How to Derive the RMS Value of Pulse and Square Waveforms

June 13th, 2010 by Adrian S. Nastase


The RMS value of a pulse waveform can be easily calculated starting with the RMS definition. The pulse waveform is shown in Figure 1. The ratio t1/T is the pulse signal duty-cycle. As shown in other articles in this website (How to Derive the RMS Value of a Trapezoidal Waveform and How to Derive the RMS Value of a Triangle Waveform), the RMS definition is an integral over the signal period as in equation (1).

pulse signalFigure 1

(1)

The pulse function, with the variable “time”, is a constant, which is the signal amplitude, between 0 and t1 and zero from t1 to T as in (2).

pulse waveform function (2)

where with u1(t) I noted the function of the waveform in Figure 1. After replacing u1(t) in equation (1) we can find the RMS value squared as in the following expression.

rms value calculation of a pulse signal (3)

Therefore, the RMS value of a pulse signal is

rms value of a pulse signal (4)

This expression can also be found as in (5)

rms value of a pulse signal as a function of duty-cycle (5)

where with D I noted the pulse signal duty cycle, D = t1/T.

What if the pulse signal is bipolar, as in Figure 2?

bipolar pulse waveform

Figure 2

In this case we should expect that the negative section of the signal to also contribute to the energy delivered to the load. To calculate its RMS value, let’s split the signal in two: from 0 to t1 and from t1 to T as in (6).

bipolar pulse waveform function (6)

where with u11(t) and u12(t) I noted the two sections of the waveform in Figure 2.

The RMS value of u11(t) is identical with the one shown in equation (3).

rms value of a bipolar pulse, the first section (7)

In a similar way, we can calculate the RMS value of u12(t):

rms value calculation of a bipolar pulse, second section (8)

The total RMS value of the bipolar pulse waveform is then calculated by applying the square root of the sum of squares of u11RMS and u12RMS.

rms value addition (9)

After calculations, the RMS value of a bipolar pulse waveform is

rms value of a bipolar pulse (10)

As you can see, the bipolar pulse RMS value does not depend on its duty-cycle, and it is equal with its amplitude.

Knowing the RMS value of a pulse waveform we can easily calculate the RMS value of a periodic square signal. The square wave in Figure 3 is a pulse signal with 50% duty-cycle. Its RMS value can be calculated from equation (5), where D = 1/2. Its RMS value is given in (11).

square wave

Figure 3

rms value of a square wave (11)

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Categories: Analog Design, Waveforms

4 Comments to “How to Derive the RMS Value of Pulse and Square Waveforms”

  1. Paul Dalkin says:

    Hi,

    I found this page very usefull for a recap but i belive there is a mistake in equation 7, which refers to equation 4, it should refer to equation 3.

    At the moment it seems that the square of the Vp has vainish, to me it should be:

    U11^2rms = Vp^2 * t1/T — or as per equation 3, U11^2= Vp^2/T * t1

  2. Rizza says:

    Hi,

    Is this also valid for current pulse?

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