Nine-Phase Induction Motor Dynamic Model Based On 3 x 9 Transformation Matrix

Analysis of the dynamic model nine-phase induction motor becomes difficult because based on a 9x9 matrix and a circuit of magnetically coupled transformer. In order to gain the qdn dynamic model of the nine-phase induction motors in easy, simple, quick and consistent way, the motor analysis will based on 3x9 transformation matrix and the equivalent circuit T model. The 3x9 transformation matrix qdn is substituted into the equation in form abc so that the matrix equation of qdn is obtained. Then qdn equation results a similar qd equivalent circuit, which has different methods n that is, based the circuit of magnetically coupled transformer. Simulation results show that the dynamic characteristics of 9-phase induction motor based on the T model and magnetically coupled circuit have similar torque and speed dynamic response.


Introduction
A dynamic model of qdn for induction motor is required to determine the characteristics of the starting conditions of motor until it gets steady state.All dynamic induction models of motor have been introduced by using the analysis based on the circuit of 1-phase transformer, which is magnetically coupled by stator and rotor frequency references [1]- [3].A single-phase transformer circuit can easily be converted into T models of electrical circuit by primary/secondary reference because it has similar frequency.Mutual reactance of transformer under steady-state conditions is constant because all primary and secondary winding of the transformer are silent.
However, an induction motor is very different from the transformer, the stator winding is stationary and the rotor winding is moving.In steady-state conditions, mutual reactance of the motor should have a function to manage rotation speed of the rotor which is then assumed to be constant to simplify the analysis.In addition, all of the parameters and frequency references refer to each part of the motor.While stator and rotor reactance are the leakage reactance as given in [4].Thus, magnetically coupled circuits for 3-phase induction motor may not produce the equivalent circuit of the transformer T models [5].Similarly, the equivalent circuit for the motor with higher phase becomes much more complex and may not be a T model of the equivalent circuit of the motor.Meanwhile, all the motor parameters obtained from testing the motor are based on the equivalent circuit T model.All parameters are expressed in the stator reference [6][7].Mutual inductance can be obtained from the mutual reactance.Inductance of the stator and rotor can be obtained from leakage reactance the motor.At T model, the frequency of the rotor follows the stator frequency.Thus, there is a difference base in the analysis and testing of induction motors.
To generate dynamic model in term qdn, a square matrix is required according to the phase number of induction motors [8].The transformation matrix 9-phase system has been introduced to anticipate the development of equipment that is considered to produce large power and small dimensions [9][13].Transformation matrix for the 9-phase induction motors has 9x9 orders.Strong effort is required to obtain dynamic model of the 9-phase induction motors in term qdn.
A 3x9 matrix transformation can be used to simplify the form the analysis of the equation of 9-phase systems and can speed up calculations with the same result.Purpose of simplification can be achieved when a small number of matrix elements are used in the analysis.Transformation matrix used in the modeling of 9-phase induction motor has a line of three.Thus, the form of the transformation matrix 9-phase system is much simpler than the 9x9transformation matrix.
Therefore, it will be simple, consistent and fast to obtain the analysis of the dynamic modeling of 9-phase induction motor using 3x9 transformation matrix based on the motor equivalent circuit T model.This dynamic model using qdn transformation is based on equivalent circuit T model by reference to the stator.Simulation is used to test the simple dynamic model by a synchronous reference frame.Motor load is made from full to half-full load.A dynamic response obtained from the simulations shows that there is no difference between the proposed model with the dynamic model by synchronous and rotor reference frame.
Electromechanical torque generated in the motor in term qdn can be expressed as: ) where: J is load inertia T L is load torque

Simulation
Equations 19 and 20 are the basic simulation of 9-phase induction motors.However, the desired result is the flux linked to the stator and the rotor, so that Equations 19 and 20 are substituted into the qdn modified equations 32 and 33 as follows: where  s expressed as: As desired outputs and inputs are respectively, the motor current and the motor flux.Therefore, the motor flux equations must be adapted to the needs so that it looks like the Equation 36.Simulations obtained from Equation 36 can be seen in Figure 4. Figure 4 has already shown the stator flux and rotor flux act as input, while the stator and rotor current act as output.Simulation of the motor is not complete before the slip is obtained.The slip that occurs in motor is according to the following equation 37: where  m is the rotor angular velocity in electrically When the input is synchronous rotor speed, the output is the slip.Simulation of slip model can be seen in Figure 5.The last simulation of 9-phase induction motors is electromechanical parts.The Equation that contributes to this section is equations 30 and 31.Both equations are not described as inputs or outputs according to the simulation.Adjustments are made in order to obtain the mechanical angular speed of the motor by giving input of an electromechanical torque and a load torque.The simulation of electromechanical torque can be seen in Figure 6.

