RFICDUDE
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I would just like to add a little to Aaron's response:
Power is useful when comparing signals because voltage and current in a circuit often depend on impedance levels depending on where you are measuring them, but power takes impedance into consideration.
Controlled impedance systems: In a "system" there may be a variety of different impedances or there may be "controlled" impedances (such as 50 or 75 ohm systems). If the impedance is "controlled" or defined then it is easier to compare signals (desired, interfering, distortion, noise, etc.) in power because all signals are reffered to defined load/source impedances. And as Aaron pointed out, a component (filter, amplifier, mixer, etc) designed for a specific load/source impedance may not work properly if it is connected to a arbitrary impedance (filter response may be off, insertion loss maybe high, or an amplifier may oscillate). Controlled impedances are also very important if the signal must be transfered through a transmission line over a distance that is comparable to the wavelength of the highest frequency of interest in the system. Impedance mismatches on transmission lines can cause large peaks in voltage or current at points along the line that could damage active components (exceed rated values).
Integrated circuits: Now, all that said, things are little different in the integrated circuit design world. Integrated circuit design deals mainly with transistors, resistors, caps and a few inductors with very close connections. So, we are not as concerned about stading waves between circuit connections on chip (for frequencies below 5-6GHz). With integrated circuits it is easier to talk about signals as voltage or current mode with voltage mode implying current sources (transistors) driving high resistance loads while current mode implies current sources driving low impedance transistor loads (this may be too loose of a description). It is no longer very meaningful to talk about load and source power because the resitive impedances change widely, but it is still meaningful to compare signals using a 20*log10(V) or 20*log10(I) scales as if we are comparing relative power at different points in the circuit. I am getting a bit fuzzy here because the justification for using voltage or current for comparisons is dictated by the way we use signals and what is meaningful at different parts of the system.
Power Applications: Power definitions and considerations are most clear when the job of the circuit is to deliver a specific amount of power to a well defined load (antenna, speaker, motor, LED, backplane, loaded cable, etc.). For power transmission the impedance and power levels are defined, so large signal voltage and current signal levels must be considered. Some applications are efficiency sensitive while others may have signal to noise and distortion as a more stringent consideration.
I probably should stop rambling ...
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