Electronics/Inductors

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[edit] Introduction

An inductor is a passive electronic component dependant on frequency, and is used to store electric energy in the form of a magnetic field. An inductor has the symbol:

Inductance is the characteristic of the Inductor to generates a magnetic field for a given current. Inductance has a letter symbol L and measured in units of Henry (H).

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

[edit] Important Qualities of Inductors

There are several important properties for an inductor that may need to be considered when choosing one for use in an electronic circuit. The following are the basic properties of a coil inductor. Other factors may be important for other kinds of inductor, but these are outside the scope of this article.

Current carrying capacity is determined by wire thickness and resistivity.
The quality factor, or Q-factor, describes the energy loss in an inductor due to imperfection in the manufacturing.
The inductance of the coil is probably most important, as it is what makes the inductor useful. The inductance is the response of the inductor to a changing current.

The inductance is determined by several factors.

Coil shape: short and squat is best
Core material
The number of turns in the coil. These must be in the same direction, or they will cancel out, and you will have a resistor.
Coil diameter. The larger the diameter (core area) the less induction.

[edit] Inductor's Characteristic

A simple inductor made from a coil of conductor with the magnetic field associated with the current i

[edit] Inductance

The following illustrates the properties of inductors using the example of a coil. Let this coil have the following properties:

Area enclosed by each turn of the coil is A
Length of the coil is 'l'
Number of turns in the coil is N
Permeability of the core is μ. μ is given by the permeability of free space, μ₀ multiplied by a factor, the relative permeability, μ_r
The current in the coil is 'i'

The magnetic flux density, B, inside the coil is given by:

$B=\frac{N \mu i}{l}$

We know that the flux linkage in the coil, λ, is given by;

$\lambda=NBA \,$

Thus,

$\lambda=\frac{N^2A\mu}{l}i$

The flux linkage in an inductor is therefore proportional to the current, assuming that A, N, l and μ all stay constant. The constant of proportionality is given the name inductance (measured in Henries) and the symbol L:

$\lambda=Li \,$

Taking the derivative with respect to time, we get:

$\frac{d \lambda}{dt}=L \frac{di}{dt} + i \frac{dL}{dt}$

Since L is time-invariant in nearly all cases, we can write:

$\frac{d \lambda}{dt}=L \frac{di}{dt}$

Now, Faraday's Law of Induction states that:

$-\mathcal{E}=N \frac{d \Phi}{dt}=\frac{d \lambda}{dt}$

We call $-\mathcal{E}$ the electromotive force (emf) of the coil, and this is opposite to the voltage v across the inductor, giving:

$v=L \frac{di}{dt}$

This means that the voltage across an inductor is equal to the rate of change of the current in the inductor multiplied by a factor, the inductance. note that for a constant current, the voltage is zero, and for an instantaneous change in current, the voltage is infinite (or rather, undefined). This applies only to ideal inductors which do not exist in the real world.

This equation implies that

The voltage across an inductor is proportional to the derivative of the current through the inductor.
In inductors, voltage leads current.
Inductors have a high resistance to high frequencies, and a low resistance to low frequencies. This property allows their use in filtering signals.

An inductor works by opposing current change. Whenever an electron is accelerated, some of the energy that goes into "pushing" that electron goes into the electron's kinetic energy, but much of that energy is stored in the magnetic field. Later when that or some other electron is decelerated (or accelerated the opposite direction), energy is pulled back out of the magnetic field.

[edit] Inductor's Voltage

$v=L\frac{di}{dt}$

[edit] Inductor's Current

$i=\frac{1}{L} \int_{t_1}^{t_2} v(t)\, dt + i \left(t_1 \right)$

Note that usually, the voltage will fall off to zero as the magnetic field is set up, reaching zero at an infinite time after the voltage is applied, so the integral of voltage tends to a finite limit, meaning that the current is also a finite quantity.

[edit] Inductor's Impedance

For an ideal lossless inductor the impedance of an inductor at any given angular frequency $ω$ is given by

Z = j ω L

where j is $\sqrt{-1}$ and L is the inductance.

Z = s L

Z = ωL /_90

In practice, all conductor has a resistance hence for non-ideal lossless inductor

Z L = R L + Z

Z = R L + j ω L

where j is $\sqrt{-1}$ and L is the inductance.

Z = R L + s L

Z = R_L/-0 + ωL /_90

[edit] Quality factor

Quality factor denoted as Q is defined as the ability to store energy to the sum total of all energy losses within the component

$Q = \frac{X}{R}$

where

Q = quality factor (no units)
R = total resistance associated with energy losses (in ohms)
X = reactance (in ohms). ( $X L = 2π f L$ for inductors; $X_C = \frac{1}{2\pi f C}$ for capacitors; f is the frequency of interest ).