Electricity/Electric circuit

Electric circuit

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

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.

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

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}}

{\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

{\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}

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

{\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

{\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}

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 )

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

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