Effects of Magnetic Saturation on Induction Machines Driven by Static Converters

EFFECTS OF MAGNETIC SATURATION ON INDUCTION MACHINES DRIVEN BY STATIC CONVERTERS

Darizon A. de Andrade∗ Marcos A. A. de Freitas†

Luciano M. Neto∗ H´elder de Paula∗

Jos´e L. Domingos‡

∗FEELT-UFUCaixa Postal 2160 CEP 38.400-902 - Uberlˆandia - MG †FEIT-UEMGCaixa Postal 431 CEP 38.302-192 - Ituiutaba - MG ‡CEFET/GORua 75, Nr. 46, CentroCEP 74.055-110 - Goiˆania - GO

ABSTRACT magnética na opera¸cão de motores de indu¸cão, quando acionados por conversores estáticos. Para estudos de The effects of magnetic saturation on the operation of in- simula¸cão um modelo matemático baseado em fun¸cões duction motors driven by static converters are analyzed. harmônicas magnéticas é utilizado para levar em conta A mathematical model based on magnetic harmonic a satura¸cão. Observa-se que distor¸cões no fluxo de en- functions is used to account for saturation. Distortions treferro devido às caracter´ısticas magnéticas não line- on the air gap flux due to non-linear magnetic charac- ares resultam em harmônicos espaciais na distribui¸cão teristics lead to appearance of space harmonics in the da densidade de fluxo resultante. Estes por sua vez, resultant flux density distribution. This causes specific causam distor¸cões nas grandezas temporais de estator e distortions in stator and rotor time quantities, which rotor, diferentes daquelas devido ao conversor estático. are different from those due to static converter. Op- Aopera¸cão do motor em condi¸cões de satura¸cão éana- eration with six pulse and sinusoidal PWM converters lisada para conversores de seis pulsos e PWM senoidal. under saturated conditions is considered. Comparisons Acompara¸cão dos resultados experimentais com os si- of experimental and simulated results are presented and mulados mostrouotima ´ concordância. found to be in very good agreement. PALAVRAS-CHAVE:Máquinas de indu¸cão, modelagem KEYWORDS: Induction Machine Drive, Magnetic Satu- matemática, satura¸cão magnética, acionamento está- ration. tico.

RESUMO 1 INTRODUCTION

Neste trabalho são analisados os efeitos da satura¸cão Induction machines driven by static converters are widespread in the industry. The majority of works Artigo submetido em 04/10/02 found in literature neglect the effects of magnetic sat- 1a. Revisão em 25/11/02; 2a. Revisão em 19/05/03 Aceito sob recomenda¸cão do Ed. Assoc. Prof. José A. Pomilio uration and air gap spatial flux harmonics, limiting the

Revista Controle & Automa¸cão/Vol.15 no.2/Abril, Maio e Junho 2004 181 study to superficial analyses. Due to the non-linear re- are determined in their own referential, providing a bet- lation between flux and magnetizing current, either dq ter physical insight about their behavior. Although the or phase quantities based models lead to inaccurate and final equations to implement the model come out very uncertain results under several operation characteristics simple, the full development is quite involved and is ex- (Levi, 1994). A great number of papers is focused on plained in detail in the above references. Following, a the development of mathematical models for a suitable simple description aiming to give a background idea of representation of the magnetic saturation effects. Thus, the modeling is presented. many modeling techniques, with different alternatives to account for such effects, have been proposed. These Considering a generic phase nthat represents a stator or techniques are based mainly on the election of the state rotor phase, and neglecting the machine iron losses, the variables (Levi 1996a and 1996b), small–signal models motor voltage equation is given by

