# Are resistors, inductors, capacitors the only possible passive components in this universe?

I wonder whether or not there is a possibility to find another passive component (with value either $$X$$ or $$Y$$) that will change the well known RLC circuit equation below

$$L\; \frac{\mathrm{d}^2 \; i}{\mathrm{d}\; t^2} + R\; \frac{\mathrm{d} \; i}{\mathrm{d}\; t}+\frac{i}{C}=f(t)$$

to either

$$L\; \frac{\mathrm{d}^3 \; i}{\mathrm{d}\; t^3} + R\; \frac{\mathrm{d}^2 \; i}{\mathrm{d}\; t^2} + \frac{1}{C}\; \frac{\mathrm{d} \; i}{\mathrm{d}\; t} + \frac{i}{X}=f(t)$$

or

$$Y\; \frac{\mathrm{d}^3 \; i}{\mathrm{d}\; t^3} + L\; \frac{\mathrm{d}^2 \; i}{\mathrm{d}\; t^2} + R\; \frac{\mathrm{d} \; i}{\mathrm{d}\; t} + \frac{i}{C}=f(t)$$

# Question

Are resistors, inductors, capacitors the only possible passive components in this universe? Is there a possibility to find another one?

From a mechanical point of view, I can rephrase the question into the equivalent mechanical question:

Are dampers, springs and masses the only possible mechanical components in this universe?

• In general, use of capital "I" is recommended, since physicists use "i" and engineers use "j" for $\sqrt{-1}$ and vice versa for current. :-) – Carl Witthoft Jul 31 at 10:54
• @CarlWitthoft: I prefer lowercase $i$ for time varying quantities. The imaginary units $i$ is clear enough from the context. :-) – Not A Zoomed Image Jul 31 at 10:58

Circuit theory traditionally defined 4 important circuit variables and 3 kinds of passive components to interrelate them. Other devices (transistors, diodes, vacuum tubes, etc) can be modeled using combinations of these "fundamental" components and dependent voltage and current sources.

Resistors couple (create a relationship between) voltage and current. For a linear resistor, the constitutive relationship is $$V=RI$$. (We might also wish to consider a nonlinear resistor with relationship $$V=R(I)$$, or its behavior due to perturbations about an operating point with relationship $$dV=r\ dI$$).

Capacitors couple charge and voltage, with the constitutive relationship $$Q=CV$$ (With similar extensions to the nonlinear case as for the resistor).

Inductors couple current and magnetic flux, with the constitutive relationship $$\Phi=LI$$ (again, with extensions for the nonlinear case).

Current and charge are in some sense inherently related (you don't need a component to give them a relationship). We express their relationship by $$I=\frac{dQ}{dt}$$ or $$Q=\int I\ dt$$.

As the duals of current and charge, voltage and flux are also inherently related. We express their relationship as $$V=\frac{d\Phi}{dt}$$ or $$\Phi = \int V\ dt$$.

That means there's one combination of circuit variables that aren't coupled by one of our traditional circuit elements: charge and flux.

To fill in this gap in circuit theory, in 1971 Leon Chua proposed the definition of an additional circuit element, the memristor that can couple charge and flux. The memristor has the constitutive relationship $$\Phi = M Q$$.

The relationships between the circuit variables, how they're coupled by the various circuit elements, and the memristor's place in filling out the possible couplings is shown in this diagram: (Image source: Wikimedia user Parcly Taxel)

(The diagram shows the circuit variables as differentials, since we often want to analyze perturbations of the operating point of a nonlinear circuit, rather than truly linear circuits that follow the linear relationships I used above for simplicity)

Since Chua's proposal, analog models of memristors have been constructed using op-amps, and a few reasonably simple physical systems have been found that act memristively, but there is still not any simple, passive, physical memristor you can buy at Digikey or Mouser, and the device remains mostly (but not entirely) a theoretical curiosity.

• What is the mechanical dual for memristor? – Not A Zoomed Image Jul 31 at 20:12
• @ArtificialStupidity, sorry I don't work with the mechanical analogy enough to work it out. – The Photon Jul 31 at 21:34
• Thank you very much. OK. – Not A Zoomed Image Jul 31 at 21:37

Look at it this way: what possible forces can be applied to an electric current in a constrained environment, e.g. a wire? Or, more specifically, to the value of the voltage at any location along the wire. About all you can do is apply a lead, a lag, or an attenuation (and all of those as a function of $$\omega$$ ) .
Since physical components can be modeled as combinations of ideal R,L and C elements, unless you can demonstrate a different action which can be taken to modify V(t), there's no need for a new kind of component (passive).