Root mean square, commonly called RMS, is an important concept in electronics.
It’s also one that can cause a bit of confusion at first.
In this video, we introduce RMS, why it is needed, and how it is used.
We also explain the difference between “averaging” RMS and “true” RMS.
RMS is a mathematical concept used to derive the average of a constantly varying value.
In electronics, RMS provides a method for calculating effective AC power.
In other words, RMS determines the heating value of AC power in a way that allows us
to compare it to the equivalent heating value of a DC system.
To help understand this, let’s begin by considering a DC power source, such as a 9V
battery.
If we plot the voltage from this battery on a graph with time as the horizontal axis,
we end up with a straight line.
The graph starts at 9V, and remains more or less constant for the useable life of the
battery.
In this case, measuring the voltage is straightforward: we simply pick any instantaneous point on
the graph, and that gives us the value.
Note that our straight line graph is somewhat idealized.
In a system with an actual load, the voltage would gradually slope downward over time.
But our central point remains the same: all we need to do to determine the voltage is
to simply take a measurement at any moment.
For AC power, it’s considerably more complicated.
When we plot clean and undistorted AC power against time, we see the familiar sine wave
graph, due to its constant cycling from positive to negative values and then back again.
If we measure the voltage or current at random times on this graph, we get different values,
depending on which points we choose.
At first glance, it may seem logical to simply take the average of these measurements to
calculate the effective power for this source.
However, if we do so, we’d end up with a voltage or current of zero.
By definition, the pure sinusoidal AC power graph spends as much time below zero as above,
therefore the negative values will cancel out the positive ones.
So no matter how high our peak power is, the average power for the archetypal AC system
would always be zero, the same as if we had no power source at all.
Obviously such a result tells us nothing about the true power flow in this system, so we
need another way to derive the effective AC voltage or current.
We can do this by applying the RMS method.
For demonstration purposes, we will consider the simplest and “cleanest” AC waveform,
an undistorted sine wave such as the one shown on the screen.
In real life an AC system can display waveform asymmetries due to factors such as non-linear
loads.
We will return to this point a little later in this video.
But for now, let’s discuss how to find RMS for a regular sine wave.
The first step is to square the AC power values.
This produces a graph similar to the shown here.
As you can see, all points on the graph are at or above zero, so we no longer have to
deal with negative values.
We can now use these positive numbers to calculate the effective power of this AC system.
A step-by-step derivation of these calculations requires a moderate amount of math; so for
the sake of brevity we’ll cut to the chase and show the final formula, which for sinusoidal
voltage is as follows:
In this equation, RMS voltage (VRMS) equals the peak AC voltage (VP) divided by the square
root of 2.
For example, suppose we have an AC system in which the peak voltage is 170 volts.
Applying the formula shown on the screen, we derive a VRMS value of approximately 120
volts.
In other words, the effective voltage of this AC system is 120 volts, the same as household
outlet voltage in the U.S.A.
This formula can also be applied to calculate effective AC current:
And for a quick approximation, you can calculate effective power by simply multiplying the
peak power by 0.707, which represents the decimal form of one over the square root of
2:
This simple RMS equation forms the basis of so-called “averaging” RMS instruments.
For electrical systems where the AC cycle is sinusoidal and reasonably undistorted,
these products can produce accurate and reliable results.
Unfortunately, for other AC waveforms, such as square waves, this calculation can introduce
significant inaccuracies.
The equation can also be problematic when the AC wave is irregular, as would be found
in systems where the original or fundamental wave is distorted by one or more harmonic
waves.
In these cases, we need to apply a method known as “true” RMS.
This involves a more generalized mathematical calculation that takes into consideration
all irregularities and asymmetries that may be present in the AC waveform:
In this equation n equals the number of measurements made during one complete cycle of the waveform.
For AEMC instruments, this is always a multiple of 2.
The higher this number is, the more accurate the RMS calculation will be.
This is because the higher the value of n, the higher the order of harmonics this formula
can accommodate.
To understand why this is so, let’s consider the following example.
Suppose we added a 3rd harmonic wave to our fundamental wave, as shown on the screen.
The combination of these two waves produces the distorted waveform shown here.
As you can see, the greatest distortion to the fundamental occurs when the 3rd harmonic
reaches either its positive or negative peak.
By definition, this occurs 6 times per cycle of the fundamental.
Therefore to accommodate the effect of the 3rd harmonic, we will take 6 measurements,
timed to correspond with the positive and negative peaks of the third harmonic.
Since in this example we have 6 measurements, the value of n is 6.
Plugging this number and the measurement values into the formula, we derive the following:
So in this example, RMS is approximately 123 volts.
This means that the harmonic distortion introduced in the waveform shown on the screen has changed
the VRMS value from 120 volts in its pure sinusoidal form to approximately 123 volts
in its distorted form.
Note that this is a very simple example.
In the real world, we would likely need to take into consideration additional orders
of harmonics, such as the 5th, 7th, 9th, 11th, and so on.
In this case the number of measurements per cycle, and correspondingly the value of n,
would be much higher.
But for the purposes of demonstration, our example should provide some insight into how
true RMS instruments perform their calculations.
We hope you have found this brief explanation of RMS helpful.
If you have further questions about this topic, or have topic suggestions for future videos,
please email us at the address shown on the screen.
And be sure to check our YouTube channel for additional videos on other topics in electronics,
including the many products offered by AEMC.