INDUCTORS AND TRANSFORMERS
Thursday September 15, 2011
This presentation is introductory and does not present all of
the different magnetic materials, their properties, vendors,
core geometries, sizes, mounting hardware etc. However
an extensive list of references is provided at end of the
presentation for further exploration.
There are over 100 slides so I will go quickly. Please
reserve all questions for the end. This presentation will be
posted on the IEEE Website in the next few days.
SOME BASIC DEFINITIONS
• Permeability- Figure of merit of a particular magnetic material representing
the ease of producing a magnetic flux for a given input
• B is the magnetic flux density and H is the magnetizing force that
generated the flux.
• In the CGS system B is measured in Gauss which is Lines of
Magnetic Flux per Square Centimeter.
• In the SI system the unit of measurement of B is the Tesla which
is Webers per Square Meter.
• A Tesla is equal to 10,000 Gauss.
=µ (Equation 1)
• H, the Magnetizing Force that produces the Flux, is measured in either
Oersteds in the CGS system or Amps per Meter in the SI system
• The Oersted is defined as 1000/4π (≈79.5774715) amperes per meter
so, the H-field strength inside a long solenoid wound with 79.58 turns
per meter, of a wire carrying 1 Ampere will be approximately 1 Oersted.
• Permeability µ is dimensionless and can be considered a figure of
merit of a particular magnetic material, since it represents the ease of
producing a field of magnetic flux for a given input. The permeability of
air or that of a vacuum is 1 (an air-core inductor for example).
• Magnetizing force is caused by current flowing through turns of wire; so
H in Oersteds can be determined from ampere-turns by
where N is the number of turns, I is the current in amperes, and mL is the
mean length of the magnetic path in centimeters.
= (Equation 2)
• The inductance of a coil is directly proportional to the number of flux
linkages per unit current. The total flux is found from
where A is the cross-sectional area in square centimeters and ml is the
mean path length in centimeters.
• The inductance proportionality may then be given by
or directly in henrys
• Observe that the inductance of a coil is directly proportional to the
permeability of the core material. If an iron core is inserted into an air-
core inductor, the inductance will increase in direct proportion to the
core’s permeability. The inductance is also proportional to N2.
• All the previous design equations make the assumption that the magnetic
path is uniform and closed with negligible leakage flux in the surrounding
air such as would occur with a single-layer toroidal coil structure.
• A toroid is the most ideal core structure and has the least leakage flux.
However, the assumption of no leakage flux is really never completely valid,
so some deviations from the theory can be expected.
• The induced voltage of an inductor can be related to the flux density by
B is the maximum flux density in Gauss, N is the number of turns, f is the
frequency in hertz, and A is the cross-sectional area of the core in square
centimeters. This important equation is derived from Faraday’s law.
• Using this equation and solving for B one can determine if the operating
conditions result in core saturation using BH curves given for the magnetic
−×= BNfAE (Equation 6)
THE BH CURVE
A plot of B vs. H (BH Curve) appears as follows. Let us start at point A
and increase the magnetizing force to obtain point B. Decreasing the
magnetizing force will pass through point C and then D and then E as
the magnetizing force is becomes negative. An increasing magnetizing
force, again in the positive direction, will travel to B through point F.
The enclosed area formed by the curve is called a hysteresis loop and
results from the energy required to reverse the magnetic molecules of
The magnitude of H between
points D and A is called
coercive force and is the
amount of H necessary to
reduce the residual magnetism
in the core to zero.
• The magnetic materials discussed in this presentation and the operating
points of relatively low level AC excitation and little or no DC bias results in
very little hysteresis. The emphasis of this part of the presentation is to
obtain High Q inductors at nominal AC excitation. The theory and design
of high power inductors is outside the context of this presentation.
The amount of flux density remaining is called the remanence (residual magnetism)
of the magnetic material. 'Soft' magnetic materials, used in the manufacture of coil and
transformer cores, have a very small remanence and 'hard' magnetic
materials, used in the manufacture of permanent magnets, have a very high remanence.
• Permeability µ is defined
as the ratio B/H and can
be obtained from the
slope of the BH curve.
• With low-level AC signals, the region
of interest is restricted to a relatively
narrow range. We can then assume
that the permeability is determined
by the derivative of the curve at the
origin. The derivative of a B/H curve
is sometimes called incremental
• BH curves are normally shown for
one quadrant for simplicity. Note
that magnetic materials’ 1 through 9
have different permeabilities (slopes
at the origin) and some saturate
(flatten) at a lower H than others.
• If a DC bias or offset is introduced, the quiescent point will move from the
origin to a point farther out on the curve.
• Since the curve tends to flatten out with higher values of H, the incremental
permeability will decrease, which reduces the inductance. This effect is
known as saturation and can also occur without a DC bias for large AC
• Severe waveform distortion usually accompanies saturation. The B/H curve
for an air core is a straight line through the origin. The permeability is unity,
and no saturation can occur.
• The following figure shows a BH curve of a Ferrite core and the associated
µ which is the slope or derivative of the BH curve.
• It is well known that inductors in parallel combine like resistors in parallel
• Inductors in series combine like resistors in series
• LT=L1+L2+ Ln
PROPERTIES OF INDUCTORS
• Two coil windings that share the same magnetic core have a Mutual
Inductance since the change in current in one inductor would cause a
flux that induces a voltage in another nearby inductor. (this is also the
mechanism of transformers).
• The mutual inductance, M, is also a measure of the coupling between
two inductors since not all lines of flux produced by one of the inductors
links with the other (called Leakage Inductance).
• So the two inductances above would add as :
M=k L1 L2
where k is the Coefficient of Coupling of flux between the two coils and
ranges from 0 to 1 (For 1 all lines of flux couple)
• Note that if L1=L2 =L and k=1, then LT = 4xL
• The Figure of Merit (quality of the inductor) is given by the Inductors’
Q, a dimensionless number. It is defined by:
Rdc = DC Resistance (copper loss)
Rac = Core Loss and Skin Effect
Rd = Dielectric Loss
• DC Resistance also known as DCR or copper loss consists strictly of the
DC winding resistance and is determined by the wire size and total length
of wire required as well as the specific resistivity of copper.
• The Core Loss is composed mostly of losses due to eddy currents and
hysteresis. Eddy currents are induced in the core material by changing
magnetic fields. These circulating currents produce losses that are
proportional to the square of the inducing frequency.
• Hysteresis is represented by the enclosed area within a BH curve and
results from the energy required to reverse the magnetic domains in the
core material. These core losses increase in direct proportion to frequency,
since each cycle traverses the hysteresis loop.
• Skin Effect increases the wire resistance above approximately 50 kHz