Keyword

1 Introduction

Induction machines as electric power equipment are widely used in various fields, including industry, transportation, offices, and households. Besides having the advantages offered, this machine also has a weakness in setting the rotational speed. The rotational speed of an induction motor can be controlled in two ways, namely direct and indirect starting. Starting directly is done by connecting the motor to the grid voltage. This method has the disadvantage of causing power losses when the motor is initially connected to a power source, because at that time the motor draws a very large starting current. The starting method is indirectly by adjusting the supply voltage, input current, stator field flux, and frequency using an electronic device known as a Variable Speed Drive (VSD), especially Inverter [1] [2]. The use of VSD in induction motors has an effect on the improvement of torque, speed, and current drawn by the motor. The study begins with modeling the character control of speed, rotor current, and motor torque [3,4,5,6,7]. Ahmed J. Ali1 modeled the transient regulation approach of a cage-rotor induction motor based on the winding function (WFA) which assumes the MMF coupling between the rotor and stator is sinusoidal and non-sinusoidal. The result is that the second model approach is better [4]. Babak Kiani in his paper writes a Limited Set Model Predictive Direct Torque control method (FS—MPDTC) to reduce torque and flux ripples, by implementing two or more Space Vector Pulse Modulation (SVPWM) voltage vectors [8]. Yassine Zahraoui said space vector modulation is a very appropriate solution to reduce high ripple levels regardless of its complexity [9]. Mihoub Youcef implemented RST and adaptive fuzzy controller (AFC) to increase variable speed Induction motor drive control system [10]. Arkan A. Kadum presents a new adaptive hysteresis band control approach used in direct torque control (DTC) of induction motor drive (IM) with a switching pattern for PWM signal generation; the simulation results under different operating conditions over a wide speed range show the validity, effectiveness, and feasibility of the proposed design [11]. Several other researchers developed induction motor drives with space vector control to obtain lower torque ripple and wider speed regulation [12,13,14,15,16,17,18,19]. Control with VSD is generally done by setting the ignition pattern of the inverter switch using switch control or vector control. Switch ignition settings with vector control have advantages such as more efficient use of energy, and produce a wide rotational speed setting. This study describes a study of induction motor control testing on various changes in load torque using the Space Vector Pulse Width Modulation method to see the character of the stator current, rotor current, and speed of a three-phase induction motor through modeling using the Simulink MATLAB tool. Previous researchers used a lot of Sinusoidal Pulse Width Modulation (SPWM) control method, but in the article the method of generating a gate ignition signal using the Space Vector Modulation (SVPWM) method has a smaller harmonic distortion and a higher output voltage than the SPWM method. The simulation results show that the design of the induction motor speed regulation using SVPWM can work well and is able to operate at changes in load torque. This design can be a technological contribution and a reference for advanced researchers to be implemented into electronic circuits.

2 Space Vector Concept

The representation of the time function of a balanced three-phase system of a voltage and current can be shown in Eq. (1):

$${v}_{\mathrm{a}}+{v}_{\mathrm{b}}+{v}_{\mathrm{c}}=0$$
(1)

The magnitude and direction of the vector \({\left[{v}_{\mathrm{a}}0 0 \right]}^{T}\) of a three-phase system is placed along the coordinates of the d-axis; the vector \({\left[0 {v}_{\mathrm{b}}0 \right]}^{T}\) is shifted by 1200 and the vector \({\left[0 0 {v}_{\mathrm{c}}0\right]}^{T}\) is shifted by 2400. Figure 1 is an illustration of this vector in imaginary coordinates d-q.

Fig. 1
A vector diagram in which a three phase vector is plotted on the q d coordinate. v reference is at an angle of alpha from the d axis. v d is along the d axis. diagram. v q is along the q axis. a, b, and c point to corresponding matrix vector forms.

Placement of the Vector of a three-phase to d-q Coordinate

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$$\alpha ={tg}^{-1}\left(\frac{{V}_{d}}{{V}_{q}}\right)=\omega =2\pi f$$
(2)

where f = fundamental frequency for Eq. (2).

