I’m talking about electrical circuits, but I’m interested in them as models of more general physical systems. Last time we started seeing how this works. We developed an analogy between electrical circuits and physical systems made of masses and springs, with friction:
|inverse capacitance:||spring constant:|
But this is just the first of a large set of analogies. Let me list some, so you can see how wide-ranging they are!
People in system dynamics often use effort as a term to stand for anything analogous to force or voltage, and flow as a general term to stand for anything analogous to velocity or electric current. They call these variables and
To me it’s important that force is the time derivative of momentum, and velocity is the time derivative of position. Following physicists, I write momentum as and position as So, I’ll usually write effort as and flow as .
Of course, ‘position’ is a term special to mechanics; it’s nice to have a general term for the thing whose time derivative is flow, that applies to any context. People in systems dynamics seem to use displacement as that general term.
It would also be nice to have a general term for the thing whose time derivative is effort… but I don’t know one. So, I’ll use the word momentum.
Now let’s see the analogies! Let’s see how displacement , flow momentum and effort show up in several subjects:
|Mechanics: rotation||angle||angular velocity||angular momentum||torque|
|Electronics||charge||current||flux linkage||voltage||Hydraulics||volume||flow||pressure momentum||pressure|
|Thermal Physics||entropy||entropy flow||temperature momentum||temperature|
|Chemistry||moles||molar flow||chemical momentum||chemical potential|
We’d been considering mechanics of systems that move along a line, via translation, but we can also consider mechanics for systems that turn round and round, via rotation. So, there are two rows for mechanics here.
There’s a row for electronics, and then a row for hydraulics, which is closely analogous. In this analogy, a pipe is like a wire. The flow of water plays the role of current. Water pressure plays the role of electrostatic potential. The difference in water pressure between two ends of a pipe is like the voltage across a wire. When water flows through a pipe, the power equals the flow times this pressure difference—just as in an electrical circuit the power is the current times the voltage across the wire.
A resistor is like a narrowed pipe:
An inductor is like a heavy turbine placed inside a pipe: this makes the water tend to keep flowing at the same rate it’s already flowing! In other words, it provides a kind of ‘inertia’ analogous
A capacitor is like a tank with pipes coming in from both ends, and a rubber sheet dividing it in two lengthwise:
When studying electrical circuits as a kid, I was shocked when I first learned that capacitors don’t let the electrons through: it didn’t seem likely you could do anything useful with something like that! But of course you can. Similarly, this gizmo doesn’t let the water through.
A voltage source is like a compressor set up to maintain a specified pressure difference between the input and output:
Similarly, a current source is like a pump set up to maintain a specified flow.
Finally, just as voltage is the time derivative of a fairly obscure quantity called ‘flux linkage’, pressure is the time derivative of an even more obscure quantity which has no standard name. I’m calling it ‘pressure momentum’, thanks to the analogy
Just as pressure has units of force per area, pressure momentum has units of momentum per area!
People invented this analogy back when they were first struggling to understand electricity, before electrons had been observed:
• Hydraulic analogy, Wikipedia.
The famous electrical engineer Oliver Heaviside pooh-poohed this analogy, calling it the “drain-pipe theory”. I think he was making fun of William Henry Preece. Preece was another electrical engineer, who liked the hydraulic analogy and disliked Heaviside’s fancy math. In his inaugural speech as president of the Institution of Electrical Engineers in 1893, Preece proclaimed:
True theory does not require the abstruse language of mathematics to make it clear and to render it acceptable. All that is solid and substantial in science and usefully applied in practice, have been made clear by relegating mathematic symbols to their proper store place—the study.
According to the judgement of history, Heaviside made more progress in understanding electromagnetism than Preece. But there’s still a nice analogy between electronics and hydraulics. And I’ll eventually use the abstruse language of mathematics to make it very precise!
But now let’s move on to the row called ‘thermal physics’. We could also call this ‘thermodynamics’. It works like this. Say you have a physical system in thermal equilibrium and all you can do is heat it up or cool it down ‘reversibly’—that is, while keeping it in thermal equilibrium all along. For example, imagine a box of gas that you can heat up or cool down. If you put a tiny amount of energy into the system in the form of heat, then its entropy increases by a tiny amount And they’re related by this equation:
where is the temperature.
