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In engineering, there is a theory called Similitude, which describes how models and testing conditions must be correctly scaled in order to behave correctly and represent the real thing.

The reason this theory exists is that the world doesn’t work the same at all scales.

For example, how do I scale a circuit generating a stepped biphasic square wave (between 900Vpk to 1500Vpk and 10 to 15A, ie ON+ = +1500V, 2sec; OFF 0V, 2sec; ON- = -1500V, 2 sec; OFF 0V 2sec) with a period of 8 seconds, to 15V (see Figure 1)?

Figure 1

Figure 1: This sequence is generated by an IGBT H-bridge from the transmitter for an electrical geophysical survey method called induced polarisation (IP).

I am looking to scale the voltage and current ? 1:100, but downstream in the circuit, the electrode is passing through the middle of a solenoid, and the magnetic field from the electrode induces a current in the solenoid, that goes into an energy-harvesting circuit.

So parameters to be examined are: voltage, current, magnetic flux, rise- & fall-time, overshoot, undershoot, ringing, and feedback compensation.

Period would remain the same in the scaled circuit. FYI, the electrode lengths in the field are typically 2 to 6km of PVC-insulated single-core building wire, in a typical mineral exploration IP survey.

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    \$\begingroup\$ What important parameters should be scaled and, what important parameters need to remain unscaled? \$\endgroup\$
    – Andy aka
    Commented Jan 6, 2023 at 12:36
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    \$\begingroup\$ Interesting question, probably "no", but I hope someone can come up with an answer explaining why it's not done in electronics (as far as I know!) \$\endgroup\$
    – pipe
    Commented Jan 6, 2023 at 12:39
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    \$\begingroup\$ @SteveSh Please write an answer so your "opinion" can be judged by the peers, instead of pushed up on top of every quality-controlled answer. \$\endgroup\$
    – pipe
    Commented Jan 6, 2023 at 14:10
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    \$\begingroup\$ @pipe - Took your suggestion and turned my comment into an answer. \$\endgroup\$
    – SteveSh
    Commented Jan 6, 2023 at 16:17

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Yes, absolutely.

Or more precisely: "Yes, but".

We must be careful doing it, as different elements of the circuit scale at different rates.

In short, as others have commented, whole systems may not be easily scaled, at least in an obvious and straightforward manner.

To illustrate, I'll give two examples, how they scale, and how they tend not to.


Consider the IGBT circuit: if you wish to model its switching behavior at smaller voltage and current, then the stray loop inductance (i.e. for a voltage-sourcing half-bridge: the loop between +V, IGBTs, GND and the nearest bypass capacitor(s) back to +V; or analogous elements for other topologies) must scale as \$L = L_0 \frac{Z}{Z_0}\frac{t}{t_0}\$, for original values naughted, and impedance Z and switching time t. (We might take Z to be, for example, the ratio of voltage to current before/after the switching event). This will give the same peak overshoot (percentage) and same relative \$\frac{dI}{dt}\$ as the original case. But already we come into a problem: IGBTs are not available with linear scaling, and even choosing the nearest component of comparable (scaled) ratings, we most likely have different voltage drop and capacitance. In particular, capacitance of semiconductors is quite nonlinear (dependent on voltage), and it matters very much where we are along that curve, to get the same dynamics -- since capacitance must be scaled in an analogous way as well: \$C = C_0 \frac{Z_0}{Z}\frac{t}{t_0}\$.

And if we want to scale an IGBT inverter to different frequencies, say, we don't have very much say over the switching speed: that's a device property, we can't magically will into existence new IGBTs (or MOSFETs or etc.) that run as fast as we want them to. So we instead have a limiting factor, where given devices in a given configuration (stray inductance, capacitance) must operate below some magnitude of V, I and F. (Generally being derated as frequency goes up, so the V*I might be nominal up to some break frequency, then dropping proportionally above there.)

