Assumption that $s=j\omega$ when using only the sine function

Question

Working in engineering we very often use the following identity:

$$s=j\omega$$

Where $s$ is the Laplace variable, $j^2=-1$ and $\omega$ is the angular frequency.

But where does that comes from. Because normally we set: $s=\sigma+j\omega$ so why do we assume that $\sigma$ equals $0$?

Answer 1

The Laplace Transform can be thought of as a generalization of the Fourier Transform into (at least the right half of) the complex plane. In the s-plane of the Laplace Transform, the Fourier Transform is along the $s = j\omega$ axis.

The Laplace Transform is defined as:

$$\mathscr{L}\{f(t)\} = \int_0^{\infty} f(t)e^{-st} dt$$

The one sided Fourier Transform can be defined as:

$$\mathscr{F}\{f(t)\} = \int_0^{\infty} f(t)e^{-j\omega t} dt$$

Which should make obvious why the substitution $s=j\omega$ gives one the Fourier Transform (in most cases).

Answer 2

Setting $\sigma$ equal to zero allows you to evaluate the frequency response of the system. The frequency response tells us how the system responds to sinusoids of varying frequency. If $\sigma$ was not equal to you would not be evaluating how the system responds to pure sinusoids. (Think about Euler's formula.)

Specifically, Bode plots are a common tool for frequency response. Bode plots consist of a magnitude plot and a phase plot, the magnitude plot tells you the magnitude of the system's transfer function and the phase plot tells you the phase. This is especially useful in control because we can use it to determine the gain and phase margins. Consider the unity negative feedback system describe by the transfer function:

\begin{equation} H(s)=\frac{G(s)}{1+G(s)} \end{equation}

This system is unstable when

\begin{equation} G(s)=-1=1\arg(180) \end{equation}

So we can use the Bode plot of open loop transfer function to determine the stability margins. First we set $s=j\omega$ and then evaluate for a range of $\omega$. Computer programs make this easy, however, you can use approximations to do this by hand. The gain margin is found by determining how the difference between the actual magnitude and 1 when the phase is 180. The phase margin is found by determining the difference between 180 and the actual phase when the gain is magnitude 1.

Summary: to obtain a pure frequency response (how the system behaves with changing input frequency) we require $s=jw$. This is especially useful for stability analysis via Bode plots.