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Tell me how can the 's' in first equation described with the 'integration' in second equation?

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This is a Laplace transform. 1/s is an integrator, is the Laplace transform of an integral. s is the Laplace transform of a derivative.

$v_1(t)=-R*C*d(v_0(t))/dt ------>V_1(s)=-R*C*s*V_0(s)$

$s$ is dimensional, its dimension is $time^{-1}$. It's basically a complex variable associated with frequency. The Laplace realm is the frequency realm, as opposed to the temporal realm.

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The first equation is the Laplace transform of the second. This is a subject that's touched on in a differential equations class (usually the second semester of a 2-semester Junior-level college class in the US), and is covered extensively in signal processing course work in Electrical Engineering.

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