How to Derive the RMS Value of a Sine Wave with a DC Offset

by Adrian S. Nastase


I noticed a question posted on one of Yahoo’s Q&A sites, asking what is the RMS value of a sine wave with a DC offset. The chosen answer as being “the best” was actually wrong. The next comment, which was trying to correct “the best” answer, was wrong too. I am not going to post the Yahoo link here. What I can do, is to show how to derive the RMS value of such waveform.

Let’s derive first the RMS value of a sine wave with no DC offset

Let’s start with the RMS value of a sine wave, with no DC offset, which is shown in Figure 1. It is well known that the RMS value of a sine wave is 0.707 times the signal peak level, but how can you prove this?

sine-wave-with-zero-offsetFigure 1

As shown in this article, MasteringElectronicsDesign.com: How to Derive the RMS Value of a Trapezoidal Waveform, or other RMS articles in this website , let’s start with the RMS definition.

rms-definition-1 (1)

The sine wave time dependency can be described by the following function:

sine-wave-2 (2)

T is the function period, or T = 1/f where f is the waveform frequency. Also, a1 is the amplitude.

Replacing (2) in (1), and calculating the integral over a full period T, we find the RMS value squared as in the following equation:

sine-wave-rms-no-offset-3 (3)

The standard method to calculate a squared sine integral is to transform it into its double angle equivalent, using a trigonometric identity usually called the power-reduction formula.

power-reduction-formula-4 (4)

So the RMS squared becomes

rms-calculation-5 (5)

If you’re wandering why the sine term is zero in the previous equation, that’s because

rms-calculation-6 (6)

Therefore, the RMS value of a sine wave with offset zero is the following well known formula,

rms-value-sine-wave-no-dc-offset-7 (7)

The RMS Value of a Sine Wave with a DC Offset

sine-wave-with-DC-offsetFigure 2

Now, let’s look at a sine wave with a DC offset. This waveform is shown in Figure 2 and is described by the following function.

sine-wave-with-DC-offset-8 (8)

where with a0 I noted the DC offset. Applying the RMS definition, the RMS squared can be written as:

rms-calculation-9 (9)

Let’s calculate the integral.

rms-calculation-10 (10)

rms-calculation-11 (11)

rms-calculation-12 (12)

rms-calculation-13 (13)

rms-calculation-14 (14)

Therefore, the RMS value of a sine wave with a DC offset is given by the following expression.

rms-value-sine-wave-with-dc-offset-15 (15)

The immediate verification of the validity of this expression is the RMS value of a sine wave with zero DC offset. Indeed, when a0 = 0 V, the RMS level reverts back to equation (7), which is 0.707 of the sine amplitude.

Expression (15) can also be verified by comparing it with Parseval’s Theorem. This theorem says that the integral of the square of a function is equal with the integral of the squared components of its spectrum. In effect, the theorem states that the total energy of a waveform can be found in the total energy of the waveform’s Fourier components. In our case, a0 is the DC level, or the frequency zero component, and a1 is the fundamental frequency. There are no other Fourier components. As such, the RMS value of a sine wave with a DC offset as given by expression (15) is correct.




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Categories: Analog Design

31 Comments to “How to Derive the RMS Value of a Sine Wave with a DC Offset”

  1. Brian Coode says:

    Helped me refresh some theory

  2. David says:

    So do “True RMS” multimeters read this value correctly if there is a DC offset on a sine wave – or are they really AC coupled and read the RMS value of the AC component only? If I owned a true RMS meter, I would try it myself – but alas, I don’t.

    • You need to check the datasheet of the multimeter. Not all of them are capable of measuring the RMS value of an AC signal riding on a DC level. I would not take that for granted, even if it says “true RMS measurement”. Most manufacturers will make sure to write this capability in the multimeter feature list.

  3. I was highly skilled in Math in a previous life. Ever since then, I forgot most of my teachings. After reading this article, my memory gradually began to be refreshed. Thank you.

  4. Theo says:

    Thanks, correct apart from the last line! Either u-squared OR root the RHS!

  5. ashish says:

    infinite thanks sir,specially why the sine term becomes zero,I searched lot on that your’s explanation is best.

  6. Javed Bukhari says:

    RMS really points to the “heating” enregy present in any waveform. So be it Sine wave or DC voltage riding on it irrespective of the polority of any component the final effect of all the components is governned by superposition theorm, provided the effective bandwidth is infinite or close to it.

  7. Jessie M says:

    Excellent article, sir and thank you for this website.

