There were historically 5 fundamental circuit elements: resistor, capacitor, inductor, and current and voltage sources. They could be described by very simple equations:

Voltage source: *V* = constant with respect to *I*

Current source: *I* = constant with respect to *V*

Resistor: \$V = I R\$

Capacitor: \$Q = V C\$

Inductor: \$\Phi = I L\$

with \$Q = \int{I dt}\$ and \$\Phi = \int{V dt}\$.

If we extend the concepts of these fundamental circuit elements to include nonlinear behavior, and multiport behavior (controlled sources, 2-port networks), we can model all other circuit elements we use. For example, a diode is a nonlinear resistor, and a transistor can be described by a nonlinear 2-port impedance matrix.

**\$\Phi\$ is considered a fundamental circuit variable because it is the variable that describes the state of an inductor**, just as stored charge defines the state of a capacitor.

The memristor, of course, is the missing "fundamental circuit element" that connects charge and flux linkage together without reference to voltage or current: \$\Phi = M Q\$ (someone correct me if I've inverted this one).

If a resistor is a device where \$\mathrm{d}v = R\mathrm{d}i\$, than resistance is EMF (\$v\$) divided by current (\$i\$), or in SI units, volts \$(V)\$ divided by amperes (\$A\$). We can define the volt in terms of SI base units:

$$ V = \frac{kg \cdot m^2}{A \cdot s^3} $$

The ampere is already an SI base unit. The volt divided by the ampere is then the ohm:

$$ \Omega = \frac{V}{A} = \frac{\frac{kg \cdot m^2}{A \cdot s^3}}{A} = \frac{kg \cdot m^2}{A^2 \cdot s^3}$$

A memristor is where \$\mathrm{d}\phi = M\mathrm{d}q\$, so memristance must be magnetic flux (\$\phi\$) divided by charge (\$q\$), or in SI units, webers (\$Wb\$) divided by coulombs (\$C\$). Defined by SI base units:

$$ Wb = V \cdot s = \frac{kg \cdot m^2}{A \cdot s^2} $$

$$ C = A \cdot s $$

Then the weber divided by the coulomb is our unit of memristance:

$$ \text{unit of memristance} = \frac{Wb}{C} = \frac{\frac{kg \cdot m^2}{A \cdot s^2}}{A \cdot s} = \frac{kg \cdot m^2}{A^2 \cdot s^3}$$

Which, one can see, is identical to the ohm. So, memristance is measured in ohms, just like resistance.

If you don't strip everything to base units, then the equivalence can be more simply expressed:

$$ \require{cancel}
\frac{Wb}{C} = \frac{V \cdot s}{A \cdot s} = \frac{V}{A} = \Omega$$

I find this the most interesting approach, because it can be seen that a memristor integrates both EMF and current (as indicated by the \$\cdot s\$ in the numerator and the denominator). It's as if it's somehow an inductor and a capacitor at the same time, and in doing so, the time terms cancel, and it becomes neither. If you were to remove just one of the \$s\$ terms, then you'd be left with the henry or the farad, but with both you get back to the ohm.

Of course, if \$M\$ in \$\mathrm{d}\phi = M\mathrm{d}q\$ is just a constant, then the time terms really do cancel and you are left with just an ordinary resistor. As Wikipedia puts it:

There is no such thing as a standard memristor. Instead, each device implements a particular function, wherein the integral of voltage determines the integral of current, and vice versa. A linear time-invariant memristor, with a constant value for \$M\$, is simply a conventional resistor.

What makes memristors fancy is that \$M\$ is a function (usually defined as a function of the time integral of current: charge, but could be defined as a function of the time integral of voltage: flux). Because \$M\$ is a function, it allows one to have a device where \$\mathrm{d}v/\mathrm{d}i\$ (usually we'd call that resistance) *changes* based on what's happened so far: how much current, or how much voltage, there has been in the past. The discrepancy is explained by physical effects I don't really understand, like the arrangement of oxides on microscopic structures, or something. They get rearranged as charge moves through, changing the voltage it will take to move additional charge (resistance).

