THE INDUCTOR
In its most basic form, an Inductor
is simply a coil of wire. For most coils the current, (i) flowing through the
coil produces a magnetic flux, (NΦ) that is proportional to it. According to Electromagnetism,
when electrons flow through a conductor a magnetic flux is developed around the
conductor producing a relationship between the direction of this flux and the
direction of the electron flow called the "Left Hand Rule". But
another important property of a wound coil is to use this magnetic flux to
oppose or resist any changes in electrical current flowing through it.
The Inductor is another passive
type electrical component designed to take advantage of this relationship by
producing a much stronger magnetic field than one that would be produced by a simple coil. Inductors are formed with wire tightly wrapped around a solid central core which can be either a straight cylindrical rod or a continuous loop or ring to concentrate their magnetic flux. The schematic symbol for an inductor is that of a coil of wire so therefore, a coil of wire can also be called an Inductor. Inductors are categorised according to the type of inner core with the different core types being distinguished by adding continuous or dotted parallel lines next to the wire coil as shown below.
producing a much stronger magnetic field than one that would be produced by a simple coil. Inductors are formed with wire tightly wrapped around a solid central core which can be either a straight cylindrical rod or a continuous loop or ring to concentrate their magnetic flux. The schematic symbol for an inductor is that of a coil of wire so therefore, a coil of wire can also be called an Inductor. Inductors are categorised according to the type of inner core with the different core types being distinguished by adding continuous or dotted parallel lines next to the wire coil as shown below.
Inductor Symbols
The current, i that flows through an inductor
produces a magnetic flux that is proportional to it. But unlike a Capacitor which opposes
a change of voltage across their plates, an inductor opposes the rate of change
of current flowing through it due to the build up of self-induced energy within
its magnetic field. In other words, inductors resist or oppose changes of
current but will pass a steady state DC current. This ability of an inductor to
resist changes in current and which also relates current, i with its magnetic
flux linkage, NΦ as a constant of proportionality is called Inductance
which is given the symbol L with units of Henry, (H) after
Joseph Henry.
Because the Henry is a relatively large unit of
inductance in its own right, for the smaller inductors sub-units of the Henry
are used to denote its value. For example:
Prefix
|
Symbol
|
Multiplier
|
Power of Ten
|
milli
|
m
|
1/1,000
|
10-3
|
micro
|
µ
|
1/1,000,000
|
10-6
|
nano
|
n
|
1/1,000,000,000
|
10-9
|
So to display the sub-units of the Henry we would
use as an example:
- 1mH = 1 milli-Henry - which is equal to one thousandths (1/1000) of an Henry.
- 100uH = 100 micro-Henries - which is equal to 100 millionths (1/1,000,000) of a Henry.
There
are many factors which determine the inductance of a coil such as
·
Shape of the coil,
·
Number of turns,
·
Number of layers,
·
Spacing between the turns,
·
Permeability of the core material,
·
Size or cross-sectional area of the core etc.
An inductor coil has a central core area, (A)
with a constant number of turns of wire per unit length, (l). So if a coil of N
turns is linked by an amount of magnetic flux, Φ then the coil has a flux
linkage of NΦ and any current, ( i ) that flows through the coil will
produce an induced magnetic flux in the opposite direction to the flow of
current. Then according to Faraday's Law, any change in this magnetic flux
linkage produces a self-induced voltage in the single coil of:
Where
- N is the number of turns
- A is the cross-sectional Area in m2
- Φ is the amount of flux in Webers
- μ is the Permeability of the core material
- l is the Length of the coil in meters
- di/dt is the Currents rate of change in amps/second
A time varying magnetic field
induces a voltage that is proportional to the rate of change of the current
producing it with a positive value indicating an increase in emf and a negative
value indicating a decrease in emf. The equation relating this self-induced
voltage, current and inductance can be found by substituting the μN2A / l
with L denoting the constant of proportionality called the Inductance
of the coil. This then reduces the above equation to give the self-induced emf,
sometimes called the back emf induced in the coil too:
The Back emf Generated by an Inductor
Where
L
is the self-inductance
di/dt is the rate of current change.
So from this equation we can say that the
"Self-induced emf = Inductance x Rate of Current
Change" and a circuit has an inductance of one Henry when an emf of one
volt is induced in the circuit when the current flowing through the circuit
changes at a rate of one ampere per second.
One important point to note about the above
equation. It only relates the emf produced across the inductor to changes in
current because if the flow of inductor current is constant and not changing
such as a DC current, then the induced emf voltage will be zero because the
instantaneous rate of current change is zero, di/dt = 0. With a
steady state DC current flowing through the inductor and therefore zero induced
voltage across it, the inductor acts as a short circuit in the presence of
continuos current.
