Inductors

1
Inductors
If a current I flows through a conductor (a wire)
a magnetic field results. Magnetic flux lines
surround the wire as shown. If the current flows
in the direction indicated by the arrow, the
magnetic flux will also have the indicated
direction. The flux is proportional to the
current doubling I doubles the flux.
I
2
Inductors
If the wire is coiled , as shown below, the flux
lines from the coils turns sum. Consider a
small vertical slice of a two turn coil
By KCL, the same current flows through both
turns, and in the same direction. The flux lines
from the first turn combine with the flux lines
from the second turn, so the flux is doubled.
Three turns triples the flux, and so on. The
flux, F, is proportional to I, and to the number
of turns.
Coiled wire
I
I
Vertical slice of a two-turn coil
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Inductors
For a multiturn coil, the flux lines arrange
themselves as shown below. This is similar to a
bar magnet. If I is constant, thats all there
is to it. If I is time-varying (AC or
transient), theres more.
Faradays law says that if a coil of N turns is
placed in a time-varying magnetic field (a region
of changing flux), a voltage will be induced
across the coil. The voltage is given by
N
I
S
Where e is the induced voltage, N is the number
of turns, and df/dt is the instantaneous rate of
change of the flux linking the coil (intersecting
the turns).
The flux lines go from the north pole to the
south pole (outside the coil).
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Inductors
Notice that if the current flowing in the coil is
constant (pure DC), the flux is unchanging. This
means df/dt is zero, so the induced voltage is
zero.
If the current is flowing into the coil in the
direction indicated at left, and is increasing,
the induced voltage has the indicated polarity.
If its decreasing (i.e., increasing in the
opposite direction) the polarity is reversed.
The bottom line is that a change in current
through the coil induces a voltage which opposes
the change in current.
I

e
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5
Inductors
The self-inductance of a coil is given by
Where N is the number of turns, f is the flux
linking the coil from its own magnetic field, and
i is the current flowing in the coil. Recall that

I

e
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This can be rewritten as
6
Inductors
If the induced voltage is symbolized by v instead
of e,
Notice that this looks very similar to the
relationship between voltage and current in a
capacitor
I

e
Except that the roles of voltage and current are
reversed, and capacitance is replaced by
inductance. Knowing how capacitive (RC) networks
work will make understanding inductive (RL)
networks easy!
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7
Inductors
Lets take a closer look at the inductors
current-voltage relationship
If i is unchanging (i.e., DC) then the derivative
(slope) of i with respect to time is zero, and v
0. In other words, for constant current (DC)
the inductor behaves like a short circuit.
Compare this to the inductors current-voltage
relationship
I

e
-
If the voltage across a capacitor is unchanging
(DC),
So for an unchanging (DC) voltage, the capacitor
behaves like an open circuit.
8
Inductive Reactance
The inductor is the dual of the capacitor
If I is constant (DC), di/dt 0, so vL 0.
This means that the inductor is a short circuit
to DC. (Duh! Its a wire!)
ic(t)

vL
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9
Inductive Reactance
If the inductor current is a sinusoid,
then
So in an inductor, current leads voltage by 90
degrees.
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Inductive Reactance
If we consider only amplitudes,
So the reactance of an inductor is (note that
its sign is positive)
ic(t)

vL
XL
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Inductors
Now for some practical considerations. Weve
already seen that an inductors inductance is
proportional to the number of turns
Inductance is also proportional to df/di. In
other words, if the effect a change in current
has on the flux, the inductance is increased.
One way this can be done is by introducing a core
of some magnetic material like iron, powdered
iron, or ferrite. The flux passes through the
core more easily than through air, so there is
more of it. This is analogous to reducing the
resistance of an electrical circuit. The ease
with which magnetic flux passes through a
material is called permeability, and is
symbolized by m.
I

e
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12
Inductors
It is possible to construct an adjustable
inductor, or variable inductor, by using a core
which can be pushed into or pulled out of the
coil, usually by a screw adjustment. The more of
the coil that has the core in it, the greater the
inductance.
Another way of increasing is df/di is to increase
the area of the coil, by increasing its diameter.
Reducing the length of the coil also increases
df/di. These effects results from the laws of
electromagnetic fields, which are beyond the
scope of this course. The bottom line is this
Inductance, permeability, and coil dimensions are
related by the following formula
I

e
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Where N is the number of turns, A is the area of
the coil, and l is its length.
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Inductors
Permeability may be given in absolute terms, or
relative to air. If m0 is the permeability of
air and mr is the relative permeability of some
material, then
The permeability of air or other nonmagnetic
materials (wood, plastic, aluminum, copper, etc.
is
I

e
-
For nonmagnetic materials, mr 1. For magnetic
materials, the value of is dependent on the
propoerties of the particular material, but other
factors as well. Because of that, tables of mr
are not published, and the method of calculating
mr are beyond the scope of this course.
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Inductors
The inductance of a coil with a core of some
magnetic material is given by
This can also be expressed in terms of the
relative permeability
I

e
-
Where L0 is the inductance of the same coil (same
dimensions and number of turns) with an air core.
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Inductors
There are several schematic symbols for inductors
of different types
Air - core
Iron - core
Variable (permeability-tuned)
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Inductors
Many network analysis problems can be solved
using the ideal model for inductors. In some
cases, though, secondary effects must be
accounted for by using a more complex (and
accurate) model
Ideal model
L
This model includes the winding resistance, which
is due to the nonzero resistivity of the wire the
inductor is made from.
L
R
17
Inductors
The model below includes stray capacitance
Adjacent turns of the coil act to a small degree,
like parallel plates of a capacitor. Thus, the
coil looks like N-1 small-value capacitors in
series. This series combination of capacitors
appears to be connected in parallel with the
inductor and its winding resistance.
The stray capacitance can often be ignored, but
not always. The effects of capacitance and
inductance are, for AC circuits, dependent on
frequency. For the model shown here, at some
frequency the effects of the inductor and
capacitor will cancel each other out. This
results in resonance, and if the frequency at
which resonance occurs unwanted consequences may
result. This will be dealt with more fully in
EET 152/207.
L
C
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