Calculate an average power of a signal

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Discussion Overview

The discussion revolves around calculating the average power of a given signal composed of multiple cosine terms. Participants explore methods for calculating average power in both the time and frequency domains, while considering the implications of signal characteristics and assumptions about load impedance.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants suggest calculating average power using the integral definition for periodic functions, while others mention complications for non-periodic signals.
  • There is a proposal that the average power can be calculated by summing the squares of the amplitudes of the cosine terms, leading to a total of 120 W, though the rationale for dividing by 2 is questioned.
  • Participants discuss the assumption of a 1 Ω load impedance in power calculations, with some arguing that this assumption may not be valid for all contexts.
  • One participant raises a concern about ignoring load impedance when calculating power from a voltage waveform.
  • There is mention of the relationship between root mean square (rms) values and peak values in the context of power calculations for sine waves.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding load impedance and the validity of certain calculations. There is no consensus on the correct approach to calculating average power, and multiple competing views remain throughout the discussion.

Contextual Notes

Limitations include the unspecified nature of the signal as purely electrical or a more general mathematical signal, as well as the unknown period T in some calculations. The discussion also highlights the dependence on assumptions about load impedance.

Adir_Sh
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Homework Statement


Given the following signal,

##v(t)=2\cos (2\pi f_{0}t)+4\cos (4\pi f_{0}t)+6\cos (6\pi f_{0}t)+8\cos (8\pi f_{0}t)##

1. Calculate the signal's average power in the time domain.
2. Calculate the signal's average power in the frequency domain.

Homework Equations


Didn't make a real attempt for a solution as yet, but I try to think of a shorter, easier way other than a straight calculation using the exact definition. If there's a hint according to the even form of the given cosines' amplitudes and frequencies that you could advise me of - it might make my way significantly easier here, I feel.

Thanks in advance for any guidelines suggested!

Adir.
 
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Adir_Sh said:

Homework Statement


Given the following signal,

##v(t)=2\cos (2\pi f_{0}t)+4\cos (4\pi f_{0}t)+6\cos (6\pi f_{0}t)+8\cos (8\pi f_{0}t)##

1. Calculate the signal's average power in the time domain.
2. Calculate the signal's average power in the frequency domain.

Homework Equations


Didn't make a real attempt for a solution as yet, but I try to think of a shorter, easier way other than a straight calculation using the exact definition. If there's a hint according to the even form of the given cosines' amplitudes and frequencies that you could advise me of - it might make my way significantly easier here, I feel.

Thanks in advance for any guidelines suggested!

Adir.

The average power is zero.

Oh wait, the average power is very high.

Oh wait, what is missing that keeps us from calculating a power given the voltage only...? :smile:
 
I think we have to presume that the "signal" is not necessarily electrical, but is a signal in the more general mathematical sense. In such a case, for a periodic function f(t) the power is defined to be (If my memory serves... it's been a while...):

$$P = \frac{1}{T} \int_0^T |f(t)|^2 dt$$

I think it gets more complicated when the signal is not periodic and you have to integrate using limits from -∞ to +∞.

There may be a trick for finding the power of a sum of sinusoids involving summing the squares of the amplitudes and dividing by two... don't quote me on this :nb), as I say, it's been a while...
 
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gneill said:
I think we have to presume that the "signal" is not necessarily electrical, but is a signal in the more general mathematical sense. In such a case, for a periodic function f(t) the power is defined to be (If my memory serves... it's been a while...):

$$P = \frac{1}{T} \int_0^T |f(t)|^2 dt$$

I think it gets more complicated when the signal is not periodic and you have to integrate using limits from -∞ to +∞.

There may be a trick for finding the power of a sum of sinusoids involving summing the squares of the amplitudes and dividing by two... don't quote me on this :nb), as I say, it's been a while...

Ok, so
##P_{av}=2^2+4^2+6^2+8^2=4+16+36+64=120\left [W \right ]##

But why you decided to divide this sum by 2 if ##T## is unknown?
 
Adir_Sh said:
Ok, so
##P_{av}=2^2+4^2+6^2+8^2=4+16+36+64=120\left [W \right ]##

But why you decided to divide this sum by 2 if ##T## is unknown?
T is the period of the periodic signal. In this case you can see that the lowest frequency is ##2 \pi f_o##, and all the other terms have frequencies that are multiples of this. That means the signal as a whole will repeat every ##1/f_o##, so that's its period.

As to why the sum of squares is divided by two, that follows from the power integral for a cosine function. Do the integral for a single cosine; you can use the angular period in place of the time period since it covers the same domain for the function:
$$\frac{1}{2 \pi} \int_0^{2 \pi} |cos(\theta)|^2 d \theta$$
 
Adir_Sh said:

Homework Statement


Given the following signal,

##v(t)=2\cos (2\pi f_{0}t)+4\cos (4\pi f_{0}t)+6\cos (6\pi f_{0}t)+8\cos (8\pi f_{0}t)##

1. Calculate the signal's average power in the time domain.
2. Calculate the signal's average power in the frequency domain.

Homework Equations


Didn't make a real attempt for a solution as yet, but I try to think of a shorter, easier way other than a straight calculation using the exact definition. If there's a hint according to the even form of the given cosines' amplitudes and frequencies that you could advise me of - it might make my way significantly easier here, I feel.

Thanks in advance for any guidelines suggested!

Adir.
For the freq. domain, what did you learn about power for a Fourier series?
For the time domain, it's (1/T)∫T v2(t) dt where T comprises an integer number of FUNDAMENTAL cycles.
The assumption in problems like this is R = 1 Ω.
Main thing is to get the same answer both ways! :smile:
 
Sorry, I don't get it. How can you ignore the load impedance when calculating the power of a voltage waveform supplied to that load?
 
berkeman said:
Sorry, I don't get it. How can you ignore the load impedance when calculating the power of a voltage waveform supplied to that load?
Yes, but in communications theory for example we talk of the power of a signal, assuming 1 ohm:
http://mathworld.wolfram.com/AveragePower.html
Note that v(t) was not specified to be a voltage, just a "signal".
 
Adir_Sh said:
Ok, so
##P_{av}=2^2+4^2+6^2+8^2=4+16+36+64=120\left [W \right ]##

But why you decided to divide this sum by 2 if ##T## is unknown?
For sine waves, rms = peak/√2 so power = rms2 = peak2/2.
 
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