# Is the association of useful work to active power a convention?

I know that there are many questions about reactive power. What I am trying to understand is what is sometimes called useful work associated to active power, compared to no useful work for reactive power.

In the picture below, an inductor and a lamp are connected in series. There is a iron bar inside the coil, that can be used to change its inductance.

The useful work in this case is supposed to be the light (and heat) of the lamp. A lot of the total work is however consumed changing the magnetic field of the inductor by the AC current. It can be seen because the difference in the glow of the lamp with and without the iron bar inside the coil is huge.

But on the other side, there is a buzz sound in the iron bar produced by the alternating magnetic field. If my intention is to produce this sound, and if the lamp is considered the inevitably resistive part of the circuit, I could call the changing magnetic field the useful work, and the heat (or glow) as a kind of loss.

My questions: is the concept of active and reactive power well described with this circuit? Is the concept of useful work associated with active power not relevant and only subjective?

• When did reactive VA become reactive power? What(!) is the unit of reactive power?
– Chu
Aug 25, 2021 at 9:59

Useful work could, I suppose, be a matter of judgement. But reactive power is well defined and is not subjective. Basically, reactive power results in storing energy in inductors and/or capacitors during the AC cycle, and delivering it back to the source at other times in the cycle. Reactive power cannot result in useful work for any reasonable definition of "useful work."

A load that is purely reactive consumes no power (when integrated over 1 or more full cycles) and therefore cannot do any physical work of any kind without violating conservation of energy.

The idea of useful work is quite subjective.

Real power is the measure of the rate of transfer or conversion of energy from a source.

Reactive power is the rate of constant circulation of energy between one storage device and another. The flow of energy changes direction every half cycle of the sine wave. The rate of energy continually lost in that process is real power that would typically be converted to heat.

Real power transfers energy that can be converted to some other form of energy like light, sound, heat, or mechanical work. It can also be stored. Real power that represents energy that accomplishes an intended purpose is called "useful power." Real power that does not directly contribute to the useful purpose is said to be a loss, wasted or inefficient.

You have lots of terms running around, and some confusion.

• Real power (your "active power") is the actual average power flowing from point A to point B.
• Reactive power is power that swishes back and forth, but goes nowhere.
• Useful work is whatever you define it to be -- that may be useful for efficiency calculations, but it won't help you understand real vs. reactive power.
• Total work as you use it: "A lot of the total work is however consumed changing the magnetic field of the inductor by the AC current." is a will-o-the-wisp. It's an illusion caused by thinking about reactive power too much to accept the simple explanations, but not enough to really grasp what reactive power means.

So -- unless you're doing efficiency calculations, forget about "useful work". There's energy flows.

If you take your electrical system, draw a boundary around it, and stipulate that what's inside the boundary can't store energy for any appreciable amount of time then you can do the following: sum up all the energy flowing out across the boundary in any form, and subtract from that the energy that's flowing in across the boundary in any form. Energy will be conserved*, and the energy outflow will equal the inflow.

That's it. No more.

Your "useful work" has to do with efficiency. If I'm supplying 50 watts of real power to your apparatus, and you note that you're only observing 40 watts of light and heat from your light bulb and who cares about the 10 watts being dissipated from the inductor in the form of heat and sound -- then your apparatus is 80% efficient at generating "useful work". But energy is still conserved, and there's still 50W going in on average, and 50W coming out, on average.

I think you're confused about what reactive power is, as indicated by your comment about total work. If you calculated the work exerted on and by the magnetic field in the inductor, and you integrated the absolute value of that work over time, then yes, there'd be a large "total". But it would be mostly meaningless. That's because in all but a few special cases you don't care about the "total" of the reactive power -- you care about the sum over time, and by definition that sum total is zero. If it wasn't zero, it'd be real power, not reactive power.

* Exceedingly pedantic disclaimer: to the extent that the universe is not expanding. If the "big rip" is real, then a hypothetical observer a hundred billion years from now would be able to observe that perpetual motion machines can too work -- at least briefly, before their atoms are disassociated from one another.

• "If you calculated the work exerted on and by the magnetic field in the inductor, and you integrated the absolute value of that work over time, then yes, there'd be a large "total"." I think that is the point where the concept (reactive power) is not clear to me. But instead of use the comments I will ask another question, using the mechanical analog of eletromagnetic equations. Aug 24, 2021 at 21:17

The production of sound involves active power; the fact that the bar is vibrating changes the characteristics of the system in such a way that you no longer have a pure inductance. The energy converted into sound appears as real power, not reactive.

• Well, it is a matter of definition. Reactive power is that of reactance, eg of an inductor. An inductor based buzzer can do useful work in the "real" world. In other words, "reactive" power is "real" power. Aug 24, 2021 at 1:13
• @tlfong01 The mathematical definition is quite clear. The energy being dissipated in the form of sound appears, from an electrical point of view, exactly like a resistance in series with the inductance, shifting the current phase closer to the voltage phase. Aug 24, 2021 at 2:28
• Ha, you confused more. (1) I agree that the mathematical definition is clear: E = L * I². (2) So you should also agree the following: if I increases, E grows; and when I decreases, E shrinks. (3) Now let us look closely at an inductor based (electromagnetic) buzzer. As you said, the buzzer can be electrically defined as a "pure" resistance in series with a "pure" inductor. (4) Please let me know if you agree with (1), (2), and (3) above, or you would like to have more comments, before I move on. Aug 24, 2021 at 7:54