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**Chapter 4 Modeling of B-H Loops under DC Bias**

**Introduction**

In [1] a model is described with the ability to generate the sigmoid shaped B-H loops that are typical of magnetic materials under symmetrical magnetic excitation conditions. Commonly referred to as the Jiles-Atherton, J-A, model, it assumes the magnetization process is anhysteretic for an ideal magnetic material. Under this condition the magnetization curve is a single valued and, therefore, reversible function of the magnetic field strength [2]. For non-ideal magnetic materials the J-A model assumes magnetization losses exist, which make the magnetization process hysteretic and, thus, irreversible. In this case the total magnetization, M, of a magnetic material will consist of an irreversible component, Mirr, in addition to a reversible component, Mrev. The J-A model mathematically describes the total magnetization process, including the anhysteretic magnetization, Man, as well as the Mrev and Mirr components, using a small number of parameters and equations formulated on physical considerations. These equations and parameters are described in this chapter, along with the rationale underlying their existence, and are used to propose a model that predicts the increase in core losses with dc bias levels. In addition the model is able to predict the figure-eight shaped B-H loop phenomenon described in Chapter 3.

**The J-A Model and the Nature of Hysteresis under DC Bias**

The magnetization process over an excitation cycle generating a B-H loop can be described in terms of a series of jumps between metastable energy states of a magnetic system [3]. In a metastable state the thermodynamic forces acting on a magnetic system are in equilibrium and a thermodynamic potential, such as the Landau free energy, GL, will be at a local minimum for a short period of time. Therefore, the metastable state is characterized by the thermodynamic relation: which states that in equilibrium the variation of GL with respect to the magnetization, M, is zero, while the H-field and temperature, T, are held constant. The time period can end if the thermodynamic forces include a contribution from an applied magnetic field, which is changed to a sufficiently large degree. Following such a change the magnetic system loses its stability, as the local energy state it occupies no longer exists as a minimum, and a jump is made to the nearest new local energy minimum. If the jump is so rapid in time that for its duration the applied magnetic field can be considered as constant, then it can be assumed that time plays no role during the jump. In this case the only important feature is the sequence of local energy equilibria that are visited [3], and the magnetic system may be described as being rateindependent.

Under this condition the applied magnetic field simply forces the magnetic system to pass from one local energy minimum to the next. However, if the rate at which the applied magnetic field varies is increased so that rate-dependent effects are incurred, such as the inducement of eddy currents in magnetic materials, then a rate-independent approach can no longer be applied.

Following a change to an applied magnetic field the time period occupied in a local energy equilibrium can also end if thermal agitation causes the magnetic system to relax to a thermodynamic equilibrium [3]. If the rate at which the applied field is varied is so slow that a sequence of thermodynamic, rather than local, energy equilibria are traced then the magnetization process will be anhysteretic. In this case losses are not incurred, disallowing the measurement of the phenomenon of hysteresis. Therefore, there exists a frequency bandwidth within which the rate of change of the applied magnetic field is sufficiently slow to avoid rate-dependent effects, yet is fast enough to prevent the magnetic system relaxing to a thermodynamic equilibrium. In [4] it is stated that the time-independent hysteresis loop of a Mn-Zn ferrite core in Ferroxcube

3C80 material is identical to that occurring at 1 kHz. Time-independence in this context is considered to be equivalent to rate-independence by the author and, therefore, 1 kHz is assumed in this Thesis to be a frequency which allows rate-independent B-H loops to be modeled for Mn-Zn ferrite materials. This assumption is reasonable if the jumps between local energy equilibria of the magnetic system of a Mn-Zn CUT are assumed to take place through Barkhausen jumps [3], which are relatively undamped due to the low eddy current losses of Mn-Zn ferrite at 1 kHz. Furthermore, the assumption is supported by the core loss results shown in Table 4.1, which were measured using the technique described in Section 5.5 in the absence of a dc bias, at an ac excitation level of 0.05 T, and on a 25x15x10 mm toroidal CUT in MMG F49 Mn-Zn ferrite material. The results show core losses increased by a factor of 1.94 as the frequency was increased from 1 kHz to 2 kHz, and demonstrate that the energy lost per cycle remained approximately constant as the frequency was doubled. Consequently, the area contained within the B-H loops at 1 kHz and 2 kHz must be approximately equal, and both B-H loops can be considered as rate-independent. A comparison between the core losses at 1 kHz and 10 kHz shows an increase by a factor of 12.36, which is greater than the factor by which the frequency increased. This demonstrates a rate-dependence, which is even more pronounced with a comparison between the losses at 1 kHz, and 20 kHz. At higher frequencies, such as 20 kHz, the approximation that the applied magnetic field is constant during Barkhausen jumps no longer holds, and modifications must be made to the rate-independent J-A model. The rate-independent J-A model has previously been modified to account for rate-dependent effects through the use of an equation of motion for moving domain walls [4]. With this equation the damping, and thus the loss, associated with domain wall movement is taken into consideration. Therefore, the issue arises as to whether the increase in core losses with dc bias should be modeled using a rate-independent, or a rate-dependent J-A model. Resolution of this issue is achieved through measuring core losses under dc bias conditions at 1 kHz, to observe whether the mechanisms causing core losses to increase with dc bias are operative at this low frequency. A core loss characteristic measured at 1 kHz under dc bias conditions is shown in Fig. 4.1. Forthese measurements the CUT was initially demagnetized before an ac excitation level of 0.05 T was applied. The dc bias was then increased monotonically from zero to a maximum, then decreased back to zero using the technique described in Section 5.5. The characteristic shows core losses decreasing with the initial application of a dc bias, before increasing significantly at higher dc bias levels. This trend is typical of other core loss characteristics measured under dc bias conditions at higher frequencies, which are presented in Chapter 6. Therefore, it shows that the mechanisms causing core losses to increase with dc bias conditions are rate-independent. This is consistent with the theory proposed in Chapter 3, which stated that the increase in core losses with dc bias can be explained by an increase in the level of irregular stress within a CUT, which in turn creates more domain wall pinning sites. Although an increase in domain wall pinning sites generates higher losses, it does not slow the speed of moving domain walls during the Barkhausen jumps between local energy equilibria of a magnetic system. Therefore, according to the definition of rate-independent magnetization in [3], the core loss mechanisms described in Chapter 3 can be accounted for within a rate-independent model. This allows the rate-independent J-A model to be used to simulate the effects of dc bias conditions on the core losses of Mn-Zn ferrite materials.Of importance to the J-A model is the nature of Man, because once a curve characterizing this quantity is determined, the model only needs to be completed with the introduction of hysteresis into the magnetization process [5]. Man is described in Section 4.3. 4.3 The Anhysteretic Magnetization

