# Electricity/Alternating current

## AC (Alternating Current)

Electricity provides a sinusoidal time varying voltage over time

$v(t)=VSin\omega t$ o---[~]---o

## Alternating Current and conductor

### 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 Voltage

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

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

$p(t)={\frac {1}{2}}Cv^{2}(t)$ 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

$p(t)={\frac {1}{2}}Li^{2}(t)$ 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