# Hysteresis Curve

A curve of the flux density $B$ versus the magnetizing force H of a material is of particular importance to the engineer. Curves of this type can usually be found in manuals, descriptive pamphlets, and brochures published by manufacturers of magnetic materials.
A typical B-H curve for a ferromagnetic material such as steel can be derived using the setup of Fig. 1.
Fig. 1: Series magnetic circuit used to define the hysteresis curve.
The core is initially unmagnetized and the current $I=0$. If the current $I$ is increased to some value above zero, the magnetizing force $H$ will increase to a value determined by $$H = {NI \over l}$$ The flux $\Phi$ and the flux density B ($B= \Phi/A$) will also increase with the current I (or H). If the material has no residual magnetism, and the magnetizing force H is increased from zero to some value $H_a$, the $B-H$ curve will follow the path shown in Fig. 2 between o and a. If the magnetizing force $H$ is increased until saturation ($H_s$) occurs, the curve will continue as shown in the figure to point b. When saturation occurs, the flux density has, for all practical purposes, reached its maximum value. Any further increase in current through the coil increasing H will result in a very small increase in flux density B.
Fig. 2: Hysteresis curve.
If the magnetizing force is reduced to zero by letting I decrease to zero, the curve will follow the path of the curve between b and c. The flux density $B_R$, which remains when the magnetizing force is zero, is called the residual flux density. It is this residual flux density that makes it possible to create permanent magnets.
If the coil is now removed from the core of Fig. 2, the core will still have the magnetic properties determined by the residual flux density, a measure of its "retentivity". If the current I is reversed, developing a magnetizing force, $-H$, the flux density B will decrease with an increase in I. Eventually, the flux density will be zero when $-H_d$ (the portion of curve from c to d) is reached.
The magnetizing force $-H_d$ required to "coerce" the flux density to reduce its level to zero is called the coercive force, a measure of the coercivity of the magnetic sample. As the force $-H$ is increased until saturation again occurs and is then reversed and brought back to zero, the path def will result. If the magnetizing force is increased in the positive direction ($+H$), the curve will trace the path shown from f to b.
The entire curve represented by bcdefb is called the hysteresis curve for the ferromagnetic material, from the Greek hysterein, meaning "to lag behind."
The flux density B lagged behind the magnetizing force H during the entire plotting of the curve. When H was zero at c, B was not zero but had only begun to decline. Long after H had passed through zero and had become equal to $-H_d$ did the flux density B finally become equal to zero.
In addition, you will find that a further application of the same magnetizing forces to the sample will result in the same plot. For a current $I$ in $H =NI/l$ that will move between positive and negative maximums at a fixed rate, the same $B-H$ curve will result during each cycle. Such will be the case when we examine ac (sinusoidal) networks in the later chapters. The reversal of the field ($Phi$) due to the changing current direction will result in a loss of energy that can best be described by first introducing the "domain theory of magnetism".