DC belonging to mf =int+fra and mf =int-fra are almost the same. mf =int±0.03 presents an interesting special case (Fig. 5). Even though Ts=100Tr but within Ts almost perfect three periods take place. Ts’=33.33 pu somehow reflects it (Table I). IV.
FUNDAMENTAL COMPONENT OF FOURIER SERIES OF vPWM,DC(k)
Using the Fourier expansion, the amplitude VPWM,DC of the fundamental component of vPWM,DC (k) was calculated by substituting vPWM,DC (k) in each Tr=1pu period by a rectangle with average value of vPWM,DC (k) (Fig. 6a) The results are in Fig. 6 from fraction = ±0.1 to fraction = ±0.5 (Fig. 6b) and from fraction = ±0.01 to fraction = ±0.05 (Fig. 6c) around any integer in mf. The numerical values of the fundamental components can approximately be checked by substituting the vPWM,DC waves with a square wave. The conclusions drawn from Fig. 6 are as follows: (k+1)Tr depends on mf. The smaller the integer in mf is, the , vPWM,DC (k)= ∫kT vPWM (t)dt≠0 (3) r higher will be. belonging to positive fraction , , is only a little bit smaller than belonging to the same , where k = 1, 2, ..., Ts but the sum of the DC components over fraction but with negative sign. increases with the , the integer quasi-subharmonic period Ts adds up to zero: absolute value of the fraction. The function (fra) is , Ts approximately linear at m =const. starting from fra=0. f ∑ vPWM,DC = ∫0 vPWM,DC dt=0 (4) The main sources of vPWM,DC wave are the harmonics of vPWM since their instantaneous values are non-zero at Tr or at its multiple provided that mf is not integer. By virtue of their frequency the term subharmonic is justified for vPWM,DC waves. By virtue of the Fourier series theory or of their origin they are not subharmonics. In order to solve the conflict it is suggested to call them “quasisubharmonic waves”. Their significance is that they generate large losses in the rotor of high speed induction machines. Table I presents the subharmonic period Ts = Ds and Ts’=1/fsideb (see (1)) for a few fractions. In the calculation it is assumed that the integer in mf is even and m=1. They do not depend on the integer in mf
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a,
V.
b, Fig. 6.
Amplitude of fundamental component of Fourier series of vPWM,DC as a function of fraction around mf parameter (ma=0.8)
Calculating the real subharmonic amplitudes of the PWM voltage vPWM by using the Fourier expansion, in theory infinitely large number of subharmonics (m=1,2,3 etc) with very small amplitudes and with different frequencies are obtained. E.g. when the integer in mf is 6 and m=1 the subharmonic amplitude A6=8.08·10-5 pu. The contribution of the real subharmonic to the quasi-subharmonic is marginal and can be neglected. Finally it must be emphasized, that the quasi-subharmonic waves can be calculated from the Fourier series of vPWM as well and the results are shown in paper [20]. The numerical values obtained from (3) and (4) and from the Fourier series are identical of course but the calculation is more time consuming using the Fourier series.
Fig. 7. Space vector trajectory Ψ ,mf =21
SIMULATION
The simulation program developed in Matlab/Simulink® environment includes a complete converter model applying the PWM technique studied in the previous section. All the basic parameters of the converter and the PWM controller having influence on the operation and accuracy of the simulation results are taken into consideration. The output voltage of the converter is fed to a Space Vector Transformation block that yields the complex time function of the output voltage space vector. The parameters of the induction machine and converters (See later: Conv-a) specified in the next chapter, presenting laboratory test results, are used also in the simulations. The study of subharmonic components is focused on the analysis of phase voltages of a three phase DC/AC converter (the phase voltage is measured between the centre point of the DC input voltage and an output terminal), however, in practice the loads are connected to the line-to-line voltages. It has to be noted that the harmonic spectra of the line-to-line and phase voltages are significantly different from each other. As it is known, there are no harmonic components in the line-to-line voltages with orders of h = 3k, where k = 1,2,3,... as in the three phase system the phase voltages with these orders form a zero sequence component. Next few samples of the simulation results are presented. In Fig. 7 the trajectory of the stator linkage flux space vector Ψ composed of the time integral of the converter phase voltage φ(t)= ∫ vPWM (t)dt (corresponding to the stator flux when the stator resistance is negligible) and its fundamental component are shown at fr = 571.43 Hz, fc = 12 kHz, mf = 21. The sixsided symmetry of the trajectory is perfect. Fig.s 8 and 9 show the trajectories of Ψ for mf = 8 and mf = 8.05, respectively.
