Dept. of Electrical Engineering Al-Azhar University Cairo, Egypt Eng_mo.hussein@yahoo.com

Abstract—Induction Motors are rugged, robust and widely used in industrial applications. Field oriented control (FOC) or Vector control is an efficient method to control either the speed or the torque or both of the three phase Induction motor. FOC converts the complex stator currents into two orthogonal components, one of which is responsible for speed control, and the other for the electromagnetic torque control, similar to DC machines. FOC has greater stability and capable of fast torque response and accurate speed control. This research explains the concept of FOC and its simulation rather than experimental implementation using Matlab simulink and MexBios softwares.

I.         INTRODUCTION

Induction motors are the preferred choice for industrial applications due to their rugged construction, maintenance free and development in power electronics, which leads to easy control of motor speed.

The three-phase Induction Motors speed is:

where;

fs: Supply frequency in Hz p: Number of pole pairs of stator

S: Slip between of rotating field and mechanical speeds

The speed of three phase Induction motor can be controlled by varying the frequency of the applied stator voltage via scalar control or vector control methods.

II.       FIELD ORIENTED CONTROL

Field oriented control or offers more precise control of ac motors compared to scalar control. FOC is therefore used in high performance drives where oscillations in air gap flux linkages are intolerable, e.g. robotic actuators, centrifuges etc. In Field oriented control method, the components of the motor stator currents are represented by two orthogonal components in a rotating reference frame. In a special reference frame, the expression for the electromagnetic torque of the smooth-air-gap machine is similar to the expression for the torque of the separately excited DC machine. The control is normally performed in a reference frame aligned to the rotor flux space vector.  

The field oriented algorithm controls the components of the motor stator currents, represented by a vector, in a rotating reference frame d, q aligned with the rotor flux and perpendicular directions. The Clarke transformation converts three-phase currents i_a, i_b and i_c into two-phase orthogonal stator currents: i_a and i_b. These two currents in the fixed coordinate stator phase are transformed to the i_d and i_q currents components in the d, q reference frame using Park transformation. 

The stator variables are translated into a flux model which is compared with a required reference values and then is tuned by PI controllers. After a back transformation from field to stator coordinates, the output voltage will be impressed to the machine with Pulse Width Modulation (PWM) technique [2, 3 and 4].

Fig. 1. Stator currents in the d, q rotating reference frame       

where;

i_a , i_b , i_c : Three-phase stator currents  i_d : Direct current i_q : Quadrature current i_α , i_β : Two-phase orthogonal current  θ   : The phase frame rotation angle

A.         Mathematical Clarke transform

The mathematical Clarke transformation converts a three-phase system to a two-phase orthogonal system is obtained from:

Fig. 2. Direct Clark Transform

B.         Mathematical Park transform

The two phase components in the α, β frame representation calculated using Clarke transformation are then fed to a vector rotation block where it is rotated over a flux angle ρ to follow the frame d, q coincides to the rotor flux [2, 3 and 4].

The rotation over a flux angle ρ is achieved according to the formulae:

Fig.3. Direct Park Transform

C.         Mathematical Inverse Park and Clarke transforms Transformation uses the Inverse Clarke transformation, Figure 4.The vectors in the d, q frame are transformed from d, q frame to the two phases α, β frame representation calculated with rotation over a flux angle ρ according to the formulae:

Fig. 4. Inverse Park Transform

The modification from a two-phase orthogonal α, β frame to a three-phase system is done using the equations:

Fig. 5. Inverse Clarke Transform

III.     SIMULINK PROGRAM

Simulink is a software add-on to MATLAB which is a mathematical tool that enables to simulate model, and analyze systems whose outputs change over time. Such systems are often referred to as dynamic systems.

S-functions (system-functions) provide a powerful mechanism for extending the capabilities of Simulink to allow addition new blocks to Simulink models [5].

A.    Simulation of Vector Control of Induction Motor using Simulink program

Vector Control of three phase Induction Motor is simulated by the block diagram and S-Function as shown in Figures. 6 and 7 with following components:

1. Three phase Induction Motor
2. Clarke and Park transforms 
3. I_d_PI controller              4- I_q_PI controller
4. Inverse Park and Clarke transforms
5. Plus width modulation (PWM)
6. Three phase Inverter   8- Speed PI controller

The simulation of the Motor starts at No load and when the Motor reaches the steady state speed, there will be two case studies of simulation:

1. Rotating the motor in the reverse direction by inversing the sign of quadrature current (Iq) with no load.
2. Increasing the load torque (Tm).

Fig. 6. Field oriented control for three phase Induction Motor schematic diagram in Simulink

Fig. 7. Matlab Subsystem schematic diagram

B.    Rotating the motor in the reverse direction

The motor starts with no load, the three phase

currents (ia, ib, ic) will start with large values (I_start) and they will have some distortions until the motor reaches it's no load speed. When the motor reaches its rated speed and its rated three phase currents, the motor continues running with no load for 0.5 seconds until we  inverse the sign of quadrature current (Iq).

So, the quadrature current (Iq) reaches its peak value with negative sign then falls down to be close zero value because the motor runs with no load and the Direct current continue with its positive value as shown in Figure 8.                       

Fig. 8. Two phase currents VS Time                      

Fig. 23: Speed curve in PU with negative quadrature current

When the motor starts, the starting Iq, Id currents reache their negative peak values and starting to increase to reach their negative No load values as shown in Figure 24 a. After the Motor reaches No load Speed, Iq, Id currents will continue at their negative No load values as shown in Figure 24 b.

Fig. 24 a): Tow phase current starting values with negative load          b): Tow phase current No  values with negative quadrature current quadrature current

The value of Theta will start from zero and jump to one. Theta starts to decrease to reach to zero and jump again to one and so on. Because the Motor starts with Sensor less control, the value of Theta will start with a lot of disturbances for only few milliseconds until the Motor feedback signals return to the control circuits as shown in Figure 25 a and Figure 25 b.

Fig. 25 a): Theta starting values with negative quadrature              b): Theta after few milliseconds with negative quadrature