Electricity/Alternating current

AC (/Alternating Current/)
Electricity provides a sinusoidal time varying voltage over time
 * $$v(t)=V Sin \omega t$$

Symbol

 * o---[~]---o

Resistors
Voltage
 * $$v(t) = i(t) Z_R$$

Current
 * $$i(t) = \frac{v(t)}{Z_R} $$

Power
 * $$p(t) = i(t) v(t)$$

Impedance
 * $$Z_R = \frac{v(t)}{i(t)} = R +X_R = R$$

Reactance
 * $$X_R = 0$$

Capacitors

 * [[Image:Capacitortheorywithuppercaseextension.JPG|500 px|An idealized capacitor.]]

Voltage
 * $$v(t) = \frac{1}{C} \int i(t)$$

Current
 * $$i(t) = C \frac{d}{dt} v(t) $$

Power
 * $$p(t) = \frac{1}{2} C v^2(t)$$

Impedance
 * $$Z_C = \frac{v_C(t)}{i_C(t)} = R_C +X_C $$
 * $$Z_C = R + \frac{1}{\omega C} \angle -90^o = R + \frac{1}{j \omega C} = R + \frac{1}{sC}$$

Reactance
 * $$X_C =  \frac{1}{\omega C} \angle -90^o =  \frac{1}{j \omega C} = \frac{1}{sC}$$

Phase angle difference
 * $$Tan \theta = \omega T$$

Time constant
 * $$T =RC$$
 * $$X_R = 0$$

Frequency respond
 * Low frequency . $$\omega=0$$, $$X_C=\frac{1}{\omega C} =00$$ . Capacitor open circuit
 * High frequency. $$\omega=00$$, $$X_C=\frac{1}{\omega C} =0$$ . Capacitor short circuit

Inductors
Voltage
 * $$v(t) = L \frac{d}{dt} i(t)$$

Current
 * $$i(t) = \frac{1}{L} \int v(t) dt $$

Power
 * $$p(t) = \frac{1}{2} L i^2(t)$$

Impedance
 * $$Z_L = \frac{v_L(t)}{i_L(t)} = R_L+X_L $$
 * $$Z_L = R+\omega L \angle 90^o = R + j \omega L = R + sL$$

Reactance
 * $$X_L =  \omega L \angle 90^o =  j \omega L =  sL$$

Phase angle difference
 * $$Tan \theta = \omega T$$

Time constant
 * $$T =\frac{L}{R}$$

Frequency respond
 * Low frequency . $$\omega=0$$, $$X_L=\omega L =0$$ . Inductor shorts circuit
 * High frequency. $$\omega=00$$, $$X_L=\omega L =00$$ . Inductor opens circuit