In Section 4.2 metastable states were described as local equilibrium energy states, which were occupied for short periods of time until the applied magnetic field was changed to a sufficient degree. However, over relatively longer time periods, and in the absence of changes to the applied magnetic field, thermal agitation allows a magnetic system to progress from a local equilibrium to a lower energy state. If the lower energy state corresponds to a global minimum it is known as a thermodynamic equilibrium [3]. The locus of points joining thermodynamic equilibria at various applied magnetic field values represents the Man curve [5], the movement along which does not depend on previous history of the magnetic system, and is reversible. It can be regarded as the path that would be followed if a magnetic material were free from the hysteresis generating phenomena that create domain wall pinning sites and, thus, domain wall movement losses. A simplified diagram comparing a Man curve to an initial magnetization curve is shown in Fig. 4.2, and it is apparent that the permeability of the Man curve is higher than that of the initial magnetization curve at magnetization levels below saturation [5]. In [6] it is stated that the expression for Man used in the J-A model is derived under the assumption that a magnetic material is composed of an array of pseudo-domains with fixed domain walls. As a consequence, the net magnetization of each pseudo-domain is only allowed to change through domain rotation, which must be reversible to avoid hysteresis. These changes are brought about by variations in the effective magnetic field strength, He. Man can be defined as a function of He by [1]: The constant, a, in (4.2) is a parameter of the J-A model that is proportional to temperature, domain density and the saturation magnetization, and can be determined through experimental measurements. The He field consists of contributions derived from a number of domain energy terms including the applied magnetic field energy, and the stress anisotropy energy. The effect of stress anisotropy on the magnetization process has previously been described in Chapter 3 and, within the framework of the J-A model, can influence the total magnetization process through acting on Man. This influence is expressed by defining the total magnetization in terms of two components according to then expressing Mrev and Mirr as variables dependent on Man. Physically, the influence of Man on M can be understood with knowledge that each point on the Man curve represents a minimum and, therefore, optimal energy state for a CUT to exist in. Consequently, domain walls will move, or bend in such a manner as to minimize the difference between the actual magnetization, and Man. The bending of domain walls can be a lossless and, therefore, reversible process and so is associated with Mrev. However, domain wall movement incurs losses and so is associated with Mirr. The importance of domain wall pinning to both these components of magnetization, is discussed further in Section 4.4.

**Chapter 1 Introduction .**

1.1 Research Background

1.2 The Characteristics of an Mn-Zn Ferrite Material

1.3 The Scope of the Thesis

1.4 References

**Chapter 2 Literature Review**

2.1 Introduction

2.2 Core Loss Measurement Circuits under DC Bias Conditions

2.3 Models and Explanations for Ferrite Core Losses under dc Bias

2.4 Summary

2.5 References

**Chapter 3 Theory of B-H Loop Phenomena under DC Bias**

3.1 Introduction

3.2 DC Bias, Vibration and Hysteresis Losses at Microscopic Levels

3.3 DC Bias, Vibration and Hysteresis at Macroscopic Levels

3.4 Other Losses Related to Vibration under DC Bias

3.5 Summary

3.6 References

**Chapter 4 Modeling of B-H Loops under DC Bias**

4.1 Introduction

4.2 The J-A Model and the Nature of Hysteresis under DC Bias

4.3 The Anhysteretic Magnetization

4.4 Magnetization and the J-A Model

4.5 The J-A model under dc Bias Conditions

4.6 The Modified J-A Model

4.7 B-H Loops under DC Bias Conditions

4.8 Simulated Figure-eight B-H Loops under DC Bias

4.9 Summary

4.10 References

**Chapter 5 Core Loss Measurement Techniques**

5.1 Introduction

5.2 Core Loss Measurement

5.3 The Mutual Inductance Neutralization Technique

5.4 The Zero-Voltage-Switching Technique

5.5 A Fast, High Accuracy Core Loss Measurement Technique

5.6 Summary

5.7 References VIChapter

**6 Vibration and Core Loss Measurements under DC Bias**

6.1 Introduction

6.2 Core Losses under dc Bias Conditions

6.3 Measurements under dc Bias for Type R Ferrite Material

6.4 Summary

6.5 References

**Chapter 7 Magnetomechanical Interactions and B-H Loop Shape**

7.1 Introduction

7.2 Figure-eight B-H Loops

7.3 Other Distorted B-H Loops under DC Bias Conditions

7.4 Summary

7.5 References

**Chapter 8 Conclusion and Suggestions for Future Work**

8.1 General Conclusions

8.2 Contributions

8.3 Suggestions for Future Work

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