Impedance response of a linear system is independent of the perturbation amplitude?

Question

I'm studyng electrochemical impedance spectroscopy. On my book there is the sentence: "Impedance response of a linear system is independent of the perturbation amplitude.".

Why? If I have a non-linear system, when I zoom the voltage-current curve, this will be linear and, if I chose a bias voltage \$V_B\$ and if I superimpose a perturbation with small amplitude \$v=v^* sin(\omega t)\$ to \$V_B\$, for the frequency response theorem, I'll get:

$$i(t)=I_B+i^*sin(\omega t+\phi)$$

The impedance is:

$$Z=\frac{\mid V \mid}{\mid i \mid}e^{j(0-\phi)}=\frac{\mid V \mid}{\mid i \mid}e^{-j\phi}$$

Maybe the impedance is independent of the perturbation amplitude because, in the region in which the V-i curve is linear, the ratio \$\frac{\mid V \mid}{\mid i \mid}\$ is constant?

Thank you for your time.

Answer 1

The response of a linear system does not depend on the perturbation amplitude.

That's the definition of what it means to be a linear system.

If I have a non-linear system,

... then you can't expect a result specified to apply to a linear system to apply to your system.

In the real world, just about all systems are non-linear, but often it is useful to approximate their behavior with a linear model.