(Melkebeek etal., 1983) and modifications in the ma- dλn Vn = Rnin + (1) chine equivalent circuit (Moreira et al., 1992 and Liao dt etal., 1994). Some works analyze the machine behavior when it is driven by vector-controlled converters, tak- where Vn, Rn, in e λn are the voltage, resistance, current ing into account the magnetic saturation. However, the and total flux linkage of the phase n, respectively. analyses are limited to steady-state conditions and only Assuming the leakage flux as linear, the total phase flux the air-gap fundamental flux component is considered linkage is calculated by (Khater etal., 1987 and Ojo etal., 1994). In Moreira etal.(1992) and Liao, etal. (1994), the influence of λn − λmn λn = Lnin + λmn ⇒ in = (2) spatial harmonics is considered. A modified dq model Ln is developed where the saturation is included through where Ln and λmn are the leakage inductance and mag- the consideration of a variable air-gap length, which is netizing flux of phase n. a function of saturation level and spatial position. The analysis presented refers only to the star connection, and Since Vn and Rn are known, the solution of (1) is pos- the third spatial harmonic is incorporated by the inser- sible using the relation between in and λn given by (2), tion of fictitious windings, thus modifying the machine provided λmnis known. In order to obtain the phase equivalent circuit. flux linkage, only the fundamental component of the resultant spatial distribution of magnetomotive force pro- The aim of this paper is to offer a contribution to the duced by the currents of all rotor and stator phases is topic induction machines driven by static converters, considered. taking into account the effects of magnetic saturation and spatial flux harmonics. By using a suitable mathe- Suppose that the maximum value (FM ) of this spatial matical model, saturation is accounted for and the pa- distribution is located at a position given by an angle α, per presents an analysis of static converter-fed saturated at a certain instant of time, defined in a θ reference axis induction-machines. Experimental and simulated re- of which origin is coincident with the phase a winding sults are obtained and analyzed, leading to important axis (figure 1). The distribution of magnetomotive force conclusions about drive operation. is written as

2 MATHEMATICAL MODELING mmf(θ)=FM cos(θ − α)= 2Knin cos(θ − θn)(3) The mathematical model uses a phase quantity based n=a,b,c,A,B,C modeling approach, developed by Bispo etal. (2001), Neto etal. (1999a and 1999b) and Resende etal. (1999), where using the concept of harmonic magnetic function. The total phase ﬂux linkages are taken as state variables. FM - amplitude of the spatial distribution of magneto- There is no use of variable transformations as commonly motive force; used in other works in order to represent the cross- θ - angular displacement along the reference axis; saturation. The eﬀects of the 3rd spatial harmonics are considered without any manipulation in the equivalent Kn - winding factor of rotor and stator phases (a, b, c, circuit and with no requirement for the machine design A, B, C); data. Only the knowledge of machine terminal quantities are required to quantify the saturation. Quantities θn - generic position of the rotor and stator phases (θa,θb,θc,θA,θB,θC );

182 Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 α - angular position of the m.m.f peak. the winding due to B(θ)is λmn = Fh FM cos [h (α − θn)] (5) aA bB cC h=1,3 FM where Fh FM is the “magnetic harmonic function”. It is obtained experimentally.

o o o 0 R 120 240

Figure 1: Spatial Distribution of the Magnetomotive Force.

The m.m.f(θ) produces a resultant distribution of magnetic flux density B(θ). Regarding the magnetizing Figure 3: Distribution of (m.m.f(θ), Flux Density and curve which characterizes the machine magnetic circuit, Winding. the m.m.f (θ) distribution produces a non-sinusoidal B(θ), but still symmetric in relation to the m.m.f (θ) axis. Manipulating (2), (3) and (5) as shown by Bispo etal. (2001), the following equations arise The application of Fourier analysis leads to λn fR (λ)= cos (θn)(6) n B (θ)= Bh cos [h (θ − α)] (4) n=a,b,c,A,B,C L h=1,3 λn fI (λ)= sin (θn)(7) n n=a,b,c,A,B,C L where 2 2 th Bh - maximum value of the h component of B(θ); f (λ)= [fR (λ)] +[fI (λ)] (8) fI (λ) h -harmonicorder. tg (α)= (9) fR (λ) Considering the winding of phase n distributed in sev- f (λ) FM F1 FM = − (10) eral coils, each one with N turns, of which central axis is AS AS situated at a θn position (figure 3), the total flux linking 3 1 1 AS = + (11) 2 Ls Lr

where Ls and Lr are the leakage inductances of stator and rotor respectively. Equations (1) to (11) comprise the motor model.

