RMS value of the square wave signal

[Pixelx]

As is the case with a square wave signal, without the RMS offset it is not dependent on the fill
And when there is an offset in the RMS signal, it depends on the fill.
How is it derived? There may even be integral calculus, but I would like to understand it in a simple and mathematical way, step by step.

for DC offset the formula works Vrms = Vmax * square root (duty).
for Ac it doesn't work

 

[albbg]

The formula you wrote is valid only for rectangular waveform that goes from 0 to a given voltage. If you apply the AC coupling you will have a zero mean waveform going from -Vm/2 to Vm/2. From the definition of RMS (root mean square) you have to consider that the negative part folds on the positive y axis (that is -Vm/2 will go to a level Vm/2) then the result will be a constant voltage of amplitude Vm/2.

 

[Pixelx]

Thanks for the answer, what if there is such a chart for a different duty cycle?
DC DC converters regulate the duty cycle, so the RMS value of the voltage that goes to the transformer winding must change.
How to determine the RMS of such a waveform for different duty cycle?

 

[KlausST]

Hi,

this is not a square wave (2 levels) but now you show a three level signal. Here one needs to now the timing.

For a square wave -4V / +4V levels (no 0V level)
--> the RMS always is 4V (independent of duty cycle)

But I don´t know how this refers to a transformer. Please explain.

Klaus

 

[FvM]

For waveforms with discrete voltage levels, integral calculus reduces to a simple square root of sum of squares.

--- Updated Oct 14, 2024 ---

E.g.

|V| = V1 for delta t = T1
|V| = V2 for delta t = T2

Vrms = sqrt((T1*V1² + T2*V2²)/(T1+T2))

 

[BradtheRad]

I would like to understand it in a simple and mathematical way

At some time the definition of RMS for an arbitrary waveform was a way of answering the question "If DC can produce an identical heating effect on a resistor, what DC voltage do you apply?" Amid math and meter readings some consensus emerged as to the proper coefficients relating RMS to Average to AC to DC.

The more recent version of Falstad's animated interactive simulator can be commanded to tell these figures onscreen in edit windows or scope traces. Example, you can choose a sine wave readout in peak amplitude or RMS value, or both.

 

[KlausST]

Vrms = sqrt((T1*V1² + T2*V2²)/(T1+T2))

referring to the picture of post#3:
I´d repalce (T1 + T2) with (Tperiod)

--> Vrms = sqrt((T1*V1² + T2*V2²)/(Tperiod))
Because it also includes the times where V = 0

Klaus

 

[Pixelx]

Thank you for your answers.

I think I calculated something wrong and I don't get the exact Vrms results from the oscilloscope.
Could you substitute the values based on these screenshots to get what the oscilloscope shows?

How to calculate the voltage generated on the oscilloscope?
I will add that the RMS depends on the given half of the filling. If the upper rectangles are at 10% and the lower ones at 40%, the RMS is also different.
What is the formula to calculate this?
If you know how to derive this formula from integral form, it would be great.

 

[KlausST]

How to calculate the voltage generated on the oscilloscope?

--> see post#5 / post#7.

It´s on you to give the timing vlaues.

What is the formula to calculate this?

--> see post#5 / post#7.

You coulld use a calculator or something like eXcel.

Klaus