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Subharmonics on Ultrahigh-Speed AC Motor Drives. Rafael K. ... renewable and waste energy recovery. After a brief .... ponents are quite low, which would experimentally be hard ..... small thermal capacity of the rotor accelerates the failure process at .... industrial drives with mechanical elasticity using nonlinear adaptive.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 58, NO. 4, APRIL 2011

A Novel Approach in Studying the Effects of Subharmonics on Ultrahigh-Speed AC Motor Drives Rafael K. Jardan, Member, IEEE, Peter Stumpf, Student Member, IEEE, Peter Bartal, Zoltan Varga, and Istvan Nagy, Fellow, IEEE

Abstract—This paper is concerned with the adverse effect of subharmonics generated by a pulsewidth-modulation-controlled voltage source converter supplying an ultrahigh-speed induction machine (USIM) and also with the calculation, simulation, and test of the subharmonic generation. It is mostly believed, except for a few publications, that the impact of subharmonics is negligible in most modern drives. Our finding is the contrary, i.e., it can cause significant additional losses that might lead even to breakdown in special cases like USIMs applied in systems developed for renewable and waste energy recovery. After a brief description of the system, the generation of subharmonics and the additional losses resulting from subharmonics in USIMs are discussed. Index Terms—Harmonic analysis, high-speed induction machines, pulsewidth-modulated power converters.

I. I NTRODUCTION

S

ERIOUS efforts have been focused on the generation of electric energy from renewable energy sources and waste energies. A novel paradigm in electric power production is its generation in small- and medium-scale power plants (Disperse Power Plants) utilizing local energy sources, thus reducing the environmental pollution and saving the cost and losses of power transmission and distribution by harvesting local renewable and waste energies and losses [1]–[3]. The system described briefly has been developed basically for utilizing a range of waste energies and renewable energy resources. An example for one of these systems includes the energy that can be extracted during the process of pressure reduction in steam and gas networks. The pressure reduction is carried out traditionally by using special throttle valves; however, in this way, an environment-friendly possibility to generate electric power is missed. The basic idea of the subject is elaborated, and the research work is described in more detail in previous papers [4]. Manuscript received September 1, 2009; revised May 13, 2010, July 28, 2010, and October 5, 2010; accepted November 15, 2010. Date of publication December 10, 2010; date of current version March 11, 2011. This paper was supported in part by the Hungarian Research Fund (OTKAK72338), by the Control Research Group of the Hungarian Academy of Sciences, by the Hungarian Science and Technology Foundation in the framework of IT-20/2007 project, and by the European Economic Area (EEA)/Norwegian Financial Mechanism—HU0121-GAN-00064-E-V1. This work is connected to the scientific program of the “Development of quality-oriented and cooperative R+D+I strategy and functional model at BME” project. This project was supported by the New Hungary Development Plan (Project ID: TAMOP-4.2.1/B-09/1/KMR2010-0002). The authors are with the Department of Automation and Applied Informatics, Budapest University of Technology and Economics, 1117 Budapest, Hungary (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TIE.2010.2098353

The electromechanical energy conversion in the system is performed by a turbine–generator set. The system applied an ultrahigh-speed induction machine (USIM) (up to 90 kr/min) to achieve reduced sizes and increased efficiency and to match the speed of the generator and the turbine. The problems of high-speed drives are not widely covered [4]–[6]. During the development, considerable difficulties have been encountered in connection with the application of USIM. Efforts to solve the problems caused by the generation of subharmonics at low mf values have been reported. In [5], an approach is described based on the application of current source inverters. In [6], another solution has been chosen for a real ultrahigh-speed (500 kr/min) application: Unmodulated square-wave converter supplied by a dc/dc converter is applied. The source of difficulties was the interaction of the voltage source converter (VSC) and the USIM. It is widely known that the higher harmonic contents of the voltages and/or currents, supplied by the converters, result in a number of undesirable effects, e.g., additional copper losses due to current harmonics, additional iron losses caused by flux harmonics, currents through the ball bearings that can reduce their lifetime, accelerated aging of the insulation due to high dv/dt, torque pulsation due to current ripples, etc. A different kind of difficulty is caused by subharmonics. It is mostly believed, except for a few publications, 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 USIMs. Investigations following several unexpected damages of USIM have revealed that phenomena connected with subharmonics generated by a pulsewidthmodulation (PWM) converter could be accounted for with these damages. The results of the research work carried out in this area are presented hereby. The basic features of the PWM-converter-fed USIMs, namely, the necessarily high fundamental or reference (fr ) frequency and the limited carrier frequency fc , result in a low mf = fc /fr frequency ratio that leads to stator voltage and current harmonic spectra far more unfavorable as compared to those obtained at standard fundamental frequencies. The widely used naturally sampled (NS) PWM applying triangular carrier signal vtri with frequency fc to compare against the sinusoidal reference wave with rated stator frequency frn = 1500 Hz was applied. The generally accepted rule of thumb in the literature is that the frequency ratio mf = fc /fr should be higher than 21 for applying asynchronous PWM technique. This would require, in our application, at least fc = mf · frn = 21 · 1500 Hz = 31.5 kHz. Such a high fc frequency cannot be