The solution is as follows: from a numerical integration method, (1) is solved. For each integration step, Vn and λn are known. From (6) – (8), f(λ)iscalculated;As is determined in (11). Knowing the magnetic function F1(FM), f(λ)andAs, a line equation is deﬁned by (10). The intersection of this line with the magnetizing curve deﬁnes the values of F1(FM )and FM in each Figure 2: Resultant spatial Distribution of Flux Density integration step. The value of F3 FM is calculated by B(θ) means of numerical interpolation with F1(FM).

Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 183 The harmonic magnetic functions F1(FM)and F3 FM are determined experimentally driving the in- 3rd duction machine at synchronous speed. Diﬀerent values 5th of balanced and distortion-free three-fase voltage are ap- 7th

Amplitude(A)

plied to the stator and the instantaneous values of Vnand Line Current (A) in are digitally stored. This current corresponds to ex- citation current. Taking the iron losses out of the mea- Time(s) Harmonic order (a) (b) sured current and using the machine parameters, the instantaneous values of the magnetizing ﬂux using (1) and rd (2) are determined. Fourier analysis provides F1(FM) 3 5th and F3 FM (ﬁgure 4), corresponding to the amplitude 7th

of its fundamental and third harmonic components. Amplitude(A)

Phase Current (A)

Time(s) Harmonic order (c) (d)

Figure 5: Results for delta connection. (a) Line Cur- rent; (b) Frequency Spectrum; (c) Phase Current; (d) Frequency Spectrum.

Figure 4: Curves Experimentally Obtained: (a) F1 vs (FM);(b)F1 vs (FM).

3rd 5th

Amplitude(A) 7th 3 SIMULATION RESULTS Line Current (A) Time(s) Harmonic order Some simulations were performed in order to verify the (a) (b) behavior of the machine supplied by six step and sinusoidal PWM converters. Analyses were carried for both 3rd 5th star and delta winding connections, with the motor at 7th

no-load. Initially, the results are presented considering Amplitude(A) a linear magnetic circuit; in the sequence, the effects of Phase Current (A) the saturation and spatial 3rd harmonic of the air-gap Time(s) Harmonic order flux are included. In the non-saturated condition, that (c) (d) is, considering the machine magnetic circuit linear, the air gap flux varies linearly with the magnetizing current Figure 6: Results for star connection. (a) Phase Volt- (figure 2). age; (b) Frequency Spectrum; (c) Line Current; (d) Fre- quency Spectrum. 3.1 Six-step motor drive

Figures 5 and 6 show results of simulation consider- and 13th . . . harmonics. ing linear magnetic circuit for the machine. Line and phase currents are shown for the delta connection under Following, the behavior of the machine considering sat- six-step driving in Figure 5. Figure 6 shows the phase uration is studied for the six-step drive. Figure 7 shows voltage and line current for the six-step drive with ma- the phase current for delta connection. Compared with chine windings connected in star. The presentation of figures 5c-5d it is noticed the presence of third harmonic results along the paper is structured such as to show the in this current besides the harmonics due to electronic corresponding frequency spectrum of the waveform un- converter. The saturation of the air gap flux is the re- der analysis. As expected from the analysis with linear sponsible for this harmonic component in the phase cur- magnetic circuit, only the harmonics originated by the rent. Turning to star connection, figure 8, the effect of switching strategy are present in the waveforms, which saturation will lead to distortion in the phase voltage, in the case of six-step drive correspond to 5th, 7th, 11th observed by the “rounding” of the stepped waveform.

184 Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 3rd 3rd 5th 5th 7th 7th 3rd

Amplitude(A)

Amplitude(A) th

Phase Current (A) 5

Line Current (A) 7th

Time(s) Harmonic order Time(s) Harmonic order (a) (b) (a) (b)

Figure 7: Results for delta connection. (a) Phase Cur- rent; (b) Frequency Spectrum. 3rd

Amplitude(A) 5th Phase Current (A) 7th

Time(s) Harmonic order (c) (d)

Figure 9: Results for delta connection. (a) Line Cur- rent; (b) Frequency Spectrum; (c) Phase Current; (d) Frequency Spectrum.

