# 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

$v_{L}+v_{C}+v_{R}=0$ $L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt+iR=0$ ${\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i$ $i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=Ae^{-\alpha t}e^{\pm j\omega t}=A(\alpha )Sin\omega t$ $\omega ={\sqrt {\beta -\alpha }}$ $A(\alpha )=Ae^{-\alpha t}$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\beta \gamma ={\frac {R}{2L}}$ $T=LC$ $\gamma =RC$ 