Electricity/Alternating current

Engineering

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Electricity provides a sinusoidal time varying voltage over time

Symbol

o---[~]---o

Mathematical Formula

$v(t)=VSin\omega t$

Alternating Current and conductor

Resistors

Voltage, Current, Power

Voltage

$v(t)=i(t)X_{R}$

Current

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

Power

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

Reactance, Impedance

Reactance

$X_{R}={\frac {v(t)}{i(t)}}=0$

Impedance

$Z_{R}=R+X_{R}=R\angle 0=R=R$

Magnetic Field, Inductance

Magnetic Field intensity

$B=Li={\frac {\mu }{2\pi r}}i$

Inductance

$L={\frac {B}{i}}={\frac {\mu }{2\pi r}}$

Heat inside and outside of resistor

Heat generated inside resistor

$W_{i}=i^{2}R(T)$

$R(T)=R_{o}+nT$

$R(T)=R_{o}e^{nT}$

Heat radiated outside resistor into the srrounding

$W_{e}=pv=mC\Delta T$

Capacitors

Voltage, Current, Power

Voltage

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

Current

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

Power

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

Reactance, Impedance, Phase angle different, Time constant

Reactance

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

Impedance

$Z_{C}={\frac {v_{C}(t)}{i_{C}(t)}}=R_{C}+X_{C}$

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

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

Charge, Capacitance, Electric field

Charge on plate

$Q=CV$

Capacitance of Capacitor

$C={\frac {Q}{V}}=\epsilon {\frac {A}{l}}$

Electric field between plates of capacitor

$E={\frac {V}{l}}={\frac {Q}{Cl}}$

Heat inside and outside capacitor

Heat energy generated inside capacitor

$W_{i}=\int QdV=\int CVdv={\frac {1}{2}}CV^{2}$

Inductors

Voltage, Current, Power

Voltage

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

Current

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

Power

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

Reactance, Impedance, Phase angle different, Time constant

Reactance

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

Impedance

$Z_{L}={\frac {v_{L}(t)}{i_{L}(t)}}=R_{L}+X_{L}$

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

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

Magentic field, Inducatance, Phase angle different, Time constant

$B=Li={\frac {N\mu }{l}}i$

$L={\frac {B}{i}}={\frac {N\mu }{l}}$

Heat inside and outside of inductor

$W_{i}=\int Bdi=\int Lidi={\frac {1}{2}}Li^{2}$

$W_{e}=pv=pC=hf$

See Also

• Capacitor
• Maxwell's Equations