Project 2 - Modeling of 3 phase Induction Motor Drive

Power SVPMW stands for Space Vector Pulse Width Modulation. It is a modulation technique used in power electronics to control the output voltage of a three-phase inverter.

The technique involves generating a space vector diagram that represents the three-phase output voltage of the inverter. The space vector is then split into six smaller vectors, each representing a different switching state of the inverter. By modulating the duty cycle of the switching signals for each vector, the amplitude and phase of the output voltage can be controlled.

Power SVPMW has several advantages over traditional PWM techniques, including higher output voltage quality, higher efficiency, and lower switching losses. It is commonly used in applications such as motor drives, renewable energy systems, and power converters.

In summary, Power SVPMW is a powerful modulation technique used in power electronics to control the output voltage of a three-phase inverter by modulating the duty cycle of the switching signals for each vector.

Simulink model is as below: 

 For the condition of Constant Torque at 10 N-M : 

Results : 

1. Stator Voltage (V).
2. Stator current (I). 

Results for : 

Speed (RPM), Vrms,Freq(Hz),Torq(N-M) 

 Outputs on display  : 

For the condition of Constant Torque at 20 N-M : 

Results : 

For the condition of Constant Torque at 20 N-M 

1. Stator Voltage (V).
2. Stator current (I). 

Results for : 

Speed (RPM), Vrms,Freq(Hz),Torq(N-M).

Outputs on display  : 

For back emf to speed relation, assume a linear rate from 0.25 to 1, with speed changing from 500 RPM to 2000 RPM

Results : 

For back emf to speed relation, assume a linear rate from 0.25 to 1, with speed changing from 500 RPM to 2000 RPM

1. Stator Voltage (V).
2. Stator current (I). 

Results for : 

For back emf to speed relation, assume a linear rate from 0.25 to 1, with speed changing from 500 RPM to 2000 RPM.

Speed (RPM), Vrms,Freq(Hz),Torq(N-M).

Outputs on display  : 

2.Use the 3 phase inductor motor theory to derive the transfer function of 3 phase Induction Motor.

Answer =

The 3-phase induction motor is a type of AC motor that consists of a stator and a rotor. The stator has three windings, typically spaced 120 degrees apart, that are energized with a three-phase AC voltage. This creates a rotating magnetic field that induces a current in the rotor, which in turn produces a torque that drives the motor.

The dynamics of induction motor (IM) is traditionally represented by differential equations.The space-vector concept  is used in the mathematical representation of IM state variables such as voltage, current, and flux. The concept of complex transfer function derives from the application of the Laplace transform to differential equations in which the complex coefficients are in accordance with the spiral vector theory which has been presented by. The complex transfer function concept is applied to the three-phase induction motor mathematical model and the induction motor root locus was presented in. Other procedures for modeling and simulating the three-phase induction motor dynamics using the complex transfer function concept are also presented in. The induction machine high performance dynamics is achieved by the field orientation control (FOC) . The three-phase induction motor field orientation control using the complex transfer function concept to tune the PI controller by using the frequency-response function of the closed-loop complex transfer function of the controlled induction machine was presented in . This strategy has satisfactory current response although stator currents had presented cross-coupling during the induction machine transients. An interesting solution was presented in [11] in which it was designed a stator-current controller using complex form. From this, the current controller structure employing single-complex zeros is synthesized
with satisfactory high dynamic performance although low-speed tests had not been shown in mentioned strategies. An alternative for induction motor drive is the direct torque control (DTC), which consists of the direct control of the stator flux magnitude λ1 and the electromagnetic torque Te. DTC controllers generate a stator voltage vector that allows quick torque response with the smallest.

The basic equations that describe the behavior of the motor are:

V_phase = R_phase * I_phase + jX_phaseI_phase ---(1)

where V_phase is the voltage applied to each phase of the motor, R_phase is the resistance of each phase winding, X_phase is the reactance of each phase winding, I_phase is the current flowing through each phase winding, and j is the imaginary unit.

The total torque produced by the motor is given by:

T = (3/2)P(synchronous speed - rotor speed)*R2/S ---(2)

where T is the torque produced by the motor, P is the number of poles in the motor, synchronous speed is the speed of the rotating magnetic field, rotor speed is the speed of the rotor, R2 is the rotor resistance, and S is the slip of the motor.

The slip of the motor is given by:

S = (synchronous speed - rotor speed)/synchronous speed

We can derive the transfer function of the motor by expressing the rotor speed in terms of the input voltage. Assuming a balanced three-phase system, the current in each phase is given by:

I_phase = (1/(jwL_phase))(V_phase - jwM_phaseI_r)

where w is the angular frequency of the AC voltage, L_phase is the inductance of each phase winding, M_phase is the mutual inductance between the stator and rotor windings, and I_r is the rotor current.

Substituting this expression for I_phase into equation (1) and solving for I_r, we get:

I_r = (1/(jwM_phase))[(V_phase/jw*L_phase) - I_phase]

Substituting this expression for I_r into equation (2) and simplifying, we get:

T = (3/2)PR2S[(V_phase/jwL_phase)/((jwM_phase)(synchronous speed - (V_phase/jwL_phase)/((jw*M_phase) + R2))))] ---(3)

Dividing both sides of equation (3) by the input voltage V_phase, we get the transfer function of the motor:

T/V_phase = (3/2)PR2S/((jwL_phase)((jwM_phase)/(synchronous speed - (V_phase/jwL_phase)/((jwM_phase) + R2))))

Simplifying this expression, we get:

T/V_phase = K/(1 + jwTau)

where K is a constant, w is the angular frequency of the AC voltage, and Tau is a time constant given by:

Tau = (L_phase*(jwM_phase + R2))/(jwL_phase*(synchronous speed - (V_phase/jwL_phase)/((jwM_phase) + R2)))

This is the transfer function of the 3-phase induction motor, which describes the relationship between the input voltage and the output speed of the motor. 

