Alternating Current

We invented AC power because they’re much easier and economical to implement than DC power.

title:Personal thoughts, making connections
I am pretty sure this is right, but I have not checked to confirm. Essentially, AC is created from the basics of [[Faraday's Law]]. 
 
We can rotate loops, make it move in a circle under a constant magnetic field (thanks to [[Magnetic Material|Permanent Magnet]]s). The loop experiences a changing magnetic field. This results an induced current that looks like a [[Simple Harmonic Motion|Sinusoidal function]], thus the invention of AC. 
 
See [[Motor]] as there are similar ideas.

12-15-2022: I had this conversation with Chester about electrocution, because my heater keeps

AC Applied To Resistors

For the resistor of an AC Circuit, there is no lag between the current and voltage (they are in-phase).

$i(t)=I_m\cos (\omega t+{\theta }_i)$ $v(t)=I_{m}R\cos (\omega t+{\theta }_i)$ We can write this in Phasor domain as $\tilde{I}=I_m{e}^{j{\theta }_i}$ $\tilde{V}=I_{m}R{e}^{j{\theta }_i}$ The relationship is given by $\tilde{V}=R\tilde{I}$

We say that the Phasor $\tilde{V}$ and $\tilde{I}$ are in phase since ${\theta }_v={\theta }_i$ →The current and voltage oscillate in sync.

AC Applied to Inductor

For the inductor of an AC Circuit, voltage leads and current follows. $i(t)=I_m\cos (\omega t+{\theta }_i)$ $v(t)=\omega L{I}_m\cos (\omega t+{\theta }_i+\frac{\pi }{2})$

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Notice that they are not in phase anymore! This is because we derived $v(t)$ from $v(t) = L \frac{di(t)}{dt}$, see [[Inductor]].

We can write this in the Phasor domain as $\tilde{V}=j\omega L\tilde{I}$

$\tilde{V}$ leads $\tilde{I}$ by $\frac{\pi }{2}=90°$ (voltage leads current)

AC Applied to Capacitor

In capacitor, current leads and voltage follows.

$v(t)=V_m\cos (\omega t+{\theta }_i)$ $i(t)=\omega C{V}_m\cos (\omega t+{\theta }_i+\frac{\pi }{2})$

title:
Notice that they are not in phase anymore! This is because we derived $i(t)$ from $i(t) = C \frac{dv(t)}{dt}$, see [[Capacitor]].

We can write this in the Phasor domain as $\tilde{V}=\frac{\tilde{I}}{j\omega C}$

$\tilde{I}$ leads $\tilde{V}$ by $\frac{\pi }{2}=90°$ (current leads voltage)

AC Frequency, Personal aha moment on Polarity!

In North America, its 60 Hz so it flips back and forth 60 times a second. aha moment! this is why we can plug both ways in the wall socket (Polarity does not matter), cuz its using AC, but DC u gotta be super careful with polarity. See Electrical Outlet