A hammer is a simple machine; you apply energy to it over a long period of time, and upon impact, all that energy is released in a short time. As an example, you might swing a 2kg hammer in a 3m-diameter circle at 2Hz, which works out to 9.4 m/s and 89 J, and it might take you 500ms over 2.4 m for the whole swing, thus an easily human-achievable 37 N and 178 W (though at least the power must vary, increasing as it does from zero); upon impact, this energy is released within, say, a millimeter, which is about 200 μs, 450 kW, and 89 kN. So in effect you have a mechanical advantage of about 2400 — your 37 N applied over 2.4 m is amplified to 89 kN applied over 1 mm.
Correspondingly, your 178 W is amplified by 2500 to get 450 kW, because it’s released over 1/2500 the time.
Springs can be used in a somewhat similar way, and also in reverse, which is a thing we don’t usually do with hammers. Springs have the feature that the input force and distance are always the same as the output force and distance, but the input and output power need not be. A bow, slingshot, or onager is the hammer-like approach — you apply a small power over a long period of time to charge up the spring, then release the energy over a short period of time. Applying the same principle in reverse, you can wind a watch, applying a large power over a short period of time, and then let it run down, applying a small power over a long period of time.
A traditional simple machine like a lever doesn’t have the variability in time that hammers and springs do — the instantaneous power in is always equal to the instantaneous power out. A rough electrical equivalent of the lever, if we disregard dc, is the transformer — the input voltage can have whatever relationship to the output voltage, but the power in is equal to the power out.
Capacitors and inductors can be used like springs or hammers in the electrical world. You can add energy to either one slowly over a period of time, then release it over a short period of time; in the case of the inductor, the voltage is arbitrarily different from the original voltage, while the current changes continuously, while in the case of the capacitor, the current is arbitrarily different form the original current, but the voltage changes continuously. As one example, engine spark plugs are fired with ignition coils that work this way, and flashtubes are fired with capacitors that work this way. Some spot-welding machines also use either capacitors or inductors in either of these two ways, and of course ordinary switching power supplies use inductors precisely in order to change voltages. And it’s easy to provoke accidental, and dangerous, energy releases from either kind of device. Heck, I just spewed sparks of vaporized copper into the air when I plugged in my power strip because it has a wall-wart plugged into it.
In this process, the non-ideal properties of the various devices, including even switching elements, come into play. The ESR and ESL of capacitors limits how quickly they can discharge (or charge), their leakage current wastes energy, and the ESR also causes energy losses during both charge and discharge; parasitic capacitance of inductors limits how quickly they can store or release energy, and their ESR causes energy losses during the whole time they are storing energy. The limited energy density of either device also
For capacitors, the resistive losses are concentrated in the high-power part of the cycle, because they are quadratic in current. Consider charging a 1μF 1Ω capacitor with a constant-current source to 1 V over 1 second and then discharging it, also at constant current, over 1 millisecond. You charge it at 1μA, eventually depositing 1μC in it, dissipating 1pW and thus finally 1pJ in its ESR (and 1μJ in its dielectric). When you discharge it at 1 mA, it’s dissipating 1μW, a million times as much, and finally 1nJ, a thousand times greater.
For inductors, instead, resistive losses are concentrated in the low-power part of the cycle. If you spin up a 1mJ 1Ω inductor with a constant-voltage source to 1 A over 1 second, then discharge it through a constant-voltage load in 1 ms, your current rises linearly from 0 A to 1 A, and the resistive loss rises quadratically from 0 W to 1 W over that second, dissipating a total of ⅓J. During the discharge cycle, it drops again from 1 W back to 0 W, playing the same curve backwards but a thousand times faster, and so dissipating only ⅓μJ.
So I was wondering about the limits of this kind of thing.
The Pulse Electronics PA4309.105NLT unshielded drum-core inductor is 1mH (±20%), 1.08 A, and 3.915Ω; it’s 7mm × 12.8mm × 12.8mm. At its full current, it stores 0.58 mJ and dissipates 4.6 W, which I guess means it only stores about 127 μs worth of power, about 0.5 J/liter. I don’t know what its self-resonant frequency or parasitic capacitance are, but Pulse measures the inductance at 100kHz, so I’m guessing it’s at least 1MHz. So you can probably get a boost or dropdown of up to about a factor of 100 or 200 higher or lower voltage and power out of these inductors, with microsecond-scale spikes of a few hundred watts and volts.
Their PA4309.104NLT is 0.1mH (±20%), 3.5 A, and 0.405Ω, and is the same size. It can store 0.61 mJ, almost the same as the previous one (which makes sense, since all that’s changed is the wire!) and dissipates even more, 5.0 W (which, again, makes sense).
The PA4309 series goes down to a 1μH 20A 19mΩ inductor of the same size, which is slightly higher power (7.6W) and lower energy (0.2 mJ).
So it seems like, for energy-storage purposes, the power, energy per unit size, and cutoff frequency don’t really vary with inductance, at least within a given core material and size.
Capacitors can handle much longer energy storage (going beyond microseconds up to seconds, hours, even longer) as a result of us having much better approximations to perfect insulators than to perfect conductors, at least around our normal temperatures. They’re also much denser.
