# Why are there only four passive elements?

I've read that there are four types of passive elements: resistances, capacitors, inductors and memristors.

The memristor was predicted 30 years before it was produced. But why couldn't you invent other type of passive element? Is there a proof?

The definition I'm using of passive elements is something with no gain, no control and linear.

• There's this spiffy graphic, which you might have seen. en.wikipedia.org/wiki/… Unfortunately I just find myself staring at it and thinking about memristors, rather than feeling like the question has been answered. Jan 29, 2014 at 20:52
• @PhilFrost Clearly I'm not the only one who likes that graphic! Jan 29, 2014 at 20:56
• I think it's important to keep in mind that every wire displays resistance, capacitance and inductance. These are ideal circuit elements but in real life they are characteristics of pretty much every circuit element. The memristor doesn't fit that mold. You can't talk about the "memristance" of a wire. In my mind, the memristor does not belong in the same set as resistance, capacitance, and inductance. Jan 29, 2014 at 21:28
• I'm using of passive elements is something with no gain, no control and linear. Then the memristor is not a passive element since it is non-linear (except for the trivial case where it is just a resistor). According to Wiki, for the memristor we have: $v = M(q)i$ where q is understood to be the time integral of $i$. If $M(q)$ is constant, $v \propto i$ and, thus, we have a resistor. Otherwise, $v$ is not a linear function of $i$. For example, if $M(q) = mq$ then $\frac{dv}{dt} = m(i^2 + q\frac{di}{dt})$ Jan 30, 2014 at 17:24
• @jinawee, if a passive element must be linear, the memristor is not a passive element. From the Wiki article "Memristor": In his 1971 paper, Chua extrapolated a conceptual symmetry between the nonlinear resistor (voltage vs. current), nonlinear capacitor (voltage vs. charge) and nonlinear inductor (magnetic flux linkage vs. current). He then inferred the possibility of a memristor as another fundamental nonlinear circuit element linking magnetic flux linkage and charge. Jan 30, 2014 at 18:18

There are four physical quantities of interest for electronics: voltage, flux, charge, and current. If you have four things and want to pick two, order not mattering, there are 4C2 = 6 ways to do that. Two of the physical quantities are defined in terms of the other two. (Current is change in charge over time. Voltage is change in flux over time.) That leaves four possible relationships: resistance, inductance, capacitance, and memristance. If you want another fundamental component, you need another physical quantity to relate to these four. And while there are many physical quantities one might measure, none seem so tightly coupled as these. I'd suppose this is because electricity and magnetism are two aspects of the same force. I'd further suppose that since electromagnetism is now understood to be part of the electroweak force, one might be able to posit some relationships between the weak nuclear interaction and our four elements of voltage, current, charge, and flux.

I haven't the first clue how this would be physically manifested, especially given the relative weakness of the weak nuclear force at anything short of intranuclear distances. Perhaps in the presence of strong magnetic or electrical fields affecting the rates of radioactive decay? Or in precipitating or preventing nuclear fusion? I'd yet further suppose (I'm on a roll) that the field strengths required would be phenomenal, which is why they're not practical for everyday engineering.

But that's a lot of supposition. I am a mere engineer, and unqualified to speculate on such things.

• I think it's more like "someone decided there are four physical quantities of interest for electronics". And really maybe there are only two, since charge is the integral of current, and flux the integral of voltage. Temperature is pretty important. So is power, or its derivative, energy. Or maybe I want to integrate flux to get a new thing, and define a component about that. Jan 29, 2014 at 21:24
• I think maybe the proof lies in the requirement (set in the question) that these passive components are linear, and that means that they have some linear relationship between current and voltage, thus there can't be other physical quantities of interest, by definition. But I'm just guessing. Jan 29, 2014 at 21:26
• Resistance is not defined as $R = \frac{dv}{di}$ but, rather, as the constant of proportionality of voltage and current, $R = \dfrac{v}{i}$ so, at best, this graphic is misleading. For example, an ideal voltage source in series with an ideal resistance $R$ satisfies $R = \frac{dv}{di}$ but such a combination is not a fundamental passive circuit element. Jan 30, 2014 at 4:05
• @AlfredCentauri There's a bit of explanation in the Wikipedia article for memristor that explains why everything was written as differential equations. I can't say I follow it (I don't speak math very well), but I understood it as "because it makes it easier to argue for memristors." Jan 30, 2014 at 12:51
• Personally I would have chosen to define M the other way round, so dq=MdΦ, then you could compare with dq=Cdv and justifiably call them flux-capacitors Jan 30, 2014 at 13:49

But why couldn't you invent other type of passive element? Is there a proof?

