# Complex Fourier series mathematical model of a three-phase inverter with improved PWM output voltage control/Trifazio inverterio, geriau valdancio IPM isejimo itampa, kompleksines Furje eilutes matematinis modelis.

Introduction

A three phase voltage source inverter permits to product an alternative three-phase voltage of variable or constant frequency on the basis of continuous input source voltage. The load of the inverter can be either passive, which does not contains any source of the voltage, or active containing some source of voltage.

The applications of the three-phase inverters are various and their utilisations are increasing more and more. They are used in most cases as a supply device for asynchronous and synchronous motors where frequency and voltage are controlled.

In the Fig.1 there the three-phase bridge connected inverter using the insulated gate bipolar transistor (IGBT) has been drawn. This device is being increasingly used in the both the single-phase and three-phase inverters [1, 2].

[FIGURE 1 OMITTED]

Mathematical model of the inverter

For inverter's operation study at steady state we consider following idealized conditions:

* Power switch, that means the switch can handle unlimited current and blocks unlimited voltage;

* The voltage drop across the switch and leakage current through switch are zero;

* The switch is turned on and off with no rise and fall times;

* Sufficiently good size capacity of the input voltage capacitors divider, to can suppose converter input DC voltage to by constant for any output currents.

This assumption helps us to analyze a power circuit and helps us to build a mathematical model for the inverter at steady state.

An improvement to the notched waveform is to vary the on and off periods such that the on-periods are longest at the peak of the wave. This form of control is known as pulse-width modulation (PWM).

It can be observed that area of each pulse corresponds approximately to the area under the sine-wave between the adjacent mid-points of the off-periods. The pulse width modulated wave has much lower order harmonic content than the other waveforms.

If the desired reference voltage is sine-wave, two parameters define the control:

* Coefficient of the modulation m--equal to the ratio of the modulation and reference frequency;

* Voltage control coefficient r--equal to the ratio of the desired voltage amplitude and the DC supply voltage. Generally to control the inverter numeric control device is used. The turn on (a) and turn off (p) angles are calculated by the discredit of the reference sine-wave. That means the reference sine-wave is by a values discreet replaced. If the coefficient of modulation m is sufficiently great, the difference between real values and discrete values is negligible.

The phase's branches are control to create the output voltages as seen in (1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [U.sub.e] -is a DC inverter's input voltage value.

To calculate a turn on ([alpha]) and turn off ([beta]) angles we compare the DC impulse area with the requested voltage area, as depicted on the Fig. 2.

[FIGURE 2 OMITTED]

For the left and right crosshatched areas of the first output transistors branch the following equations are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

After the calculus we obtain for the turn-on and turnoff angles of the first transistors branch the following expressions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

It will be similarly for the second and third transistor branch.

The inverter's output voltage of the first branch can be mathematically expressed as a complex Fourier series of the form [3-6]

[u.sub.01] = [U.sub.e] [[infinity].summation over (k=-[infinity])] [m.summation over (n=1)], (4)

where the Fourier coefficients take a form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The Fig. 3 depicts a branch voltage waveform for supply voltage of [U.sub.e] = 100 V, output frequency f = 50Hz and r = 1; m = 10;

[FIGURE 3 OMITTED]

Similarly for the other two phases the following equations are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

The terminal voltages calculation

Assuming, that the load is balanced and wye connected, as given in the Fig. 4. Consider the load of each phase consists of series connected RL elements (R = 2[ohm]; L = 50mH) .

[FIGURE 4 OMITTED]

The line voltages are given by a difference between two voltages of the branches as follow :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

There is in the Fig. 5 shown the waveform of the output line-to-line voltage [U.sub.12] (similarly for other line-to-line voltages).

[FIGURE 5 OMITTED]

For the balanced load and insulated neutral node, following relations between line and phase voltages are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The Fig. 6 depicts waveforms of a phase voltages, calculated on basis of (7).

[FIGURE 6 OMITTED]

For any of the phases the voltage equations are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

The phase currents calculation

Based on (8) the analytical solution takes a form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

In the Fig. 7 is given calculated phase load current wave-form. The waveform was calculated on a basis of (9).

