International Journal of Advance Engineering and Research Development

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1 Scientific Journal of Impact Factor (SJIF): 4.7 International Journal of Advance Engineering and Research Development Volume 4, Issue 5, May-07 e-issn (O): p-issn (P): Mathematical modeling of three phase Induction motor and its speed control by SPWM and SVPWM techniques Chetan S, A. Kumar M.Tech student, EEE, BNMIT, Bengaluru,Associate Professor, Dept. of EEE, BNMIT, Bengaluru Abstract-Three phase induction motors are the work horse of industry, and they are one of the most commonly used type. The analysis of it can be simplified if the three phase quantities like flux, current field lines which rotate in space can be transformed into an equivalent two phase quantities, this transformation can be brought about using the concept of arbitrary reference frame theory. Once the model is developed it ensures that the induction motor can be replaced by it for study purposes. The speed control of Induction motor is possible by many methods but the most efficient ways is the Sinusoidal pulse width modulation (SPWM), and Space vector pulse width modulation (SVPWM) technique. In this paper the mathematical modeling of three phase Induction motor is done and ensuring its correctness, speed control is carries out on Induction motor block available in MATLAB. Keywords- Mathematical modeling, Reference frame, MATLAB, pulse width modulation, space vector. I. INTRODUCTION The dynamic modeling /simulation is one of the major steps in the acceptance of the design process of any drive system [3], due to this there is elimination of mistakes during design process and during the construction of prototype. From the fundamental equations of transformation the dynamic model can be developed in dq0 (direct, quadrature and zero-sequence) axis. For the analysis of induction machines arbitrary reference frame theory is extensively used, making its useother reference frames can be developed. The modeling sets all equations for inertia, torque, and speed vs time. Differential voltages, current and also flux linkages between the moving rotor and stationary stator can be modeled. THEORETICAL BACKGROUND Before deriving the mathematical model for a three phase induction machine, few assumptions are made and they are as follows Air gap is uniform Squirrel cage rotor construction Stator and rotor windings are balanced, and have sinusoidal winding distribution Parameter change and saturation are neglected. For studying the performance of the induction machine the equivalent circuit is necessary, which is as shown in Fig. Figure.dq0 equivalent of an induction All rights Reserved 393

2 II. INDUCTION MOTOR MODEL Concept of reference frame - The reference frames are similar to observer platforms, wherein each platform gives an unique and distinct view of the system being studied, and simplifies it in terms of analysis. Generally, for controlling purpose, although the actual variables are sinusoidal in nature it is desirable to have them as DC quantities. This can be achieved by having a revolving reference frame whose speed is same as that of the sinusoidal quantity. Since both,the quantity as well as the reference frame is revolving at the same speed their differential speed is zero which means that they are stationary with respect to each other. Once the general transformation is derived for an arbitrary reference frame, then any required particular frame can then be obtained by substituting the appropriate frame speed. Conversion from three-phase to two-phase - For deriving the dynamic model for an induction motor, first equivalence has to be established between three-phase and two-phase. The mmf produced in the two-phase and the three-phase must be equivalent. If a three-phase winding hasnumber of turns as Ns per phase and with equal current magnitudes, then for two-phase winding the number of turns should be 3Ns/ for mmf equivalence. The d and q axis mmf is found by resolving the three-phase mmf s. The number of turns being common in both sides of equation gets cancelled leaving current equalities. Under balanced conditions the three-phase stator voltages of an induction machine is given as follows, V a = Vrms sin(ωt) () V b = Vrms sin(ωt π 3 ) () V c = Vrms sin(ωt 4π 3 ) (3) Where, V a, V b and V c are the line voltages. The relation between αβ and abc is as follows, Vα Vβ = Va Vb Vc (4) The direct axis and quadrature axis voltages are given as, Vd Vq = cos θ sin θ sin θ cos θ Vα Vβ (5) The instantaneous values of the stator and rotor currents in a three-phase system are calculated using the transformation, iα iβ cos θ = sin θ sin θ cos θ id iq (6) Transformation to abc axis is given as, ia ib ic = iα iβ (7) Equations of flux All rights Reserved 394