Fig. 8. Space vector trajectory Ψ ,mf=8
Fig.
9. Space vector trajectory Ψ ,mf =8.05
It can be seen in Fig.8 that for mf < 21, the trajectory of Ψ has only a two-sided symmetry. In Fig.9 mf is not an integer, thus the trajectory closes on itself only after 20 cycles and a subharmonic component with frequency fsideb = fsub = 8.058 = 0.05 pu (see (1)) is generated. In Fig.10. time function of the time integral of the converter phase voltage vPWM is shown for mf = 6.7. The simulation program makes it possible to take samples periodically at instants of Tr, 2 Tr, 3 Tr. Storing these values, diagrams, similar to those presented in Figs. 3 – 5. can be plotted by using Excel program. We have found that these Fig. 10. Time function φ(t), mf=6.7
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simulation results and the calculation results are in very good is close to the one presented in Fig. 8. It is visible that the agreement (see later Fig.15 and 16). trajectory misses the six-sided symmetry. In Fig. 13. the trajectory of the same variable is shown for mf = 8.05, here the trajectory is moving around and closes on itself after 20 cycles VI. TEST RESULTS of the fundamental, demonstrating the presence of quasiSignificant amount of laboratory tests have been carried out subharmonics (see Fig.9.). on a turbine-generator system using a 2 pole, 4.5 kW In Fig. 14 time function of the converter phase voltage time induction machine with a rated speed of 90 krpm, thus the integral φ(t) is plotted for mf = 6.7 (see Fig.10.). A test method rated stator frequency is f1n = 1500 Hz. The maximum has been devised and applied to verify the results of our fundamental frequency of the converter is frmax = 3 kHz, the numerical calculation of quasi-subharmonics with integer possible carrier frequency can be selected from 3 kHz, 8 kHz, period T . The waveform of φ(t) is displayed and stored in s 12 kHz and 16 kHz. The maximum carrier frequency numerical form by the help of a digital CRO and the results are produced less stable operation and higher level of processed by a computer. From the large set of data the subharmonics thus for the turbine-generator system the instantaneous values vPWM,DC measured at the instants of 0, Tr, carrier frequency of fc = 12 kHz has been chosen, i.e. at the 2 Tr, 3 Tr... k Tr, are selected, where Tr is the period of the rated frequency frn the value of mf = 8. The converters tested fundamental and, thus we obtain results similar to those gained (CONV-1 and CONV-2) are identical, 7.5 kW units (Conv-a) by calculations. These results confirm the typical waveform of connected back-to-back, both applying the PWM technique the calculated Vsec areas i.e. Σ vPWM,DC (Tr). Furthermore the test results, i.e. the measured amplitudes show good agreement studied. To study the generation of subharmonics, further tests have with those obtained by numerical calculation. In Fig.s 15 and 16 measured values of Vsec areas i.e. also been completed using two other converters from different manufacturers: Conv-b : power level, Pn = 1.4 kW, ΣvPWM,DC(Tr) are plotted in the diagrams together with the frmax = 500 Hz and fc = 2900 Hz or 5900 Hz, Conv-c: Pn = 7.5 calculated values for mf = 6.1 and mf = 7.05, respectively. The kW, frmax = 2 kHz and fc = 12 kHz.
Fig.11. Space vector trajectory Ψ, mf = 21
Fig. 12. Space vector trajectory Ψ, mf = 8
Fig. 14. Space vector trajectory Ψ, mf = 6.7
test results are in good agreement with those obtained by numerical calculations. The test results presented in Fig.s.15 and 16 were obtained by using Conv-c.