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III. T HEORETICAL BACKGROUND

Fig. 1.

Waste and renewable energy recovery system.

set in the converters available on the market even though the recent development in insulated-gate bipolar transistor technology would make it technically feasible. The maximum value of fc is typically about 16 kHz, i.e., the frequency ratio in this case is mf = 10.66, much lower than it would be necessary. PWM methods have been discussed in two excellent papers [7], [8] and, later, in more detail in a book [9]. In [8], the authors stress that the presence of subharmonics can have “dangerous consequences to electric motor” and offer to “use a modulation method called “uniform sampling.” We could not follow this advice since we used converters that are available on the market. The theory of PWM techniques is well covered in the literature [10]–[45]. However, the laboratory tests sometimes produce unexpected results like, in our case, when we detected subharmonics with levels that cause problems in the operation of special machines, e.g., USIMs. One of the main reasons for generating subharmonics is the imperfect operation of the microcontrollers, commanding the PWM pattern at low mf values. The subharmonics generated at low mf values have significant effects on the operation of high-speed machines enhanced by their unusual parameters, e.g., by their very small stator resistance. Here, the additional rotor copper losses resulting from subharmonics will be treated in Section V after a brief description of the system in Section II, followed by a short overview of the theoretical background, simulation, and test results in Sections III–V, respectively. The conclusion is found in the last section.

II. S YSTEM The system has four main components (Fig. 1): turbine (T), induction generator (USIM), AC/DC Converter 1, and DC/AC Converter 2. The inlet pi and outlet po pressure difference of the turbine, Δp = pi − po , is approximately proportional to the turbine power that is converted to electric power by the three-phase USIM. The electric power is fed to the mains by the PWM-controlled converters through a dc link. The DC/DC Converter 3 absorbs the kinetic energy when the system is turned off in fault conditions. Three control loops are shown in Fig. 1, from left to right: outlet pressure po , speed Ω, and vDC voltage control loop. The controllers are as follows: Pr.C. is the pressure, S.C. is the speed, and V.C. is the voltage controller. In the last control loop, there is a fourth controller (P-Q. C.), i.e., the active power P and the reactive power Q controller. Due to high speed, some basic parameters of USIM are considerably different from those of conventional machines [4].

One of the most common forms of NS PWM that uses a triangular carrier signal with amplitude Vˆtri and frequency fc to compare against the sinusoidal reference waveform vsine = Vˆr sin ωr t with frequency fr is considered. The modulation produces the PWM signal vPWM at the output terminal of the converter measured to the center of the dc supply voltage. The signal vPWM contains a single fundamental component with frequency fr and the groups of sideband harmonics around the carrier and multiple carrier harmonics. The frequency of the sideband harmonics grouped around the multiples of the carrier frequency fc is fsideb = ±(m · mf ± n)fr

(1)

where mf = fc /fr , m = 1, 2, . . ., and n = 1, 2, . . .. One constraint is given as follows: When m is odd, then n = 2, 4, . . ., and when m is even, then n = 1, 3, . . .. How the generation of subharmonics can be understood? There are two ways to explain the development of subharmonics. The first explanation is based on the theory resulting in (1), and it states that subharmonics are the lower sideband harmonics intruding into the frequency range that is lower than fr = 1 p.u.[9]. In such a case, fsideb = fs , and the subharmonic period is Ts = 1/fs . The subharmonic period Ts can be either integer or integer + fraction. The second explanation does not need any sophisticated theory, and it is applicable when mf is not an integer but a rational number, e.g., mf = 7.02. The reason is that, now, mf can be written always as mf = N/D = Ns /Ds

