# Equation for the average power of a signal

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 $\tau_r$ = $\tau_f$ :

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.

• Maybe I'm missing something, but what's the actual waveform? Is it a square wave? Is it all above zero or symmetrical? Jan 24, 2016 at 16:20
• I've added a diagram of the waveform in my OP. Jan 25, 2016 at 5:29
• 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. Jan 25, 2016 at 17:48
• How do I get the specific formula from the trapezoidal waveform? Jan 26, 2016 at 5:29

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.

• I don't know what $x(t)$ is in this case. Is it the trapezoidal shape? Is there a formula for that? Jan 25, 2016 at 5:30
• 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). Jan 25, 2016 at 5:37
• 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 $t$. Jan 25, 2016 at 5:43