1. ME 335 Boğaziçi University A Study on Motor Speed Control

2. ME 335 Boğaziçi University System Equations KVL and Newton’s 2’nd law gives : Let : Output : y = Input : u = v a Disturbance w = T l Laplace transforming, we obtain :

3. ME 335 Boğaziçi University Transfer Functions Where:

4. ME 335 Boğaziçi University System Parameters For this example, the following values for the physical parameters will be assumed. moment of inertia of the rotor (J) = 0.0001 kg.m 2 /s 2 * damping ratio of the mechanical system (b) = 0.0 Nms * electromotive force constant (K=K e =K t ) = 0.01 Nm/Amp * electric resistance (R) = 0.1 ohm * electric inductance (L) = 0.0005 H * input (V): Source Voltage * output ( ): angular velocity of shaft * disturbance (T l ) : Load torque * The rotor and shaft are assumed to be rigid Reference : http://www.engin.umich.edu/group/ctm/index.html

5. ME 335 Boğaziçi University Numerical Values J=0.0001; b=0.0; Ke=0.01; Kt=0.01; R=0.1; L=0.0005; A=Kt/(b*R+Kt*Ke); A=100 B=1/(b*R+Kt*Ke); B=10000 den=[J*L/(b*R+Kt*Ke) (J*R+b*L)/(b*R+Kt*Ke) 1]; r = roots(den); tau1= -1/r(1); tau2=-1/r(2); tau1 = 0.0947 s.: mech. time constant tau2 = 0.0053 s.: electrical time constant.

6. ME 335 Boğaziçi University Open Loop Step Response step(A,den,0:.05:1.5) title('Step Response for the Open Loop System')

7. ME 335 Boğaziçi University Steady State Model y ss = 100 u + 10000 w where y is in rad/s, u is in Volts and w is in N/m. For example, u = v a = 1.5 Volts gives  ss = 150 rad/s = 1432 rpm. If w = T l = -0.002 N m., then  ss = 150 - 20 = 130 rad/s = 1241 rpm. Result : Without control, the speed decreases (or increases) proportional to w. Controller must increase (or decrease) u, to compensate for the effect of disturbance.

8. ME 335 Boğaziçi University Feedback Control Measurement : Tachometer gain : 0.028 Volts/rad/s Simulink block diagram Input saturation : Input voltage v a can be between 0 and 2.5 volts Reference signal : 150 rad/s * 0.028 Volts/rad/s = 4.2 Volts; is compared with tachometer output.

9. ME 335 Boğaziçi University Proportional Control y ref is increased from 150 to 160 rad/s. There is ss. error, ss. error decreases as K P is increased

10. ME 335 Boğaziçi University Disturbance Rejection A step torque disturbance of T l = -0.002 Nm. is applied Compared to no control (130 rad/s), there is improvement, still there is ss. error.

11. ME 335 Boğaziçi University PID Controller Control signal is a linear combination of error, integral of error and time rate of change of error.

12. ME 335 Boğaziçi University PID Control of DC Motor No steady state error, good response.