3 dφqs dt dφds dt = [Vqs ωe = [Vds ωe φds ( Rs )(φmq φqs)] (8) φqs ( Rs )(φmd φds)] (9) dφqr dt = [Vqr ωe ωr φdr ( Rr )(φmq φqr)] (0) Xlr dφdr dt = [Vdr ωe ωr φqr ( Rr )(φmd φdr)] () Xlr Where, φds q axis component of stator flux in weber, φqs q axis component of stator flux in weber Vqs q axis component of stator voltage in volts, Vds d axis component of stator voltage in volts Speed of stator supply frequency rad/s, ωe speed of stator reference frame rad/s ωr Speed of rotor in rad/s, φmd d axis component of mutual flux in weber, φmq q axis component of mutual flux in weber, Rs, Rr- stator and rotor resistance in Ω, Xlr stator and rotor reactance respectively in Ω, Xml mutual leakage reactance in Ω Xml = /[ Xm + + Xlr ] (), φmq= Xml [ φmd= Xml [ φ qs φ ds + φqr Xlr ] (3), + φdr Xlr ] (4) Then, substituting the flux linkage equations to find the qd stator and rotor currents, iqs = ids = (φqs φmq) (5), iqr = Xlr (φds φmd) (7), idr = The electromagnetic torque developed is given as, (φqr φmq) (6) Xlr (φdr φmd) (8) Te = 3 P (φds iqs φqs ids) (9) ωr = P J (Te Tl) (0) Where, P Number of poles, J Inertia in Kg m, Tl- Load torque in Nm III. MATLAB Simulations The developed model was simulated using MATLAB All rights Reserved 395

4 Figure.Machine parameters given to MATLAB Figure3.Overall model of three phase induction motor The model was developed in stationary reference frame wherein the speed between the reference frame and stator is zero, meaning the frame is fixed to the stator. The angle is taken as zero while converting the voltage equations from abc - dq. The result of the model developed was compared with the induction motor block available in MATLAB, for checking its All rights Reserved 396

5 Results of the developed model Figure 4.a)Stator currents. I as = 4.8A, I bs = 3.97 A, I cs = 3.93 A, for load torque T l = 5 Nm b) Torque, Te = 5.079Nm, Speed, Nr = 503 RPM The developed model was compared with the Induction motor block available in the MATLAB with the same machine parameters and the results were similar. SPWM method of speed control - This is one of the most commonly used technique in VFD application, it has low THD at the output of inverter and is easy to implement. Here, sine wave is compared with triangular wave and when the magnitude is greater than the magnitude of triangular wave, a pulse is All rights Reserved 397

6 MATLAB simulation of SPWM technique - Blocks inside 50Hz 500 RPM subsystem generates the pulses required for switching the switches for inverter are given below. Repeating sequence block - frequency =.7Khz, Amplitude =,Sine wave block Amplitude =, frequency = 50Hz, Phase difference is set as 0, 0 and 40 for the three blocks All rights Reserved 398

7 Results obtained for SPWM technique for speed control - Figure 5. Speed and torque waveforms for M.I = FFT analysis was carried out on the waveform of current waveform obtained from the inverter, and it showed a Total Harmonic Distortion (THD) of 7.3% SVPWM Technique In Space vector pulse width modulation(svpwm) technique, the poly phases (three) (A, B, C) axis system are transferred to two frames axis (X, Y), which are a complex axis frame the (X-axis) represents a real axis, and (Y-axis) represents a direct imaginary axis. The Vxy Is referring to voltage vector, this vector rotating at a constant angular frequency (ω), in SVPWM (V, V, V3, V4, V5, V6) Represent vectors in plan, this plan divide into six sectors, each sector, having (π/3)or(60deg), a reference voltage Vxy Is synthesized by adjacent two nulls zero vector (VtoV6), and two zero vector (V0&V7), as shown in Fig. 5 The radius of inside circle is ( 3 VDC). [] The following are some of the advantages of SVPWM -\ The modulation range is wide and linear Low losses due to switching Low total harmonic distortion Easy to implement. Figure 6. Hexagon boundary with six sectors The switching times of each switch is calculated mathematically and is given in All rights Reserved 399

8 where, T s = /f s T 0 =T s -T -T T = 3 Vxy Vdc T s sin( nπ θ T = 3 Vxy T s sin( θ (n )π ) Vdc 3 3 ) MATLAB implementation of SVPWM technique - Table.Switching time for each All rights Reserved 400

9 Results obtained The THD for SVPWM model was 3.54%. Figure7.Speed and Torque waveforms for SVPWM technique IV. Conclusion and future work The dynamic modeling of three phase induction motor helps in understanding of the machine s parameter response to changes in operating conditions. PWM and SVPWM techniques are the widely used techniques for speed control, however due to low harmonic content SVPWM is preferred. I have used only one method of SVPWM for other types also similar model can be developed. REFERENCES. Comparison of SPWM VSI and SVPWM VSI FED Induction Machine Using Volt per Hertz Control Scheme by Neha Sharma, Vijay Kumar Garg.. Simulation and Study of SVPWM Inverter for (VFD) Applications by Ahmed K. Ali. 3. Mathematical Modelling of an 3 Phase Induction Motor Using MATLAB/Simulink by Mr. Punit L. Ratnani, Dr. A. G. All rights Reserved 40