[FIGURE 7 OMITTED]

In electrical engineering, the Clarke [alpha], [beta], 0 mathematical transformation is employed. It employment very often simplifies the analysis of three-phase circuits.

Conceptually the Clarke transformation present particular part of Park transformation. It converts ordinary 3-phase system into orthogonal 2-phase one [2, 4]. One of very useful application of the Clarke transformation is the generation of the reference signal used for space vector modulation control of the three-phase inverters.

The Clarke transformation applied to the three-phase quantities is shown below in a matrix form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Note also, that [x.sub.0] is just a scaled version of the zero sequence term from symmetrical components. We can also write equation (11):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Then for transformed phase voltages equations (12) are valid '3, 7, 8]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

The Fig. 8 bellow shows the trajectory of the space vector of the transformed voltage components [u.sub.[[alpha], [u.sub.[[beta], calculated on the basis of (12).

[FIGURE 8 OMITTED]

On the basis of the (11) and (12) can be done the currents transformations. For transformed phase currents (Fig. 9) the following equations are valid:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

[FIGURE 9 OMITTED]

The trajectory (Fig. 9) will be a circle for infinite number of harmonics very high switching frequency.

Conclusions

In this study, an article about the more accurate mathematical model of a three-phase voltage source inverter with PWM output voltage control has been presented.

The method is based on use of complex Fourier series calculation. The results carried-out by modelling are adequate to the presented mathematical model.

Comparison of results to those of carried-out by classical method ones will be done in next paper.

Acknowledgements

The financial support of the Slovak Research and Development Agency under the contract No. APVV-013810 is acknowledged.

References

[1.] Buhler H. Power Electronics (Electronique de puissance in French) // Traite d'electricite.--Laussane, Switzerland, 1987.

[2.] Mohan N., Undeland T. M., Robbins W. P. Power Electronics: Converters, Applications, and Design, 2nd ed.--John Wiley & Sons, Inc., 1993.

[3.] Zaskalicka M., Zaskalicky P., Benova M., Mahmud A.R., Dobrucky B. Analysis of complex time function of converter output quantities using complex Fourier transform/series // Communications-Scientific letters.--University of Zilina, Slovakia, 2010.--Vol. 12.--No. 1.--P. 23-30.

[4.] Dobrucky B., Spanik P., Kabasta M. Power Electronic Two/Phase Orthogonal System with HF Input and Variable Output // Electronics and Electrical Engineering.--Kaunas: Technologija, 2009.--No. 1(89).--P. 9-14.

[5.] Marcokova M. Equiconvergence of Two Fourier Series // Journal of Approximation Theory, 1995.--Vol. 80.--No. 2. P. 151-163.

[6.] Zaskalicky P., Zaskalicka M.: Analytical Method of the Calculation of the Torque Ripple of the Universal Motor Supplied by a IGBT Chopper // Acta Technica CSAV.--Prague, Czech Republic, 2010.--Vol. 55.--No. 3.--P. 275286.

[7.] Dobrucky B., Marcokova M. Complex Time Function of Converter Output Quantities as Orthogonal Polynomials and Their Analysis // Proc. of OPFSA Conf.--Madrid, Spain, 2011.--Vol. (Chap.) 5.--P. 43-48.

[8.] Marcokova M., Guldan V. On One Orthogonal Transform Applied on a System of Orthogonal Polynomials in Two Variables // Proc. of APLIMAT Int'l Conf.--Bratislava, Slovakia, 2009.--Vol. X.--P. 621-626.

Accepted after revision 2012 05 31

Faculty of Electrical Engineering & Informatics, Technical University of Kosice, Letna 9, 042 00 Kosice, Slovakia, e-mail: pavel.zaskalicky@tuke.sk

B. Dobrucky

Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 010 26 Zilina, Slovakia, e-mail: branislav.dobrucky@fel.uniza.sk

cross ref http://dx.doi.Org/10.5755/j01.eee.123.7.2376
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