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Can anyone pls explain how complex numbers are used in alternating current,I was reading about rl circuit and found out they changed voltage $$V \sin \omega t \tag{1}$$ to $V e^{i\omega t}$. as we know Euler equation was $$ e^{i\theta} = \cos \theta +i \sin \theta \tag{2} $$ So how can we convert $\sin \omega t$ to $e^{i \omega t}$? As we don't have $i$ in equation (1) and also how complex numbers work for waves? what do the imaginary and real parts symbolize for the wave?

GiorgioP-DoomsdayClockIsAt-89
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Xyz
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This is really just a mathematically convenience.

A lot of circuit analysis tasks are a lot easier to do with complex sine waves and complex impedances. Instead of slogging through the analysis with real sine waves and time-domain differential equations you simply add an imaginary part, do the math in complex numbers. When you are done you get the real solution simply by throwing away the imaginary part of the complex solutions.

So how can we convert $\sin \omega t$ to $e^{i \omega t}$?

By simple convention. Any real signal $x(t) = cos(\omega t + \varphi)$ can be represented as $x(t) = \Re[{e^{i\omega t + \varphi}}]$.

what do the imaginary and real parts symbolize for the wave?

The imaginary part has no physical meaning. It's simply a mathematical convenience. You discard it once you are done with the math.

Hilmar
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