(2)

where N , D, Ns , and Ds are integers. Ns /Ds is the simplest form, i.e., 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 = T /Tc = Ns /Ds , it results in Ds Tr = Ns Tc . We conclude that, in Ds number of reference period, the number of carrier period is exactly Ns . Consequently, now, the subharmonic period is Ts = Ds , and it is integer as Tr = 1 p.u. The relation between the two subharmonic periods is Ts = pTs , and since Ts is a subharmonic period as well, p must be integer. With both Ns and Ds being integers, the voltage vPWM is in frequency-locked state. (When mf is irrational, we have quasiperiodic state, but it is out of the scope of this paper.) IV. S IMULATIONS 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 (SV) transformation block that yields the complex time function of the output voltage SV. The simulations were performed to lay the foundation for the evaluation of the test results presented in

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Fig. 2. Two components of the flux SV. (a) Ψ1 > Ψsub . (b) Ψ1 < Ψsub .

Section V. The actual levels of the subharmonic voltage components are quite low, which would experimentally be hard to extract from the output voltage time function, consisting of a train of high-voltage pulses. In the tests of subharmonic components, an efficient and especially convenient approach is to study the SV of the time integral of the converter output voltage SV. Its dimension is in volt–seconds; thus, we call this variable “flux” and denote it by Ψ. It is approximately the stator flux of the induction machine at higher frequencies. When the flux trajectory is observed on the display of a Cathode Ray Oscilloscope (CRO), the most important features connected with the PWM of the convertermachine system are seen at a glance even if the reasons are not clear. The approach is particularly useful in our case because it articulates the subharmonics of very small frequency we are interested in. First, the basic properties of flux SV are clarified (Fig. 2). Let us add subharmonic flux SV Ψsub = Ψsub · ejωsub t to the fundamental component of flux SV Ψ1 = Ψ1 · ejω1 t , where ω1 and ωsub are the fundamental and the subharmonic angular frequency, respectively. Furthermore, ω1 = 2πf1  ωsub = 2πfsub . In Fig. 2(a), Ψ1 > Ψsub , while in Fig. 2(b), Ψ1 < Ψsub . As long as Ψsub completes one full turn, Ψ1 makes large number of turns. This explains why a ring-shape image can be seen on the display of the CRO when it is switched to “persist” mode. The absolute value of the subharmonic flux is equal to the half of the thickness of the ring in Fig. 2(a), and it equals the mean radius of the inner and outer circles in Fig. 2(b). We know that the subharmonic component of the output voltage vPWM of VSC vsub = Vsub · ejωsub t is much smaller than its fundamental component v1 = V1 · ejω1 t , i.e., Vsub /V1 is very small. However, after calculating the respective flux components, the ratio of Ψsub /Ψ1 is greatly enhanced due to the integration of time function Vsub and V1 . The integration equals the multiplication of the voltage SV by 1/jω Ψsub = −j(Vsub /ωsub ) · ejωsub t Ψ1 = −j(V1 /ω1 ) · ejω1 t .

(3) (4)

The ratio of the peak value of flux components is Ψsub Vsub ω1 = Ψ1 V1 ωsub

(5)

or, in per unit, Vsub = ωsub Ψsub . If ωsub is small enough, Ψsub > Ψ1 can be the result. In the case studied, the frequency of the significant subharmonic component is two to three orders of magnitude lower as compared to

