Parallel Capacitors-Theoretical Question

03-22-11: Paulsax
Shouldn't the tolerances for cap add in quadrature? (0.1^2 + 0.1^2)^0.5 = 0.14 or 14%?

Hi Paul,

You ask a good question, as usual.

If the two tolerances are the same in percentage terms, then as I stated the tolerance of the parallel combination will be that same percentage. That can be seen by calculating the worst case values. For example, if a 10uF 10% capacitor is paralleled with a 5uF 10% capacitor, the minimum possible value of the combination is 9uF + 4.5uF = 13.5uF. The maximum possible value of the combination is 11uF + 5.5uF = 16.5uF. In both cases the deviation from the nominal value of the combination (15uF), is 1.5uF, or 10%.

My statistics courses are now a (very) distant memory, but I believe that combining inaccuracies on an rss (root sum square) basis such as you described would be applicable to standard deviation and related calculations, that involve the PROBABILITY that a combined inaccuracy will fall within limits that are NARROWER than the worst-case limits.

That in turn would typically involve situations where tolerances are being combined that act on the same nominal value, not on nominal values that sum together.

Best regards,
-- Al

Speaking theoretically, the most predictable performance will be from two capacitors of the same manufacturer and series, and close in value - that is, the 6uF and the 7uF together. This is because their residual inductance and ESR will also be close as well . . . which will give improved characteristics (over a single 13uF capacitor) without the possibility of a secondary HF resonances as can happen when an electrolytic is bypassed with a high-Q film cap.

As far as tolerance goes, Al is correct for the worst-case scenario - two 10% tolerance components in parallel have a maximum deviation of 10%. But in reality the tolerance does indeed get better the more components are placed in parallel, the extent of how much is dependent on their probability density function: http://en.wikipedia.org/wiki/Probability_density_function

The "normal" or Gaussian function can be used to derive an approximate tolerance, but some manufacturers will actually have documentation that specifies this. Frequently the required testing and documentation for these sorts of things is what comprises the difference between mil-spec and standard components, and is what drives up the cost of the former.

Kirkus, please elaborate on the statement that 6uF & 7uF caps together will be better than one 13uF. Isn't there a benefit to having two less solder joints in the filter?
Thanks,
Jay

Hi Jay . . . the most basic model of a real-world capacitor is an ideal capacitor in series with both an inductor and a resistor. For a high-quality film cap, the inductance comes from the leads and the fact that it's impossible to acheive a perfect "non-inductive" wind to the film/foil itself. Series resistance comes from the leads and the (usually crimped) connection from the leads to the foil. Virtally all of these non-ideal properties increase with the physical size of the capacitor, especially inductance. Larger caps also tend to be more suceptible to crosstalk from nearby inductors.

For electrolytics there are additional losses and parasitic properties from the electrolytes, which can be signal-modulated and heat-modulated. And while these properties do get better with physically larger capacitors, doubling the number tends to make much larger improvements than merely increasing the physical size, and it does so without incrasing inductance. And if there is significant ripple current (causing heat), two caps will disipate the heat much better than one, given the greater surface area.

Compared to all of this, any solder-joint of reasonable workmanship is but a tiny grain of sand on the beach. The biggest vulnerability would be for i.e. a production wave-soldered PC board, which frequently has "thermal" pads of significant resistance . . . but even in this case, paralleling two smaller caps would mean less weight (hence less vibration stress) on each connection, and the resistance of the board traces and thermal pads would be in parallel. So, still an improvement.

Thanks, Kirkus :)