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Problem 3.56PP
The transfer function of a typical tape-drive system is given by
__________________________
j[(j + OJ)(j+ l)(j2 + 0 ^ + 4 )] ’
where time is measured in miiiiseconds. Using Routh’s stability criterion, determine the range of 
K for which this system is stabie when the characteristic equation is 1 + KG(s) = 0.
Step-by-step solution
step 1 of 6
Step 1 of 6
Consider the transfer function of a typical tape drive system.
, , __________g ( j + 4)__________
s + 0.5)( j + 1)( + 0.4j + 4)J
Step 2 of 6
Consider characteristic equation.
l + ̂ TG(j) = 0 .......(2)
Step 3 of 6
Substitute Equation (1) in Equation (2).
, K(s+4)
1 + f ^ ~ Q
4|^(»+ 0 .5 )(4 + l ) ( s + 0 .4 s + 4 ) ]
j [ ( j + 0 . 5 ) ( j + l) (» ’ + 0 .4 s + 4 ) ] + ^ r ( j+ 4 ) = 0 
s ’ + l .9 s‘ + 5 .ls ’ + 6 .6 j ’ + 2 s + j A :+ 4A: = 0 
s ’ + 1.9s‘ + 5 .1 i’ + 6 .2 s’ + ( 2 + A : ) i+ 4 X = 0
Step 4 of 6
Thus, the characteristic equation is +1.9s*+5.L$^+6.2»^ + ( 2 + J l)4+4AT = 0 
Apply Routh array for this polynomial
Step 5 of 6
5.1 2 + K
1.9 6.2 4K
F ig u re 1
Step 6 of 6
The system is stable if the equation satisfies the following conditions;
• All the terms in the first column of the Routh’s array should have a positive sign.
• The first column of Routh’s array should not posses any sign change.
From the above statement, the stability conditions are,
/ :+ 3 .6 3 > 0
^ > - 3 . 6 3
And
-8 .4 3