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In circuit analysis, I understand the use of Laplace Transforms to obtain the impedance of a linear RLC circuit, ie transforming from the time domain to the frequency domain. In most texts I have seen and classes I have taken, the Bode plot looks at the rational transfer function of $H(j\omega)$, which is effectively the Fourier Transform, and not $H(s)$.

Since $\omega$ is considered to be the variable that represents frequency, what is $\sigma$ if $s = \sigma + j\omega$? Does $\sigma$ have a physical interpretation?

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Note that each complex pole $s_{\infty}=\sigma + j\omega$ of the transfer function $H(s)$ contributes to the system's impulse response a complex exponential of the form

$$e^{s_{\infty}t}=e^{\sigma t}e^{j\omega t}$$

The term $e^{\sigma t}$ is real-valued and represents exponential damping (assuming that the system is causal and stable, i.e. $\sigma<0$), and the complex term $e^{j\omega t}$ represents an oscillation at frequency $\omega$. So $\sigma$ is a damping constant.

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