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Assume that there is a current which consists of a sum of sinusoids flowing through a capacitor with capacitance C.
How to calculate its reactive power?
It is quite simple with a pure sine wave but I wonder how would you do it for the waveform of multiple sine waves as above as the superpostion is not allowed to use here.
If you use RMS current, Q = IRMS^2*Xc then how would you calculate reactance as it is frequency dependent?

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    \$\begingroup\$ Complex power (and therefore reactive power) is only well-defined for a single sinusoidal frequency since it's defined via phase angles. You can only calculate it for each frequency individually; adding them would furthermore be meaningless. Therefore there is no single "reactive power" number in a multi-frequency system. What are you supposed to calculate exactly? The apparent power, maybe? \$\endgroup\$
    – Jonathan S.
    Commented Jan 3, 2022 at 21:06
  • \$\begingroup\$ @JonathanS. the question is to calculate the reactive power of the capacitor. \$\endgroup\$
    – emnha
    Commented Jan 3, 2022 at 21:08
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    \$\begingroup\$ "Reactive power," or reactive volt-amperes is not a very useful concept for waveforms other than un-distorted sine waves. It may be somewhat useful to calculate the reactive VA for the fundamental current in an AC power system that has some harmonic distortion, but the effects of the harmonic components need to be considered separately. \$\endgroup\$
    – user80875
    Commented Jan 3, 2022 at 21:09
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    \$\begingroup\$ I assume this is university homework... Reactive power is simply not defined in this context so you can't calculate it either. You should ask your professor to clarify. \$\endgroup\$
    – Jonathan S.
    Commented Jan 3, 2022 at 21:21
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    \$\begingroup\$ I think that reactive power in this context could be defined as the rate of energy transfer for a charge-discharge cycle. If the series of currents have frequencies that are a fundamental and a series of harmonics, I think that the energy transfer could be calculated for one cycle of the fundamental. Normally the angles given would be the angles between the current and voltage. Are a corresponding series of voltages given? \$\endgroup\$
    – user80875
    Commented Jan 3, 2022 at 21:41

2 Answers 2

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You may find this reference handy:

"Power measurement techniques for non-sinusoidal conditions", S. Svensson, doctoral thesis, Chalmers University of Technology, Electric Power Engineering, Göteborg, Sweden, 1999
https://core.ac.uk/download/pdf/70557608.pdf
A review of reactive power accounting methods is provided in the first section.

Related: What is the correct relationship between active, reactive and apparent power in non-linear AC systems?

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  • \$\begingroup\$ Thank you for sharing the link. It would be great if you could copy or capture and paste the relevant content here, so that it will be permanently available. While links can be added for reference, they can sometimes become broken over time. \$\endgroup\$
    – emnha
    Commented Mar 19, 2023 at 9:17
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    \$\begingroup\$ @emnha While I would like to, the relevant sections cover a dozen pages, which the author or other rightsholder would probably not appreciate. Hence I provide the full citation, which can always be found at the university, or other academic publishing services. If anyone wishes to excerpt or summarize, edits are welcome. \$\endgroup\$
    – Tim Williams
    Commented Mar 19, 2023 at 10:31
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It is quite simple with a pure sine wave but I wonder how would you do it for the waveform of multiple sine waves as above as the superposition is not allowed to use here.

Superposition certainly is allowed to be used here. Think again; this is the way to do it.

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  • \$\begingroup\$ As I know superposition does not work for power. \$\endgroup\$
    – emnha
    Commented Jan 3, 2022 at 21:14
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    \$\begingroup\$ This is wrong: Assume f1=f2, I1=I2, phi1=phi2. Using superposition, you get a power P for each of these components, resulting in a total power of 2P. The correct answer is, however, 4P (I²Z). Superposition doesn't work as soon as there are nonlinear components in a system, which includes the quadratic function contained in the formula for calculating power. \$\endgroup\$
    – Jonathan S.
    Commented Jan 3, 2022 at 21:18
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    \$\begingroup\$ @Andyaka users.ece.cmu.edu/~jhoe/course/ece100/supplementals/spop.pdf \$\endgroup\$
    – emnha
    Commented Jan 3, 2022 at 21:18
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    \$\begingroup\$ @emnha that example is for real power in a real resistor at DC (a common frequency) and not for some "version" of reactive power in a capacitor at different frequencies. The example is miles away. If you want an answer that is mathematically correct, you must define what you mean by reactive power when multiple frequencies are involved. \$\endgroup\$
    – Andy aka
    Commented Jan 4, 2022 at 0:50
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    \$\begingroup\$ There seems to be a fog around this topic. This paper examines Budeanu's concept and concludes, "powers [do] not always meet superposition property in circuits with non-sinusoidal voltage and current waveforms". Also, mention of Budeanu was removed from IEEE 1459 in the 2010 revision. In other words, i'm not sure this is a cut and dried topic. +1 to @Andyaka for his comment above. \$\endgroup\$
    – relayman357
    Commented Jan 5, 2022 at 18:56

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