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ECE Control syControl Tutorials for MATLAB and Simulink - Motor Position_ System Modeling

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DC Motor Position: System Modeling Key MATLAB commands used in this tutorial are: tf , ss Contents Physical setup System equations Design requirements MATLAB representation Physical setup A common actuator in control systems is the DC motor. It directly provides rotary motion and, coupled with wheels or drums and cables, can provide translational motion. The electric equivalent circuit of the armature and the free-body diagram of the rotor are shown in the following figure. For this example, we will assume the following values for the physical parameters. These values were derived by experiment from an actual motor in Carnegie Mellon's undergraduate controls lab. (J) moment of inertia of the rotor 3.2284E-6 kg.m^2 (b) motor viscous friction constant 3.5077E-6 N.m.s (Kb) electromotive force constant 0.0274 V/rad/sec (Kt) motor torque constant 0.0274 N.m/Amp (R) electric resistance 4 Ohm (L) electric inductance 2.75E-6H In this example, we assume that the input of the system is the voltage source (V) applied to the motor's armature, while the output is the position of the shaft (theta). The rotor and shaft are assumed to be rigid. We further assume a viscous friction model, that is, the friction torque is proportional to shaft angular velocity. System equations 12/21/2015 Control Tutorials for MATLAB and Simulink- Motor Position: System Modeling http://ctms.engin.umich.edu/CTMS/index.php?example=MotorPosition§ion=SystemModeling 2/4 (4) (7) (8) (9) (10) (1) (2) (3) (5) (6) In general, the torque generated by a DC motor is proportional to the armature current and the strength of the magnetic field. In this example we will assume that the magnetic field is constant and, therefore, that the motor torque is proportional to only the armature current i by a constant factor Kt as shown in the equation below. This is referred to as an armature-controlled motor. The back emf, e, is proportional to the angular velocity of the shaft by a constant factor Kb. In SI units, the motor torque and back emf constants are equal, that is, Kt = Ke; therefore, we will use K to represent both the motor torque constant and the back emf constant. From the figure above, we can derive the following governing equations based on Newton's 2nd law and Kirchhoff's voltage law. 1. Transfer Function Applying the Laplace transform, the above modeling equations can be expressed in terms of the Laplace variable s. We arrive at the following open-loop transfer function by eliminating I(s) between the two above equations, where the rotational speed is considered the output and the armature voltage is considered the input. However, during this example we will be looking at the position as the output. We can obtain the position by integrating the speed, therefore, we just need to divide the above transfer function by s. 2. State-Space The differential equations from above can also be expressed in state-space form by choosing the motor position, motor speed and armature current as the state variables. Again the armature voltage is treated as the input and the rotational position is chosen as the output.

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