If \({v}_{\mathrm{a}},{v}_{\mathrm{b}},{v}_{\mathrm{c}}\) is specified as a reference, the vector \({v}_{\mathrm{ref}}\) can be determined by Eq. (3):

$${v}_{\mathrm{ref}}=\frac{2}{3}\left[{v}_{\mathrm{a}}+{v}_{\mathrm{b}}{e}^{j\left(2/3\right)\pi }+{v}_{\mathrm{c}}{e}^{-j\left(2/3\right)\pi }\right]$$
(3)

The vector \({v}_{\mathrm{ref}}\) (three-dimensional space) is then transformed into a two-dimensional d-q space base system into an algebra for the sum of real elements and imaginary elements in Eq. (4):

$$\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}1& \frac{-1}{2}& \frac{-1}{2}\\ 0& \frac{\sqrt{3}}{2}& \frac{-\sqrt{3}}{2}\end{array}\right]\left[\begin{array}{c}{v}_{a}\\ {v}_{b}\\ {v}_{c}\end{array}\right]$$
(4)

The next step is the transformation of the d-q coordinate space to the \(\alpha -\beta\) coordinates obtained by Eq. (5):

$$\left[\begin{array}{c}{v}_{\alpha }\\ {v}_{\beta }\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\left(\omega t\right)& \mathrm{cos}\left(\frac{\pi }{2}+\omega t\right)\\ \mathrm{sin}\left(\omega t\right)& \mathrm{sin}\left(\frac{\pi }{2}+\omega t\right)\end{array}\right]\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\end{array}\right] =\left[\begin{array}{cc}\mathrm{cos}\left(\omega t\right)& -\mathrm{sin}\left(\omega t\right)\\ \mathrm{sin}\left(\omega t\right)& \mathrm{cos}\left(\omega t\right)\end{array}\right]\left[\begin{array}{c}{v}_{d}\\ {v}_{q}\end{array}\right]$$
(5)

If \({v}_{\mathrm{a}},{v}_{\mathrm{b}},{v}_{\mathrm{c}}\) is a three-phase voltage-balanced system with a maximum value of \({V}_{\mathrm{m}},\) then the vector \(v\left(t\right)\) can be denoted by Eq. (6):

$$v\left(t\right)= {V}_{m}{e}^{j\omega t}$$
(6)

Thus, \(v\left(t\right)\) is a vector with a maximum amplitude of \({V}_{\mathrm{m}}\) rotating at a constant angular velocity of \(\omega\) rad/sec.

2.1 Eight Vector Switching Inverter Combinations in Space Vector Pulse Modulation (SVPWM)

The performance of the three-phase inverter switch can be grouped into eight states. Switches a, b, and c have complementary switch pairs a', b', and c' where when switch a is connected or ON then the complementary switch pair a' is open or OFF. Likewise, switch b and switch c work in reverse with their complementary switches b' and c'.

Figure 2 describes a three-phase inverter having eight switching conditions that can be formed in eight binary compositions from 000 to 111. Each composition is denoted by \({V}_{0}\), \({V}_{1}\), \({V}_{2}\), \({V}_{3}\), \({V}_{4}\), \({V}_{5}\), \({V}_{6}\), and \({V}_{7}\) hereinafter referred to as vector switching. \({V}_{1}\), \({V}_{2}\), \({V}_{3}\), \({V}_{4}\), \({V}_{5}\), and \({V}_{6}\) are called non-zero vectors because they produce an output voltage that is not equal to zero, while \({V}_{0}\) and \({V}_{7}\) are called zero vectors where this vector produces an output voltage equal to zero.

Fig. 2
A chart presents the 8 switching conditions of a three phase inverter. Each has a different binary composition of 100, 110, 010, 011, 001, 101, 111, and 000. Each composition is denoted by v 1, 2, 3, 4, 5, 6, 7, and 0.

Eight combinations of inverter switch conditions

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Figure 3 shows 6 vectors mapped to six sectors (sectors 1 to 6) in a position around the central point forming a hexagonal shape while two vectors are mapped to the center point (origin) [9, 20,21,22]. Each sector has a vector angular distance of 600. In one rotation on the complex coordinate plane d-q, \({V}_{\mathrm{ref}}\) will pass through the six non-zero vectors and zero vectors. The inverter output voltage is also affected by the SVPWM modulation index. Selection of the right modulation index will assist in the production of the rated voltage. If the modulation index is less, the pulse duration on time will be less and therefore, the device conduction time will also be less, so the inverter output voltage is reduced [23, 24].

Fig. 3
A diagram has a circle with a hexagon inside, with each division numbered from 1 through 6 in an anticlockwise direction. It is representative of 6 vectors mapped to six sectors in a position around the central point, forming a hexagonal shape. 2 vectors are mapped to the central point.

The position of each vector on the coordinate plane of two-dimensional space, d-q

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Table 1 shows the state of the output voltage (\({V}_{0}\)–\({V}_{7}\)) at various switch conditions for each sector.

