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# Time constants of batteries ?

Hi,

I'm trying to figure out the way solar battery charger work and I'm a bit confused about the following point.

I read that for batteries you could consider the PWM voltage to be equivalent to DC. This I understand well for average voltage (obviously it's the same), but what about the dynamics ?

If your PWM frequency is something like 100 kHz, that's a 10 µs period. Does the battery has time to follow and make big voltage and current peaks or does it just averages all that to more decent values ?

One way to put it is : Are the battery time constants indeed much bigger than the PWM period ?

Thanks for any idea on that topic !

## Comments

2 years ago

All this replies are correct but i must emphasize thet one argument is missing. A battery actually is not an RC-circuit. One has to consider the chemistry in the battery. Will the chemistry follow those quick PWM impulses or will they be more or less averaged out by it?

2 years ago

Well, I think if you ask first yourself, "What is the R*C time constant of a first order, low-pass, RC filter,

https://en.wikipedia.org/wiki/RC_circuit

that can effectively attenuate 100 KHz?", and then consider the fact this filter could be built with a small R and a small C, and then ask yourself if R and C of this size, could exist inside the battery, then I think that will maybe work as an effective "hand waving" argument, to prove that a battery really does act like a low pass filter.

Another way to look at the problem would be to ask, how inexpensive would it be to actually built an RC circuit, outside the battery, where R consists of the real copper resistance in the cables leading to the battery. I mean, this is really small, but if it were as large as 1 milliohm (1/1000 of an ohm), then corresponding C needed to make a filter with R*C = 10 microsecond, would only be 0.01 F = 10 000 uF.

Changing focus, to this time imagine what is inside the battery, I think it is not hard to imagine there is an internal series resistance, inside the battery.

I mean, that is one of the popular ways to model real battery behavior; i.e. model a battery as a resistor in series with an ideal voltage source.

https://en.wikipedia.org/wiki/Th%C3%A9venin%27s_th...

https://en.wikipedia.org/wiki/Electrical_load

From there, I do not think it is too crazy to imagine that series resistance, Rs, really is in there, inside the battery. How much stray capacitance C would be needed to make an effective Rs*C filter?

Reply 2 years ago

Well, I can imagine that the plates have a capacitive effect, so my question would be more to have some typical values of that time constant. Thanks for your insight

Reply 2 years ago

I have no idea what typical values would be. But maybe this could be discovered experimentally. I mean, supposing you put an oscilloscope across the battery's terminals, and measure the magnitude of the ripple, the AC part, of the voltage measured there, and compare that to the voltage magnitude of the PWM signal, which I guess is (Von-Voff), when measured peak to peak.

Also you do this experiment at number of different frequencies, and try to construct something resembling a Bode plot.

https://en.wikipedia.org/wiki/Bode_plot

And this would essentially tell the story of how well the battery, and the wires feeding current to it, are behaving as a filter.

Assuming this plot resembles a

low-passfilter, then the frequency it starts to "roll off", or "cut off", occurs at a frequency that is the reciprocal of its approximate, "time constant".For a simple RC filter, the approximation is exact. The roll off frequency occurs at,

omega= 2*pi*f = 1/(R*C)

The Wikipedia article for, "Low-pass filter", has a picture of this Bode plot, the amplitude plot, for an RC filter, here, in the section titled, "Continuous time low-pass filters", here:

https://en.wikipedia.org/wiki/Low-pass_filter#Cont...

2 years ago

The battery acts like a very large capacitor, since, however much charge you remove from it, the voltage on its terminals stays the same. Likewise, it has a low series resistance, because when you draw the load charge, the terminal voltage doesn't change then either.

Reply 2 years ago

Nice way to see it, but since the time constant must be something like RC, small resistance and big capacity can be higher or lower than my PWM typical period. I guess it'll work but I'd really like some numbers.