Inductors and capacitors do this also, but unlike a capacitor, where getting the voltage to \$0V\$ requires bringing the time integral of current, charge, to \$0C\$, and unlike inductors, where getting the current to \$0A\$ requires bring the time integral of voltage, flux, to \$0Wb\$, a memristor can go to \$0A\$ and \$0V\$ without requiring charge or flux to go to zero. Thus, they don't lose their memory of the past when no current or voltage is applied.

That also means if you graph for a memristor voltage on one axis, and current on the other, you don't get a straight line (that would be a resistor), but you do get a hysteresis loop that always passes through the origin, \$0V\$ and \$0A\$. Just where that loop goes at other points depends on the kind of memristor.

## Best Answer

Fig. 1 in the paper by Chua that you provided a link to, published in 1971, also shows what a mutator does and extends the concept to the

memristor - the fourth basic circuit element: It is an active circuit that uses a known circuit element and mutates it into one that has memristive properties. (Knownas of the paper's publishing date - today, HP has discovered something memristive.) This may still sound a bit confusing, so let's do a little detour.You can use a gyrator when you need an element that behaves like an inductance, but all you have available is a capacitor. You pay for this "magic" with the cost of an active circuit (read: transistors, tubes, ...). A practical example for this may be integrated design: Transistors are almost free, and it's possible to build capacitors on silicon, but you can hardly wind inductors on a chip - except extremely tiny ones for very, very high frequencies. Also, inductors for audio filters often would have to be fairly large, both in Henries and in physical size; this is why you will find gyrators or (somewhat inaccurately named)

solid state inductors^{1)}on the internets. Using a gyrator, you can use a capacitor and transmogrify ("mutate", maybe even "gyrate"^{2)}) some of its its properties into one of an inductor^{3)}. You could also go the other way round; wind an inductor and use a gyrator to obtain a capacitive circuit element (although this will hardly ever be a practical and cost-effective solution). A gyrator, in this example, and using the words from Chua's paper on memristors, would thus be something like a mutator between capacitors and inductors.Anyone familiar with Calvin and Hobbes will, I am sure, appreciate this picture. It shows two gyrators in the shape of transmogrifiers; one makes an inductor behave like a capacitor, the other one makes a little 100 nF ceramic capacitor act like an inductor.

Now back to the memristor. Until recently, when an element with memristive properties was developed by Hewlett-Packard, you had not only the trouble of not being able to use memristors in certain processes (like IC design) - you had no memristors at all. Thus, the memristor shown in fig. 1a of Chua's paper was plain theory. The only way to get something memristic onto the screen of an actual curve tracer was using a known element like a (nonlinear) resistor (fig. 1b), a (nonlinear) inductor (fig. 1c) or a (nonlinear) capacitor (fig. 1d), and to put it through a circuit that would mutate its properties into those of the proposed memristor.

In Chua's paper, fig. 1 shows three types of black-box-like, two-port mutators. Fig. 2 shows a practical circuit that gives you a memristic behaviour looking into port 1 when you offer a (nonlinear) resistive element on port 2, say a resistor (linear) or a diode (nonlinear resistive). Therefore, fig. 2 would be a more detailed view into the black box of fig. 1b.

Aside from fig. 2 in Chua's paper, here are some other examples that show what practical implementations of mutators might look like:

From left to right: A 1N4148 diode (nonlinear resistive element) acts like a memristor (similar to figs. 1b or 2 in the paper), an inductor transmogrified into a memristor (Chua: Fig. 1c), and a 100 nF ceramic capacitor transmogrified into a memristor (Chua: Fig. 1d).

I have to admit that the examples in the pictures are non-perfect: They are white-box models instead of the original black-box models. Other that that, I promise that they actually do work! As a proof, screenshots taken with my Tek 575 curve tracer will follow ;-)

^{1)}I have actually never seen a non-solid inductor. Gas? Liquid? Plasma? Only when a solid inductor blows, but that's on the odd side of reasoning. I guess calling these gyratorstransistorizedwould be less misleading.^{2)}see here for a well-known, cardboard-box-like, real-world (!), non-electronic (?) transmogrifier. As the cited source says, when it comes to transmogrification, "Scientific Progress Goes 'Boink'" - use with caution!^{3)}See this question about energy storage in a gyrator for some hints on a gyrator's limitations.