Results and Discussion
In this paper, the induction motor is a nine-phase induction motor with very low voltage.This motor was developed from a 3-phase induction motor for electric car need with the goal of a high level of safety.Motor data are taken from previous research as presented in Table 1.The number of poles and the inertia motors respectively are 2 and 0.025 kg-m 2 .
To determine the dynamic characteristics of 9-phase induction motor it is done by making a simulation of the analytical results.Then the motor data in Table 1 are  frames that have been developed previously are used as validation.Source voltage of 9-phase induction motor is 9-phase voltage symmetry that is divided in groups of 3-phase, each different phase angle of 40°as shown in Figure 8.After that, the source voltage is transformed to the coordinates of the reference that the stator qdn can generate v q = 225 V, v d = 0 V and v n = 0 V as shown in Figure 9.
Stator and rotor produced currents in term qdn can be seen in Figures 10 and 11.Rotor currents show a negative value to the stator current, which means that the actual current is unidirectional due to the opposite rotor current to the direction given.Just at the start, the stator current (i d ) is higher than rotor current (i q ).Then when approaching steady-state current, for a moment, the stator current (i d ) turns lower than the rotor (i q ) current.At steady-state with full load, the stator current (i d ) is higher than the iq current.When the stator current (i d ) is still lower than i q current and does not change position despite the motor load, the (i d ) is lowered to halffull load.Characteristics of 9-phase induction motor obtained from the simulation for synchronous reference frame can be seen in Figure 12 and 13.The transient response of angular speed of the proposed method is no over-shoot and in the same time achieving steady-state conditions at full load compared to models using synchronous reference and rotor reference.The load decrease does not cause oscillations in response of angular speed.Meanwhile, the transient response of the oscillating torque equally and the same time achieve steady-state conditions at full load compared to the synchronous reference frame.The load decrease does not cause oscillations in response of torque and angular speed.The proposed model has the same maximum torque of the models using synchronous reference.At steady-state with full load, the torque generated is similar between the proposed motor model with a model that uses synchronous reference and rotor reference.In areas of change, from full to half load, the torque on both motors is down and the speed is faster.Both torque and speed responses have a large value.Both motors have no overshoot and oscillation when the load changes.They remain the same in next steady-state conditions.

Conclusion
Analysis of qdn dynamic model of 9-phase induction motor indicates that at the qd equivalent circuit has no difference with the previous method, but in the neutral equivalent circuit (n), there is a parameter mutual inductance between the stator to the rotor.From the simulations, it can be seen that the synchronous reference frame, compared to the previous method, shows the characteristic torque and angular speed of 9-phase induction motor reaching steady-state in the same time to produce the same maximum torque.The transient response does not occur of oscillation and over-shoot.Dynamic characteristics of 9-phase induction motor based on the T model and magnetically coupled circuit have similar torque and speed dynamic response.In the circuit-based model of T, torque and angular speed response with similar maximum torque achieve steady-state rather than the model based on magnetically coupled circuit.

Figure
Figure 1.Motor equivalent circuit T model

Figure 4 .
Figure 4. Simulation of motor current as a function of the motor flux

Figure 5 .
Figure 5. Simulation of slip as a function angular velocity

ISSN 273 Figure 6 .Figure 7 .
Figure 6.Simulation of the mechanical angular velocity as a function of torque included in the simulation by different loading treatment in the stator reference frame.The motor load is in fullload conditions at start and then it is changed to a half-full load.Stator and rotor reference  ISSN: 1693-6930 TELKOMNIKA Vol.11, No. 2, June 2013: 265 -276 274

Figure 8 .Figure 9 . 275 Figure 11 .
Figure 8. v abc output voltage which is the result of the transformation of the term v qdn by  = 0

Figure 12 .Figure 13 .
Figure 12.Characteristics of torque and angular speed vs. time in the refference frame  s )