Transfer Function Modeling of Parallel Connected

Transfer Function Modeling of Parallel Connected Two Three-Phase Induction Motor Implementation Using LabView Platform R.Gunabalan, Member IEEE, P. Sanjeevikumar, Senior Member, IEEE, Ahmet H.Ertas, Member, IEEE, Viliam Fedák, Frede Blaabjerg, Fellow IEEE, V.Subbiah, Senior Member, IEEE Abstract– This paper presents the transfer function modeling and stability analysis of two induction motors of same ratings and parameters connected in parallel. The induction motors are controlled by a single inverter and the entire drive system is modeled using transfer function in LabView. Further, the software is used to perform the stability analysis of the parallel connected induction motor drive under unbalanced load conditions. It is very simple compared with the methods discussed so far to study the performance of the drive under unbalanced load conditions. Control design and simulation toolkits are used to model the drive system and to study the stability analysis. Simulation is done for various operating conditions and the stability investigation is performed for different load conditions and difference in stator and rotor resistances among the two motors.

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Index Terms– Three-Phase Induction Motor, LabView Software, Matlab Simulation, Transfer function model.

I. INTRODUCTION MatLab (Matrix Laboratory) and LabView (Laboratory Virtual Instrument Engineering Workbench) are the modeling softwares developed by Math Works and National Instruments respectively. Matlab is preferred by most of the engineers and scientists around the world for modeling and simulation of various power electronic circuits, electrical drives, digital signal processing, soft computing and power system analysis because of plenty of additional libraries and the simulink add-on. LabView is a graphical programming language preferred for measurement and data acquisition systems. It is also preferred for real time data processing and for interaction with the hardware. It was employed to simulate the direct torque control (DTC) of asynchronous motor using fuzzy logic [1]. It was used to identify the induction motor parameters [2-3], on line determination of single/two phase induction motor drive characteristics [4], to study the asynchronous motor functional characteristics [5], dc-dc converter [6], vector control of induction motor drive [7], power quality monitoring [8] etc. Inverter fed induction motor drive is used in major industries for simple control. Induction motors are connected in parallel and driven by a single inverter to reduce the cost, size and maintenance. Unbalance load condition arises in parallel operated drives whenever there is a change in wheel diameter or motor slip-torque characteristics [9] and is mandatory to know the stability of the dual drive. To make the system stable for unbalanced load conditions, different

control methodologies were presented in the literature [10– 11]. In this paper, two induction motors connected in parallel and controlled by a single inverter is modeled using transfer function and the performance of the drive system is proved for different operating conditions in LabView. II. TRANSFER FUNCTION MODELING OF VECTOR CONTROLLED INDUCTIONMOTOR DRIVE INPARALLEL The transfer function model developed for vector control of single induction motor drive [12] is extended for two induction motors connected in parallel and operated by a single inverter. The transfer function of the induction motor drive is derived under the assumption of constant rotor flux linkages. The block diagram representation of the vector controlled drive system is framed by applying the transfer functions of various subsystems, such as the induction motor, inverter, speed controllers and feedback transfer functions. The simulation of transfer function model of the parallel connected induction motor drive system is performed for various load conditions and mismatch in stator and rotor resistances. The speed response of the transfer function model is similar to the response obtained by constructing the drive system with conventional state space model. The transfer functions of various subsystems are represented as follows: The approximate transfer function model of induction motor is (1) The q-axis stator current which produces the electromagnetic torque is derived from the d-q model of the induction motor. = − (2) =

where,

,

=

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=

Electromagnetic torque is given by = where, torque constant =

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(3) (4)

A Proportional Integral (PI) controller is used to process the speed error between the speed reference signal and filtered speed feedback one. The transfer function of PI speed controller is obtained as: (

)

(5)

Fig. 1. Configuration of parallel-connected induction motor drives.

