Inductor

Inductor is a electric component that can store magnetic field energy

Construction

Capacitor is made from a straight line conductor of several circular turns

${\displaystyle L=N\mu {\frac {l}{A}}}$

Magnetic Intensity Strength

Measurement of magnetic strength

${\displaystyle B=LI}$

Current

${\displaystyle I={\frac {B}{L}}}$

Inductance

Inductance has a symbol L measured in Henry unit H

${\displaystyle L={\frac {B}{I}}}$

For a staright line conductor

${\displaystyle L={\frac {\mu }{2\pi r}}}$

For a circular loop

${\displaystyle L={\frac {\mu }{2r}}}$

For a coil of N circular loops conductor

${\displaystyle L={\frac {N\mu }{l}}}$

Voltage

${\displaystyle v_{L}=L{\frac {di_{L}}{dt}}}$

Current

${\displaystyle i_{L}={\frac {1}{L}}\int {dv_{L}}{dt}}$

Reactance

Resistance to the AC current flow

${\displaystyle X_{C}={\frac {v_{L}}{i_{L}}}={\frac {L{\frac {di_{L}}{dt}}}{{\frac {1}{L}}\int {dv_{L}}{dt}}}}$

In frequency domain

${\displaystyle X_{L}(j\omega )=j\omega L}$

In phasor domain

${\displaystyle X_{L}(\omega \theta )=\omega L\angle 90}$

Impedance

Resistance to the AC current flow

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

In frequency domain

${\displaystyle Z_{L}=R_{L}+j\omega L}$

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

In phasor domain

${\displaystyle Z_{L}=R_{L}\angle 0+j\omega L\angle 90}$

${\displaystyle Z_{L}={\sqrt {R_{L}^{2}+(j\omega L)^{2}}}\angle \omega T}$

Time Constant

${\displaystyle T={\frac {L}{R_{L}}}}$

Power

Power of the capacitor

${\displaystyle p={\frac {1}{2}}Li^{2}}$

Inductors in Series

For n inductor connected adjacent to each other as shown

The total resistance

${\displaystyle L_{t}=L_{1}+L_{2}+...+L_{n}}$

For 2 series resistor of same value

${\displaystyle L_{t}=L_{1}+L_{2}=L+L=2L}$

Inductors in parallel

For n inductors connected facing each other as shown

The total inductance

${\displaystyle {\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+...+{\frac {1}{L_{n}}}}$

For 2 parallel resistor of same value

${\displaystyle {\frac {1}{L_{t}}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}={\frac {L_{1}+L_{2}}{L_{1}L_{2}}}={\frac {L+L}{LL}}={\frac {2L}{LL}}={\frac {2}{L}}}$

${\displaystyle L_{t}={\frac {1}{2}}L}$