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

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?

• Yes, exactly that. You must assume that or else all linear theory is busted and you have to solve a nonlinear system. – Janka Jun 12 '18 at 17:36
• Your question goes in circles. The definition of impedance absolutely requires linearity, and the passage you quote is about a linear system. You then try to apply it to a non-linear one. You cannot do that, you can only use a linear model to approximate your system to the degree to which your system's behavior in a limited regime is sufficiently linear for the result to have meaning. – Chris Stratton Jun 12 '18 at 17:37
• Well that is pretty much the definition of a linear system. – Brian Drummond Jun 12 '18 at 21:54