Root mean square versus root sum square


Root mean square values and root sum square values are related concepts which have mathematical similarity and their respective areas of applicability, but they should never be confused with each other.

Imagine some load resistance to which we apply a voltage E1 at time T1. There will be some power dissipation going on while we do that. That power will be P = E1² / R. Then, moving on to time T2, we will apply E2 to obtain E2² / R and then at time T3 we will apply E3 to obtain E3² / R and so forth and so forth.

If we keep the time marker closely spaced and we do this for time from T1 to Tn, we take the numerical sum of the squares of E1 through En meaning E1² + E2² + E3² + …. + En² which is the sum of the power deliveries to that resistance R. Having obtained that sum, we divide that sum by n to obtain the mean value, the average value, which we may call the “sum square”. Next, we take the square root of the “sum square” to obtain the “root sum square” and with that we’ve hit pay dirt having found the “root sum square” or the RMS of the voltage being applied to R.

If the sequence of voltage applications is repeated and repeated and repeated and… then the RMS value is that voltage which if it were steadily applied to R, it would have the same power delivery, the same heating effect, as the application of the sequence of voltages we’ve been discussing. The applicable equation is something very familiar:

The “root sum square”, or RSS, is similar in appearance but quite different and apart as a concept from the “root mean square” or RMS. The equation for the root sum square is as follows:

The difference between the two concepts is that we do, or we do not, divide the sum or our square terms by “n”. Root sum square does not relate to power levels, but it does have an application in microwave system analyses.

Any microwave system will consist of some cascade of devices where at each inter-device interface, there will be some standing wave ratio. If each of those devices is lossless of power, the standing wave ratios in the cascade will contribute to a total standing wave ratio for the composite system which is nominally the product of those individual ratios.

Unfortunately, when we seek to estimate a worst case for the total SWR calculation when taken as the product of the separate ratios, the resulting numbers can become implausibly high. It is sometimes decided in such cases to estimate the worst case for the total SWR as the root sum square of the separate contributing ratios.

Consider the following illustration in Figure 1:

Figure 1 Setup where the standing wave ratios are products of inter-device SWRs. Source: John Dunn

Just making up some ludicrous device names, calculations for one through five inter-device interfaces are shown and then those values are compared to root sum squared numbers in Figure 2.

Figure 2 Comparisons of products versus root sum squares for one through five inter-device interfaces. Source: John Dunn

The root sum square calculation of the SWR total isn’t always smaller than the product calculation result, but sometimes it is. For the illustrated inter-device SWRs, the root sum square result is the smaller for the cases of having four interfaces and for having five interfaces.

An SWR of 6.928:1 is so unlikely that such a result may be seen as unrealistic in the real world. By comparison, the SWR of 3.413:1 is very likely a true measure of the worst-case real world SWR value.

I have had clients not just accept this interpretation, but actually insist upon it.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

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