Fig. 14. Space vector trajectory Ψ, mf = 6.7
Illustrative examples of test results are presented next. The test results shown in Fig.s.11…14 were obtained using Conv-a. In Fig. 11 the space vector trajectory of the converter output voltage time integral is recorded at fr = 571.43 Hz, fc = 12 kHz, i.e. mf = 21 (see Fig.7.). The six sided symmetry of the trajectory is nearly ideal. In Fig. 12 the recorded trajectory of the converter output voltage time integral is shown for f1n = 1500 Hz, fc = 12 kHz, mf = 8. The shape of the trajectory
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Fig. 15.
Measured and calculated values of ΣvPWM,DC(Tr), mf = 6.1
[4] [5]
[6]
[7]
[8]
Fig. 15.
[9]
Measured and calculated values of ΣvPWM,DC(Tr), mf = 6.1
VII.
CONCLUSIONS
The quasi-subharmonic wave vPWM,DC generated by PWM voltage of noninteger frequency ratio mf can cause serious overheating in the rotor of high speed induction machine [21]. The quasi-subharmonic voltage wave can be calculated from either the time integral of the PWM votlage vPWM taken for its fundamental period or from the Fourier series of vPWM and of course the results are identical. By virtue of the frequency fsub=1/Ts of the quasi-subharmonic voltage the term subharmonic is clearly justified since its period Ts is multiple of the fundamental period. On the other hand, from the viewpoint of Fourier series theory the name subharmonic is misleading because the real subharmonic of the output in the PWM inverter is only its one component and its contribution is marginal in the wave vPWM,DC with peak value Vsub . In fact its main source is the contributions of the carrier and sideband harmonics. As a compromise the name ”quasi-subharmonic” was proposed and adopted. The ratio of /fsub, which is fairly proportional to the quasi-subharmonic stator flux, can be significant and responsible for the very high additional rotor copper loss and the failures of USIM.The results of the new approach were verified by simulation and by tests.
VIII.
ACKNOWLEDGEMENT
[1]
[2]
[3]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
The authors wish to thank the Hungarian Research Fund (OTKA K72338) and the Control Research Group of the Hungarian Academy of Scinces (HAS) furthermore the Hungarian Science and Technology Foundation in framework of JP-25/2006 project and IT20/2007 project as well as the EEA / Norwegian Financial Mechanism, HU0121-GAN-00064-E-V1 for the financial support.
IX.
[10]
[19]
[20] [21]
REFERENCES
C. Zwyssig,S. D. Round., J.W. Kolar : An Ultrahigh-Speed, Low Power Electrical Drive System, IEEE Transactions on Industrial Electronics, February 2008, Vol 55., No 2. pp 577-585 A. Cataliotti, F. Genduso, A. Raciti, G.R. Galluzo: Generalized PWMVSI Control Algorithm Based on a Universal Duty-Cycle Expression: Theoretical Analysis, Simulation Results, and Experimental Validations, IEEE Transactions on Industrial Electronics, June 2007, Vol 54., No 3. pp 1569-1580 Stumpf P., Varga Z., Bartal P.: Járdán R.K., Nagy I., Effect of Subharmonics on the Operation of Ultrahigh Speed Induction Machines IECON’09, IEEE, Porto, Portugal, 2009, pp 431-436
[22] [23]
[24]
