The frequency response or impedance spectrum of an electrochemical cell obtained under potentiostatic conditions can be used to predict the stability features of the electrochemical cell under different electrical conditions, such as galvanostatic control for example. It is shown that the particular condition for which the system is resonant with the perturbing signal implies the existence of a so-called Hopf bifurcation to sustained spontaneous oscillations in the time domain. The ability of impedance spectroscopy to make predictions about the occurrence of oscillations leads in a natural way to a classification of electrochemical oscillators into two different categories, depending on the kinetic nature of the negative faradaic impedance. The theory is illustrated by experimental and model examples.