Why do we use $s=jomega$ in AC analysis instead of $s=sigma+jomega$?

In AC analysis, $s=j\omega$ when we deal with $sL$ or $1/sC$. But for a Laplace transform, $s=\sigma+j\omega$.

Sorry for being ambiguous but I would like to connect the questions below:

• Why is sigma equal to zero?


• Is neper frequency connected to this?


• Is sigma equal to zero as the input signal is a sinusoid of constant $\pm V_{max}$?

Answer

Of course, $s = \sigma + j\omega$, by definition. What's happening is that $\sigma$ is being ignored because it is assumed to be zero. The reason for it is that we are looking at the response of the system to periodic (and thus non-decaying) sinusoidal signals, whereby Laplace conveniently reduces to Fourier along the imaginary axis. The real axis in the Laplace domain represents exponential decay/growth factors that pure signals do not have, and which Fourier does not model.