Magnetic Field

When a voltage is applied across the inductor, current generates Electric Field . Change of Electric Field in the turns generates Magnetic Field perpendicular to Electric Field

${\displaystyle B=IL}$

Inductance

Inductance is the ability to generartes Magnetic Field B for a given Current

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

Voltage

${\displaystyle V_{L}=L{\frac {dI}{dt}}={\frac {dB}{dt}}}$

Current

${\displaystyle I_{L}={\frac {1}{L}}\int Vdt}$

Reactance

Reactance is defined as the ratio of Voltage over current

${\displaystyle X_{L}={\frac {L{\frac {dI}{dt}}}{I}}}$

${\displaystyle X_{L}=\omega L}$/_90

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

${\displaystyle X_{L}=sL}$

Impedance

Impedance is defined as the sum of Reactance and Resistance of Inductor . Since all conductor has Resistance

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

${\displaystyle Z_{L}={\sqrt {R_{L}^{2}+(\omega L)^{2}}}}$/_Tan^-1 ${\displaystyle \omega {\frac {L}{R}}}$

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

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

Frequency Respond

Inductor is a device depends on frequency ${\displaystyle \omega }$

• ${\displaystyle \omega =0,X_{L}=0}$, Inductor Closed circuit, I ≠ 0
• ${\displaystyle \omega =00,X_{L}=00}$, Capacitor Opened circuit, I = 0
• ${\displaystyle \omega ={\frac {R_{L}}{L}}}$

${\displaystyle X_{L}=R_{L}}$ ,

${\displaystyle Z_{L}=[R_{L}]^{\frac {1}{3}}}$ ,

${\displaystyle V_{L}={\frac {V}{2}}}$

${\displaystyle I_{L}={\frac {V}{2R_{L}}}}$

With the value of I at three frequency points ω = 0, 00 , 1 / CR_C I - f curve can be drawn to give a picture of current in the inductor over time

Phase Angle

When a Voltage is applied across inductor, current generates magnetic field. Change in current generate change in magnetic field which generate voltage across inductor. Therefore, current will lead voltage

For ideal losses inductor which has no internal resistance, Current will lead Voltage an angle 90 . For Non - Ideal inductor which has an internal resistance, Current will lead Voltage an angle θ

${\displaystyle \tan \theta =\omega {L}{R_{L}}=2\pi f{\frac {L}{R_{L}}}}$

Phase angle relates to time frequency or time and the value of R and L . When there is a change in phase angle Time and frequency also change

${\displaystyle f=(\tan \theta /2\pi ){\frac {R_{L}}{L}}}$

${\displaystyle t=(2\pi /\tan \theta ){\frac {L}{R_{L}}}}$

If choosing L = 1 and R = 10ⁿ then the formulas above become

${\displaystyle f=(\tan \theta /2\pi )\times 10^{n}\approx 0.3\times 10^{n}\tan \theta }$

${\displaystyle t=(2\pi /\tan \theta )\times 10^{-n}\approx 6\times 10^{-n}(1/\tan \theta )}$

Induced Voltage

Induced Voltage is defined as the voltage of the turns which oppose the current flow

${\displaystyle -\xi =N{\frac {dB}{dt}}={\frac {d\phi }{dt}}{\text{ where }}\phi =NB}$

Coil

For a straight wire with the following dimensions Length l, Area A, and Permitivity u and number of Turns N

${\displaystyle L=uN{\frac {l}{A}}}$

Parallel Connection

${\displaystyle {\frac {1}{L_{\mathrm {eq} }}}={\frac {1}{L_{1}}}+{\frac {1}{L_{2}}}+\cdots +{\frac {1}{L_{n}}}}$

Series Connection

${\displaystyle L_{\mathrm {eq} }=L_{1}+L_{2}+\cdots +L_{n}\,\!}$