Electricity/Electric circuit

Electric circuit

Electric components are connected in a closed loop to form an electric circuit

Electric circuit's Laws

Kirchhoff's Voltage Law	The algebraic sum of the voltages around a closed circuit path must be zero.
Kirchhoff's Current Law	The sum of the currents entering a particular point must be zero.
Ohm's law	The current through a conductor between two points is directly proportional to the potential difference across the two points.
Watt's law	The power through a conductor between two points is directly proportional to the potential difference and its current across the two points.

Electric circuit's configuration

Series circuit	components are connected in adjacent to each other
Parallel circuit
2 port network

RL circuit

RL series

$v_{L}+v_{R}=0$

$L{\frac {d}{dt}}i+iR=0$

${\frac {d}{dt}}i=-{\frac {1}{T}}i$

$i=Ae^{-{\frac {t}{T}}}$

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

2 port LR

${\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega L}}={\frac {1}{1+j\omega T}}$

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

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=v_{i}$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=0$

2 port RL

${\frac {v_{o}}{v_{i}}}={\frac {j\omega L}{R+j\omega L}}={\frac {j\omega T}{R+j\omega T}}$

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

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=0$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=v_{i}$

RC circuit

RC series

$v_{C}+v_{R}=0$

$C{\frac {d}{dt}}v+{\frac {v}{R}}=0$

${\frac {d}{dt}}v=-{\frac {1}{T}}v$

$v=Ae^{-{\frac {t}{T}}}$

$T=RC$

2 port RC

${\frac {v_{o}}{v_{i}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}$

$T=RC$

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=v_{i}$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=0$

2 port CR

${\frac {v_{o}}{v_{i}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega T}{R+j\omega T}}$

$T=RC$

$\omega _{o}={\frac {1}{T}}$

$\omega =0$ . $v_{o}=0$

$\omega _{o}=\omega _{o}$ . $v_{o}={\frac {v_{i}}{2}}$

$\omega _{o}=00$ . $v_{o}=v_{i}$

LC circuit

Circuit at equilibrium

$v_{L}+v_{C}=0$

$L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt=0$

${\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i$

$i=Ae^{\pm j{\sqrt {\frac {1}{T}}}t}=Ae^{\pm j\omega t}=ASin\omega t$

$\omega ={\sqrt {\frac {1}{T}}}$

$T=LC$

Circuit at resonance

$Z_{L}+Z_{C}=0$

$j\omega L+{\frac {1}{j\omega C}}=0$

$\omega =\pm {\sqrt {\frac {1}{T}}}$

$T=LC$

$V_{L}+V_{C}=0$

$v(\theta )=ASin(\omega +2\pi )-ASin(\omega -2\pi )$

RLC circuit

RLC series

$v_{L}+v_{C}+v_{R}=0$

$L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt+iR=0$

${\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i$

$i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=Ae^{-\alpha t}e^{\pm j\omega t}=A(\alpha )Sin\omega t$

$\omega ={\sqrt {\beta -\alpha }}$

$A(\alpha )=Ae^{-\alpha t}$

$\beta ={\frac {1}{T}}={\frac {1}{LC}}$

$\alpha =\beta \gamma ={\frac {R}{2L}}$

$T=LC$

$\gamma =RC$