# Electricity/Alternating current

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 Educational level: this is a secondary education resource.
 Educational level: this is a tertiary (university) resource.

## AC (Alternating Current)

Electricity provides a sinusoidal time varying voltage over time

${\displaystyle v(t)=VSin\omega t}$

o---[~]---o

## Alternating Current and conductor

### Resistors

Voltage

${\displaystyle v(t)=i(t)Z_{R}}$

Current

${\displaystyle i(t)={\frac {v(t)}{Z_{R}}}}$

Power

${\displaystyle p(t)=i(t)v(t)}$

Impedance

${\displaystyle Z_{R}={\frac {v(t)}{i(t)}}=R+X_{R}=R}$

Reactance

${\displaystyle X_{R}=0}$

### Capacitors

Voltage

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

Current

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

Power

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

Impedance

${\displaystyle Z_{C}={\frac {v_{C}(t)}{i_{C}(t)}}=R_{C}+X_{C}}$
${\displaystyle Z_{C}=R+{\frac {1}{\omega C}}\angle -90^{o}=R+{\frac {1}{j\omega C}}=R+{\frac {1}{sC}}}$

Reactance

${\displaystyle X_{C}={\frac {1}{\omega C}}\angle -90^{o}={\frac {1}{j\omega C}}={\frac {1}{sC}}}$

Phase angle difference

${\displaystyle Tan\theta =\omega T}$

Time constant

${\displaystyle T=RC}$
${\displaystyle X_{R}=0}$

Frequency respond

Low frequency . ${\displaystyle \omega =0}$  , ${\displaystyle X_{C}={\frac {1}{\omega C}}=00}$  . Capacitor open circuit
High frequency. ${\displaystyle \omega =00}$  , ${\displaystyle X_{C}={\frac {1}{\omega C}}=0}$  . Capacitor short circuit

### Inductors

Voltage

${\displaystyle v(t)=L{\frac {d}{dt}}i(t)}$

Current

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

Power

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

Impedance

${\displaystyle Z_{L}={\frac {v_{L}(t)}{i_{L}(t)}}=R_{L}+X_{L}}$
${\displaystyle Z_{L}=R+\omega L\angle 90^{o}=R+j\omega L=R+sL}$

Reactance

${\displaystyle X_{L}=\omega L\angle 90^{o}=j\omega L=sL}$

Phase angle difference

${\displaystyle Tan\theta =\omega T}$

Time constant

${\displaystyle T={\frac {L}{R}}}$

Frequency respond

Low frequency . ${\displaystyle \omega =0}$  , ${\displaystyle X_{L}=\omega L=0}$  . Inductor shorts circuit
High frequency. ${\displaystyle \omega =00}$  , ${\displaystyle X_{L}=\omega L=00}$  . Inductor opens circuit