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 induce a voltage V

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 = N2* A/l        (physical definition for inductance)

where   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   , 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* I2       where W stands for the energy on the inductor.

 

 

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 Unitsare 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 Ltotal = L1 + L2 + L3 + ... + Ln,

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!

DC Electronics

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