The Time Constant of an Inductor
We now know that the current can not change
instantaneously in an inductor because for this to occur, the current would
need to change by a finite amount in zero time which would result in the rate
of current change being infinite, di/dt = ∞, making the induced
inductor emf infinite aswell but infinite voltages do no exist. However, if the
current flowing through an inductor changes very rapidly, such as with the
operation of a switch, high voltages can be induced across the inductors coil.
Consider the circuit of an inductor, with the
switch, (S1) open no current flows through the inductor so the rate of current
change (di/dt) is equal to zero and therefore, zero self-induced emf exists
across the inductor. If we now close the switch (t = 0), a current
will flow through the circuit and slowly rise to its maximum value at a rate
determined by the inductance of the inductor. This rate of current flowing
through the inductor multiplied by the inductors inductance in Henry's results
in some fixed value self-induced emf being produced across the coil as
determined by Faraday's equation above, V = LdΦ/dt. This self-induced
emf across the inductors coil fights against the applied voltage until the
current reaches its maximum value and a steady state condition is reached. The
current which now flows through the coil is determined only by the resistance
of the coil because the inductance of the coil has decreased to zero, a short
circuit condition as a steady state condition now exists.
Likewise, if switch, (S1) is opened, the current
flowing through the coil will start to fall but the inductor will again fight
against this change and try to keep the current flowing at its previous value
by inducing a voltage in the other direction. The slope of the fall will be
negative and related to the inductance of the coil as shown below.
Current and Voltage in an Inductor
The amount of voltage induced by the inductor
depends upon the rate of current change. An induced emf will always OPPOSE the
motion or change which started the induced emf in the first place. So with a
decreasing current the voltage polarity will be acting as a source and with an
increasing current the voltage polarity will be acting as a load. So for the
same rate of current change through the coil, either increasing or decreasing
the magnitude of the induced emf will be the same.
POWER AND ENERGY IN AN INDUCTOR
POWER
We know that an inductor in a circuit opposes the
flow of current, ( i ) through it because the flow of this current
induces an emf that opposes it, Lenz's Law. Then work has to be done by the
external battery source in order to keep the current flowing against this
induced emf. The instantaneous power used in forcing the current,
( i ) against this self-induced emf, ( VL ) is
given from above as:
An ideal inductor has no resistance only
inductance so R = 0 Ω's and therefore no power is dissipated within
the coil, so we can say that an ideal inductor has zero power loss.
Energy
When power flows into an inductor, energy is
stored in its magnetic field. When the current flowing through the inductor is
increasing and di/dt becomes greater than zero, the instantaneous power in the
circuit must also be greater than zero, ( P > 0 ) ie,
positive which means that energy is being stored in the inductor. Likewise, if
the current through the inductor is decreasing and di/dt is less than zero then
the instantaneous power must also be less than zero, (P < 0) ie
negative which means that the inductor is returning energy back into the
circuit. Then by integrating the equation for power above, the total magnetic
energy which is always positive, being stored in the inductor is therefore
given as:
Energy stored by an Inductor
Where
- W is in joules, L is in Henries and i is in Amperes
The energy is actually being
stored within the magnetic field that surrounds the inductor by the current
flowing through it. In an ideal inductor that has no resistance or capacitance,
as the current increases energy flows into the inductor and is stored there
within its magnetic field without loss, it is not released until the current
decreases and the magnetic field collapses. Then in an alternating current, AC
circuit an inductor is constantly storing and delivering energy on each and every
cycle. If the current flowing through the inductor is constant as in a DC
circuit, then there is no change in the stored energy as
P = LI(di/dt) = 0.
So inductors can be defined as passive components
as they can both stored and deliver energy to the circuit, but they cannot
generate energy. An ideal inductor is classed as lossless, meaning that it can
store energy indefinitely as no energy is lost. However, real inductors will
always have some resistance associated with the windings of the coil and whenever
current flows through a resistance energy is lost in the form of heat due to Ohms Law,
( P = I2 R ) regardless of whether the
current is alternating or constant.
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.
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.
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.
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.
MUTUAL INDUCTANCE
When the emf is induced in the same
circuit in which the current is changing this effect is called Self-induction,
(L). However, when the emf is induced into an adjacent coil situated within the
same magnetic field, the emf is said to be induced magnetically, inductively or
by Mutual-induction, symbol (M). Then when two or more coils
are magnetically linked together by a common magnetic flux they are said to
have the property of Mutual Inductance.
