Current & Voltage vs. Time and Power Calculation

steveb

Joined Jul 3, 2008
2,436
Do you know how to display an integral sign with limits in tex? I can't find a description of how to include the limits.
I just use the superscript and subscript operators, "^" and "_"; as follows.

\( \int^{\;\;\;\; t_2 }_{t_1} f(t)\; dt \)

If there is another way, I don't know about it.
 
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That's what I was afraid of. There doesn't seem to be a version of the integral sign with built-in limits.

I guess the super- sub-script method will have to do, even though the superscript needs to be a little to the right.
 
The integration of a function squared is not equal to the square of the integral of that same function: at least, not in general.

i.e. \( \int f^2(t)\;dt \ne \Biggl(\int f(t)\; dt\Biggr)^2 \)
It occurred to me that the condition for equality follows from Parseval's theorem:

http://mathworld.wolfram.com/ParsevalsTheorem.html

Parseval's theorem as applied to electrical engineering says that the energy in a signal (a voltage waveform for example) is equal to the sum of the energies of the harmonics (taking the DC component to be a harmonic of zero order).

The left hand side of the inequality above is proportional to the total energy in the waveform. The right hand side is proportional to the energy of the DC component.

The only way they can be equal is if there are no other harmonics beyond the DC component. That is, equality only holds for DC; otherwise the two expressions are unequal.
 

Thread Starter

Management

Joined Sep 18, 2007
306
You guys have been a great help.

So basically when in doubt do it the integral way by multiplying instantaneous voltages and currents and integrating. Using the second method is just a special case.

Thanks a lot!
 
Wish I'd had this thread some years ago when I was doing my first PSPICE dissipation sims - it can get horribly confusing.

You have to interpret the answer with a bit of intelligence and applied experience, as strictly speaking it's the rate of energy flow that's being given, not necessarily power consumed or dissipated. For instance, take the favoured equation and apply it to a light bulb. Some of the answer will be power that goes into heat, some power is optical, maybe a tiny amount makes the bulb vibrate, and a few more joules/sec leak away as radio energy, and so on.

Here's an interesting one: Put an AC signal across a perfect capacitor, and work out the power of that. We all know it's zero, but PSPICE doesn't know that and it will give a definite non-zero answer. What's being measured here is the energy per second flowing backwards and forwards, even though no heat is generated and not one solitary electron crosses from one plate to the other.

Don't be tempted to apply any averaging or r.m.s. techniques*, they're meaningless when applied to power, and make as much sense as using seconds r.m.s. You can look upon the integral(V*I)/t technique as giving a rolling average of sorts, so the choice of sampling window should be judiciously applied.

* Manufacturers of cheap hi-fi and speakers can take a running jump. Normally what they mean is the power derived from an r.m.s. voltage into a constant load. A sine wave at best, or a square wave if they're being particularly cheeky.
 

beenthere

Joined Apr 20, 2004
15,819
* Manufacturers of cheap hi-fi and speakers can take a running jump. Normally what they mean is the power derived from an r.m.s. voltage into a constant load. A sine wave at best, or a square wave if they're being particularly cheeky.
It was worse when the measurement was in "music power". That was the ability of the amp to pass one period of a 1 KHz sine wave without clipping.
 
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