Figure 8: Results for star connection. (a) Phase Voltage; (b) Frequency Spectrum.

3rd th Comparison of figures 8 and 6a-6b show the difference Amplitude(V) 5 Phase Voltage (V) 7th caused by the presence of third harmonic in this wave- Time(s) Harmonic order form. It is important to observe that, as expected, the (a) (b) line currents for both star and delta connection do not show any distortion caused by the third harmonic component of air-gap flux either for delta or star connection.

3rd

Amplitude(A) 5th 3.2 Sinusoidal PWM motor drive Line Current (A) 7th

Simulations were also conducted with the machine be- Time(s) Harmonic order (c) (d) ing fed by a sinusoidal PWM converter. As before, ﬁrst the results for delta and star connections taken from linear magnetic circuit modeling are shown, and in the se- Figure 10: Results for star connection. (a) Phase Volt- quence those corresponding to saturated machine model age; (b) Frequency Spectrum; (c) Line Current; (d) Fre- are placed for comparison. quency Spectrum.

Figure 9 shows the line and phase current for delta connection and ﬁgure 10 shows the phase voltage and line current for star. As noticed, the PWM switching leads to improved waveforms, with negligible presence of low order harmonics. 3rd th Amplitude(A) 5

Phase Current (A) th With simulation including the effects of saturation, the 7 phase currents for delta will show the distortions caused Time(s) Harmonic order by the non-linearity of the magnetic circuit, which is (a) (b) confirmed by the corresponding frequency spectrum as seen in figure 11. For the star connection case, the dis- Figure 11: Results for delta connection. (a) Phase Cur- tortion is caused in the phase voltage as observed in rent; (b) Frequency Spectrum. figure 12.

Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 185 Figure 12: Results for star connection. (a) Phase Volt- age; (b) Frequency Spectrum.

Figure 14: Eletromagnetic Torque.

3rd th Amplitude(A) 5

Phase Current (A) 7th

Time(s) Harmonic order (a) (b)

Figure 13: Rotor Current and Frequency Spectrum. 3rd 5th

Amplitude(A) 7th

Phase Current (A) Rotor quantities, as well as the electromagnetic torque, Time(s) Harmonic order will also be affected by the air-gap flux saturation. Figs. (c) (d) 13 and 14 show the simulated waveforms of the rotor current, its frequency spectrum and the steady-state electromagnetic torque. The “flattened” air gap flux wave Figure 15: Phase current, delta connection. Operation induces distorted voltage in the rotor, which in turn es- with voltage boost - (a) Linear Model; (b) Frequency tablishes “flattened” rotor currents. Machine torque be- Spectrum; (c) Non-linear Model; (d) Frequency Spec- ing the result of interaction of air-gap flux and rotor trum. current spatial distributions will be affected resulting in low frequency oscillations as shown in Fig. 14. 20% higher). This indicates a considerable error when Another situation analyzed is the operation with voltage using a linear model for this operation condition. Fur- boost. A voltage 20% greater than the rated V/f value thermore, the increased saturation leads to a stronger was supplied to the motor. This can occur in real cases flattening in the flux distribution, which is reflected in when the stator voltage drop compensation for lower op- higher third harmonic component in the phase current, eration frequencies is excessive. In such situations, since as noticed in figure 15. Comparing with figure 11a, a the motor operates at a higher saturation level, the fun- threefold increase in the magnitude of the third har- damental component of the stator current is increased, monic is observed. as well as the presence of the spatial third harmonic. The results are observed in figure 15. Comparing fig- 4 EXPERIMENTAL RESULTS ures 15a-15b, that correspond to linear model, with figures 15c-15d, from the saturated model, one can note In order to confirm and validate the previous studies, the higher value of fundamental component of current experimental results with six-step and sinusoidal PWM that occurs in the saturated condition (approximately modulation techniques are shown in the following. The

