**Trapezoidal Profile for a de Motor**

ih many applications we want to accelerate the motor to a desired speed and allow it to ru

at that-speed for some time before decelerating to a stop. Investigate whether an appli

‘. – voltage having a trapezoidal profile will accomplish this. Use the values R = 0.6 Q.,

L = 0.002 H, Kr = 0.04 N· miA, K. = 0.04 V· s/rad, c = 0, and I = ~x JO-s kg· m~

The applied voltage in volts is given by

{

lOOt 0::5 t < 0.1

JO 0.1 < t < 0.4

vet) = -100(1 – 0.4) + JO 0.4:; t ~ 0.5

,0 t > 0.5

•• This function Is shown in the top graph in Figure 8.6-6 .

j.

• Solution

First find the time constants using the .eig function. Use the following script file:

‘,R = O.o;L = 0.002;c = 0rJ I

K_T =.0.04;K_e = 0.04;1 = 6e-5;

A = [-R/L, -K_elL;” K,51I, ,-CIl];’ •.

% compute the characteristic roots and

disp(‘The characteristic roots are:’)

eig(A) 10′

disp(‘The time constants are:’j

-l./real(e~g(b)}·

time constants.

The roots~ s = ~”245.7427 and s = -54.2573. The time constants are T) = 0.0041

. and’ T2.=,fOI-8’4 s:the largest time CU!l~arit’indicates that the motor’s response time is

approximately 4(0.0184) = 0.0736 t:&c;ause this time is less than the time needed for

the applied voltage to reach 10.V, the motor should be able to follow the desired trapezoidal

profile reasonably-well. To know for certain. we must solve the motor’s differential

Voltage input and resulting velocity response of a de motor.

equations. Use the following derivative function file:

function xdot = dcmotor(t,x)

% dc motor model with trapezoidal voltage profile.

% First variable is current; second is velocity.

global c I K_T K_e L R

A = [-R/L, -K_e/L; K_T/l, -ell);

B = [l/L; 0);

ift<O.l

v = 100*t;

elseif t <= 0.3

v = 10;

elseif t <= 0.4

v -100*(t – 0.4) + 10;

else

\

v 0;

end

xdot A*x + B*v;

Using the initial conditions x,(O) = 0, X2(O) = 0, the sol~tr is called as follows:

global c I K_T K_e L R

R = 0.6;L = 0.002;c.,= 0;

K_T = 0.04;K_e = 0:04;l =

[t, x) = ode23(‘dcmotor’,

6e-5;

[ 0 , O. 5), [ 0 , O·n;

The results are plotted in Figure 8.6-6. The motor’s velocity follows a trapezoidal profi e

as expected, although there is some slight deviation because of its electrical resistance a

inductance and its mechanical inertia.

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