that of the fundamental component; thus, a subharmonic component with an amplitude of 10−3 p.u. magnitude results in a subharmonic flux that is comparable to the fundamental component. This recognition is the basis of the laboratory test method applied for measuring subharmonic voltage components. Figs. 3 and 4 show the simulation results, Ψ1 > Ψsub in Fig. 3 and Ψ1 < Ψsub in Fig. 4. The frequency ratio is mf = 8.1, and the amplitude ratio of the fundamental reference and the triangular carrier wave is ma = 0.8 in Fig. 3(a)–(c). The period of the subharmonic Ts = 10 p.u. in all three figures. Subharmonic voltage is added to the output voltage vPWM with amplitudes of 0.0075 and 0.015 p.u. in Fig. 3(b) and (c), respectively, with period Ts = 10 p.u., while no subharmonic voltage is added to vPWM in Fig. 3(a). The trajectories rotate ten times in all figures before they close on themselves. The subharmonic voltage widens the “ring” of the flux trajectory pictures. The fundamental component of flux Ψ1 = 0.816/2πf1 = 0.13 p.u. (VLL = 1 p.u. and f1 = 1 p.u.), which is the radius of the flux pattern in Fig. 3(a). From (3), the values of Ψsub = 0.0075 · 10/2π = 0.012 p.u. in Fig. 3(b) and 0.015 · 10/2π = 0.024 p.u. in Figs. 3(c) and 2(a) explain the structures in Fig. 3(b) and (c). The slowly rotating subharmonic is imposed on the fast circulating flux having no added subharmonic voltage. Similar flux patterns are shown in Fig. 4, where mf = 27.985 and ma = 0.8 [also see Fig. 5(b)]. The subharmonic period Ts = 200 p.u. in all figures. The amplitudes of the subharmonic voltage added to vPWM are 0.0075 and 0.035 p.u. in Fig. 4(b) and (c), respectively, and no subharmonic is added in Fig. 4(a). From (4), Ψ1 = 0.816/2π ≈ 0.13 p.u. is the radius of the flux pattern in Fig. 4(a). From (3), the Ψsub in Fig. 4(b) is Ψsub = 0.0075 · 200/2π ≈ 0.2387 p.u., and in Fig. 4(c), Ψsub = 0.035 · 200/2π ≈ 1.1 p.u.. In both figures, Ψ1 < Ψsub . Fig. 2(b) describes the structures in Fig. 4(b) and (c). It means that the trajectory of the flux without added subharmonics [Fig. 4(b)] is imposed on the slowly circulating trajectory of the flux generated by the added subharmonic voltage. The width of the ring in Fig. 4(b) and (c) is the diameter of the flux pattern in Fig. 4(a), i.e., 2Ψ1 = 2 · 0.13 = 0.26 p.u.. V. T EST R ESULTS A significant number of laboratory tests have been carried out on the system using a two-pole 4.5-kW induction machine with a rated speed of 90 kr/min; thus, the rated stator frequency is frn = 1500 Hz. The converters (CONV-1 and CONV-2, Fig. 1) are identical, 7.5 kW units (Type: Conv-a) connected back-to-back, both applying the PWM technique mentioned right at the beginning in Section III, i.e., NS carrierbased subharmonic modulation. Their maximum fundamental frequency is fr max = 3 kHz, and the carrier frequency is 12 or 16 kHz. Selecting carrier frequency fc = 12 kHz and rated stator frequency frn = 1500 Hz makes mf = 8. To study the generation of subharmonics, further tests have also been completed using two other converters from different manufacturers, namely, Conv-b (power level Pn = 1.4 kW, fr max = 500 Hz, and fc = 2900 or 5900 Hz) and Conv-c (Pn = 7.5 kW, fr max = 2 kHz, and fc = 12 kHz). Our main

JARDAN et al.: NOVEL APPROACH IN STUDYING THE EFFECTS OF SUBHARMONICS ON AC MOTOR DRIVES

Fig. 3.

Flux trajectories. mf = 8.1 and ma = 0.8. (a) Vsub = 0 p.u. (b) Vsub = 0.0075 p.u. (c) Vsub = 0.015 p.u.

Fig. 4.