186 Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 1) Line Current (A): 3 Volt 10 ms 1) Line Current (A): 2 Volt 5 ms

3rd 5th 7th 3rd th

Amplitude(A) Amplitude(A) 5

Line Current (A)

Line Current (A) 7th

Time(s) Harmonic order Time(s) Harmonic order (a) (b) (a) (b)

1) Phase Current: 2 Volt 10 ms 1) Phase Current (A): 1 Volt 5 ms

3rd 5th 7th 3rd

Amplitude(A) Amplitude(A) 5th

Phase Current (A) Phase Current (A) 7th

Time(s) Harmonic order Time(s) Harmonic order (c) (d) (c) (d)

Figure 16: Results for delta connection. (a) Line Cur- Figure 18: Results for delta connection. (a) Line Cur- rent; (b) Frequency Spectrum; (c) Phase Current; (d) rent; (b) Frequency Spectrum; (c) Phase Current; (d) Frequency Spectrum. Frequency Spectrum.

1) Phase Voltage (V): 50 Volt 10 ms third harmonic is present in the phase current, as con- ﬁrmed by the frequency spectrum. Phase voltage and

3rd line current from star-connected operation are shown in 5th ﬁgure 17. The “rounded” shape indicated by the simula- Amplitude(V) 7th Phase Voltage (V) tion is conﬁrmed in this measurement, and the amount of third harmonic present due to saturation is indicated Time(s) Harmonic order (a) (b) in the waveform frequency spectrum.

1) Line Current (A): 2 Volt 10 ms

3rd Phase voltage and line current from star-connected op- 5th eration are shown in ﬁgure 17. The “rounded” shape 7th indicated by the simulation is conﬁrmed in this measure-

Amplitude(A)

Line Current (A) ment, and the amount of third harmonic present due to saturation is indicated in the waveform frequency spec- Time(s) Harmonic order trum. (c) (d) 4.2 Sinusoidal PWM motor drive Figure 17: Results for star connection. (a) Phase Volt- age; (b) Frequency Spectrum; (c) Line Current; (d) Fre- Measurements with a 4 kHz sinusoidal PWM commer- quency Spectrum. cially available converter were also taken as shown in figure 18. There is a clear presence of the harmonics in the phase current due to saturation for the delta con- motor is driven at no-load. A six-step converter with nection (figures 18c-18d). As observed in the simulation, IGBT module was built in the laboratory for experi- the line current is free from the saturation harmonic. ments. PWM modulation driving was realized with a Distorted phase voltage is observed in the measurement commercial converter. shown in figure 19 for the star connected motor, con- firming the predicted by simulation. 4.1 Six-step motor drive Finally the measurement for the operation with voltage Figure 16 presents the voltage and six-stepped line and boost is shown in figure 20a. A V/f 20 % higher than phase currents waveforms for the delta-connected motor. rated ratio was applied to the motor. Measured results It is observed that the results are in very good agreement are quite in accordance with those predicted by satu- with those originated by simulation. As expected, the rated model, as show in fig. 15.

Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 187 1) Phase Voltage (V): 200 Volt 5 ms tions in the motor torque.

REFERENCES

3rd Amplitude(V) 5th

Phase Voltage (V) Bispo, D., et al., (2001). A New Strategy for Induction 7th Machines Modeling Taking into Account the Mag- Time(s) Harmonic order (a) (b) netic Saturation. IEEE Transactions on Industry 0 1) Line Current (A): 1 Volt 5 ms Applications. Vol. 37, N . 6, pp. 1710–1719.

Khater, F. M., et al. (1987). Selection of Flux Level in

3rd Field-Oriented Induction Machine Controllers with th Amplitude(A) 5 Consideration of Magnetic Saturation eﬀects. IEEE

Line Current (A) 7th Transactions on Industry Applications. Vol. IA-23, 0 Time(s) Harmonic order N . 2, pp. 276 – 282. (c) (d) Levi, E. (1994). Applications of the current state space model in analyses of saturated induction machines. Figure 19: Results for star connection. (a) Phase Volt- 0 Electric Power Systems Research. Vol. 31, N .3, age; (b) Frequency Spectrum; (c) Line Current; (d) Fre- pp. 203-216. quency Spectrum. Levi, E. (1996a). Main Flux-Saturation Modelling in d- q Axis Models of Induction Machines Using Mixed 1) Phase Current (A): 2 Volt 5 ms Current-Flux State-Space Models. ETEP.Vol.6, N0. 3, pp. 207–215.

