Electricity/Alternating current

Subject classification: this is an engineering resource.

Educational level: this is a secondary education resource.

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

Electricity provides a sinusoidal time varying voltage over time

o---[~]---o

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

Voltage

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

Current

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

Power

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

Reactance

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

Impedance

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

Magnetic Field intensity

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

Inductance

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

Heat generated inside resistor

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

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

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

Heat radiated outside resistor into the srrounding

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

Voltage

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

Current

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

Power

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

Reactance

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

Impedance

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

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

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

Charge on plate

${\displaystyle Q=CV}$

Capacitance of Capacitor

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

Electric field between plates of capacitor

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

Heat energy generated inside capacitor

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

Voltage

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

Current

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

Power

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

Reactance

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

Impedance

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

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

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

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

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

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

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

• Capacitor
• Maxwell's Equations