Table 1 Inverter output voltage in vector switching state
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2.2 Time Calculation and Gate Trigger Pattern

The trigger of the inverter switch gate with the SVPWM method is the most important part of this research. Figure 4 explains the basis for calculating the trigger timing of the inverter gate switch based on vector analysis in sector 1. \({V}_{\mathrm{ref}}\) is the resultant of the length of vector \({V}_{\mathrm{A}}\) and vector \({V}_{\mathrm{B}}\), and can be represented by Eq. (7):

Fig. 4
A vector diagram in the complex plane has vector V 1 corresponding to 0 0 0 along the real axis. Vector V 2 corresponding to 1 1 0 is at an anticlockwise angle of pi by 3 from the real axis. V A and V B are along vectors V 1 and V 2, respectively. Their resultant V ref is at an angle alpha from the real axis. This entire system forms a parallelogram. The length of V A is marked as T 1. The length of V 1 is marked as V D C.

Calculation of \({V}_{\mathrm{ref}}\), \({T}_{1}\), and \({T}_{2}\) in sector 1

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$${V}_{\mathrm{ref}}={V}_{\mathrm{A}}+{V}_{\mathrm{B}}$$
(7)

In each sector, the value of is always in the interval \(0\le \alpha \le \frac{\pi }{3}\). The voltage \({V}_{\mathrm{ref}}\) in sector 1 is of angle 600, the value \({V}_{\mathrm{A}}\) = \({V}_{1}\), \({V}_{\mathrm{B}}\) = \({V}_{2}\), then the voltage \({V}_{\mathrm{ref}}\) in sector 2, \({V}_{\mathrm{A}}\) = \({V}_{2}\) and \({V}_{\mathrm{B}}\) = \({V}_{3}\). In the same way for every other sector, \({V}_{\mathrm{ref}}\) shifts or moves left counterclockwise with a time lapse of \({T}_{\mathrm{z}}\). \({T}_{1}\) is the time lapse for \({V}_{\mathrm{A}}\), \({T}_{2}\) is the time lag for \({V}_{\mathrm{B}}\), and \({T}_{0}\) is the time interval for \({V}_{0}\) and \({V}_{7}\). In sector 1 mathematical analysis, the duration of each T time is determined by Eq. (8), Eq. (9), and Eq. (10):

$${T}_{1}={T}_{\mathrm{z}}.a.\frac{\mathrm{sin}\left(\frac{\pi }{3}-\alpha \right)}{\mathrm{sin}\left(\frac{\pi }{3}\right)}$$
(8)
$${T}_{2}={T}_{\mathrm{z}}.a.\frac{\mathrm{sin}\left(\alpha \right)}{\mathrm{sin}\left(\frac{\pi }{3}\right)}$$
(9)
$${T}_{\mathrm{z}} ={T}_{1}+{T}_{2}+{T}_{0}$$
(10)

where \(0\le \alpha \le {60}^{0}\); \({T}_{\mathrm{z}}=\frac{1}{{f}_{\mathrm{z}}}\); and \(a=\frac{\left|{\overline{V} }_{\mathrm{ref}}\right|}{\frac{2}{3}{V}_{\mathrm{dc}}}\)

Figure 4 explains each sector the value of is always in the interval \(0\le \alpha \le \frac{\pi }{3}\), in sector 1 \({V}_{\mathrm{ref}}\) angle of \(\alpha ={0}^{0}\) is the same as \({V}_{1}\), for angle of \(\alpha =\frac{\pi }{3}\) is parallel to \({V}_{2}\). In sector 2, \({V}_{\mathrm{ref}}\) angle of \(\alpha ={0}^{0}\) is the same as \({V}_{2}\), for angle of \(\alpha =\frac{\pi }{3}\) is parallel to \({V}_{3}\). And so on for calculations in other sectors [25] using Eq. (11) and Eq. (12).

Time duration in sector n

$${T}_{1}=\frac{\sqrt{3}{T}_{z}\left|{\overline{V} }_{ref}\right|}{{V}_{dc}}\left(sin\left(\frac{\pi }{3}-\alpha +\frac{n-1}{3}\pi \right)\right)$$
(11)
$${T}_{2}=\frac{\sqrt{3}{T}_{2}\left|{\overline{V} }_{\mathrm{ref}}\right|}{{V}_{dc}}\left(\mathrm{sin}\left(\alpha -\frac{n-1}{3}\pi \right)\right)$$
(12)

where n = 1, 1 through 6 (that is, sectors 1 through 6, 0 ≤ α ≤ 600).

3 Method

The working method with the tool of Simulink MATLAB software through analyses parameters of each section in this paper such as voltage, gate pulse, stator currents, rotor currents, and much more positively impacted this study. It is the result of the design developed by the researcher (space vector modulation), which also produces various effects that can be used as a reference for other researchers due to the data analyzed.

