GATE 2015 EE – SET 2 – Question 24

Solution :

c(t)=$$ e^{-2t}$$(sin5t+cos5t)
L$$\left [ c(t) \right ] $$= L$$\left [ e^{-2t}sin5t + e^{-2t} cos5t\right ]$$

G(s) = $$\frac{5}{(s+2)^{2}+5^{2}} + \frac{(s+2)}{(s+2)^{2}+5^{2}}$$

Formulae :

L$$\left [ e^{at} sinbt\right ] $$= $$\frac{b}{(s-a)^{2}+b^{2}}$$

L$$\left [ e^{at} cosbt\right ] $$= $$\frac{(s-a)}{(s-a)^{2}+b^{2}}$$

G(s) = $$\frac{5}{(s+2)^{2}+5^{2}} + \frac{(s+2)}{(s+2^{2})+5^{2}}$$

G(s) = $$\frac{5}{(s+2)^{2}+25} + \frac{(s+2)}{(s+2^{2})+25}$$

G(s)= $$\frac{s+7}{(s+2)^{2}+ 25}$$

DC gain  = $$\lim_{s\rightarrow 0}$$G(s)

$$\lim_{s\rightarrow 0}$$ $$\frac{s+7}{(s+2)^{2}+ 25}$$ = $$\frac{7}{4+25}$$ = $$\frac{7}{29}$$ = 0.241