Self Inductance
Inductance is the name given to the property of a component that opposes the change
of current flowing through it and even a straight piece of wire will have some inductance. Inductors do this by generating
a self-induced emf within itself as a result of their changing magnetic field. When the emf is induced in the same circuit
in which the current is changing this effect is called Self-induction, (L) but
it is sometimes commonly called back-emf as its polarity is in the opposite direction to the applied voltage. When the emf
is induced into an adjacent component situated within the same magnetic field, the emf is said to be induced by
Mutual-induction, (M) and mutal induction is the basic operating principal of
transformers, motors, relays etc. Self inductance is a special case of mutual inductance, and because it is produced within
a single isolated circuit we generally call self-inductance simply, Inductance. The basic unit of inductance
is called the Henry, (H) after Joseph Henry, but it also has the units
of Webers per Ampere (1 H = 1 Wb/A).
Lenz's Law tells us that an induced emf generates a current in a direction which opposes the change
in flux which caused the emf in the first place, the principal of action and reaction. Then we can accurately define
Inductance as being "A circuit will have an inductance value of one Henry when an emf of one volt is
induced in the circuit were the current flowing through the circuit changes at a rate of one ampere per second" and
this definition can be presented as:
Inductance is actually a measure of an inductor’s "resistance" to the change of the current flowing
in the circuit and the larger is its value in Henries, the lower will be the rate of current change.
We know from the previous tutorial about the
Inductor, that inductors are devices
that can store their energy in the form of a magnetic field. Inductors are made from individual loops of wire combined to
produce a coil and if the number of loops within the coil are increased, then for the same amount of current flowing through
the coil, the magnetic flux will also increase. So by increasing the number of loops or turns within a coil, increases the
coils inductance. Then the relationship between self-inductance, (L) and the number of turns,
(N) and for a simple single layered coil can be given as:
Self-inductance of a Coil
- Where:
- L is in Henries
- N is the Number of turns
- Φ is the Magnetic Field linkage
- Ι is in Amperes
This expression can also be defined as the flux linkage divided by the current flowing through
each turn. This equation only applies to linear magnetic materials.
Example No1
A hollow air cored inductor coil consists of 500 turns of copper wire which produces a magnetic
flux of 10mWb when passing a DC current of 10 amps. Calculate the self-inductance of the coil in milli-Henries.
Example No2
Calculate the value of the self-induced emf produced in the same coil after a time of 10mS.
The self-inductance of a coil or to be more precise, the coefficient of self-inductance also
depends upon the characteristics of its construction. For example, size, length, number of turns etc. It is therefore
possible to have inductors with very high coefficients of self induction by using cores of a high permeability and a
large number of coil turns. Then for a coil, the magnetic flux that is produced in its inner core es equal to:
If the inner core of a coil is hollow "air cored", the magnetic induction in its air core will
be given as.
Then by substituting these expressions in the first equation above for Inductance will give
us:
Finally giving us an equation for the coefficient of self-inductance for an air cored coil of:
- Where:
- L is in Henries
- μο is the Permeability of Free Space (4.π.10-7)
- N is the Number of turns
- A is the Inner Core Area in m2
- l is the length of the Coil in metres
As the inductance of the coil is due to the magnetic flux around it, the stronger the magnetic flux for a given
value of current the greater will be the inductance. So a coil of many turns will have a higher inductance value than one of only a
few turns so the equation above will give inductance L as being proportional to the number of turns squared
N2. As well as increasing the number of coil turns, we can also increase inductance by increasing
the coils diameter or making the core longer. In both cases more wire is required to construct the coil and therefore, more lines of
force exists to produce the back emf. The inductance can be increased further if the coil is wound onto a ferromagnetic core than one
wound onto a non-ferromagnetic or hollow air core.
If the inner core is made of some ferromagnetic material the inductance of the coil would increase because
for the same current flow the magnetic flux would be much greater. This is because the lines of force would be more concentrated
through the ferromagnetic core material as we saw in the
Electromagnets tutorial. For example,
if the core material has a relative permeability 1000 times greater than free space, 1000μο such as soft iron or
steel, than the inductance of the coil would be 1000 times greater so we can say that the inductance of a coil increases proportionally
as the permeability of the core increases. Then for a coil wound around a former or core the inductance equation above would need to be
modified to include the relative permeability μr of the new former material.
If the coil is wound onto a ferromagnetic core a greater inductance will result as the cores permeability
will change with the flux density. However, depending upon the ferromagnetic material the inner cores magnetic flux may quickly
reach saturation producing a non-linear inductance value and since the flux density around the coil depends upon the current
flowing through it, inductance, L also becomes a function of current, i.
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