A memristor (/ˈmɛmrɪstər/; a portmanteau of memory resistor) is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which also comprises the resistor, capacitor and inductor.

Chua and Kang later generalized the concept to memristive systems. Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably ReRAM.

The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.

Chua in his 1971 paper identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). From this symmetry he inferred the characteristics of a fourth fundamental non-linear circuit element, linking magnetic flux and charge, which he called the memristor. In contrast to a linear (or non-linear) resistor, the memristor has a dynamic relationship between current and voltage, including a memory of past voltages or currents. Other scientists had proposed dynamic memory resistors such as the memistor of Bernard Widrow, but Chua introduced a mathematical generality.

The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φ_m(t) and the amount of electric charge that has flowed, q(t): $${\displaystyle f(\mathrm {\Phi } _{\mathrm {m} }(t),q(t))=0.}$$ The magnetic flux linkage, Φ_m, is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here. Its physical meaning is discussed below. The symbol Φ_m may be regarded as the integral of voltage over time.

In the relationship between Φ_m and q, the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge: $${\displaystyle M(q)={\frac {\mathrm {d} \Phi _{\rm {m}}}{\mathrm {d} q}}\,.}$$ Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are: $${\displaystyle M(q(t))={\cfrac {\mathrm {d} \Phi _{\rm {}}/\mathrm {d} t}{\mathrm {d} q/\mathrm {d} t}}={\frac {V(t)}{I(t)}}\,.}$$ To relate the memristor to the resistor, capacitor, and inductor, it is helpful to isolate the term M(q), which characterizes the device, and write it as a differential equation.

The above table covers all meaningful ratios of differentials of I, q, Φ_m, and V. No device can relate dI to dq, or dV to dΦ_m, because I is the time derivative of q and V is the time derivative of Φ_m.

It can be inferred from this that memristance is charge-dependent resistance. If M(x) is a constant function (i.e. has the same value for all x), then we obtain Ohm's law: R(t) = V(t)/I(t). If M(x) is nontrivial, however, the equation is not equivalent because q(t) and thus M(q(t)) vary with time. Solving for voltage as a function of time produces $${\displaystyle V(t)=\ M(q(t))\cdot I(t)\,.}$$ This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Non-zero current implies time-varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum value of q does not cause much change in M compared to the initial value M(0).