Some of the parameters analyzed in this paper include the generation of reference voltages, the turn-on time of the inverter switch for each sector, gate pulses, stator currents, rotor currents, and the relationship between torque and DC motor speed. Figure 5 describes the research method used by building a path or wiring step from the designed system design. The working steps are as follows: The three-phase power source is a three-phase sine wave generator at a frequency of 50 Hz, and variable amplitude with a phase difference of 1200 for each wire. These three voltages are then transformed into a 2-coordinate system or abc—αβ which is called the park transformation. The reference voltage and the sector selection angle are used to determine the sector of the plane on the d-q coordinate axis where the stress vector is selected. There are six sectors in the d-q coordinate axis; each sector is 600. The next step is to determine the values of \({T}_{\mathrm{z}}\), \({T}_{1}\), \({T}_{2}\), and \({T}_{0}\) using Eqs. 7, 8, 9, and 10 to get the timing pattern of the inverter ignition pulse. The inverter pulse generator is obtained by comparing the space vector modulator signal with a triangular carrier signal with a frequency of 4 kHz. The inverter is composed of 3 leg pairs of IGBT switches as Variable Speed Driver (VSD) activated by a pulse ignition pulse. A three-phase induction motor as a load is simulated with parameters 54 HP (4 kW), 400 V, 50 Hz, and 1430 rpm. The value of the DC link voltage on the inverter is 400 V. When the position vector is at \({V}_{0}\) and \({V}_{7}\) then no current flows in the load or \({I}_{\mathrm{inv}}\) = 0 A. All voltages in each sector are multiplied by \({V}_{\mathrm{dc}}\). To change the magnitude of abc—αβ or better known as park transformation, the three-phase source voltage is lowered to a level of 1 V, 50 Hz with different phase angles of 00, 1200, and 1200. The system design in Fig. 5 is then modeled in Simulink MATLAB. The results of the modeling shown in Fig. 6 is a three-phase two-level inverter with a three-phase induction motor load.

Fig. 5
A flow diagram describes the method used for building a path or wiring step from the designed system. It starts with a three-phase sine wave generator, then goes through a park transformation to determine the sector of the plane on the d-q coordinate axis and values of T z, T 0 to T 2 to get the timing pattern of the inverter ignition pulse. This is then fed to an inverter composed of 3 leg pairs of I G B T switches to get V D C.

Design of a space vector modulation system for control of a three-phase induction motor

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Fig. 6
A Simulink MATLAB model of a three phase two level inverter with a three phase induction motor load. Rotor speed, electromagnetic torque, stator current, and rotor current are the outputs.

MOSFET two-level inverter as a three-phase induction motor drive

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4 Results and Discussion

Observations were made by running the modeling and then analyzing the simulated waves at the observed modeling image points. Using Eq. (1), the angle can be obtained in degrees in the form of a sawtooth wave with an amplitude of 3 \({\mathrm{V}}_{(\mathrm{p}-\mathrm{p})}\) (in the top figure). Figure 10 is the center of the angle in radians with an amplitude of 180 \({\mathrm{V}}_{(\mathrm{p}-\mathrm{p})}\), while the bottom picture is the value of \({V}_{\mathrm{beta}}\)/\({V}_{\mathrm{alpha}}\) in the form of a sinusoidal wave with an amplitude of 1 \({\mathrm{V}}_{(\mathrm{p}-\mathrm{p})}\) with a phase difference of 900. Figure 7 is a graph of changes in load torque, electromagnetic torque, and induction motor speed with time. In the model built, the induction motor bears the load at different times. Load of three-phase induction motor with parameters 54 HP (4 kW), 400 V, 50 Hz, and 1430 rpm. So the relationship between motor torque and load torque can be analyzed as follows using Eq. (13):

Fig. 7
3 line graphs trace the variation in signals for torque, electromagnetic torque, and motor speed with time. The torque time curve is stepped. It first ascends until 28 and then descends. The electromagnetic torque curve fluctuates about zero initially, then moves horizontally, ascends, and then descends again. The motor speed curve dips to 1200 initially, ascends to 1800, dips again, and then fluctuates.

Load torque curve, electromagnetic torque, and induction motor speed

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$${T}_{a}={T}_{L}+{T}_{sh}$$
(13)

\({T}_{\mathrm{a}}\) = torque (armature) or motor torque.

\({T}_{\mathrm{sh}}\) = shaft torque = net power shaft torque.

