Electricity/Alternating current

Engineering

AC (/Alternating Current/)

Electricity provides a sinusoidal time varying voltage over time

$v(t)=VSin\omega t$

Symbol

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

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

$Z_C=\frac{v_C(t)}{i_C(t)}=R_C+X_C$

$Z_C=R+\frac{1}{\omega C}\mathrm{\angle }-90^o=R+\frac{1}{j\omega C}=R+\frac{1}{sC}$

Reactance

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

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

$Z_L=\frac{v_L(t)}{i_L(t)}=R_L+X_L$

$Z_L=R+\omega L\mathrm{\angle }90^o=R+j\omega L=R+sL$

Reactance

$X_L=\omega L\mathrm{\angle }90^o=j\omega L=sL$

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

See Also

• Capacitor
• Maxwell's Equations