Flux trajectories. mf = 27.985 and ma = 0.8. (a) Vsub = 0 p.u. (b) Vsub = 0.0075 p.u. (c) Vsub = 0.035 p.u.

objective with these tests is to obtain authentic data for the calculation of the additional rotor copper losses generated by the subharmonics. Furthermore, illustrative examples of test results are presented to prove the existence of significant subharmonics. First is to compare the simulation and test results as shown in Fig. 5(a) and (b). The trajectory of Ψ is moving around the center point and closes on itself after 20 cycles of the fundamental. Its thickness demonstrates the presence of significant subharmonics (see Fig. 3). Another convincing example is shown in Fig. 5(b) using Conv-c. The trajectory of the stator flux is recorded for an extended period of time at fr = 428.8 Hz and fc = 12 kHz, i.e., mf = 27.985. Due to the presence of subharmonic component, the trajectory moves inside a wide ring with a width of ΔD. The diameter of the path of the fundamental component is D. The ratio ks = ΔD/D is proportional to the resulting subharmonic component in per unit. It was obvious from the time function of the phase quantities that Ψ1 > Ψsub [Fig. 2(a)]. The laboratory tests have shown that, due to imperfect operation of the PWM controller of VSC and distortion of the output voltage waveform vPWM caused by side effects (like rippled dc input voltage, voltage drops and limited switching times of the switching devices, blanking times, etc.), one or more of them can generate subharmonics that are significantly different from those obtained in ideal conditions and much more significant.

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Fig. 5. SV trajectory of Ψ. (a) fc = 12 kHz and mf = 8.05. (b) fc = 12 kHz and mf = 27.985.

The laboratory test results of the actual system have shown that subharmonic components can be measured both at frequencies corresponding to theoretical predictions (let us call them regular components) and some irregular components with frequencies and magnitudes completely unexpected, unpredicted by theory based on ideal conditions. Regular components with levels worth mentioning can be detected in the range of low mf values having a form of integer.fraction. Irregular subharmonic components (ISCs) can be recorded in a wide range of mf , including high values, where theoretically no subharmonic components could occur or only with totally negligible levels.

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Fig. 6. Test results: Subharmonic component ks versus mf .

A. ISC Testing the level of subharmonics this way in the full range of operation, a map of the dangerous points of irregular subharmonics can be obtained. The results of these tests revealed considerable levels of subharmonics. Up to now, we could not find its theoretical explanation; thus, we called them ISCs. A map of this kind, based on the test results, is shown in Fig. 6 (carrier frequency fc = 12 kHz). These tests are quite labor intensive, as the subharmonics appear with high amplitudes in a very narrow range around the integer mf values; thus, a high number of working points have to be measured. Athough the parameters of the ISCs are generated by the imperfect operation of the PWM converter unit, the behavior of the resulting subharmonics shows very interesting and consistent relationships. In the close vicinity of a given mf , where high level of ISC is detected, we have changed the fundamental frequency in 0.1-Hz steps and measured the subharmonic flux versus mf . In order to measure the amplitude of the subharˆ sub,m , first, the real and the imaginary components monic SV Ψ of the inverter output voltage SV VPWM are generated. Next, the time integrations of the two components are performed by two separate R–C circuits. They are the real and imaginary components of a voltage SV Vi , which is proportional to the flux SV Ψ. Next, the Fourier series of the last two components is produced. We can read from it the amplitudes both of the fundamental Vˆi1 and the subharmonic Vˆi,sub components of Vi (a recorded spectrum is shown in Fig. 8). By keeping in mind the approximate R–C integration, the measured amplitude Vˆsub,m of the subharmonic voltage in VPWM can be calculated from  Vˆsub,m = Vˆi,sub 1 + (ωsub T )2 (6) where T = RC, the time constant of the RC integrator. When T  1/ωsub Vˆsub,m ≈ ωsub T Vˆi,sub .

(7)

Even if T is comparable to 1/ωsub , by using (6) rather than (7), the calculated subharmonic amplitude will be correct. Taking (3) into consideration, the amplitude of the subharmonic ˆ sub,m = T Vˆi,sub . flux Ψ

Fig. 7. Test results: Amplitude and frequency of the ISC stator flux and additional rotor loss.

ˆ sub,m , as well as calculation Test results for fsub and Ψ results for λ obtained in the range of mf = 14 ± 0.03 (carrier frequency fc = 16 kHz), are shown in Fig. 7. λ is the additional copper loss in per unit, λ(mf ) will be explained in detail in Section VI. The frequency of the ISC is a linear function of the difference between the actual mf and mf = 14, while the resulting subharmonic flux and voltage components can be approximated by a higher order function. The amplitude and the frequency of the subharmonic component can be taken from the Fourier spectrum of the real or imaginary component of flux SV time function as shown in the display of the CRO. These values were confirmed also by measuring the frequency and the amplitude of the envelope of the time function of the flux SV for a full subharmonic period. These two measurements gave close results, but reading the fast Fourier transform values yields more accurate measurement.

