# Fundamental Physics/Electronics/Capacitors

## Capacitor

Capacitor is a electric component that can store electric field energy . Capacitor is made from 2 parallel plates separated by an insulator

### Capacitance

Capacitance is a characteristic of capacitor which indicates capacitor's capability to store energy under Electric field

${\displaystyle C=\epsilon {\frac {A}{l}}}$

## Capacitor in DC Circuit

### Capacitance

Capacitance has a symbol C measured in Farat unit F

${\displaystyle C={\frac {V}{Q}}}$

### Charge

${\displaystyle Q={\frac {V}{C}}}$

### Current

${\displaystyle I={\frac {Q}{t}}}$

### Voltage

${\displaystyle V={\frac {W}{Q}}}$

### Power

${\displaystyle P_{C}=IV={\frac {Q}{t}}{\frac {W}{Q}}={\frac {W}{t}}}$

## Capacitor in AC Circuit

### Voltage

${\displaystyle v_{C}={\frac {1}{C}}\int idt}$

### Current

${\displaystyle i_{C}=C{\frac {dv}{dt}}}$

### Reactance

Resistance to the AC current flow

In time domain

${\displaystyle X_{C}={\frac {v_{C}}{i_{C}}}={\frac {{\frac {1}{C}}\int idt}{C{\frac {dv}{dt}}}}}$

In frequency domain

${\displaystyle X_{C}(j\omega )={\frac {1}{j\omega C}}}$

In phasor domain

${\displaystyle X_{C}(\omega \theta )={\frac {1}{\omega C}}\angle -90}$

### Impedance

Resistance to the AC current flow

${\displaystyle Z_{C}=R_{C}+X_{C}}$

In frequency domain

${\displaystyle Z_{C}=R_{C}+{\frac {1}{j\omega C}}}$
${\displaystyle Z_{C}={\frac {j\omega T+1}{j\omega C}}}$

In phasor domain

${\displaystyle Z_{C}=R_{C}\angle 0+{\frac {1}{\omega C}}\angle -90}$
${\displaystyle Z_{C}={\sqrt {R_{C}^{2}+({\frac {1}{\omega T}})^{2}}}\angle {\frac {1}{\omega T}}}$

### Time Constant

${\displaystyle T=CR_{C}}$

### Power

Power of the capacitor

${\displaystyle p={\frac {1}{2}}Cv_{C}^{2}}$

## Capacitor Configuration

### Capacitor in Series

For n capacitors connected adjacent to each other as shown

The total resistance

${\displaystyle {\frac {1}{CR_{1}}}+{\frac {1}{C_{2}}}+...+{\frac {1}{C_{n}}}}$

For 2 series capacitor of same value

${\displaystyle {\frac {1}{C_{t}}}={\frac {C}{R_{1}}}+{\frac {1}{C_{2}}}={\frac {C_{1}+C_{2}}{C_{1}C_{2}}}}$
${\displaystyle C_{t}={\frac {C_{1}C_{2}}{C_{1}+C_{2}}}={\frac {c^{2}}{2C}}={\frac {1}{2}}C}$

### Capacitors in parallel

For n capacitors connected facing each other as shown

The total resistance

${\displaystyle C_{1}+C_{2}+...+C_{n}}$

For 2 parallel capacitor of same value

${\displaystyle {\frac {1}{C_{t}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}={\frac {C_{1}+C_{2}}{C_{1}C_{2}}}={\frac {C+C}{CC}}={\frac {2}{C}}}$
${\displaystyle C_{t}={\frac {1}{2}}C}$