Electricity/Electric circuit

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

Electric circuit's Laws

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 * 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. 
 * 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. 
 * 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. 
 * 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. 


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Electric circuit's configuration

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 * Series circuit || components are connected in adjacent to each other
 * Parallel circuit ||
 * 2 port network ||
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 * 2 port network ||
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RL series

 * $$v_L + v_R = 0$$
 * $$L \frac{d}{dt} i + iR = 0$$
 * $$\frac{d}{dt} i = - \frac{1}{T} i$$
 * $$ i = A e^{-\frac{t}{T}}$$
 * $$ T = \frac{L}{R}$$

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 series

 * $$v_C + v_R = 0$$
 * $$C \frac{d}{dt} v + \frac{v}{R} = 0$$
 * $$\frac{d}{dt} v = - \frac{1}{T} v$$
 * $$ v = A e^{-\frac{t}{T}}$$
 * $$ T = RC$$

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
 * $$v_L + v_C = 0$$
 * $$L \frac{d}{dt} i + \frac{1}{C} \int i dt= 0$$
 * $$\frac{d^2}{dt^2} i = - \frac{1}{T} i$$
 * $$ i = A e^{\pm j \sqrt{\frac{1}{T}}t} = A e^{\pm j \omega t} = A Sin \omega t$$
 * $$ \omega = \sqrt{\frac{1}{T}}$$
 * $$ T = LC$$

Circuit at resonance
 * $$Z_L + Z_C = 0$$
 * $$j \omega L + \frac{1}{j \omega C} = 0$$
 * $$\omega = \pm \sqrt{\frac{1}{T}}$$
 * $$T = LC$$
 * $$V_L + V_C = 0$$
 * $$v(\theta) =A Sin(\omega + 2 \pi) - A Sin(\omega - 2 \pi)$$

RLC series

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