# 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 LawThe algebraic sum of the voltages around a closed circuit path must be zero. Kirchhoff's Current LawThe sum of the currents entering a particular point must be zero. Ohm's lawThe current through a conductor between two points is directly proportional to the potential difference across the two points. Watt's lawThe 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 circuitcomponents 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

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