An integral controller is used to improve the transient

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An integral controller is used to improve the transient response of a first order system. If G (s) = 1/1 + s and the system is operated in closed-loop with unity feedback, what is the value of T_i if integral controller transfer function is 1/T_i s to provide damping ratio of 0.5?

1. 0.5
2. 2
3. 1
4. 4

Correct Option: C

According to question C.E. of the system 1 + G(s) H(s) = 0

1 +	1	×	1	= 0
	T_is		1 + s

T_is + T_i s² + 1 = 0
or
s² + s + 1/T_i = 0 …(i)
on comparing equation (i) with the standard equation
s² + 2ξω_n s + ω_n² = 0
2ξω_n = 1
and
ω_n² = 1/T_i
or
ω_n = 1/√T₂
or
ξ = 1/2ω_n
or
ξ = √T_i/2
or
0·5 = √T₁/2
or
T_i = 1