Practical Electronics/Inductors

Intro
An inductor is a device made from a wire conductor with several turns. This device generates a magnetic field as current passes through it similar to the magnetic field of a magnet. An inductor stores electrical energy in the form of a magnetic field.

Inductance is defined as the capability to store electrical energy in the form of a magnetic field for a given current which is directly proportional to permeability, length of inductor, and number of turns and inversely proportional to cross-sectional area.
 * $$L = \frac{B}{I} = \mu N^2\frac{A}{l}$$

The symbol for inductance is L and is measured in Henry which has the symbol H.

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
 * $$B = IL$$

Inductance
Inductance is the ability to generartes Magnetic Field B for a given Current
 * $$L = \frac{B}{I}$$

Voltage

 * $$V_L = L \frac{dI}{dt} = \frac{dB}{dt}$$

Current

 * $$I_L = \frac{1}{L} \int V dt $$

Reactance
Reactance is defined as the ratio of Voltage over current
 * $$X_L = \frac {L \frac{dI}{dt}}{I} $$
 * $$X_L = \omega L$$/_90
 * $$X_L = j \omega L$$
 * $$X_L = s L$$

Impedance
Impedance is defined as the sum of Reactance and Resistance of Inductor. Since all conductor has Resistance
 * $$Z_L = R_L + X_L$$
 * $$Z_L = \sqrt{R_L^2 + (\omega L)^2}$$/_Tan-1 $$\omega \frac{L}{R}$$
 * $$Z_L = R_L + j \omega L$$
 * $$Z_L = R_L + s L$$

Frequency Respond
Inductor is a device depends on frequency $$\omega $$


 * $$\omega = 0, X_L = 0$$, Inductor Closed circuit, I ≠ 0
 * $$\omega = 00, X_L = 00$$, Capacitor Opened circuit, I = 0
 * $$\omega = \frac{R_L}{L}$$
 * $$X_L = R_L$$ ,
 * $$Z_L = [R_L]^\frac{1}{3}$$ ,
 * $$V_L = \frac {V}{2}$$
 * $$I_L = \frac {V}{2R_L}$$

With the value of I at three frequency points ω = 0, 00, 1 / CRC 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
 * $$\tan\theta = \omega {L}{R_L} = 2\pi f \frac{L}{R_L}$$

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
 * $$f = (\tan\theta/2\pi) \frac{R_L}{L}$$
 * $$t = (2\pi/\tan\theta) \frac{L}{R_L}$$

If choosing L = 1 and R = 10n then the formulas above become
 * $$f = (\tan\theta/2\pi) \times 10^n \approx 0.3\times 10^n \tan\theta$$
 * $$t = (2\pi/\tan\theta) \times 10^{-n} \approx 6 \times 10^{-n} (1/\tan\theta)$$

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


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

Coil
For a straight wire with the following dimensions Length l, Area A, and Permitivity u and number of Turns N
 * $$L = u N \frac{l}{A}$$

Network
Inductors can be connected in series to increase inductance or in parallel to decrease inductance

Parallel Connection

 * [[Image:inductors in parallel.svg|A diagram of several inductors, side by side, both leads of each connected to the same wires]]


 * $$ \frac{1}{L_\mathrm{eq}} = \frac{1}{L_1} + \frac{1}{L_2} + \cdots + \frac{1}{L_n}$$

Series Connection

 * [[Image:inductors in series.svg|A diagram of several inductors, connected end to end, with the same amount of current going through each]]


 * $$ L_\mathrm{eq} = L_1 + L_2 + \cdots + L_n \,\! $$

Reference

 * wikipedia Inductor