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Problem 4.32PP
The DC motor speed control shown in Fig. is described by the difTerential equation
y + 60y = 600va-1500w,
where y is the motor speed, î a is the armature voltage, andw is the load torque. Assume the 
armature voltage is computed using the PI control law
Va = - ^*j>e + t / j T ed t^ ,
where e = r - y .
(a) Compute the transfer function from W toYas a function of kP and kl.
(b) Compute values for kP and k l so that the characteristic equation of the closed-loop system
(b) Compute values for kP and k l so that the characteristic equation of the closed-loop system 
will have roots at -60 ± 60/.
Figure DC Motor speed-control block diagram
Step-by-step solution
step 1 of 4
(a)
Consider the differential equation of the DC motor.
+60;> = 600v, - l.SOOw 
Apply Laplace transform on both sides. 
jK(j)+60y (s) = 600K.(5)- 1.500FK(5)
Consider the armature voltage value in PI control.
Apply Laplace transform on both sides.
Step 2 of 4
Substitute
equation.
for in the Laplace transform of the differential
(i + 60)r(i)-600^-*,,£(j)-^£(j)j-l,500»'(s) 
{s + 60)Y(s) = - 6 0 0 ^ ^ * , j £ ( j ) j - l ,5 0 0 » '( i )
( j + 6 0 ) r ( j ) + 6 0 0 ^ ^ * , + ^ j £ ( j ) j = - l , 5 0 0 » '( j ) 
^ s + 60 + 6 0 0 t , + 6 0 0 ^ j £ ( s ) = - l ,5 0 0 » '(» )
E (s) -1 ,5 0 0
+ 6 0 + 600* , + 6 0 0 ^ j
- l ,5 0 0 i
” ( i “ + 6 0 j + 60 0 * ,* + 600*,)
Consider the input R as zero. The error function is,
£(s)=-r(s)
y ( j ) ________ l,5 0 to __________
W{s) « '+ 6 0 ( l + IO * ,)s+ 6 0 0 * ,
Thus, the transfer function from 1/Yto Y is
1,5005
*’ +60(1+ 10*,)i +600*,
Step 3 of 4 ^
The roots of characteristics equation are -6 0 + 6 0 y ,-6 0 -6 0 y 
Write the characteristic equation from the roots. 
(s + 6 0 + 6 0 y ){ i+ 6 0 -6 0 y ) = s’ + 120*+7200 
The characteristic equation of the system is,
, * + 6 0 ( l + 1 0 * ,) i + 6 0 0 *,= 0
Compare the two characteristic equations.
600Jt,»7200 
60(U10Jfcp)»120 
Calculate integral constant kf ■
_ 7200
600
-12
Thus, the value of kf is m
Calculate proportional constant kp ■
60(1+ 10*,) = 120 
1 + 10*, = 2 
* ,= 0 .1
Thus, the value of kf, is I P
Step 4 of 4