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
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 = µ N2* 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* 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
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 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!
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