Can Euler's Formula Connect Sine Waves and Phasors in AC Circuits?

[Calpalned]

Homework Statement

Prove that current ##I = I_0\sin{\omega t}## can be rewritten as ##I = I_0 e^{i\omega t}##

Homework Equations

Euler's formula ##re^{i\theta} = r(\cos{\theta}+i\sin{\theta})##

The Attempt at a Solution

If ##r = I_0## and ##\theta = \omega t ## then ##re^{i\theta} = r(\cos{\theta}+i\sin{\theta}) ## will be ##I = I_0e^{i\omega t} = I_0(\cos{\omega t}+i\sin{\omega t})##. The left hand side ##I = I_0 e^{i\omega t}## is correct, but the right hand side is not equal to ##I = I_0\sin{\omega t}##
Thank you

 

[gneill]

I think what you're looking at is the introduction of phasors to your toolkit. As you have spotted, e^iωt is not formally congruent to sinωt. However, if one considers a phasor to be a rotating vector (a vector that rotates in direction around the origin with a constant angular frequency ω), then a projection of that vector on the imaginary axis does indeed follow sinωt. Similarly, its projection on the real axis follows cosωt. Now, e^iωt can be considered to be a rotating vector, as t is a time variable. So, for example,##I_0 e^{i\omega t}## would be a phasor of magnitude ##I_0## rotating at a rate of ##\omega## radians per second.

Typically (but not always) all the phasors in a given circuit have the same angular frequency, and the convention is to drop the common ##e^{i \omega t}## from the notation. Declare that ##I## is a phasor and the ##e^{i \omega t}## is implied.

Does that help?