# 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