# Analysis of an induction motor supplied by H-bridge multilevel inverters.

1 INTRODUCTION

During last decades, the multi-level inverter has become a classical drive competing academically as well as industrially other topologies. The multilevel topology has emerged bringing new features like the possibility to overcome the problem related to the maximum voltage drop of the power switches [1, 2]. It also limits voltage and current harmonics thus generate waves very close to a sinusoid with a switching frequency equal to the fundamental [3, 4, 5, 6]. Nowadays, multilevel inverters are increasingly used in research areas related to power applications including renewable energy , urban rail , ship propulsion  and feeding AC machines .

The main purpose of this work is to conduct a quantitative study to show the advantage of supplying an induction machine by a multilevel inverter. The voltage and current total harmonic distortion and the torque ripple rate are the main targets.

This paper is organized into four sections. After the introduction, the second section includes development of mathematical models concerning the induction motor and the H-bridge multilevel inverter. The third presents a comparative study between three, five and seven inverter levels when feeding an induction motor. For this purpose, simulations were carried out to obtain phase currents harmonic spectra and torque ripple rates in steady state. Finally, a conclusion is drawn in section 4.

2 MATHEMATICAL MODELS OF THE ASSOCIATION H-BRIDGE INVERTER-INDUCTION MOTOR

The cascaded full bridge inverter is the first attempt to multilevel inverters, also called series connected H-bridge or cascaded H-bridge [9,10]. RH Baker and LH Bamnister  proposed this topology in 1975. Its first application was in plasma stabilization in 1988 [12, 13]. It consists on connecting in series single-phase H-bridges. The selection of H-bridge inverter is supported by many advantages such as simple modularized structure, reduced number of components and balanced distribution of voltage stress across switches.

The structure of an H-bridge leg with m levels, figure 1, is defined by the number of:

* bridges: S = m - 1/2.

* switches: K = 2 (m - 1).

Every switching device is designated by S(i,j,k) where i represents the switch number of the jth cell belonging to the kth phase.

The DC source of each cell is noted Ej.k supplying the jth cell of the kth phase. ([V.sub.aj], [V.sub.bj], [V.sub.cj]) are the output voltages of the jth cell respectively of the first, second and third phase.

([V.sub.an], [V.sub.bn], [V.sub.cn]) are respectively the phase voltages of the first, second and third arms.

The stepped voltage waveform is composed by m levels which depends on DC sources numbers such that: m = 2 S +1. Thus, whatever is the type of cascaded multilevel inverter, the output voltage levels number is always odd (3,5,7,11, ...). The different H cells are connected in series so that the resulting voltage of a leg is equal to the sum of all the voltages generated by each cell as follows:

[V.sub.an] = [s.summation over (j=l)] [V.sub.aj] (1)

For S = 2, the three output phase voltages can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The three line to line output voltages can also be written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The modulation technique applied in the following are based on sinusoidal Pulse Width Modulation (PWM) with multiple carriers. This method involves comparing a sinusoidal signal also called modulating waveform with triangular signals or carrier waves. The switches of each leg require (m-1) triangular signals having the same frequency [f.sub.p] and same magnitude [A.sub.p] , figure 2.

The reference sine waves are phase shifted by 2[pi]/3. Their amplitude and frequency are denoted respectively [A.sub.m] and [f.sub.m]. The sinusoidal PWM technique is characterized by two ratios defined as follows:

* [m.sub.c] = [f.sub.p]/[f.sub.m]

Where [m.sub.c] is the frequency modulation index.

* [m.sub.a] = [A.sub.m]/(m-1)[A.sub.p]

Where [m.sub.a] is the amplitude modulation ratio.

Regarding the induction machine, the adopted model is established in a d-q stationary reference frame whose equations are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Where [[omega].sub.e] is the electrical rotor speed.

3 SIMULATION RESULTS OF THE INDUCTION MOTOR FED BY A CASCADED MULTILEVEL INVERTER

We aim, in this section, to compare the quality of various wave quantities, namely the line to line voltages, winding currents and electromagnetic torque.

For each m level inverter, the simulation results will include the waves previously mentioned and their total harmonic distortion. Figure 3 shows the block diagram necessary to undertake the simulation.

The simulation parameters related to H-bridge multilevel inverters are the following:

* Amplitude modulation depth: [m.sub.a] = 0.4

* Frequency ratio: [m.sub.c] = 21

* Sinusoidal wave: [f.sub.p] = 50 Hz, [A.sub.p] = 0.8

* Carrier: [f.sub.m] = 1050 Hz, [A.sub.m] = 1

Parameters of the squirrel cage induction motor where all rotor quantities are referred to the stator are:

* 85/140 V

* 3.5/6 A

* f = 50 Hz

* Rs = 3.45 [OMEGA], Rs: Stator phase resistance.

* Ls = 0.1442 H, Ls: Stator phase inductance

* Rr = 2.95 [OMEGA], Rr: Rotor phase resistance.

* Lr = 0.1442 H, Lr: Rotor phase inductance.

* M = 0.1342 H, M: Magnetizing inductance.

* P = 2 pole pairs

* J = 0.01 Kg m2, J: Motor inertia.

As previously mentioned, the selected transient concerns the starting from the rest, followed by an abrupt application of a constant torque load [T.sub.l] of 2 Nm at t= 0.5s. These steps are illustrated in figure 4 giving the evolution of the speed versus time.

The inverters output voltages [V.sub.an] or winding voltages are displayed in figure 5. The number of levels generated by each inverter can be easily identified from the waveforms.