  3. Model 3 phase induction motor with a 3 phase inverter using Direct Torque control using an Average Model, Implement both torque and flux control to achieve desired power.

Design Parameters:

1. Initial Speed = 0 RPM
2. Flux Target = 2.5 webers
3. DC Voltage = 500 V;

• Fundamental Frequency = 60 Hz

1. Speed Step Time = 0.01 secs
2. Determine fundamental speed based upon fundamental frequency
3. Determine the switching sequence for direct torque control
4. Moment of Inertia = 0.1 Kgm^2
5. Load Torque Initial = 0 Nm
6. Final Load Torque = 50 Nm
7. Torque Step Time = 0.03 secs

Motor Parameters: (Create an Average Model)

1. Stator Resistance = 0.5 Ohms
2. Rotor Resistance = 0.2 Ohms
3. Stator Leakage Inductance = 20 mH
4. Rotor Leakage Inductance = 10 mH
5. Magnetizing Inductance = 200 mH
6. Number of Poles = 4
7. Torque to Speed 

DTC motor drive:

 DTC is emerging technique for controlling PWM inverter -fed induction motor IM drives. it allows the precies and quick control of the IM flux and torque without calling of complex control algoritham.

To implement both torque and flux control to achieve the desired power, we need to control the stator flux and the electromagnetic torque of the motor. This can be done using two control loops: one for the stator flux and the other for the electromagnetic torque. The control loops use the measured values of the stator currents, stator voltages, and rotor speed to control the motor.

The following steps can be followed to implement DTC using an average model for a three-phase induction motor with a three-phase inverter:

1. Determine the motor parameters: We need to determine the motor parameters such as the stator resistance, rotor resistance, leakage inductance, magnetizing inductance, and the number of poles.

2. Calculate the average values of the motor variables: We need to calculate the average values of the stator voltages, stator currents, and flux in each phase. This can be done using the measured values of the voltage and current in each phase.

3. Calculate the electromagnetic torque: The electromagnetic torque can be calculated using the following equation:

Te = 3/2 * p * (Ld - Lq) * iqs * ids

where Te is the electromagnetic torque, p is the number of poles, Ld and Lq are the direct and quadrature-axis inductances, and iqs and ids are the stator currents in the quadrature and direct axes, respectively.

4. Calculate the stator flux: The stator flux can be calculated using the following equation:

ψs = sqrt(ψsd^2 + ψsq^2)

where ψsd and ψsq are the direct and quadrature-axis fluxes, respectively.

5. Control the stator flux: We need to control the stator flux to achieve the desired power. This can be done using a proportional-integral (PI) controller that regulates the stator current in the direct axis (ids). The PI controller adjusts the magnitude and phase of the stator current to control the stator flux.

6. Control the electromagnetic torque: We need to control the electromagnetic torque to achieve the desired power. This can be done using a PI controller that regulates the stator current in the quadrature axis (iqs). The PI controller adjusts the magnitude and phase of the stator current to control the electromagnetic torque.

7. Generate the switching signals for the inverter: Once the desired stator current in the direct and quadrature axes is obtained, the switching signals for the inverter can be generated. The switching signals determine the switching states of the inverter switches.

8. Implement the control algorithm: Finally, the control algorithm can be implemented in a microcontroller or a digital signal processor. The microcontroller or the DSP reads the measured values of the stator currents, stator voltages, and rotor speed, and calculates the switching signals for the inverter. The inverter switches are then controlled using the switching signals to regulate the stator current and the electromagnetic torque.

 Simulink model: 

Simulink of 3 phase induction motor with a 3 phase inverter using Direct Torque control using an Average Model.

Block Parameters : 

Asynchronous Machine parameters : 

M file : 

imulink of 3 phase induction motor with a 3 phase inverter using Direct Torque control using an Average Model.

 Braking choper : 

Braking choppers, sometimes also referred to as braking units, are used in the DC voltage  intermediate circuits of  frequency converters to control voltage  when the load feeds energy back to the intermediate circuit.

A braking chopper is an electrical switch that limits the DC bus voltage  by switching the braking energy to a resistor  where the braking energy is converted to heat. Braking choppers are automatically activated when the actual DC bus voltage exceeds a specified level depending on the nominal voltage of the variable frequency drive.

Results : 

VSUpply and  Isupplu and VDC:

From the above graph we get the Grid Voltage (Top Plot) , grid Current (Middle Plot) and Dc voltage (Bottom plot).

Grid Voltage/ supply voltage  (Top Plot):-

The grid will gives us the three phase voltage source with Following Statistics.

Signal 01 :  Yellow phase.

From the graph the signal one is yeello phase with following statistics.

Maximum voltage is 3.756e+02 V at the time interval 0.570 sec

Miniimum voltage is -3.756e+02  V At the time interval 0.562 sec

Signal 02 : Blue phase 

From the graph the signal two is Blue phase with following statistics.

Maximum Current is 1.205e+01 A at the time interval 0.561 sec

Minimum voltage is -9.656e+00 A At the time interval 0.570 sec

Conclusion : In summary, to model a three-phase induction motor with a three-phase inverter using DTC, we need to use an average model and control the stator flux and the electromagnetic torque of the motor using two control loops. The control algorithm can be implemented in a microcontroller or a digital signal processor to regulate the motor.