B. RSC The test results of the regular subharmonic component (RSC) are described briefly in the following part. By definition, we were testing the subharmonic component that can be calculated from mf , i.e., fsub = 0.5fr at mf = 8.5. Also at this subharmonic component, interesting relationships have been found. In the vicinity of mf = 8.5, the frequency and the amplitudes of the subharmonic component were measured similarly as described in the previous point. The level of this subharmonic flux component resulting from RSC is significantly lower compared to that of ISC, and their effect is negligible.

JARDAN et al.: NOVEL APPROACH IN STUDYING THE EFFECTS OF SUBHARMONICS ON AC MOTOR DRIVES

Fig. 8. Subharmonic range of the harmonic spectrum of the flux (mf = 8.5185).

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Fig. 10. Torque–speed characteristics for rated (curve A) and for subharmonic (curve B) quantities.

torque–slip characteristics that were used to calculate losses. In general, the torque τ forms the Kloss formula τ=

2τb ssb (1 + ) s2 + s2b + 2ssb

(8)

where s is the slip, τb and sb are the breaking torque and breaking slip, respectively, and

Fig. 9.

τb =

3V 2  ph   2Ω1 (1 + σ)2 Rs + Rs2 + Xr2

σ=

Xls Xm

Frequencies of the RSC in the vicinity of mf = 8.5.

As expected from theory, at mf = 8.5, the frequency of the flux SC, obtained by measurement, is precisely the half of the frequency of the fundamental component. However, when deviating from mf = 8.5, the SC is split into two separate components, with frequencies fsub1 and fsub2 , like upper and lower sidebands. They are apparently resulting from sidebands of higher harmonic components intruding below the fundamental [9]. A harmonic spectrum showing the subharmonic range of the flux is shown in Fig. 8, where the fundamental component, the ISC, and two sidebands of RSC can be seen at mf = 8.5185. The frequency of the first component is increasing, the second component is decreasing linearly with mf , and the two curves are crossing over at mf = 8.5, as shown in Fig. 9. The amplitudes of the two SCs are practically equal.

VI. A DDITIONAL L OSSES D UE TO S UBHARMONICS To calculate the rotor copper losses, the operation of the machine is assumed to be in the generating mode at the rated working point Pn on the rated torque–speed characteristics (A) of the machine (Fig. 10). In addition to the fundamental stator voltage, we add to it the low-frequency ISC Vsub calculated from the measured flux Ψsub,m and from the angular frequency ωsub by using (6) or (7). We have used the refined form of the Kloss formula to determine the working points on the

sb = 

Rr + Xr2

Rs2

Xr = Xlr + =

Xls , 1+σ

Rs . + Xr2

Rs2

Vph is the rms phase voltage. The rated quantities and parameters of the USIM applied in our experimental system are used in the calculation (see Appendix). Equation (8) was used to calculate Ω(τ ) belonging to the fundamental (curve A) and to the subharmonic (curve B) voltage. The torque–speed characteristics of a subharmonic component (B) have a breaking torque τbsub at point P2 (Fig. 10). The torque generated by the subharmonic component is τsub in the working point (P3 ) at the rated speed Ωn . The speed difference ΔΩsub = Ωn − Ω1sub is the same as the angular frequency of the rotor current (as p = 1) generated by the subharmonics. The power absorbed by the machine from the turbine through the shaft is Prsub = τsub ΔΩsub . It is assumed that this power is completely turned into copper loss in the rotor. The rated copper loss resulting from the

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TABLE I R ATED DATA OF USIM

R EFERENCES

fundamental component is Prn = τ1 Ωn sn , and the ratio of the two copper losses is λ = Prsub /Prn

(9)

which is actually the resulting additional loss in per unit. The calculated loss ratio λ versus mf based on measurements was shown in Fig. 7. λ increases near 0.3, explaining convincingly the failures of USIM. For the sake of illustration, the slip values in Fig. 10 are exaggerated.

