Inverse Laplace Transform via Circuit Analysis [HELP]

Let us denote by $a$ the value $1/\sqrt 2$ to have an easy typing. The the partial fraction decomposition over complex numbers of the given fraction is indeed $$ \begin{aligned} \frac{1}{s^2 + \sqrt{2}s + 1} &= \frac{1}{(s^2 + 2as + a^2)+a^2} = \frac1{2ia}\left(\frac 1{s+a-ia}-\frac 1{s+a+ia}\right) \\ &=\frac K{s+a-ia}-\frac {K^*}{s+a+ia} \ . \\[3mm] \text{Here:} & \\ K &=\frac 1{2ia} =-\frac 1{2a}i =\frac 1{2a}\left(\cos\left(-\frac\pi2\right) +i\sin\left(-\frac\pi2\right)\right) \\ &=a\left(\cos\left(-\frac\pi2\right) +i\sin\left(-\frac\pi2\right)\right)\ , \\ |K| &= a\ ,\\ \theta &=-\frac \pi 2\ . \end{aligned} $$ The inverse Laplace transform is thus using the table: $$ 2ae^{-at}\cos(at+\theta) = 2ae^{-at}\sin(at) = \sqrt 2\cdot e^{-t/\sqrt 2}\sin(t/\sqrt 2) \ . $$

Tags:

Laplace Transform

Partial Fractions

Inverse Laplace

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