Inductance and inductors

iSC
Consider a short circuit that is carrying a
current iSC — we have seen shorts often and
know that the no voltage developed,
regardless of current level.

However, we also recall from basic

electromagnetic that a current in a wire will
μoisc
have a magnetic field circulating around it. B=
The field pattern is circular and the strength 2πr
of the field diminishes with increasing
distance from the wire.

Whenever charges are moving in a conductor, there will be magnetic

fields present. Since is takes some energy to generate the field, it can
viewed as having some energy stored in it.
In a typical wire in a typical EE 201 circuit, the magnetic field and
associated energy are very small and can probably be neglected in most
instances.
EE 201 inductors – 1
 Magnetic fields — a very short review
Magnetic fields are characterized by two different field quantities,
H → magnetic field strength, units of amperes /meter (A/m)
B → magnetic flux density, units of tesla (T) [ = V·s /m2 }
The two are related and in common materials, either can be used as the
primary description of a magnetic field. In general, both are vector
quantities. (We will ignore the vector nature of the magnetic fields in
our simplified discussion of inductors.) The relationship between B and
H is given by the constitutive equation:
B = μH
where µ is the permeability of the material where the magnetic field
exists. The units of permeability are henries/meter (H/m). The free
space value of permeability is µo = 4π ×10–7 H/m. Most materials have
the free-space value of permeability, but some materials — usually
alloys that contain iron, nickel, cobalt, or gadolinium — have higher
permeabilities. A higher permeability is characterized by a relative
permeability, µr, which is the factor by which the permeability is
EE 201
increased over the free-space value. inductors – 2
 Magnetic flux density
In discussing inductors, we will focus on B, the magnetic flux density.
Visualizing EM fields and making sense of the different quantities is
always difficult. To develop a more intuitive feel for B, we can think
about the field in terms of field lines. The magnetic flux density indicates
how tightly packed the magnetic field lines are in a region of space.
The more tightly packed they are (i.e. the higher the density), the
stronger the field.
B1 B2 > B1 B3 > B2

end views

EE 201 inductors – 3
 Magnetic flux
Given a magnetic flux density, we can define an area in the space
where the magnetic field exists. Within that area, there will be some
number of magnetic field lines. The number within the area is called
the magnetic flux.

ΦB = B ⋅ A Units of webers, Wb, (= V·s)

ΦB1 ΦB2 > ΦB1 ΦB3 > ΦB2

ΦB1 = ΦB2 !!

EE 201 inductors – 4
 Faraday’s Law of Induction
Consider a magnetic field with magnetic flux density, B. A loop
of wire defines an area A that has lines of magnetic field crossing
through it. The loop defines a magnetic flux.

ΦB = B ⋅ A + –
vF
Faraday discovered that if a wire loop has magnetic field lines cutting through
i — defining a magnetic flux in within the loop and if the magnetic flux is
changing, there will be voltage a induced across the terminals of the wire.
(More rigorously, an electric field is produced between the open ends of the
wire. Of course, we know that if there is an electric field, there will be a
corresponding voltage difference.)
The negative sign comes from “Lenz’s Law”, which
dΦB says that the magnetic field created by the current in
vF = −
dt the loop caused by the induced vF must be opposite
the original magnetic field.

The induced voltage can be the result of the magnetic field itself changing (as
with inductors — to be seen shortly) or by having the area change by moving
the loop within the field (as in the case of electric generators). Or both.
EE 201 inductors – 5
 Coil
Wrapping N turns of wire around a core creates a coil. If the turns of the
coil are packed tightly together, the magnetic fields curling around each
individual wire will merge to create an approximately uniform magnetic
field extending down the length of the interior core. The interior
magnetic flux density will be

( l )
μ⋅N
Bcore = icoil

where l is the length of the core, µ is permeability of the core material

and icoil is the current in wire.
If the core has a cross-sectional area of A, the corresponding magnetic
flux is

( l )
μN 2 A
ΦB = N ⋅ Bcore ⋅ A = icoil.

(The N turns increases the effective area intersecting the magnetic flux
density.)
EE 201 inductors – 6
 Inductance

( l )
μN 2 A
ΦB = icoil

The magnetic flux is directly proportional to the L

current. We can define the proportionality constant
as the self-inductance.
μN 2 A
ΦB = L ⋅ icoil where L = . Inductor circuit
l symbol.
For a given amount of current, the field can be increased by: increasing
the number of turns, increasing the diameter of the coil (increasing A),
decreasing the length of the coil, or by having the core made of a
material with higher permeability.
The units of inductance are henries or H (after American scientist Joseph
Henry.) 1 H = 1 V·s/A. Typical values used in circuits range from about
1 µH upwards to as much 0.1 H. (In power systems, much bigger
inductors are possible.)
EE 201 inductors – 7
 Inductor voltage ΦB = L ⋅ icoil
A straight-forward application of Faraday’s Law to
the magnetic flux of an inductor gives: dΦB diL
− = vL = L
dt dt
leading to the all-important current-voltage relationship for an inductor:
diL
vL = L
dt +
A consideration of Lenz’s Law leads to the indicated relationship iL vL
between current direction and voltage polarity — identical to
–
relationship in a resistor or capacitor.
If the current is not changing, then there is no voltage and the inductor
behaves like a simple (if somewhat lengthy) short circuit. When the
current is changing, the changing field causes a voltage to be
developed that is proportional to the time derivative of the current.
Conversely, if there is a voltage on the inductor, then current must be
changing. The current can be calculated by turning the above
equation around:
1 t
L ∫0
iL (t) − iL (0) = vL (τ) dτ
EE 201 inductors – 8
 Example iL vL

+ 5A 4V

IL L vL
– 1A
t 0 t
1 ms 2 ms 1 ms 2 ms
For t < 1ms, diL /dt = 0 → vL = 0.
For 1 ms < t < 2 ms, diL /dt = 4000 A/s → vL = (0.001 H)(4000 A/s) = 4 V.
For t > 2 ms, diL /dt = 0 → vL = 0.
iL
Due to the derivative in the
inductor equation, the current 5A
cannot change instantaneously. diL
→∞ vL → ∞!
This would require an infinitely dt
large voltage and an infinitely large 1A
amount of power. t

However, the inductor voltage can change instantaneously, — the slope

change abruptly, as long as the current is continuous.
EE 201 inductors – 9
 Inductor energy
The induced magnetic field requires energy in order for it to build up —
the magnetic field represents energy stored in the inductor.
To determine the stored energy, start with power. When the inductor is
“amping up”, the power is
diL
PL (t) = iL (t) vL (t) = LiL
dt
The change in energy due to a change in current can be found by
integrating the power over time.
iL(t) iL(t)
1
∫i (0) ∫i (0)
ΔE = PL (t) dt = iLdiL = L [iL2 (t) − iL2 (0)]
L L
2
If we choose iL(0) = 0, which corresponds to no stored magnetic energy,
the inductor energy corresponding to given a current can written as
1 2
EL = LiL
2
Typical value: For L = 1 mH and iL = 10 A, E = 0.05 J — not a whole lot.
EE 201 inductors – 10
 Series and parallel inductor combinations
Parallel is = i1 + i2 + i3
ieq
= + +
iL1 iL2 iL3 +
Leq L1 L2 L3 vL
= + +
–

= + + = + +

Series
iL + vL1 –
= + +
+ L1 +
Leq veq L2 vL2 = + +
– L3 –
=( + + )
– vL3 +
= + + Inductor combinations are identical
to those for resistors.
EE 201 inductors – 11