First Principles AI

Solution

Given:

V=120\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right),\quad Z=\frac{1+i\sqrt{3}}{2}

Using Ohm's law $V=IZ$ :

I=\frac{V}{Z}

First write $Z$ in polar form:

Z=\frac{1}{2}+\frac{\sqrt{3}}{2}i

|Z|=\sqrt{\left(\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}=1,\quad \arg(Z)=\tan^{-1}(\sqrt{3})=\frac{\pi}{3}

So:

Z=\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}

Now divide in polar form:

\begin{aligned} I &=\frac{120}{1}\left[\cos\left(\frac{\pi}{4}-\frac{\pi}{3}\right)+i\sin\left(\frac{\pi}{4}-\frac{\pi}{3}\right)\right]\\ &=120\left[\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right] \end{aligned}

\boxed{I=120\left(\cos\left(-\frac{\pi}{12}\right)+i\sin\left(-\frac{\pi}{12}\right)\right)}

Equivalent form:

\boxed{I=120\left(\cos\frac{\pi}{12}-i\sin\frac{\pi}{12}\right)}