DC Motor - refresher

DC Motor - refresher - Matlab and Simulink for Modeling and...

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Matlab and Simulink for Modeling and Control Robert Babuˇska and Stefano Stramigioli November 1999 Delft University of Technology Delft Control Laboratory Faculty of Information Technology and Systems Delft University of Technology P.O. Box 5031, 2600 GA Delft, The Netherlands
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1 Introduction With the help of two examples, a DC motor and a magnetic levitation system, the use of MATLAB and Simulink for modeling, analysis and control design is demonstrated. It is assumed that the reader already has basic knowledge of MATLAB and Simulink. The main focus is on the use of the Control System Toolbox functions. We recommend the reader to try the commands out directly in MATLAB while reading this text. The examples have been implemented by the authors and can be downloaded from http://lcewww.et.tudelft.nl/˜et4092. The implementation is done in MATLAB version 5.3 and has also been tested in version 5.2. 2 Modeling a DC Motor In this example we will learn how to develop a linear model for a DC motor, how to analyze the model under MATLAB (poles and zeros, frequency response, time-domain response, etc.), how to design a controller, and how to simulate the open-loop and closed-loop systems under SIMULINK. 2.1 Physical System Consider a DC motor, whose electric circuit of the armature and the free body diagram of the rotor are shown in Figure 1. V T R L + V b = K ϖ b ϖ - + - Figure 1: Schematic representation of the considered DC motor. The rotor and the shaft are assumed to be rigid. Consider the following values for the physical parameters: moment of inertia of the rotor J = 0.01 kg · m 2 damping (friction) of the mechanical system b = 0.1 Nms (back-)electromotive force constant K = 0.01 Nm/A electric resistance R = 1 electric inductance L = 0.5 H The input is the armature voltage V in Volts (driven by a voltage source). Measured variables are the angular velocity of the shaft ω in radians per second, and the shaft angle θ in radians. 2.2 System Equations The motor torque, T , is related to the armature current, i , by a constant factor K : T = Ki. (1) The back electromotive force (emf), V b , is related to the angular velocity by: V b = = K dt . (2) 1
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From Figure 1 we can write the following equations based on the Newton’s law combined with the Kirchhoff’s law: J d 2 θ dt 2 + b dt = Ki, (3) L di dt + Ri = V - K dt . (4) 2.3 Transfer Function Using the Laplace transform, equations (3) and (4) can be written as: Js 2 θ ( s )+ bsθ ( s )= KI ( s ) , (5) LsI ( s RI ( s V ( s ) - Ksθ ( s ) , (6) where s denotes the Laplace operator. From (6) we can express I ( s ) : I ( s V ( s ) - ( s ) R + Ls , (7) and substitute it in (5) to obtain: 2 θ ( s bsθ ( s K V ( s ) - ( s ) R + Ls . (8) This equation for the DC motor is shown in the block diagram in Figure 2. Velocity Torque Armature Ts () Load Back emf Voltage Angle w( ) s Vs + - 1 Js+b q( ) s K b K Ls+R 1 s Figure 2: A block diagram of the DC motor.
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DC Motor - refresher - Matlab and Simulink for Modeling and...

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