The feeding voltages [V.sub.ds] and [V.sub.qs] of the motor, represented in (d,q) axes, are function of the line to line multilevel inverter voltages shown in figure 5. These latter will be concerned by the following harmonics study. In order to compare the quality of [U.sub.ab] waves, figure 6 details harmonic spectrums of 3, 5 and 7 multilevel inverter output voltages. It also displays the Total Harmonic Distortion (THD) corresponding to different voltage levels up to m=7.

It can be noticed that as the level number increases the distortion rate decreases. The 3 level H-Bridge inverter has a THD = 41.97% while the 7 level H-Bridge has a THD about 14 %. Therefore, the quality of the output wave at the highest level (seven levels here), obtained by the multi carrier PWM, is obviously the best.

Figure 7 gives the general shape of the absorbed current during the transient state. In addition, the steady state current, its spectral analysis and its total harmonic distortion are detailed respectively in figure 8 and 9.

According to figure 9, if the voltage level changes from m = 3 to m = 7, the total harmonic distortion drops from 4.03% to 2.12%. As can be seen, the THD varies slowly upper to 5 levels.

During the starting, figure 10 shows the electromagnetic torque [T.sub.em] oscillating in the first moments. The transient torque may reach a peak of 5 Nm. At t = 0.5s, a load torque of 2 Nm is applied.

Besides, the impact of the inverter output voltage quality on the rotor torque is Especially visible during the steady state, figure 11.

In fact, figure 11 clearly illustrates that the torque ripple during steady state period decreases gradually and progressively as the inverter number of levels increases. Relative results are resumed in Table 1.

It has to be emphasized that less torque ripple leads to better stability operation with minimum mechanical noise.

4 CONCLUSION

The association of an induction machine with an H-bridge multilevel inverter was analyzed. A simulation study was performed to show consequent improvements in the machine electrical and mechanical wave forms. Special interest was focused on the THD of the stator voltages and currents as well as the oscillations rate of the electromagnetic torque as functions of the inverter level amount. Consequently, when an induction motor is fed by an H-Bridge multilevel inverter, it would be not necessary to go beyond 5 levels to get suitable current, voltage and torque waves.

REFERENCES

[1.] Kouro. S, Rebolledo. J, Rodriguez. J, "Reduced Switching Frequency Modulation Algorithm for High Power Multilevel Inverters", IEEE Trans. on Industrial Electronics; 54, 5(2007), 2894-2901.

[2.] Tolbert. LM, Peng. FZ, Habetler. TG, "Multilevel Converters for Large Electric Drives". IEEE Trans. on Industry Applications, 35, 1(1999), 36-34.

[3.] Cecati. C., Ciancetta. F and Siano. P, "A Multilevel Inverter for Photovoltaic systems with Fuzzy Logic Control", IEEE Trans. on Ind. Electronics, 57, 12(2010), 4115-4125.

[4.] Babaei. E, "A Cascaded Multilevel Converter Topology with reduced number of switches", IEEE Trans. Power Electronics, 32, 6(2008), 2657-2664.

[5.] Rodriguez. J, Lai. JS, Peng. FZ, "Multilevel inverters: A Survey of Topologies, Controls, and Applications", IEEE Trans. on Ind. Electronics,49, 4(2002), 724-738.

[6.] Rodriguez. J, Franquelo. LG, Kouro. S, Leon. JI, Portillo. RC, Prats. MAM, Perez. MA, "Multilevel Converters: An Enabling Technology for High-Power Applications", in Proc. IEEE, 97, 11(2009), 1786-1817.

[7.] Cortes. P, Wilson. A, Rodriguez. J, Abu Rub. H, "Model Predictive Control for Multilevel Cascaded H-Bridge Inverters", IEEE Trans. Ind. Electronics, 57, 8(2010), 2691-2699.

[8.] Sasikumar. M, Pandian. SC, "Modeling and Analysis of Cascaded H-Bridge Inverter for Wind Driven Isolated Self Excited Induction Generators", International Journal on Electrical Engineering and Informatics 2011; 3 (2): 132-145.

[9.] Malinwoski. M, Gopakumar. K, Rodriguez. J, Perez. A, "A Survey on Cascaded Multilevel Inverters", IEEE Trans. on Ind. Electronics, 57, 7(2010), 2197-2206.

[10.] Hammond. PW, "A New Approach to Enhance Power Quality for Medium Voltage AC Drives", IEEE Trans. on Ind. Applications, 33, 1(1997), 202-208.

[11.] Baker. RH, Bannister. LH, "Electric Power Converter", U.S. Patent. 3 867 643, Feb. 1975.

[12.] Cleanovic. N, " Space Vector Modulation and Control of Multilevel Converters, Ph.D in Electrical Engineering and Computer Engineering 2000, The Faculty of the Virginia Polytechnic Institute and State University.

[13.] Ostojic. D , "A multilevel converter structure for grid-connected PV plants", Ph.D in Power Systmes Engineering 2010, Facolta Di Ingeneria.

[14.] McGrath. BP, Holmes. DG, "Multicarrier PWM Strategies for Multilevel Inverters", IEEE Trans. on Ind. Electronics, 49, 4(2002), 858-867.

Abir Rehaoulia, Mahmoud Hamouda, Habib Rehaoulia, Farhat Fnaeich

Laboratory of Signal, Image and Intelligent Control of Industrials Systems

High School of Sciences and Technologies (ESSTT) University of Tunis

5 Avenue Taha Hussein-Tunisia, abirrehaoulia@gmail.com, mahmoudhamouda@yahoo.fr, r.habib@planet.tn, fnaiech02@yahoo.com
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Table 1. Relative torque ripple rate as function of
inverter levels number

Levels number      Relative torque ripple in %

m = 3               15%
m = 5               7.75%
m = 7               4.3%
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