The inverter is modeled as a gain Kin with a time lag of Tin. The gain is derived from the given DC voltage to the inverter. The time lag in the inverter is equal to the average carrier switching cycle time. The transfer function of inverter is (6) where, (7) (8) Transfer function of mechanical system is

(9)

The feedback signals are current and speed, which are processed through first-order filters. The feedback gains are assumed as unity. The block diagram of the vector controlled parallel connected induction motor drive is illustrated in Fig. 1. With the measured speeds of both motors, torque reference is calculated from the speed error using PI controllers. The reference speed which is common to both induction motors is compared with the actual running speed. The speed error is processed in the PI controller and the torque reference is

developed from the average output of the PI controller. The output of the inverter is proportional to the torque reference and it drives the induction motors in parallel. III. SIMULATION OFTRANSFER FUNCTIONMODELING OF INDUCTION MOTORS CONNECTEDINPARALLELWITH LABVIEW The transfer function model of the parallel connected induction motor drive is developed in LabView. It is framed by the following block diagram codes in control and simulation loop:  Step signal in signal generation.  Gain, summation in signal arithmetic.  Transfer function in continuous linear systems.  Simulation time waveform in graph utilities.  Numeric indicator in front panel. Table I presents the parameters and ratings of the induction motor. The developed block diagram codes after substituting the ratings and parameters in Eqs. (1) - (9) are shown in Fig. 2. It is constructed with minimum number of components in

Fig. 2. Block diagram codes for modeling of two induction motors connected in parallel (LabView Software Developments).

LabView using transfer function and is very simple. The conventional state space model of induction motor with controller equations for constructing the parallel connected motor drive modeled in Matlab/Simulink. It is very complex and involves many subsystems, simulink libraries and power system blocksets. Once the drive is modeled; its performance is obtained under diverse operating conditions to confirm its performance. III. a) Step Change in Speed The speed command is set at 750 rpm and at t=4s, speed is increased from 750 rpm to 1000 rpm (step command). The speed responses of the motors for step change in speed attained in LabView is illustrated in Fig. 3. It is inferred that both motors follow the speed command with zero steady state error. The torque reference is generated by averaging the output of the PI speed controllers of motor 1 and 2 respectively. The obtained speed waveform is identical to the response obtained from the conventional state space model with speed estimation [10] as shown in Fig. 4. It indicates that the speed response is identical in both cases, which are state space complex model and the simple transfer function model. Maximum overshoot exists in the speed responses in both software packages. III. b) Balanced Load The command speed is set at 1000 rpm and motors run at no load initially. At t=4s, a load of 2.5 Nm is applied to both motors. The speed responses of the motors acquired in LabView for balanced load are depicted in Fig. 5. It is observed that both motors follow the speed and load command with zero steady state error. For verification, the transfer function model of the drive system is constructed in Matlab and the obtained speed response is shown in Fig. 6.

Matlab are illustrated in Fig. 7 and Fig. 8, respectively. It is observed that speed of heavily loaded motor drops to 927.5 (≈928) rpm from the command speed of 1000 rpm and the speed of the motor under no load conditions raises to 1071.6 (≈1072) rpm. The speed difference between the induction motors is 144 rpm and run at steady speed with stable running conditions. The speed deviation exists between the induction motors because the average parameters are considered and differential parameters are not used IV. CONCLUSIONS The performance of two induction motors operated in parallel and controlled by a single inverter based on transfer function approach is discussed for the very first time for various operating conditions. Induction motors, speed controllers, inverter and feedback gains are modeled using transfer function. The models are constructed using simple VIs in control design simulation toolkit in LabView. It is very simple and complicated d-q model, inverter circuits and PI blocks are not required. The average parameters are used and differential parameters between the induction motors are not considered. It is possible to obtain speed, current, flux and torque waveforms in state space model. But, in transfer function model, only the speed response is obtained. The speed response is the required parameter to study the performance under unbalanced load conditions. Complete set of both Matlab and LabView results are provided and show good agreement with theoretical equations.

III. c) UnBalanced Load The command speed is set at 1000 rpm and motors run at no load initially. At t=4s, a load of 2.5 Nm is applied to motor 2 alone. The speed responses obtained in LabView and 1200

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Fig. 4. Speed response by conventional method in Matlab. 1200

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Time (s) (c) Speed response of motor 2 Fig. 7. Speed response for unbalanced load in LabView.

REFERENCES [1]

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Fig. 8. Speed response for unbalanced load in Matlab.Lab

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