Or if we want to scale it physically i.e. by length dimension, we may find that power goes as the cube of length (~constant power density), but frequency goes as inverse length -- ultimately because \$\epsilon_0\$ and \$\mu_0\$ have units of F or H per length, so we are constrained by the above limits.

Perhaps the single most effective example of electrical scaling is electronic filters. These are calculated from first principles, wholly abstract -- solutions to systems of polynomial equations -- and what comes out are values in ratio to the system impedance \$Z_0\$ and transition frequency \$\omega_0\$. That is to say: we can prepare normalized tables of component values, typically \$Z_0 = \omega_0 = 1\$. We then scale L and C as per above, and out drops the values we can use in circuit.

But again, there are responsibilities we cannot ignore: it is very easy to design a filter that should use absurd (impractical or nonphysical) component values. For example, we might want a filter designed for 10kΩ, to save power; and we might just as well want a filter with a very narrow bandwidth, and maybe a very low center frequency (audio application, say?), or very high (maybe for a direct conversion radio receiver?). And suddenly we'll find we need 1kH inductors or 1fF capacitors respectively. Or 1nH inductors surrounded by 1uF capacitors (that themselves are 2nH or more i.e. 0603 SMT chip type), etc.


So, applying scaling to your problem: to reduce the output network by 100 in voltage and current, the impedance will stay the same, but the frequency is -- same I presume?, and length is undefined. If you will be using the same output and pickup coils (this sounds like a transformer arrangement for testing purposes?) as the full scale system, that can be fine, but beware if the pickup/receiver is, say, rectified with diodes, they'll drop only moderately less voltage -- not proportionally so, so the loss there will already be different (relatively higher). If you are physically scaling the components as well, mind that as you reduce length, inductance and capacitance go down, so the dynamics on various time scales will change -- and if you change the number of turns to keep inductance the same, say, beware that leakage inductance may not vary at the same rate!

And if you are studying switching dynamics of the IGBTs, you will need an accurate model or measurement, first of all, of the full scale model, so that it can be scaled proportionally; and then you will need some stand-in for the IGBTs themselves, because there's no such thing as a 15V IGBT. (And the capacitance curves of 30V and 600V MOSFETs are very different, so it can be tricky to select a suitable analog there, too.)


And, mind I haven't invoked square-cube scaling yet; in electronics, that's mostly applicable to thermal management, and at least for hand-waving purposes while scaling everything else -- one might assume proportionate heat-extraction methods are chosen independently. Say, one might scale a given converter topology from 1W to 10kW, using convection at the low end, forced air in the middle, water cooling at the high end. If the scaling however should include such aspects as this, the design may end up looking quite different -- in particular, the higher heat output of a larger design may warrant a more efficient power conversion technique. Compare commercial power supplies using inefficient (~70%) flyback converters in 1-10W ratings vs. LLC resonant converters (>90%) in 500W+ ratings. What could be more scalable than a power supply, but here as ever, we see topological changes to the solution as multiple goals are optimized simultaneously.


So, what, isn't this a problem? What do we do? Do we just test everything at scale, all the time? ...Isn't that expensive?

Well, sometimes, and it can be.

Power converters can be modeled in stages, approaching the final design; this helps alleviate the effect of hard-to-estimate nonlinearities due to component properties, construction techniques, etc.

Full-size converters can also be tested at extremes of voltage and current, somewhat independently (i.e., high voltage / low current; low voltage / high current; preferably by suitable choice of load), or the applied voltage can simply be ramped up a bit at a time. All the while, the important dynamics are to be measured: peak voltage and current, power dissipation / temp rise / efficiency, etc.

It's a common occurrence that a switching converter develops unexpectedly high peak voltages or currents when operated at scale, that didn't show up on a smaller test; well, as mentioned, inductance and capacitance scale with size, so one must be careful to adjust the time/frequency and length accordingly. And, again, often these can't be done very well -- not just to mention that semiconductors aren't infinitely variable, but they're often packaged in very limited ways, too -- you can buy TO-220 devices rated for anything from 20V 100A, to 1500V 5A; but each and every one of them has the exact same ~7.5nH lead inductance!

