Understanding the Definition of Average Power in Sinusoidal Functions

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Discussion Overview

The discussion revolves around the definition of average power in sinusoidal functions, particularly focusing on the implications of choosing different periods for calculating average power. Participants explore the relationship between the chosen period and the resulting average power values, questioning whether the definition leads to approximations of a "real mean." The scope includes conceptual clarifications and technical reasoning related to power calculations in sinusoidal contexts.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about why only one period is considered for average power calculations, noting discrepancies when averaging over different intervals.
  • Another participant suggests that while any period can be chosen for integration, using a single cycle is standard practice to avoid arbitrary outcomes.
  • A participant questions whether the average power derived from the definition is merely an approximation of a "real mean," indicating a lack of understanding of the implications of the definition.
  • One participant provides an analogy involving a 1kW electric heater to illustrate how averaging over a short interval could yield a lower average power, implying that the choice of period affects the results.
  • A later reply reiterates that the average over one period approximates the long-time average, emphasizing the importance of the period length in determining the average power.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the definition of average power. There are competing views on the significance of the chosen period for averaging and whether it leads to approximations of a "real mean."

Contextual Notes

Some participants highlight that the average power can vary significantly depending on the period chosen for integration, suggesting that the definition may not capture all nuances of power behavior over different intervals.

jakey
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I am slightly confused by the definition of average power if the power function $p(t)$ is sinusoidal. Why is it that only one period is considered?

I mean I know that it simplifies calculations but if we assume that the period of $p(t)$ is $T$ and I compute the average power over $[0,\sqrt{2}T]$, I do not get the same result had I computed the average power over $[0,T].$

Case in point: If $p(t)=\frac{1}{2}(1+\cos(2x))$ then the average power is not the same for both cases mentioned...
 
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If you want to know the instantaneous power then you would use VI. For a mean power value, you could choose any period you liked, to integrate over, but you would need to state that period (absolute phase intervals). It seem perfectly reasonable to me to choose a single cycle (or any integral number) because it's the most likely thing that anyone else would do. Any other period would be arbitrary and could introduce a massive range of possible outcomes (as you seem to be finding).
 
so are you saying that this is simply a definition? I'm sorry but I still can't seem to understand it...so the average power based on this definition, then, is merely an approximation of the "real mean"?
 
Think how you'd tackle a 1kW electric heater. If you took the first 10ms of one cycle, the average power would be somewhat less. Would that make sense?
 
jakey said:
so are you saying that this is simply a definition? I'm sorry but I still can't seem to understand it...so the average power based on this definition, then, is merely an approximation of the "real mean"?

The "real" mean depends on how long you are averaging over the power. Averaging over one period gives the long-time average. If you were given a power function and asked the average power over a certain period of time, then you would average over just that time. If the period (of the power function) is short, you are probably interested in the average over many periods rather than one small random interval. The average over one period approximates this quite well.
 

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