3rd 5th Levi, E. (1996b). Main Flux Saturation Modelling in

Amplitude(A) th Phase Current (A) 7 Double-Cage and Deep-bar Induction Machines. IEEE Transactions on Energy Conversion. Vol. 11, Time(s) Harmonic order 0 (a) (b) N 2, pp. 305–311. Liao, Y., et al. (1994). Eﬀect of Saturation Third Har- Figure 20: Operation with voltage boost. (a) Phase monic on the Performance of Squirrel-Cage Induc- Current (delta connection); (b) Frequency Spectrum. tion Machines. Electric Machines and Power Sys- tems. Vol. 22, pp. 155-171.

5 CONCLUSIONS Melkebeek, J. A. A., et al. (1983). The Influence of Saturation on Induction Machine Drive Dynamics. This work has presented a theoretical-experimental an- IEEE Transactions on Industry Applications.Vol. 0 alyzis of the induction motor behavior fed by static con- IA-19, N 5, pp. 671-681. verters considering the magnetic circuit saturation. Six- Moreira, J. C., et al. (1992). Modelling of Saturated ac step and sinusoidal PWM modulation techniques were Machines Including Air Gap Flux Harmonic Com- studied and implemented for both delta and star wind- ponents. IEEE Transactions on Industry Applica- ing connections. The comparison between the exper- tions. Vol. 28, N0 2, pp. 343–349. imental and simulated results obtained with the non- linear model shows very good agreement. Analyzes Neto, L. M., et al. (1999). Analysis of a Three-Phase based on linear magnetic circuit are inherently inaccu- Induction Machine Including Time and Space Har- rate, since the harmonic components are not accounted monics Effects: The A, B, C Reference Frame. for neither in the stator and rotor quantities nor in the IEEE Transactions on Energy Conversion. Vol. 14, electromagnetic torque. It was observed that the ef- N0 1, pp. 80–85. fects of the air gap flux harmonics are manifested in the phase currents for the delta-connected machine, while Neto. L. M., et al. (1999). Magnetic Saturation Ef- the phase voltages are affected when the machine is star- fect on a Three-Phase Induction Machines. IEEE connected. In addition, saturation effects are also ob- SDEMPED’99 – International Symposium on Di- served in rotor quantities and electromagnetic torque, agnostics for Electrical Machines, Power Electron- as third-order harmonics in rotor currents and oscilla- ics and Drives, Spain, pp. 1 - 3.

188 Revista Controle & Automa¸cão/Vol.15 no.2/Abril, Maio e Junho 2004 Ojo, O., et al. (1994). Steady-State Performance Evalua- tion of Saturated Field Oriented Induction Motors. IEEE Transactions on Industry Applications,Vol. IA-30, N0 6, pp. 1638–1647. Resende, J. T. (1999). Modelagem da Máquina de In- du¸cão Trifásica, incluindo a Satura¸cão Magnética –Análise Dinâmica do Gerador de Indu¸cão Auto- Excitado. Tese de Doutoramento, UFU, Uberlân- dia.

APPENDIX I - MOTOR PARAMETERS

Table 1: Induction Motor Data. Output rated power 1.5 kW Rated speed 1720 r/min Line voltage (∆/Υ) 220/380 Volts rms Stator resistance 3,11 Ω Rotor resistance 3,83 Ω Stator leakage inductance 8,4 mH Stator leakage inductance 8,4 mH Magnetization inductance 127 mH Rotor inertia 0.0074 Kg-m2

Revista Controle & Automa¸c˜ao/Vol.15 no.2/Abril, Maio e Junho 2004 189