VII. C ONCLUSION This paper has been concerned with the source of unexpected failures of a USIM supplied by a three-phase asynchronous PWM converter applied in a system converting renewable and waste energies into electrical energy. Due to the imperfect output voltage pattern of the PWM converter, subharmonic stator flux develops in the USIM, resulting in significant additional rotor copper loss overheating the machine. Based on the experimental results, the rotor copper losses were estimated by an approximate calculation. At the working point selected for illustration, approximately 30% of the rated rotor loss was calculated as additional rotor copper loss. The relatively small thermal capacity of the rotor accelerates the failure process at particular speeds where substantial subharmonic flux develops. Sweeping the fundamental frequency over the full range up to the rated frequency at various carrier frequencies, high levels of ISCs can be found, and the shapes of the resulting trajectories exhibit a great variety with extremely large subharmonic flux levels. The high level of the additional rotor loss that can be calculated from the results of these tests provides us convincing evidence that, in the field of USIM applications, these losses can be responsible for the damages of the sensitive ultrahighspeed machine.

A PPENDIX See Table I.

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Rafael K. Jardan (M’61) received the Dipl. Ing., “Dr. Techn.,” and Ph.D. degrees in electrical engineering from Budapest University of Technology and Economics (BUTE), Budapest, Hungary, in 1961, 1975, and 1998, respectively. From 1961 to 1991, he was with the Computer and Automation Institute, Hungarian Academy of Science, Budapest. During 1967–1968, he was with the University of Toronto, Toronto, ON, Canada. From 1975 to 1976, he was with the University of Manchester Institute of Science and Technology, Manchester, U.K. From 1991 to 1997, he was the Managing Director of Deltronic Power Electronics Manufacturing and Development Ltd. Since 1997, he has been with BUTE as an Associate Professor. His contributions include two university lecture notes and over 100 papers in professional journals and conference proceeding. He is the holder of 19 Hungarian and European patents. Dr. Jardan has been a member of the Hungarian Electrotechnical Association and the IEEE Industry Application Society since 2003. Peter Stumpf (S’09) was born in Budapest, Hungary, in 1985. He received the M.Sc. degree from Budapest University of Technology and Economics, Budapest, in 2009, where he is currently working toward the Ph.D. degree. He is the coauthor of nine technical papers. His research interests focus on power electronics, nonlinear systems, and modulation strategies.

Peter Bartal received the M.Sc. degree from the University “Politechnica” Timisoara, Timisoara, Romania, in 2004. He was awarded a scholarship to work toward the Ph.D. degree in the Department of Automation and Applied Informatics, Budapest University of Technology and Economics (BUTE), Hungary, Budapest. He is currently an Assistant Lecturer with the Department of Automation and Applied Informatics, BUTE. He works part-time for a private R&D company, MFKK Kft., as a Research Engineer. His research interests include power electronic converters and nonlinear systems. Zoltan Varga was born in Budapest, Hungary, in 1982. He received the M.Sc. degree from Budapest University of Technology and Economics (BUTE), Budapest, in 2008, where he is currently working toward the Ph.D. degree. He is the coauthor of six technical papers. His research interests focus on power electronics, ultrahigh-speed drive systems, and converters.

Istvan Nagy (M’92–SM’99–F’00) received the M.Sc. and Ph.D. degrees from Budapest University of Technology and Economics (BUTE), Budapest, Hungary. He was with the Ganz Electric Factory. He then joined the Hungarian Academy of Sciences (HAS), Budapest, as the Department Head in the Research Institute. Since 1975, he has been a full Professor with BUTE. He was a Visiting Professor with universities in Germany, India, New Zealand, Italy, Canada, U.S., and Japan. He was a Leader or participant in R&D work of considerable number of industrial products. He is the author or coauthor of 8 textbooks, 5 handbooks, and around 300 technical papers. He is the holder of 13 patents. His current research interests include power electronics, automatic electric drives, variable-structure nonlinear systems, and nonlinear dynamics. Dr. Nagy is a member of the HAS and the Hungarian Electrotechnical Association, Chairman of Power Electronics and Motion Control (PEMC) Council, President of the Hungarian CIGRÉ (International Council on Large Electric Systems) Committee.