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I'm reading the book "Introduction to Electromagnetic Compatibility" by CR Paul, and in Chapter 3, I come across this equation for the average power in the waveform assuming a 50% duty cycle and \$\tau_r\$ = \$\tau_f\$ :

enter image description here

Is this a general way to compute the average power? So far I've only computed the average power using the area under the waveform, and I can't understand how this equation is derived. Can anyone help me?

BTW, the waveform in this section of the book is a trapezoid.

ETA: Added the diagram of the waveform.

enter image description here

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    \$\begingroup\$ Maybe I'm missing something, but what's the actual waveform? Is it a square wave? Is it all above zero or symmetrical? \$\endgroup\$
    – Transistor
    Jan 24, 2016 at 16:20
  • \$\begingroup\$ I've added a diagram of the waveform in my OP. \$\endgroup\$
    – Rayne
    Jan 25, 2016 at 5:29
  • \$\begingroup\$ The "area under the curve" is the general way to compute the average power for any waveforem x(t). The specific formula with the 1/2 and the 1/3 happens when you plug this specific trapezoidal waveform into the general equation. \$\endgroup\$
    – davidcary
    Jan 25, 2016 at 17:48
  • \$\begingroup\$ How do I get the specific formula from the trapezoidal waveform? \$\endgroup\$
    – Rayne
    Jan 26, 2016 at 5:29

1 Answer 1

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The integral given is a general formula for the average power of a periodic signal:

$$P_{av}=\frac{1}{T}\int_0^T x^2(t) \mathrm{d}t$$

This equation does not depend on the duty cycle being 50% or the rise and fall times being equal.

It does depend on you working out a form for \$x(t)\$ that gives the instantaneous power when squared. For example, if you're talking about a voltage signal applied to a resistive load, you'd use \$x(t) = \dfrac{v(t)}{\sqrt{R}}\$ instead of just \$v(t)\$.

Only the second equation, where he reduces the integral to a formula based on the actual signal parameters, depends on those assumptions.

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  • \$\begingroup\$ I don't know what \$x(t)\$ is in this case. Is it the trapezoidal shape? Is there a formula for that? \$\endgroup\$
    – Rayne
    Jan 25, 2016 at 5:30
  • \$\begingroup\$ It could be either voltage or current (but you'd use a different normalization to get a correct power value in each case). If you have a graph of the voltage waveform then you could use that as your x(t). \$\endgroup\$
    – The Photon
    Jan 25, 2016 at 5:37
  • \$\begingroup\$ The formula for a trapezoidal waveform is usually just written as a piecewise linear function. Different linear formulas (some of which are just constant values) for different ranges of \$t\$. \$\endgroup\$
    – The Photon
    Jan 25, 2016 at 5:43

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