# What is the physical meaning of reactive power in non-sinusoidal steady-state (with harmonics)?

In sinusoidal steady-state (linear loads, no harmonics), I understand what is reactive power $$\Q = V_{\text{rms}} I_{\text{rms}} \sin{(\theta)}\$$ where $$\\theta = \theta_v - \theta_i\$$: its absolute value is the maximum rate of transfer of the energy that's oscillating between the device and the external circuit. This conceptual interpretation is in accordance with the mathematical definition above. I haven't seen/read this interpretation anywhere, but I proved it.

When there're harmonics due to non-linear loads, there're various mathematical definitions of reactive power. This article shows some. One definition I've seen quite often is the following (in the article, it's called Budeanu reactive power in section 3.1):

$$\ Q = \displaystyle \sum_{n=1}^{\infty} V_{n,\text{rms}} I_{n,\text{rms}} \sin{(\theta_n)} \tag*{}\$$

This mathematical definition is really an "extrapolation" of the definition in sinusoidal SS. But unlike the latter case, here I can't understand what this infinite sum with units of watts represents. (Yes, I know reactive power is measured in VAR, but that unit is dimensioanlly the same as watts and joules per second and volt-amperes.)

Do you know what this means? Can you explain why your interpretation is correct? I'm not asking for analogies, since they at some point break.

I read a textbook in which the author used the same mathematical definition as above, but he said the significance of that was unclear.

• Are you familiar with function orthogonality? sin() of different frequencies (e.g. harmonics) can form an infinite orthogonal basis, so the extrapolation to infinite-sum makes sense. Oct 19 '20 at 9:30
• @Vicente Unfortunately the only time where I was taught about orthogonal functions was in a communications class, but not in an ODE class or any other. I somewhat understand that the integral of the functions would be zero; and this fact simplifies the expression for average/active power in non-sinusoidal steady-state, for example. But how does this apply to my question? Would you explain it, please? Oct 19 '20 at 15:51
• Just saying that defining a power as an infinite sum of harmonic components (as the article you linked says Budeanu postulated) is not an unnatural thought and simetrically fits the active power definition. Have you read the reference mentioned [3] ? It explictely says "The usefulness of QB (Budeanu reactive power) for quantifying the flow of harmonic nonactive power has been questioned by many engineers". I think this was just an intuitive mathemathical definition to decompose apparent power. Oct 19 '20 at 16:59
• "Have you read the reference mentioned [3] ?" Nope but I will, thanks. // "I think this was just an intuitive mathemathical definition to decompose apparent power" I'm trying to understand what Q means in the equation (non-sinusoidal SS) without needing to talk about apparent power. For example, look at my interpretation of Q in sinusoidal SS; I didn't mention apparent power. I'll try to use the orthogonality as you pointed earlier to see if I come up with something. Oct 20 '20 at 4:44

The only thing that I can contribute that might be helpful is that the reactive component of harmonic power (harmonic VARs) is orthogonal to both the fundamental real power and the fundamental reactive power. The power triangle becomes a three dimensional diagram.

Any harmonic power in the system would presumably add a 4th dimension to the diagram since that would seem to be orthogonal to the other quantities. Considering the harmonic power might require considering each harmonic individually with components orthogonal to those of the fundamental and all of the other harmonics

The harmonic VARs can not be compensated by pf compensation capacitors. Harmonic compensation capacitor values must be determined using only the fundamental VARs. It is possible to build harmonic filters that mitigate harmonic distortion while also compensating fundamental VARs.

Harmonic distortion in a power system leads to high harmonic VARs, but not a lot of harmonic power. If the source voltage is undistorted, no real harmonic power is transmitted. All of the real harmonic power is the result of harmonic I squared R dissipated in the power system conductors, transformers, motor windings and the ESR of the pf compensation capacitors. That is certainly important, but it needs to be minimized by filtering or by preventing it at the source.

Look at IEEE Standard 1459-2010. 

• I already knew the geometrical interpretation in your first and second paragraph, though another person may not; thanks anyways Charles. // Regarding the third paragraph, does "harmonic VAR" mean the second and higher-order terms in the equation of my question? I agree with you in the filtering, I've read caps can even amplify harmonics and lead to resonance, but I don't see it relevant to my question. // Regarding the fourth paragraph, I found that more helpful; what's the difference between "harmonic VARs" and "harmonic power"? Does the latter refer to distortion power? Oct 20 '20 at 4:36
• I will read the standard you and Vicente suggested me. Oct 20 '20 at 4:36

$$\n\$$ denotes the various harmonics of the fundamental sinewave. For example $$\n=3\$$ is the third harmonic.

The sum $$\ \displaystyle \sum_{n=1}^{\infty}\$$ is saying that the total power is found by adding the powers for each and every individual harmonic.

• Which integral? Oct 20 '20 at 0:06
• $\displaystyle \sum_{n=1}^{\infty}$. Sorry, I should have said sum not integral, I'll make that change. Oct 20 '20 at 6:44