Root Mean Square versus Root Sum Square - John Dunn, Consultant, Ambertec, P.E., P.C.

Consider some repetitive voltage waveform applied across a resistance of R:

At time T1, we have voltage E1 across resistance R which yields a power of E1² / R.

At time T2, we have voltage E2 across resistance R which yields a power of E2² / R.

At time T3, we have voltage E3 across resistance R which yields a power of E3² / R.

.... and so forth and so forth and .....

At time Tn, we have voltage En across resistance R which yields a power of En² / R.

First consider the root-mean-square (RMS) of thee voltages as follows:

We find the average power applied to R as:    ( E1² / R + E2² / R + E3² / R + . . . + En² / R ) / n.

Factoring out 1 / R, we re-write the average power as:  ( ( E1² + E2² + E3² + . . . + En² ) / n ) / R.

We know of course that power is the square of a voltage divided by resistance, so in this circumstance and with malice aforethought, we will call that squared voltage Erms².

We therefore have:

Erms² = ( E1² + E2² + E3² + . . . + En² ) / n  where this Erms² is the mean, the average, of the sum of the squares of E1, E2, E3 and so on up to En.

We then take the square roots of both sides of this equation:

Erms² = ( E1² + E2² + E3² + . . . + En² ) / n    and    Erms = sqrt ( ( E1² + E2² + E3² + . . . + En² ) / n ) )

Lo and behold, we call Erms the root-mean-square or the RMS voltage. It is the square root of the mean of the squares of the individual voltages.

The power dissipation in R for all of those individual voltages over their applied time span is the same as the power dissipation in R for the application of Erms over that same time span.

Hold this thought and look next at the root-sum-square of these voltages.

Instead of taking the average of the sum of the squares, we take just the sum of the squares and call that Erss². When we take the square root of that Erss², we get the root-sum-square, or Erss. We have:

Erss² = E1² + E2² + E3² + . . . + En²      .....and then......      Erss = sqrt ( E1² + E2² + E3² + . . . + En² )

Now that we've looked at root-mean-square (RMS) calculation and a root-sum-square (RSS) calculation apropos of voltage, we realize that we can do this mathematics for any parameter we choose.

These two calculations can be done for voltages, for currents, for standing wave ratios, for tolerance values or whatever you happen to be devoting your attention to at the moment. Physical meanings are something else to consider, but the equations themselves are valid.

RMS and RSS each have their own roles in this space time continuum. Most commonly, RMS applies to applied voltage or current going to a load while RSS is used in voltage standing wave ratio estimates in RF designs.

Just be careful to not mistake one for the other.

 

Comments

[Tom]

RMS is used primarily to find the "average" value of a continuous, periodic process.

RSS is used to find the "average" of a statistical process, especially things like random noise.

[earl]

Erss = sqrt ( E1² + E2² + E3² + . . . + En² )

is essentially what you do with vectors such as the scallar value of say impedance.

[Agreement Templates]

Thanks for clarifying this. This makes sense to me.