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).
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).
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.
Therefore, the RMS value of a pulse signal is
This expression can also be found as in (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?
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).
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).
In a similar way, we can calculate the RMS value of u12(t):
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.
After calculations, the RMS value of a bipolar pulse waveform is
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).