Well, there is a proof, but it's circular. If you take "the four fundamental electronic variables", there are only six ways to combine them linearly. Four of the ways are components, and the other two are definitions. Stephen's answer explains this well. There are only four passive components because whoever made that claim only allowed four variables.

I can "invent" more "missing components" by introducing more variables. Current is the derivative of charge with respect to time:

$$i = \frac{\mathrm dq}{\mathrm dt}$$

I'm going to define a new term: surgingness. It's the derivative of current with respect to time:

$$s = \frac{\mathrm di}{\mathrm dt}$$

Mind blown? Put it back together. We do this all the time in physics. These sequences are analogous:

• position, velocity, acceleration
• charge, current, surgingness

We can differentiate variables as many times as we want and give the results names, if we want. Physics even has a name for the derivative of acceleration: jerk.

Now we can stick surgingness in that graphic from Stephen's answer. It goes below and to the left of current.

Now we can ask, what's the component that connects surgingness with voltage? It would be a component that obeys:

$$\mathrm dv = P \mathrm ds$$

I'm going to call $P$ Philistance. The component is called a Philator.

What's the utility of this component? I haven't a clue, but I predict it exists. In a few decades, when it's invented, I'll say "I told you so" and be famous.

• I think you're just a Philistine. Jan 30, 2014 at 1:55
• If $s = \frac{di}{dt}$ and $dv = P ds$ then $v = P\frac{di}{dt} + V$, i.e., the Philator is just an inductor in series with a constant voltage source. Jan 30, 2014 at 4:53
• @AlfredCentauri Which means that if you make a passive Philator, you will indeed be very famous.
– Buhb
Jan 30, 2014 at 6:51
• @Buhb, a passive Philator would be like a married bachelor. Jan 30, 2014 at 12:41
• @AlfredCentauri If you say so. I never was very good at math :) I was wondering, what if I integrate charge and integrate flux, then imagine there is some passive component there. Perhaps a "Forgistor"? Or is that also some combination of things we already have? Jan 30, 2014 at 12:45

jinawee,

I think there are a large number of "passive" components yet to be both discovered and invented. "Passive" is a somewhat deceptive and ambiguous term we use in electronics. In electronics we have a lot of loose terminology that throws beginners a curve ball. You would think that for an exact science we would use more exact language. Not so.

As other posters have indicated the big three passives are resistors, capacitors and inductors. I don't know about this memristor gizmo. In my 50+ years of electronics experience I never held one in my hand or had one come up in a circuit design I've worked on.

Nevertheless, I think if you could come up with a device which could convert frequency to a proportional DC voltage, like a thermocouple converts temperature to voltage, you might join the likes of Michael Faraday in EE Heaven.

Likewise, if you could invent a device which converts electron flow directly to sound without the use of a magnet and coil, you might be onto something big as well.

Or for that matter an elastic material that directly converts current to motive force - the elusive artificial muscle tissue. That would forever change the world of pornography as much as Michael Faraday's vibrating coil did.

It's been quite a while since the EE world has enjoyed a new passive component. Keep us posted on your progress.

• convert frequency to a proportional DC voltage - you mean like a low pass filter commonly used to convert a PWN output to a voltage for a cheap,DAC? Jan 30, 2014 at 4:53
• a device which converts electron flow directly to sound without the use of a magnet and coil - or what about a device that bends light without the use of a material lens? Jan 30, 2014 at 4:56
• an elastic material that directly converts current to motive force - or a holo-diode? Jan 30, 2014 at 4:57
• @Michael "what about a device that bends light without the use of a material lens?" Look up in the sky on a clear day and you'll see one shining bright.
– JAB
Jan 30, 2014 at 13:45
• @JAB I'd be more worried about what would happen if the planet fell on the device. Jan 30, 2014 at 23:22