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Grahame Holmes, Thomas A. Lipo: Pulse Width Modulation for Power Converters Principles and Practice. John Wiley & Sons, Inc., USA, 2003 Halász S.: PWM Strategies of Multiphase Inverters, IEEE IECON’08, Orlando, USA, 10-13 November 2008, ISBN: 978-1-4244-1766-7, pp. 916-921 F. Baalbergen, P. Bauer,V. Rerucha: Energy Storage and Power Management for Typical 4Q-Load, IEEE Transactions on Ondustrial Electronic, 2009,Vol 56, Issue 5, pp 1485-1498 Jaroslav Dudrik, Vladimr Ruscin: Voltage Fed Zero-Voltage ZeroCurrent Switching PWM DC-DC Converter, 13th International Power Electronics and Motion Control Conference, EPE-PEMC08, Sept. 1-3, 2008 Poznan, Poland, pp. 295-300 J. Leuchter, P. Bauer, P, V. Rerucha: Dynamic Behavior Modeling and Verification of Advanced Electrical-Generator Set Concept, IEEE Transactions on Ondustrial Electronic, 2009, Vol 56, Issue 1, pp266-279 S. Ryvkin, D. Izosimov, S. Belkin: Commutation Laws Transfer Strategy for the Feedforward Switching Losses Optimal PWM for Three-phases Voltage Inverter, IEEE Industrial Electronics Society, IECON’98, 31 Aug.-4 Sept. 1998, Vol.2, pp. 768-773 V. Oleschuk, F. Profumo, A. Tenconi: Simplifying Approach for Analysis of Space-Vector PWM for Three-Phase and Multiphase Converters, EPE’07, Aalborg, Denmark, 2-5 Sept. 2007, CD-ROM ISBN: 9789075815108 M.R Baiju, K.K Mohapatara, K.Gopakumar: Space Vector Based PWM Method for Dual Inverter-Fed Open-End Winding Induction Motor Drive Using only the Instantaneous Amplitudes of Reference Phase Voltages, IEEE PEDS 2003 Conference, Singapore M. Cirrincione, M. Pucci, G. Vitele: Direct Control of Three-Phase VSIs for the Minimalization of Common-Mode Emissions in Distributed Generation Systems, ISIE07, Vigo, Spain, 4-7, June, 2007, pp. 2532 2539, CD-ROM ISBN: 1-4244-0755-9 V. Oleschuk, V. Ermuratski, F. Profumo, A. Tenconi, R. Bojoi, A.M. Stankovic: Novel Schemes of Syncronous PWM for Dual Inverter-Fed Drive with Cancellation of Zero Sequence Currents, Speedam06, Taormina, Italy, 23-26 May, 2006, pp. S22-30 S22-35, CD-ROM ISBN: 1-4244-0194-1 A. Sikorski, T. Citko: Current Controller Reduced Switching Frequency for VS-PWM Inverter Used with AC Motor Drive Applications, IEEE Transactions on Industrial Electronics, October 1998, Vol 45., No 5. pp 792-801 C.-Yi Huang, C.-Peng Wei,J.-Tai Yu, Y.-Jent Hu: Torque and Current Control of Induction Motor Drives for Inverter Switching Frequency Reduction, IEEE Transactions on Industrial Electronics, October 2005, Vol 52., No 5. pp 1364-13 P. Stumpf, Z. Varga, P. Bartal, R. K. Járdán, I. Nagy: Impact of PWM Technique on the Operation of Ultrahigh Speed Induction Machines, EPE’09, 2009, Barcelona, Spain L. Bernardo, V. Moreno-Font, A. El Aroudi, R. Giral: Single Inductor Multiple Outputs Interleaved Converters Operating in CCM, EPEPEMC08 Poznan, Poland, 1-3 September, CD-ROM ISBN: 978-1-42441742-1 J. F. Baalbergen, P. Bauer: Energy Storage and Power Management for Typical 4Q-Load, IECON 2008, Orlando, Florida, USA, 10-13 November, 2008, pp. 2332-2337, CD-ROM ISBN: 978-1-4244-1766-7 A. Aroudi, B. Robert, L. Martinez-Salamero: Modelling and Analysis of Multi-Cell Converters Using Discrete Time Models, Proceedings IEEE International Symposum on Circuits and Systems, 2006, article number 1693046, pp. 2161-2164 I. Nagy, P. Bartal: Quasi-Subharmonics in PWM Inverters, EPE-PEMC, 2010, Ohrid, Macedonia P. Stumpf, R. K. Jardan, I. Nagy: Ultrahigh Speed AC Motor Drive Applied in the Utilisation of Renewables and Recovering Waste Energy, PCIM 2010, Nürnberg, Germany, 4-6 May 2010 Felix Jenni, Dieter Wüest: Steuerverfahren für Selbstgeführte Stromrichter, ISBN: 3-519-06176-7 W. Koczara, N-Alkhayat: “Variable Speed Integrated Generator VSIG as a Modern Controlled and Decoupled Generation System of Electrical Power“, EPE 2005, Dresden, Germany, 11-14 Sept, 2005, CD-ROM ISBN: 90-7581-08-05 M. Orabi, T. Ninomiya: “Analysis of PFC Converter Stability Using Energy Balance Theory”, IECON’03, 2-6 November 2003, Roanoke, Virginia, USA, pp 544-549
