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The transfer function of a system is \(\frac{{Y\left( s \right)}}{{R\left( s \right)}} = \frac{s}{{s + 2}}\). The steady state output \(y\left( t \right)\) is \(A\;cos( {2t +\phi })\) for the input \(cos( {2t})\). The values of \(A\;and\;\phi\), respectively are
1. \(\frac{1}{{\sqrt 2 }}, - 45^\circ\)
2. \(\frac{1}{{\sqrt 2 }}, + 45^\circ\)
3. \(\sqrt 2 , - 45^\circ\)
4. \(\sqrt 2 ,\; + {45^o}\)

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Correct Answer - Option 2 : \(\frac{1}{{\sqrt 2 }}, + 45^\circ\)

Concept:

For an LTI system, the sinusoidal input produces a sinusoidal output of the same frequency but different amplitude and phase. The amplitude and phase depend upon the transfer function of the system.

For example:

Let, r(t) = cos(ωt) then

y(t) = a cos(ωt + ϕ)

where a = |H(jω)|, ϕ = ∠ H(jω)

\(H(jω)=\frac{Y(jω)}{X(jω)}\)

Calculation:

\(H(jω)=\frac{jω}{jω+2}\)

r(t) = cos (2t)

ω = 2

\(H(jω)=\frac{j2}{j2+2}\)

\(|H(jω)|=\frac{{1}}{{\sqrt 2}}\)

∠ H(jω) = 90° - 45°

= +45° 

\(y(t)=\frac{1}{\sqrt 2}~cos (2t + 45^{\circ})\)

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