Basic Hysteresis Model (Based On Loop Width)

Jiles-Atherton Hysteresis Model

At present there a few developed theories used to model the magnetization
of a transformer core accurately. These models are generally based on
the observed behaviour of real materials, and include magnetic hysteresis
effects. Two of the best-known and widely used empirical methods to model
hysteresis are the *Jiles-Atherton*
and Preisach models, which both
provide an accurate representation of the hysteresis loop. Jiles and Atherton
used a soft magnetic material for their investigation and modeled the
saturation characteristic using a modified Langevin
function. Preisach theory is based on the fact that a ferromagnetic material
is made up of dipoles
that can be represented as having a magnetic characteristic with two statistically
distributed parameters, a coercivity and a field due to neighboring dipoles
[7],
[8],
[9].

The primary drawback of the above methods is that the core material data (required for magnetization modeling) is not usually available. However, a mathematical technique may be employed to approximate the hysteresis loop behaviour for recoil loops and renascence effects. This technique is general and sufficiently accurate to model the hysteresis phenomena, by using a lookup and interpolation method [10]. A set of curves representing the major (or parent) loop of the l-I characteristic are generated. These curves, along with a second set of curves representing flux differences between the two paths of the parent loop, are then used to determine the current to be injected for any given value of flux. In this technique, the minor loop travel, sustained and initial flux remanence, and realistic inrush phenomena are all considered [11].

In EMTDC, a current source is used to represent the saturation of the core in classical transformers (see Core Saturation), where the saturation current is calculated using Equation 6-22. This equation represents a l-I characteristic curve which is asymptotic to the specified air core reactance line, and to the vertical axis, passing through a point defined by the 1 pu. Magnetizing current and flux are shown in Figure 6-6.

In order to generate the major loop of the l-I characteristic, first a single-valued saturation curve, between plus and minus values of the specified knee point limits, is obtained as explained earlier. Then, this curve is shifted according to the specified hysteresis loop width in the positive direction of current. The other side of the major loop is then generated as the negative of the negative function described by the positively offset curve. Figure 6-8 represents the major (parent) loop of the l-I characteristic [10].

Figure 6-9 - Hysteresis Parent Loop

This maximum loop is the trajectory only when the inductor element is
driven from saturation in one direction to saturation in the opposite
direction. The UPPER curve or *g(**l**)* is for increasing values of flux
linkage and the DOWNER curve or *f(**l**)* is for decreasing values of the
flux linkage. The points of confluence for the two curves making up the
major loop are at the defined saturation knee point values. The characteristic
is assumed to be single valued beyond the points of confluence (Figure
6-9).

Figure 6-10 - A Typical Hysteresis Loop Including Saturation

A reversal or turnaround point occurs when the flux linkage derivative
*d**l**/dt* changes sign. The coordinate
(*l**T**,I**T* ), represents
a generalized flux linkage turnaround point of operation in the non-saturated
region, as shown in Figure 6-9. If a flux turn around occurs at any point
in the non-saturated region, the path of travel must change along a trajectory,
which tends towards the confluence point and travels along the opposite
parent curve. In the saturated region, if a flux turn around point occurs,
the travel path does not change since only one path exists there. At a
turnaround point the trajectory leading from any present point on the
correct new point is defined by two quantities, namely [10];

- The difference in flux between the present point and the point directly above/below the parent curve, toward which it must travel.
- The point of confluence, toward which it must travel.

Figure 6-11 - A Mapping of *f(λ)*
into *F(λ)* Showing a Typical Turnaround
Point and a Generalized Point on the Downer Curve for Discrete Time *t**n+l*.

These trajectory curves can be generated in the non-saturated region
by translating a variable displacement *d
= D(**l**)* from the *f*
or *g* curves. The simplest relationship
for *D*, and the one chosen for
this example of family pair mapping, is to make the displacement *d* vary linearly from:

(6-25) |

A typical downer curve *F* resulting
from a variable displacement *d*
from curve *f* is shown in Figure
6-10.

The Jiles-Atherton theory describes the relationship between *M*,
the magnetic moment, and *H*, the
magnetic field intensity [9].
Electrical engineers are more accustomed to dealing with a B–H relationship
in the form of a hysteresis loop. Conversion from a*
B-H* and an* M-H* loop is
straightforward using the relationship:

(6-26) |

The magnetization relationship between B and H is replaced by the anhysteretic
magnetization curve between* H**e* and *M:*

(6-27) |

where *H**e** = H + **a**M* and *a*
is a parameter that represents inter-domain coupling. *M**s *is the saturation
magnetization. Jiles and Atherton used a modified Langevin function to
produce the familiar sigmoid type curve for f(*H**e*) as expressed
below:

(6-28) |

By differentiating Equation 6-28 with respect to *H**e**
*,

(6-29) |

The Jiles-Atherton theory leads to two components in *M*
as follows:

(6-30) |

The first component, *M**irr*, is due
to pinning of the magnetic domains by discontinuities in the material
structure and can be likened to a friction effect. The second term, *M**rev*,
is due to domain wall bending in an elastic manner. The authors then build
on these fundamental relationships to arrive at the final set of equations
to be solved to construct *M-H**e* and *B-H* loops [9]:

(6-31) |

*A*, *a*,
*c* and *k*
are constants for the material being used and *d*
takes the value *1* or *-1* based on the sign of *dH/dt*.

This equation takes a modified form when *(M**an**
- M)**d*
is negative as described in [9]:

(6-32) |

In power transformers at every time step, the changes in magnetizing current should be calculated based on the changes in magnetizing flux. The magnetizing flux is also calculated by integrating the magnetizing branch voltage.

Now, these equations are iteratively solved for the unknown values of
*D**H* and *D**M* in each time step.