I proceeded to find out if it could have any effect by finding the impedance of a Li-Poly at frequencies such as those used in this application. The ESC uses 8KHz PWM when under 100%, so if battery ESR degrades after 8Khz a capacitor could, potentially, help. Sadly information about the impedance of Li-poly batteries vs frequency is not that easy to find. In total I only found 2 graphs, but luckily they agreed pretty well. In them the impedance starts rising at just about 10Khz, interestingly enough. That could mean the higher harmonics of the 8KHz PWM could potentially use the capacitor.
Testing:
Next comes the testing part. I used the RC plane with an adapted 1000uF low ESR capacitor and a switch in between the battery and ESC . It uses it's own piggyback connectors which I usually use to measure current. Overall the leads are still very small, about 10 cm. The battery is 7.4 300mA, a rather small size for RC. The motor draws max 3.5A. The ESC still has it's 100uF capacitor.I used partial throttle and measured RPM to check if any difference exists when the 1000 uF capacitor is switched in and out. I was surprised that a difference in RPM can actually be seen, but it was very small and hard to measure since the the RPM changes by itself to some degree as well. I eventually obtained several values so that the improvement can be quantized.
After plugging the numbers in various wab pages and programs, I came up with the following: The RPM is incresed about 0.15% to 0.30% at partial throttle. I checked only the lower 1/2 of throttle range since measurement accuracy was too low for the top half.
The RPM increase is very ( or is it fairly) small, but the result gets slightly better when the thrust vs rpm formula is used. I looked up the propeller used ( 6030 GWS copy ) and it's thrust (g) = 6.3e-007*(RPM^2.1) .
After some juggling the thrust percentages work out to approximately double: 0.24 to 0.67% . Note this includes errors so the real answer is probably about 0.3 - 0.5% increase. This hardly justifies the addition of a capacitor where lead size is small, but at least is shows an actual amount of improvement exists.
Calculating optimal capacitor size:
The battery internal resistance and the capacitance form a filter, which I wanted under 4KHz so that it may possibly improve impedance at the ESC. As such the capacitor is proportional to the battery ( and leads) internal resistance as in the formula : C = 1 / ( 2* PI * Rint * frequency )
The frequency chosen is arbitrary since the battery impedance is not known. As such you could use the actual pwm frequency, or a lower value such as PWM/3 for more extreme cases. If capacitor ESR is not low enough compared to the battery internal resistance ( it should be similar to it or lower)
using larger capacitance values will not do much, as the ESR will be the limiting factor. If cap ESR is higher then say, 2x battery internal resistance the capacitors will likely not do anything.
Scope comes out:
After all this I wanted to see what actually happens, so I hooked up the scope and looked at the waveforms trying to connect what is happening to where. A look at the (unsteady) motor waveform did not show anything changing. I soon realized I was not going to see a 0.5% change on a scope screen. But I reluctantly connected the probe to the battery, where the capacitor is connected. (not a very good place, really) . I could then see a 80mV sawtooth-like voltage drop reduce to a 50mV drop.
As luck would have it, the change is of the right magnitude, i. e. zero point something percent.
I will accept this result due to the fact this has used up more time then necessary for the potential improvements.
Overall, it looks like the cap provides some more voltage , like I assumed, by reducing the voltage drop. Since the cap has to charge from the battery on the off cycle , it will not do anything at full throttle, where no PWM is used. But, since the cap increasees thrust at partial throttle, it's effect is no different then increasing the throttle manually by 0.5% . As such , in my case the weight and size of the cap is not useful ( in a plane) in exchange for a throttle offset.
This does not account for long leads, where voltage spikes could damage the ESC ( or the cap). It also does not cover reverse current flow (into battery) from regenerative breaking. Such applications do, and could, respectively, benefit form capacitors.
In other applications, where batteries with much lower internal resistance are used, it is hard to believe the capacitor ESR can account for a high enough fraction to make any difference. In such cases good quality, low ESR capacitors could be paralleled together.
It is likely the capacitor/s could help more in aging, higher internal resistance batteries, but, overall, since there is no efficiency improvement, the capacitor can't overcome the bigger problem that, after all, partial throttle is, well , something arbitrary.
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from "Fast Estimation of State of Charge for Lithium-Ion Batteries Shing-Lih Wu , Hung-Cheng Chen and Shuo-Rong Chou " |
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| A schematic of the resistances involved, the 47uF cap is actually 100uF . |
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