Another way to say this is
where is time. On the left we have the power put into the system in the form of heat. But since power should be ‘effort’ times ‘flow’, on the right we should have ‘effort’ times ‘flow’. It makes some sense to call the ‘entropy flow’. So temperature, must play the role of ‘effort’.
This is a bit weird. I don’t usually think of temperature as a form of ‘effort’ analogous to force or torque. Stranger still, our analogy says that ‘effort’ should be the time derivative of some kind of ‘momentum’, So, we need to introduce temperature momentum: the integral of temperature over time. I’ve never seen people talk about this concept, so it makes me a bit nervous.
But when we have a more complicated physical system like a piston full of gas in thermal equilibrium, we can see the analogy working. Now we have
The change in energy of our gas now has two parts. There’s the change in heat energy , which we saw already. But now there’s also the change in energy due to compressing the piston! When we change the volume of the gas by a tiny amount we put in energy
Now look back at the first chart I drew! It says that pressure is a form of ‘effort’, while volume is a form of ‘displacement’. If you believe that, the equation above should help convince you that temperature is also a form of effort, while entropy is a form of displacement.
But what about the minus sign? That’s no big deal: it’s the result of some arbitrary conventions. is defined to be the outward pressure of the gas on our piston. If this is positive, reducing the volume of the gas takes a positive amount of energy, so we need to stick in a minus sign. I could eliminate this minus sign by changing some conventions—but if I did, the chemistry professors at UCR would haul me away and increase my heat energy by burning me at the stake.
Speaking of chemistry: here’s how the chemistry row in the analogy chart works. Suppose we have a piston full of gas made of different kinds of molecules, and there can be chemical reactions that change one kind into another. Now our equation gets fancier:
Here is the number of molecules of the ith kind, while is a quantity called a chemical potential. The chemical potential simply says how much energy it takes to increase the number of molecules of a given kind. So, we see that chemical potential is another form of effort, while number of molecules is another form of displacement.
But chemists are too busy to count molecules one at a time, so they count them in big bunches called ‘moles’. A mole is the number of atoms in 12 grams of carbon-12. That’s roughly
atoms. This is called Avogadro’s constant. If we used 1 gram of hydrogen, we’d get a very close number called ‘Avogadro’s number’, which leads to lots of jokes:
(He must be desperate because he looks so weird… sort of like a mole!)
So, instead of saying that the displacement in chemistry is called ‘number of molecules’, you’ll sound more like an expert if you say ‘moles’. And the corresponding flow is called molar flow.
The truly obscure quantity in this row of the chart is the one whose time derivative is chemical potential! I’m calling it chemical momentum simply because I don’t know another name.
Why are linear and angular momentum so famous compared to pressure momentum, temperature momentum and chemical momentum?
I suspect it’s because the laws of physics are symmetrical
under translations and rotations. When the assumptions of Noether’s theorem hold, this guarantees that the total momentum and angular momentum of a closed system are conserved. Apparently the laws of physics lack the symmetries that would make the other kinds of momentum be conserved.
This suggests that we should dig deeper and try to understand more deeply how this chart is connected to ideas in classical mechanics, like Noether’s theorem or symplectic geometry. I will try to do that sometime later in this series.
More generally, we should try to understand what gives rise to a row in this analogy chart. Are there are lots of rows I haven’t talked about yet, or just a few? There are probably lots. But are there lots of practically important rows that I haven’t talked about—ones that can serve as the basis for new kinds of engineering? Or does something about the structure of the physical world limit the number of such rows?
Mildly defective analogies
Engineers care a lot about dimensional analysis. So, they often make a big deal about the fact that while effort and flow have different dimensions in different rows of the analogy chart, the following four things are always true:
• has dimensions of action (= energy × time)
• has dimensions of energy
• has dimensions of energy
• has dimensions of power (= energy / time)
In fact any one of these things implies all the rest.
These facts are important when designing ‘mixed systems’, which combine different rows in the chart. For example, in mechatronics, we combine mechanical and electronic elements in a single circuit! And in a hydroelectric dam, power is converted from hydraulic to mechanical and then electric form:
One goal of network theory should be to develop a unified language for studying mixed systems! Engineers have already done most of the hard work. And they’ve realized that thanks to conservation of energy, working with pairs of flow and effort variables whose product has dimensions of power is very convenient. It makes it easy to track the flow of energy through these systems.