A random capacitor I have lying around (in a broken CFL bulb) is a can 8mm in diameter, 17mm long, 3.3 μF, 400 V. This works out to 264 mJ in 0.854 mℓ, 309 J/ℓ, 600 times denser than the inductors I was looking at. I don’t know how fast it can discharge and whether it’s electrolytic (as it appears) or polyester (it says “PET” all over one side). But I think electrolytics are good up to about 10kHz or so, so it might not be able to do under 100 μs or so. In the CFL, it’s being used to smooth the ripple in rectified 50Hz ac.
A random inkjet printer board I had sitting around has a Nichicon supercapacitor on it: 8mm diameter, 20mm long, 2.7 V, and an astounding 2.2F, which works out to an astounding 7.3 J in 1.0 mℓ, and thus 7.3 kJ/ℓ. However, supercapacitors have high ESR and much lower capacitance over short timescales.
Checking out datasheets and stock at Digi-Key, the Nichicon JUWT1105MCD costs 92¢ down to 30¢ (quantity 5000). It’s a 1-farad 2.7V 6.3mm-diameter 10.5mm-long EDLC rated at 4Ω ESR at 1kHz. The datasheet lists a 4Ω “DCR”, but I don’t know what that is. The capacitance rating is based on discharging in, I guess, 270 seconds after a 30-minute (!) charge cycle. (See Capacitors: some notes on tradeoffs.) This works out to 0.33 mℓ, 3.6 J, and 11.1 kJ/ℓ. The datasheet gives 4Ω ESR at 1kHz, so its maximum energy output at around that timescale would be into a 4Ω load, which would work out to 460 mW.
Another inkjet printer power supply has an electrolytic that is 16mm in diameter, 35mm long, 50 V, and 2200 μF, which works out to 2.75 J in 7 mℓ, 390 J/ℓ.
At the other end of some kind of scale, I have a microwave oven capacitor which is 25 μF and 450 V, 50 mm diameter and 70 mm long, 137 mℓ, 2.5 J, 18 J/ℓ. This is presumably designed for high currents and low ESR, although its extensive surface markings do not provide any indication of how high and how low.
Comparing to batteries, a lead-acid Yuasa GYZ16H holds 16 amp-hours at nominally 12 volts (690 kJ), is 150 mm × 87 mm × 145 mm (1.89 ℓ), weighs 12.4 lbs (5.6 kg), and can deliver 240 amps (nominally 2.9 kW, but I bet the voltage dips) to start your motorcycle. This works out to 365 kJ/ℓ or 120 kJ/kg, orders of magnitude better than even the supercapacitor. Li-ion and Li batteries are even better, and combustible fuels are orders of magnitude better again.
Murata has a line of “GR7” ceramic capacitors specifically optimized for camera flashes, the largest of which (GR731CW0BB473KW03#) is 3.2 mm × 1.6 mm × 1.8 mm. These are ceramic capacitors rated for 350VDC, 10nF to 47nF, with 50mA max discharge current, which works out to at least a megavolt per second, except that its capacitance droops below 50% at full rated voltage, so it’s more like two megavolts per second. The power output of this device is an astounding 17.5 watts, but its energy content is only 2.2 mJ, which works out to 234 J/ℓ. I don’t know what their ESR or ESL is; the datasheet lists their “dissipation factor” as 0.025 max at 1V and 1kHz.
Most promising are tantalum capacitors like the AVX TAJB226M010RNJ, which costs 69¢ in quantity 1, 23¢ in quantity 1000. It’s 3.5 mm × 2.8 mm × 2.8 mm, 22μF, 10V, has an ESR of 2.4Ω at 100kHz, and has a leakage current of 2.2 μA. This works out to 2.2 mJ, 27 μℓ, and a rather poor 80 J/ℓ. The leakage current will steal 100 mV/s, so these are only good for storing energy for timescales of up to a minute or two. Discharging through a 2.4Ω load, it can theoretically deliver 2100 mA and thus 20 W.
So, suppose I want to dump 1000 J into a spark that lasts 1 ms — a rather high power of 1 MW. If electrolytic capacitors are my medium of choice, I need 3.2 ℓ of electrolytic capacitors! If I were to use tiny ones like the one from the CFL, I’d need 3800 electrolytic capacitors. Using the bigger one from the inkjet, I’d need 360 capacitors, totaling 2.5 ℓ.
(Putting this power into context, arc welding on steel commonly uses a spark of 40 V at 150 A, melting a weld puddle a few millimeters across, which works out to 6 kW. This spark would generate heat 167 times as rapidly.)
Supercapacitors are apparently capable of discharges over a timescale of milliseconds, but their ESR is too high; using the one whose datasheet I examined above, you’d need 2.2 million capacitors, containing a total energy of 7.9 MJ, to get an instantaneous 1 MW out of them. Also it would cost you US$650 000 and they would occupy 720 liters.
Using Murata’s camera-flash capacitor, on the other hand, you could get 1 MW out of only 57000 capacitors, which is actually a more reasonable quantity than it sounds like — the capacitors are tiny, so it’s only 530 mℓ. But the total energy capacity would be only 126 J, which is an eighth of what I specified above, even if it is a bit larger than the literal 89J hammer blow I started the explanation with. To get to 1000 J you need 450 000 capacitors, but they “only” occupy 4.2 liters. I don’t have the price handy but I imagine it would add up to around US$45000.
The AVX tantalums seem like perhaps a better balance: they hold the same 2.2 mJ as the ceramic capacitors, but can deliver a higher maximum discharge current despite the lower voltage, so the output power ends up being almost the same. Unfortunately, they occupy even more space and are probably more expensive.