Do note that controls can be dimensionless. That is, we might have a control that reads input and output voltages and currents, etc., and outputs gate drive pulses, status signals, etc. Well, generally controllers don't like to be exposed to high and arbitrary voltages; instead they're designed around a nominal input, a nice comfortable 5V range, say. We use voltage dividers to sense high voltages, and current transducers (shunt resistor, Hall effect sensor, etc.) to sense currents, converting them into representative voltages. Such a control doesn't care at all if it's embedded in a system handling 5V or 5000, or 1A or 1000. All that matters is the surrounding system responds in the expected manner / with the expected dynamics. Scaling such a system might involve different interface elements (not just the sensors, but gate drivers / isolators, as well as transistors and etc.), but can still have the same operating behavior across a wide scaling range.


Anyway, I'm sure there are even elements in my examples that I've omitted -- given a particular worked example I could probably enumerate most of the differences, and offer alternatives that might be preferable for the various reasons; as you can see, it's a complicated task, and a simple "no" avoids a lot of particular questions. But there are interesting details within those particulars, and it would be an even more peculiar mystery if these things couldn't be scaled at all, if the "no" were more fundamental. Well, I just want to say, it's not fundamental -- but it is complicated, I'm afraid.

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In addition to the arcing that Scott Seidman mentions, there are also some cases where a first- or second-order approximation is no longer sufficient to describe real-world behaviour. For instance, subthreshold conduction in FETs is better approximated as exponential in gate voltage, unlike normal saturation-mode operation which is best approximated as quadratic in gate voltage.

Likewise, saturation in magnetic components causes the inductances to be a function of the current, no longer constant as in the first-order approximation.

And as Scott mentioned, dielectric breakdown requires very different math to calculate compared to conventional electronics--it's not even that much like other common types of breakdown, such as Zener or avalanche breakdown. Not just arcing, but also corona discharge and glow discharge.

Though even something as banal and everyday as avalanche breakdown requires a shift in models away from the Shockley diode equation and to something more accurate; the standard Shockley diode equation has no provision for any significant reverse conduction.

I guess what I'm trying to get at is, there are plenty of effects that aren't modelled by the standard equations because they aren't encountered to any significant degree when dealing with standard devices. I didn't even mention all the ones I can think of!

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IMO the answer is NO. This opinion is based on 53+ years experience in defense and space industries.

One of the reasons is that things do not, in general, scale linearly in electrical engineering disciplines, except perhaps over small ranges. As you change parameters like voltage/current/speed over a wide range, the devices needed to meet those requirements change, at times, in a step-wise manner. Take switching power supplies. Early ones used bipolar transistors, then moved to Si MOSFETs, and now are starting to use GaN. This type of change is hard to put into a model.

We tried to do something like this 20-30 years ago to try and parameterize cost estimates. The idea was that once you built the model, you entered things like throughput, equations, IO requirements, etc, and out would pop a SWaPC (size, weight, power, and cost) estimate. While the model did work, somewhat, on designs that were similar to what were used to build the model, it failed pretty badly when completely new systems were trying to be estimated, and so dies a quiet death.

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For me, the red flag is when some dimensionless parameter is used to decide when to switch the scale of the model. I haven't seen that in electronics.

The only cases I can think of where you need to use a "different physics" (if you allow me the inaccurate shorthand) are situations like determining if a high voltage will arc.

Perhaps the differences between current and electron drift velocity describe another such situation, and there may be some examples is semiconductor physics as well.

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    \$\begingroup\$ I can think of some: small signal approximation and electrical length \$\endgroup\$
    – D Duck
    Commented Jan 6, 2023 at 15:48

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