Chapter 3.3.2.2 - Saturation And Hysteresis

In useful magnetic materials, this nice relationship is not correct and
a more general view is taken. The microscopic picture is not dealt
with here, except to note that the magnetization is due to the alignment
of groups of magnetic dipoles—the groups often called domaines.
There are only so many magnetic dipoles available in any given material,
so that once the flux density is high enough, the material is said to saturate,
and the relationship between magnetic flux density and magnetic field
intensity is nonlinear.

Shown in Fig. 3.6, for example, is a “saturation curve” for a magnetic
sheet steel that is sometimes used in electric machinery. Note the
magnetic field intensity is on a logarithmic scale. If this were plotted on
linear coordinates, the saturation would appear to be quite abrupt.

At this point, it is appropriate to note that the units used in magnetic
field analysis are not always the same, nor even consistent. In almost

all systems, the unit of flux is the weber (W), which is the same as a
volt-second. In SI, the unit of flux density is the tesla (T), but many
people refer to the gauss (G), which has its origin in CGS. 10,000 G=1 T.
There is an English system measure of flux density generally called
kilo-lines per square inch, in which the unit of flux is the line. 10⁸ lines
is equal to a weber. Thus, a Tesla is 64.5 kilo-lines per square inch.

The SI and CGS units of flux density are easy to reconcile, but the units
of magnetic field are a bit harder. In SI, H has dimensions of amperes/
meter (or ampere-turns per meter). Often, however, magnetic field is
represented as Oersteds (Oe). One Oe is the same as the magnetic field
required to produce one gauss in free space. So 79.577 A/m is one Oe.

In most useful magnetic materials, the magnetic domaines tend to be
somewhat “sticky”, and a more-than-incremental magnetic field is
required to get them to move. This leads to the property called
“hysteresis”, both useful and problematical in many magnetic systems.

Hysteresis loops take many forms: a generalized picture of one is
shown in Fig. 3.7. Salient features of the hysteresis curve are the
remanent magnetization B_r and the coercive field H_c. Note that the actual
loop that will be traced out is a function of field amplitude and history.
Thus, there are many other “minor loops” that might be traced out by

out by the B-H characteristic of a piece of material, depending on just
what the fields and fluxes have done and are doing.

Now, hysteresis is important for two reasons. First, it represents the
mechanism for “trapping” magnetic flux in a piece of material to form a
permanent magnet. We will have more to say about that anon. Second,
hysteresis is a loss mechanism. To show this, consider some arbitrary
chunk of material for which one can characterize an MMF and a flux

Energy input to the chunk of material over some period of time is

Now, imagine carrying out the second (double) integral over a continuous
set of surfaces which are perpendicular to the magnetic field H. (This IS
possible!.) The energy becomes

and, done over a complete cycle of some input waveform, that is

That last expression simply expresses the area of the hysteresis loop for
the particular cycle.

Generally, most electric machine applications use magnetic material
characterized as “soft”, having as narrow a hysteresis loop, and therefore
as low a hysteretic loss as possible. At the other end of the spectrum are
“hard” magnetic materials which are used to make permanent magnets.
The terminology comes from steel, in which soft, annealed steel material
tends to have narrow loops and hardened steel tends to have wider loops.
However, permanent magnet technology has advanced to the point where
the coercive forces, possible in even cheap ceramic magnets, far exceed
those of the hardest steels.

In useful magnetic materials, this nice relationship is not correct and
a more general view is taken. The microscopic picture is not dealt
with here, except to note that the magnetization is due to the alignment
of groups of magnetic dipoles—the groups often called domaines.
There are only so many magnetic dipoles available in any given material,
so that once the flux density is high enough, the material is said to saturate,
and the relationship between magnetic flux density and magnetic field
intensity is nonlinear.

Shown in Fig. 3.6, for example, is a “saturation curve” for a magnetic
sheet steel that is sometimes used in electric machinery. Note the
magnetic field intensity is on a logarithmic scale. If this were plotted on
linear coordinates, the saturation would appear to be quite abrupt.

At this point, it is appropriate to note that the units used in magnetic
field analysis are not always the same, nor even consistent. In almost

all systems, the unit of flux is the weber (W), which is the same as...