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I'm studying the transfer function of an aircraft that relates the pitch angle, $\theta$, with the horizontal stabilizer angle, $\delta$. $$ H(s) = \dfrac{\theta (s)}{\delta (s)}$$

I have computed the frequency response by using Matlab, and I plotted the Bode Diagram: amplitude plot and phase plot.

Bode Diagram

I understand what is the amplitude curve indicating and I'm able to comment the result obtained. But when it comes to studying the phase curve (second figure above) I don't know what is the meaning and how it is related with the other curve.

So, my question is: what can I say from the phase plot by looking at the Bode Diagram of the studied transfer function, $H(s)$?

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The Bode diagram shows you the properties of the transfer function $H(s)$ for $s=j\omega$, i.e. the complex frequency response. Since $H(j\omega)$ is generally a complex function it has a magnitude and a phase:

$$H(j\omega)=|H(j\omega)|e^{j\phi(\omega)}$$

The plot shows the principal value of the phase in the interval $[-\pi,\pi)$, that's why you see a jump (of $2\pi$), which is no actual phase jump but it just comes from mapping the phase back to the interval $[-\pi,\pi)$.

The magnitude of $H(j\omega)$ tells you the relative amplification/attenuation of the different frequency components, whereas the phase shows you the relative alignment of the different components.

For example, a sinusoid $x(t)=A\sin(\omega_0t+\theta)$ will appear at the output of the system as

$$y(t)=A|H(j\omega_0)|\sin(\omega_0t+\theta+\phi(\omega_0))$$

where $\phi(\omega_0)$ is the system's phase response evaluated at the input frequency $\omega_0$.

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