  8. DRB says:

    For the purposes of an actual answer, doesn’t “a1″ need to be the peak of the non offset sinusoid, not the peak of the shifted sinusoid(as shown in your figure 2)?

  9. J. Hu says:

    I just calculate the RMS value using integration method, by hand, immediately after I saw this title and then scroll to the bottom of your article to check whether I’ve a correct answer. But when “spectrum power” shows, I realize I wasted my time, :)

  10. dox says:

    Wouldn’t it be easier to explain the answer as “the square-root of the sum of squares” of the RMS components?
    RMS of DC = DC or a0
    RMS of AC = ACpk/sqrt(2) or a1/sqrt(2)
    RMS of combination = sqrt(a0^2 + a1^2/2)

    This approach makes more complex functions easier to address. For example, add a triangular wave with amplitude a2 riding on this offset sine:
    RMS of triangle = Peak/sqrt(3)
    RMS of combo = sqrt(a0^2 + a1^2/2 + a2^2/3)

    • Your comment is correct regarding the calculation of RMS values based on the square root of the sum of squares. I am not sure that it is simpler though. In any case, this subject is part of a planned article I am writing in this series about RMS calculation. Not everybody understands the root square of the sum of squares and I plan to explain it in the near future. In this article I showed the derivation of the RMS value starting from the definition.

      Thank you for your input.

  11. Musician says:

    Hey!

    I am musician and am interested how a digital DAW would read pure DC offsets RMS and peak dbs. Any ideas on this? I think a sine with DC offset would end up being louder on the meter and with higher peak.

    • By DAW you mean a Digital Audio Workstation, right? From my time in the professional audio industry I know that offsets are dreaded in this business. So, I am not sure why you would have DC offsets getting into your DAW. But, if this is the case, DAW will digitize whatever it “sees” in the input, like a digital oscilloscope would. Then it will calculate the RMS of the signal with the square root of the sum of squared samples. I did not write an article about this method for calculating the RMS but I will in the near future. Generally, I do not think a signal with DC offset will sound louder. And that’s because a speaker does not generate sound due to DC, just a loud pop when the system is turned on.

  12. anatman says:

    why the square in u(t)^2?

    • The RMS value represents the signal power, so the integral has to be of the u(t)^2. Without that square, you would calculate the average of the signal. In case of a simple sine wave the average is 0. If the sine wave has a DC offset as in this article, the average value is the DC offset.

  13. Stefan says:

    Thank you.

  14. Daniel says:

    Thanks very much, but I have one question that keeps bugging me.

    Why is the RMS calculated as the sqrt of the avg of the squares as oppose to the avg of the sqrt of the squares.

    In other words why use sqrt((1/n)*(x1^2+x2^2…..xn^2))

    Why not use (1/n)*(sqrt(x1^2)+sqrt(x2^2)…….sqrt(xn^2))

    • sqrt(x1^2) = x1, right? So, in effect, you want to use the signal average (1/n)(x1 + x2 +…+ xn) to calculate the RMS, which is not correct. RMS represents power. Therefore, first calculate the sum of each component power. Once the power is calculated perform a square root of this sum to calculate the actual voltage.

      • Daniel says:

        Ok, so if x^2 is the power of the signal and you get them mean of x^2 ie the mean of the power than why do we than take the sqrt of the mean.

        Thanks

        • Because the mean of xk^2 results in Volts to the power of two, which is equivalent with Watts, the units of power. So, to bring this back to rms voltage, you need to perform the square root of the result.

  15. Pradeep says:

    Very convincing and clear explanation. Thank you very much!
    Extending this idea, the RMS of a waveform made by summing known simple waveforms has an RMS that is the square root of the sum of the squares of the component RMS values if the component waveforms are orthogonal.

    • Correct. That will be the subject of another article. Thank you for your comment.

    • Thomas Groover says:

      This article is very good and another article can generalize the results illustrated by the following example with the necessary proofs.

      If two time functions x and y are absolutely uncorrelated (r[xy] = 0) then the rms value of their sum can be found in the way so described above.

      If two time functions x and y are absolutely correlated (r[xy] = 1) then the rms value of their sum can be found as the linear sum of the two rms values of x and y.

      If two time functions x and y are partially correlated (e.g. r[xy] = .5) then the rms value of their sum can be found as the linear sum of the two rms values calculated in the two cases above divided by two for r[xy] = .5

  16. aravind bose says:

    helped a lot …

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