phasors

Impedance and Phasors:

AC, or alternating current, power is generally expressed as the following:

v(t) = V_m * cos( ωt + θ )
Where V_m is the max voltage, ω is the angular frequency, and θ is the phase angle.

Components in AC power can be simplified by converting them into phasor form, then combining them. The conversion to phasor form is as follows:

Resistors:
No conversion, remains as is. (Ω)

Inductors:
j*ω*L (in Ω) (j represents the imaginary number, like i)

Capacitors:
- 1 / (j*ω*C) (in Ω) (j represents the imaginary number, like i)

For capacitors, some writings do not include the negative, but add it in afterwards. However, capacitors are inverse of inductors for sure, so that's why there's a negative there.

All the previous rules, such as source transformations, node-voltage analysis, mesh-current analysis, etc. are all applicible in phasor form. Phasors combine exactly like resistors, as all of the components are in Ω. This means that when a question asks to find the equivalent impedance or something, you'd convert all inductors and capacitors into phasor form, then combine it.

Inductors and capacitors in phasor form are generally imaginary numbers. This means that we have to be employing complex numbers. Complex numbers have multiple formats:

a + bi (Rectangular form)

V_m |_ θ (Polar form)

To combine them easily, check this out:
http://www.tc3.edu/instruct/sbrown/ti83/complx83.htm

There's also the function ">Rect" or ">Polar" inside the Math menu, which will convert your answer to what you need.

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Complex Power:

Complex power is a combination of Average/Real Power and Reactive Power. It is expressed as the following:

S = P + jQ

Where P is the average/real power, and Q is the reactive power.
P = (V_m*I_m/2)*cos(θ_v - θ_i)
Q = (V_m*I_m/2)*sin(θ_v - θ_i)

Then this is a matter of plugging variables in the equation. Generally the variables are given to you, in the form of v(t) = V_m * cos( ωt + θ ) or something to that degree.