\({T}_{\mathrm{L}}\)= torque (loss).

\({T}_{sh}= \frac{{P}_{out}}{\omega }\),

where \(\omega =\frac{2\pi N}{60}\)

N = full load motor speed.

so \(\omega\) = \(\frac{2\times 3.14\times 1430}{60}\)  = 149.62 rad/s.

\({T}_{\mathrm{sh}\left(\mathrm{max}\right)}\)= \(\frac{4000}{149.62}\)

\({T}_{\mathrm{sh}\left(\mathrm{max}\right)}\)= 26.73 Nm.

Table 2 shows that between 0 and 0.2 s the induction motor has not been loaded or when the motor is starting the load torque is at zero level and the speed of the induction motor is at a maximum of 1499 rpm. When the time is in the range 0.2 s–0.4 s, the motor is loaded and the maximum torque is 26.73 Nm; at this time, the speed of the induction motor drops and is at a minimum speed of 1334 rpm. The time is between 0.4 s and 0.7 s and between 0.7 s and 0.9 s the load is reduced so that the speed increases according to the decrease in load torque.

Table 2 Net torque change of induction motor load against time
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Figure 8 describes the change in stator current at the start, zero loads, maximum load and 1/2 maximum load, 1/4 maximum load torque, and 1/8 maximum load torque. It appears that at the start of the motor running, the motor draws a large current of about 50 \({\mathrm{A}}_{(\mathrm{p}-\mathrm{p})}\). After the motor rotates for a while, the current approaches zero when the time is between 0.1 s and 0.2 s at which time the motor has not been loaded and rises to a maximum when it is at 0.2 s < t < 0.4 s maximum load torque. The current decreases over time between 0.4 s and 0.9 s following the decrease in load torque. The stator current pattern is in the form of a sinusoid with the same amplitude for each phase, and this magnitude affects the load torque.

Fig. 8
A line graph traces the variation in stator current with time at various load torque changes. It plots 3 overlapping waveforms that are noisy in 4 parts and have peaks and troughs that decrease in height.

Stator current against changes in load torque at various times

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Figure 9 shows the rotor current waveform when starting the motor, zero loads, 1/2 peak load, 1/4 peak load, and 1/8 peak loads. When starting the rotor current, a large current of about 45 \({\mathrm{A}}_{(\mathrm{p}-\mathrm{p})} \mathrm{is drawn}\), and then it drops to zero at zero loads. The rotor current reaches the maximum when full load torque is obtained when 0.2 s < t < 0.4 s. Then the current slopes as the load torque decreases when 0.4 s < t < 0.7 s and t > 0.9 s. The current drawn by this rotor causes a change in the load torque. When the minimum rotor current, the load torque is close to zero, resulting in maximum motor speed.

Fig. 9
A line graph traces the variation in rotor current with time at various load torque changes. It plots 3 overlapping waveforms that are noisy until 0.1 seconds and fluctuate thereafter with prominent peaks and troughs that decrease in height.

Rotor current at various load torque changes

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Fig. 10
A simulation waveform chart of v a, v b, and v c. The waveforms are composed of rectangular blocks connected at different heights.

Inverter output voltage \({V}_{\mathrm{a}}\), \({V}_{\mathrm{b}}\), \({V}_{\mathrm{c}}\) with SVPWM control

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Figure 10 is a simulation waveform of two voltage levels between three-phase lines \({V}_{\mathrm{a}}\), \({V}_{\mathrm{b}}\), \({V}_{\mathrm{c}}\) and amplitude 400 \({\mathrm{V}}_{(\mathrm{p}-\mathrm{p})}\) at a frequency of 50 Hz with SVPWM control. This voltage has a THD which is still quite large. Reducing this level of harmonics can be overcome by increasing the number of levels at the output voltage by adding an inverter arranged in a cascade.

5 Conclusion

The Space Vector Pulse Width Modulation method is a pulse width modulation technique that is very effective in controlling the performance of asynchronous machines. Inverter trigger pulse generation is based on the performance component of the inverter switch which is divided into 6 sectors to get the reference voltage and sector angle values. The difference in the angle of the sector is 600 for each sector. The duration and time pattern of the switching on of the inverter switch is the same, but the phase is different by 600. The load torque of the induction motor is influenced by the rotor current which has ripples when the load changes at the start and when the motor experiences a load change. The two inverter output voltage levels still have a high Total Harmonics Distortion (THD) level, so a higher number of levels is needed so that the THD content can be reduced. This model is easy to implement into the actual circuit so that it can help researchers to make Variable Speed Drives (VSD) for induction motors with good control.