# 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

${\displaystyle v_{L}+v_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i+iR=0}$
${\displaystyle {\frac {d}{dt}}i=-{\frac {1}{T}}i}$
${\displaystyle i=Ae^{-{\frac {t}{T}}}}$
${\displaystyle T={\frac {L}{R}}}$

### 2 port LR

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+j\omega L}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$ . ${\displaystyle v_{o}=v_{i}}$
${\displaystyle \omega _{o}=\omega _{o}}$ . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$ . ${\displaystyle v_{o}=0}$

### 2 port RL

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {j\omega L}{R+j\omega L}}={\frac {j\omega T}{R+j\omega T}}}$
${\displaystyle T={\frac {L}{R}}}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$ . ${\displaystyle v_{o}=0}$
${\displaystyle \omega _{o}=\omega _{o}}$ . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$ . ${\displaystyle v_{o}=v_{i}}$

## RC circuit

### RC series

${\displaystyle v_{C}+v_{R}=0}$
${\displaystyle C{\frac {d}{dt}}v+{\frac {v}{R}}=0}$
${\displaystyle {\frac {d}{dt}}v=-{\frac {1}{T}}v}$
${\displaystyle v=Ae^{-{\frac {t}{T}}}}$
${\displaystyle T=RC}$

### 2 port RC

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{1+j\omega T}}}$
${\displaystyle T=RC}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$ . ${\displaystyle v_{o}=v_{i}}$
${\displaystyle \omega _{o}=\omega _{o}}$ . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$ . ${\displaystyle v_{o}=0}$

### 2 port CR

${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega T}{R+j\omega T}}}$
${\displaystyle T=RC}$
${\displaystyle \omega _{o}={\frac {1}{T}}}$
${\displaystyle \omega =0}$ . ${\displaystyle v_{o}=0}$
${\displaystyle \omega _{o}=\omega _{o}}$ . ${\displaystyle v_{o}={\frac {v_{i}}{2}}}$
${\displaystyle \omega _{o}=00}$ . ${\displaystyle v_{o}=v_{i}}$

## LC circuit

Circuit at equilibrium

${\displaystyle v_{L}+v_{C}=0}$
${\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i}$
${\displaystyle i=Ae^{\pm j{\sqrt {\frac {1}{T}}}t}=Ae^{\pm j\omega t}=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$

Circuit at resonance

${\displaystyle Z_{L}+Z_{C}=0}$
${\displaystyle j\omega L+{\frac {1}{j\omega C}}=0}$
${\displaystyle \omega =\pm {\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$
${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle v(\theta )=ASin(\omega +2\pi )-ASin(\omega -2\pi )}$

## RLC circuit

### RLC series

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