Mutual Inductance is the basic
operating principle of transformers, motors, generators and any other electrical
component that interacts with anothers magnetic field. But mutual inductance
can also be a bad thing as "stray" or "leakage" inductance
from a coil can interfere with the operation of another adjacent component by
means of electromagnetic induction, so some form of electrical screening to a
ground potential is required.
The amount of mutual inductance that links one
coil to another depends very much on the relative positioning of the two coils.
If one coil is positioned next to the other coil so that their physical
distance apart is small, then nearly all of the magnetic flux from the first
coil will interact with the turns of the second coil inducing a large emf and
therefore producing a large mutual inductance value. Likewise, if the two coils
are farther apart from each other the amount of induced magnetic flux from the
first coil will be weaker producing a much smaller induced emf and therefore a
much smaller mutual inductance value. So the effect of mutual inductance is
very much dependant upon the relative positions or spacing, (S) of the two
coils and this is shown below.
Mutual Inductance
The mutual inductance that
exists between the two coils can be greatly increased by positioning them on a
common soft iron core or by increasing the number of turns of either coil as
would be found in a transformer. If the two coils are tightly wound one on top
of the other over a common soft iron core unity coupling is said to exist
between them as any losses due to the leakage of flux will be extremely small.
Then assuming a perfect flux linkage between the two coils the mutual
inductance that exists between them can be given as.
Where:
- µo is the permeability of free space (4.π.10-7)
- µr is the relative permeability of the soft iron core
- N is in the number of coil turns
- A is in the cross-sectional area in m2
- l is the coils length in meters
We remember that the self inductance of each
individual coil is given as:
Then by cross-multiplying the two equations
above, the mutual inductance that exists between the two coils can be expressed
in terms of the self inductance of each coil.
Giving us a final and more common expression for
the mutual inductance between two coils as:
MUTUAL INDUCTANCE BETWEEN COILS
However, the above equation
assumes zero flux leakage and 100% magnetic coupling between the two coils, L 1
and L 2. In reality there will always be some loss due to leakage
and position, so the magnetic coupling between the coils can never reach or
exceed 100%, but can become very close to this value in some special inductive
coils. If some of the total magnetic flux links with the two coils, this amount
of flux linkage can be defined as a fraction of the total possible flux linkage
between the coils. This fractional value is called the Coefficient of
Coupling and is given the letter k. Generally, the amount of inductive
coupling that exists between the two coils is expressed as a fractional number
between 0 and 1 instead of a percentage (%) value, were 0 indicates zero or no
inductive coupling and 1 indicates full or maximum inductive coupling. Then the
equation above which assumes unity coupling can be modified to take into account
this coefficient of coupling, k and is given as:
COUPLING FACTOR BETWEEN COILS
When the coefficient of
coupling, k is equal to 1, (unity) such that all the lines of flux of one coil
cuts all of the turns of the other, the mutual inductance is equal to the
geometric mean of the two individual inductances of the coils. So when the two
inductances are equal and L 1 is equal to L 2, the mutual
inductance that exists between the two coils can be defined as:
SMD Inductor
GuideA Glossary of Important Inductor Parameters forHigh Frequency Applications
INDUCTANCE.
High inductance values are
not a factor in most high frequency applications. 1.8nH to 39nH is typically
the range of interest. The critical factors are the stability and tolerance of
the inductance at operating frequency. Note that accurate measurement of these low
values is not trivial. Even at low frequencies (~10MHz), instruments available
today exhibit measurement accuracy of no better than about ±5% for an
inductance of 10nH. At typical application frequencies (³450MHz), measurement
accuracy may be even further degraded by parasitic capacitance of inadequately
characterized test fixtures.
PARASITIC CAPACITANCE
All inductors include a
certain capacitance characteristic. This capacitance is derived from the mutual
proximity of the coil windings. The capacitance is further magnified by the
inductor structural materials, especially if these are of high dielectric constant.
It is critical for high frequency inductors that the parasitic capacitance be
minimized since it determines the device SRF.
SELF RESONANT FREQUENCY:
Self resonant frequency of
the inductor is that frequency at which parallel resonance is achieved between
the device inductance and parasitic capacitance. Inductor Q drops to zero at
SRF. It is quite important, therefore, that inductor SRF be much higher than
the application frequency.
RESISTANCE.
In addition to inductance and parasitic capacitance, the inductor also
exhibits resistance. At low frequencies, the resistance of the conductor track
is the determining factor in inductor Q. At very high frequencies, the
resistivity of the conductor material is more important than the resistance.
(resistance = resistivity x
conductor length /cross section )
This is due to the skin effect whereby high
frequency current flow is largely restricted to the surface layer of the
conductor. For high Q at high frequency, it is therefore important that the
coil conductor be constructed of a metal with low resistivity.
