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P rob lem 6.35PP
(a) to determine the range of values of K (positive and negative) for which the system will be
For the system in Fig., determine the Nyquist plot and apply the Nyquist criterion
(b) to determine the number of roots in the RHP for those values of K for which the system is 
unstable. Check your answer by using a rough rootlocus sketch.
Figure Control system
S tep -by-s tep s o lu tio n
step 1 of 14
(a)
Refer to the feedback system in Figure 6.95 in the textbook.
The closed loop transfer function for the above feedback system is:
R I+G(s)N(s)
3
j ( j + I ) ( j+ 3 )
1+ a: 3 | i
« ( j+ l) (« + 3 )J
3R
5 ( j + 1) (j + 3 )+ 3 ^
From the above transfer - 4 , Z = 0 : that is there are no characteristic equation roots in the RHP.
So the system is stable for > - 4 .
When j ^ 3
Step 10 of 14 ^
(b)
The number of roots in the RHP for those values of x tor which system is unstable can be 
determined from the rough root locus sketch.
Draw the root locus for equation (1).
Procedure to draw root locus:
RULE 1 : Number of poles of the feedback system shown in Figure 1 are three. So, there are 
three branches to the locus, all three poles approach asymptotes.
RULE 2 : The real-axis segment defined bv j s o ^doesn ’t 
represent break away point.
The root - 0.45 lies in the range —]