Gyroscopic Couple: The rate of change of angular momentum () = (In the limit).
= Moment of Inertia.
= Angular velocity
= Angular velocity of precession.
Overview
Key facts
Magnetic hysteresis is a characteristic of ferromagnetic materials, consisting of the lack of retraceability of the initial magnetization curve when the magnetic field is relaxed.
The hysteresis power loss of iron is given by:
where is a constant, is the frequency of operation, and is the magnetic flux density. This equation is also known as Steinmetz's law.
Magnetic hysteresis is a property which affects ferromagnetic materials.
In order to better understand hysteresis, consider an unmagnetized sample found at the origin of a magnetic flux density - magnetic field strength plot (point in Figure 1).
Figure 1
As the magnetic field strength is increased in the positive direction, the magnetic flux density begins to grow non-linearly until the sample will be found at point on the curve. This path is called the initial magnetization curve. If the magnetic field is then relaxed, instead of following back on the initial magnetization curve, the magnetic flux density falls more slowly. In fact, even when the applied magnetic field is returned to zero, there will still be a remanent magnetic flux density (point in Figure 1). This lack of retraceability of the initial magnetization curve is called hysteresis. In order to drive the magnetic flux density back to zero (point in Figure 1), the applied magnetic field must be reversed. If the reversed magnetic field is further increased, the magnetic flux density will continue to decrease (point in Figure 1). If the magnetic field is then again increased in the positive direction, the magnetic flux density will pass through zero (point in Figure 1), and then return to the value corresponding to point in Figure 1. The resulting loop is called a hysteresis loop.
Hysteresis is a property of ferromagnetic materials with great applicability as a magnetic memory device. Indeed, as shown in Figure 1, when the driving magnetic field drops to zero (path ), the ferromagnetic material retains an important degree of magnetization. Certain compositions of ferromagnetic materials will actually retain an imposed magnetization indefinitely and can be used as permanent magnets.
A widely used application of the magnetic memory (usually that of iron and chromium oxides) is for the magnetic storage of data on computer disks.
Hysteresis Power Loss
Figure 2
Figure 3
In order to define the power losses due to hysteresis, we first have to discuss the different paths of the hysteresis loop in terms of energy. We know that for a magnetic field of volume , the energy stored can be written as:
where is the area between the curve and the axis in a plot as the one highlighted in Figure 1 (for a more detailed discussion on the energy stored in a magnetic field see Stored Energy ). Thus, for a hysteresis loop as the one exemplified in Figure 1, and also taking into account that the volume is constant, the energy required to move the ferromagnetic sample from point to point ( is the blue shaded area in Figure 2) is higher than that which returns when moving from point to point ( is the blue shaded area in Figure 3).
Taking into account the shaded areas from Figures 2 and 3, we can deduce that the power lost over one complete cycle is proportional to the area within the hysteresis loop. Because a hysteresis loss occurs each time the core cycles from positive to negative values of , the total loss rate is directly proportional to the frequency of operation . In fact, for iron, we can write that the hysteresis power loss (expressed in ) is given by:
where is a constant, is the magnetic flux density, and is the Steinmetz exponent for iron. Equation (3) is also known as Steinmetz's law, or Steinmetz's approximation.
For most electrical machines, the hysteresis power loss is a waste, and the hysteresis effect is thus kept to a minimum. For example, the iron used in silicon steel has a low magnetic hysteresis (for further references see also Alloy Steels ).
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) = 
(In the limit).
= Moment of Inertia.
= Angular velocity
= Angular velocity of precession.
where 
is a constant, 
is the frequency of operation, and 
is the magnetic flux density. This equation is also known as Steinmetz's law.
plot (point 
in Figure 1).
on the 
curve. This path is called the initial magnetization curve. If the magnetic field is then relaxed, instead of following back on the initial magnetization curve, the magnetic flux density falls more slowly. In fact, even when the applied magnetic field is returned to zero, there will still be a remanent magnetic flux density (point 
in Figure 1). This lack of retraceability of the initial magnetization curve is called hysteresis. In order to drive the magnetic flux density back to zero (point 
in Figure 1), the applied magnetic field must be reversed. If the reversed magnetic field is further increased, the magnetic flux density will continue to decrease (point 
in Figure 1). If the magnetic field is then again increased in the positive direction, the magnetic flux density will pass through zero (point 
), the ferromagnetic material retains an important degree of magnetization. Certain compositions of ferromagnetic materials will actually retain an imposed magnetization indefinitely and can be used as permanent magnets.
A widely used application of the magnetic memory (usually that of iron and chromium oxides) is for the magnetic storage of data on computer disks.
, the energy stored can be written as:
where 
is the area between the 
(expressed in 
) is given by:
where 
is the Steinmetz exponent for iron. Equation (3) is also known as Steinmetz's law, or Steinmetz's approximation.
For most electrical machines, the hysteresis power loss is a waste, and the hysteresis effect is thus kept to a minimum. For example, the iron used in silicon steel has a low magnetic hysteresis (for further references see also Alloy Steels ).