**Inductance and Inductors**

**I. Elementary Characteristics **

The coil in the figure simulates an inductor.

The main issue is how the magnetic field lines go

across the inductor (lines with arrows). There is

some magnetic field at the top bottom of the coil

too.

The current I going through the inductor generates a magnetic field which is perpendicular to I.

The Magnetic Field **H **is given by the loops that surround the

current **I. **The direction of the Magnetic Field is given by the

arrows around the loops. If the current was to flow in the

opposite direction the Magnetic Field arrows would be

reversed. For a practical display of this phenomena see: **
Magnetic field on
wire ** .

It is the Magnetic Field which contains the current through the coil which by the principle called

__ Self-Induction __will

More specifically speaking, the voltage __V across the inductor L __ is
given by: __ V = ΔΦ/ΔT
__which reads - the

voltage V is caused by the change in flux over the correspondent change in time, but since the change in flux is

given by the inductance L and the change in
current across the coil** ΔI,
**the voltage V becomes:

** V = L*ΔI/**__ΔT
__ (electrical definition for inductance)

On the other hand the physical definition of inductance L is given by:

__ L = µ N ^{2}* A/l__
(physical definition for inductance)

where ** µ **stands for the relative
ease with which current flows through the inductor or __Permeability __of the

medium. ** N** stands for the number of
turns in the coil, **A** stands for its cross-sectional area, and the length

of the coil is given by **l.
**Hence this formula tells us that the more number of turns the larger the
inductance

(i.e.: current can be contained better), also the larger the cross-sectional area the larger the inductance (since

there is more flux of current that can be contained) and the longer the coil the smaller the inductance (since

more current can be lost through the turns). L
is also proportional to ** µ ,** since the better the permeability

current will flow with more ease.

**
Inductance and Energy.**

By containing the current via the magnetic field the inductor is capable

of storing Energy. A Transformer such as the one on the Figure will certainly

remind us of the ability of storing Energy associated with Inductors.

Whereas for a capacitor the Energy stored depends on the Voltage across

it, for the inductor the Energy stored depends on the current being held,

such that:
** W = 1/2*L* I ^{2 }**

**Types of Inductors**

Although the most common type of inductor is the Bar Coil type which has been already presented, there is

also surface mount inductors, Toroids (ring-shaped core) , Thin film inductors and Transformers (which are

actually a combination inductor elements and will be dealt with in AC Electronics). The choice of a particular

kind of inductor depends on the space availability, frequency range of operation, and certainly power

requirements.

Bar-Coil Surface Mount Thin Film Toroid Type

The surface mount type inductors are very small in size and therefore deserve to be considered when space

becomes and issue. The Thin Film inductors are fabricated by several processing steps similar to the fabrication

of transistors and diodes (They are very small in size too).

__II. Inductor
Circuits__

__1. Basic Inductor Circuit__

The electrical parameters V and L (the inductance -measured in

Henrys-H - review
DC Basics or go to __ Table of Units)
__are given.

The current I is implicitly given by the relationship: V = Ldi/dt

In a similar case as with the basic capacitor circuit we are implying that at time 0 a switch closes connecting

the battery to the coil and the inductor starts to get charged. Also, in all real cases there will be a small

resistance in series with the inductor, but we will get to this case in the discussion of R-L circuits.

__At
a specific point of time __the voltage across the inductor is expressed by
** V = Ldi/dt** which is basically the

electrical definition of inductance, except that since we are just focusing at a point in time and not at an interval

of time delta = ΔT we will need to use the term dt and similarly for the current di instead of ΔI.

The electrical definition still holds, since all we are saying is that the flux or change in current over time times

the inductance is the __Induced Voltage across the
Inductor__.

**
2. Inductors in Series**

The parameters given in the circuit are the total voltage V

and the voltage across L1- namely V1 and across L2-

namely V2. The current I is the same throughout since this

is a series circuit.

The total voltage V must equal the total inductance Ltotal * ΔI/ΔT hence since V = V1 + V2 we have:

__V = Ltotal*ΔI/ΔT = L1* ΔI/ΔT + L2 *
ΔI/ΔT = ( L1 + L2) * ΔI/ΔT __and therefore :

__ Ltotal = L1 + L2 __and in case of
more than two inductors

where n stands for the total number of inductors in the circuits.

We Note that as in the case of resistance in series inductances in series add up!!!

**
3. Inductors in Parallel**

We know that for parallel circuits the voltage across

the elements (in this case the inductors L1 and L2) is

the same. The total current It will split into I1 and I2

such that It = I1 + I2.

Notice that this is exactly the same scenario that we have for resistors in parallel and henceforth:

** 1/Ltotal = 1/L1 + 1/L2
or Ltotal = L1*L2/(L1 + L2)
** as for two resistors in parallel, and

for more than two inductors we have that:

** 1/Ltotal = 1/L1 + 1/L2 + 1/L3 + ...
+ 1/Ln, ** where n is the total number
of inductors.

Again the comparison with resistors holds true in the case of D.C. Circuits, but it is not true for A.C.

circuits since frequency is an issue and both capacitance and inductance depend on frequency whereas

resistors don't !!

We are ready to discuss R-L and R-C series circuits from a D.C. point of view. For the sake of

simplicity we will omit discussing R-L and R-C parallel circuits and R-L-C circuits, the student should

refer to appropriate sources for these cases!