However, people have tried to extend the analogy chart to include ‘mildly defective’ examples where effort times flow doesn’t have dimensions of power. The two most popular are these:
|Heat flow||heat||heat flow||temperature momentum||temperature|
|Economics||inventory||product flow||economic momentum||product price|
The heat flow analogy comes up because people like to think of heat flow as analogous to electrical current, and temperature as analogous to voltage. Why? Because an insulated wall acts a bit like a resistor! The current flowing through a resistor is a function the voltage across it. Similarly, the heat flowing through an insulated wall is about proportional to the difference in temperature between the inside and the outside.
However, there’s a difference. Current times voltage has dimensions of power. Heat flow times temperature does not have dimensions of power. In fact, heat flow by itself already has dimensions of power! So, engineers feel somewhat guilty about this analogy.
Being a mathematical physicist, a possible way out presents itself to me: use units where temperature is dimensionless! In fact such units are pretty popular in some circles. But I don’t know if this solution is a real one, or whether it causes some sort of trouble.
In the economic example, ‘energy’ has been replaced by ‘money’. So other words, ‘inventory’ times ‘product price’ has units of money. And so does ‘product flow’ times ‘economic momentum’! I’d never heard of economic momentum before I started studying these analogies, but I didn’t make up that term. It’s the thing whose time derivative is ‘product price’. Apparently economists have noticed a tendency for rising prices to keep rising, and falling prices to keep falling… a tendency toward ‘conservation of momentum’ that doesn’t fit into their models of rational behavior.
I’m suspicious of any attempt to make economics seem like physics. Unlike elementary particles or rocks, people don’t seem to be very well modelled by simple differential equations. However, some economists have used the above analogy to model economic systems. And I can’t help but find that interesting—even if intellectually dubious when taken too seriously.
Beside the analogy I’ve already described between electronics and mechanics, there’s another one, called ‘Firestone’s analogy':
• F.A. Firestone, A new analogy between mechanical and electrical systems, Journal of the Acoustical Society of America 4 (1933), 249–267.
Alain Bossavit pointed this out in the comments to Part 27. The idea is to treat current as analogous to force instead of velocity… and treat voltage as analogous to velocity instead of force!
In other words, switch your ’s and ’s:
|Electronics||Mechanics (usual analogy)||Mechanics (Firestone’s analogy)|
This new analogy is not ‘mildly defective': the product of effort and flow variables still has dimensions of power. But why bother with another analogy?
It may be helpful to recall this circuit from last time:
It’s described by this differential equation:
We used the ‘usual analogy’ to translate it into classical mechanics problem, and we got a problem where an object of mass is hanging from a spring with spring constant and damping coefficient and feeling an additional external force
And that’s fine. But there’s an intuitive sense in which all three forces are acting ‘in parallel’ on the mass, rather than in series. In other words, all side by side, instead of one after the other.
Using Firestone’s analogy, we get a different classical mechanics problem, where the three forces are acting in series. The spring is connected to source of friction, which in turn is connected to an external force.
This may seem a bit mysterious. But instead of trying to explain it, I’ll urge you to read his paper, which is short and clearly written. I instead want to make a somewhat different point, which is that we can take a mechanical system, convert it to an electrical one following the usual analogy, and then convert back to a mechanical one using Firestone’s analogy. This gives us an ‘auto-analogy’ between mechanics and itself, which switches and
And although I haven’t been able to figure out why from Firestone’s paper, I have other reasons for feeling sure this auto-analogy should contain a minus sign. For example:
In other words, it should correspond to a 90° rotation in the plane. There’s nothing sacred about whether we rotate clockwise or counterclockwise; we can equally well do this:
But we need the minus sign to get a so-called symplectic transformation of the plane. And from my experience with classical mechanics, I’m pretty sure we want that. If I’m wrong, please let me know!
I have a feeling we should revisit this issue when we get more deeply into the symplectic aspects of circuit theory. So, I won’t go on now.
The analogies I’ve been talking about are studied in a branch of engineering called system dynamics. You can read more about it here:
• Dean C. Karnopp, Donald L. Margolis and Ronald C. Rosenberg, System Dynamics: a Unified Approach, Wiley, New York, 1990.
• Forbes T. Brown, Engineering System Dynamics: a Unified Graph-Centered Approach, CRC Press, Boca Raton, 2007.
• Francois E. Cellier, Continuous System Modelling, Springer, Berlin, 1991.
System dynamics already uses lots of diagrams of networks. One of my goals in weeks to come is to explain the category theory lurking behind these diagrams.