Memristor (2)

memristor symbol.

The memristor is formally defined^[4] as a two-terminal element in which the magnetic flux Φ_m between the terminals is a function of the amount of electric charge qthat has passed through the device. Each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge.

$M(q)=\frac{\mathrm d\Phi_m}{\mathrm dq}$

Noting from Faraday’s law of induction that magnetic flux is simply the time integral of voltage,^[9] and charge is the time integral of current, we may write the more convenient form

$M(q(t))=\cfrac{\cfrac{\mathrm d\Phi_m}{\mathrm dt}}{\cfrac{\mathrm dq}{\mathrm dt}}=\frac{V(t)}{I(t)}$

It can be inferred from this that memristance is simply charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohm’s Law R(t) = V(t)/ I(t). If M(q(t)) is nontrivial, however, the equation is not equivalent because q(t) and M(q(t)) will vary with time. Solving for voltage as a function of time we obtain

$V(t) =\ M(q(t)) I(t)$

This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Of course, nonzero 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 change in q does not cause much change in M.

Furthermore, the memristor is static if no current is applied. If I(t) = 0, we find V(t) = 0 and M(t) is constant. This is the essence of the memory effect.

The power consumption characteristic recalls that of a resistor, I²R.

$P(t) =\ I(t)V(t) =\ I^2(t) M(q(t))$

As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a resistor. If M(q(t)) increases rapidly, however, current and power consumption will quickly stop.

Magnetic flux in a passive device

In circuit theory, magnetic flux Φ_m typically relates to Faraday’s law of induction, which states that the voltage in terms of electric field potential “gained” around a loop (electromotive force) equals the negative derivative of the flux through the loop:

$\mathcal E = \frac{-\mathrm d\Phi_\mathrm m}{\mathrm dt}$

This notion may be extended by analogy to a single passive device. If the circuit is composed of passive devices, then the total flux is equal to the sum of the flux components due to each device. For example, a simple wire loop with low resistance will have high flux linkage to an applied field as little flux is “induced” in the opposite direction. Voltage for passive devices is evaluated in terms of energy lost by a unit of charge:

$V = \frac{\mathrm d\Phi_\mathrm m}{\mathrm dt\,}$

$\Phi_\mathrm m = \int V\mathrm dt$

Observing that Φ_m is simply equal to the integral over time of the potential drop between two points, we find that it may readily be calculated, for example by an operational amplifierconfigured as an integrator.

Two unintuitive concepts are at play:

Magnetic flux is generated by a resistance in opposition to an applied field or electromotive force. In the absence of resistance, flux due to constant EMF increases indefinitely. The opposing flux induced in a resistor must also increase indefinitely so their sum remains finite.
Any appropriate response to applied voltage may be called “magnetic flux.”

The upshot is that a passive element may relate some variable to flux without storing a magnetic field. Indeed, a memristor always appears instantaneously as a resistor. As shown above, assuming non-negative resistance, at any instant it is dissipating power from an applied EMF and thus has no outlet to dissipate a stored field into the circuit. This contrasts with an inductor, for which a magnetic field stores all energy originating in the potential across its terminals, later releasing it as an electromotive force within the circuit.