Why are complex numbers so important in the study of AC circuits?

An AC voltage or AC current contains more than one dimension.





DC voltage or current has one dimension -- 100 Volts is 100 volts DC; 10 Amps is 10 Amps DC -- that's it, but with AC, you need to know:


- the voltage,


- the frequency


- the phase


- the waveform


That's 4 different things -- 4 dimensions.





Even if you limit the AC waveform to sine-waves, you still have phase and frequency, besides voltage (or current). Furthermore, if you limit the frequency (to say, 60 Hz), you still have a 2-dimensional number to work with: voltage and phase.





Complex numbers are 2-dimensional, so it makes sense that something with 2 dimensions would need 2-dimensional numbers to make calculations possible.





If you limit an AC waveform to be a sine-type wave and fix the frequency, the 2-dimensional complex number that is used is called a ';phasor';.





.Why are complex numbers so important in the study of AC circuits?
complex numbers are needed to describe relative phase of two signals. Why care about phase? If the current and voltage are exactly 90 degrees apart, the power dissipation is zero. If they are 0 degrees apart, the power dissipation is maximized.





If you care about power dissipation in ac circuits (motors, light bulbs, speakers) then you need to understand how to use complex numbers!Why are complex numbers so important in the study of AC circuits?
Beause It can tell us how much is the combination of indutance and Capacity and how much is the resister.

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