a fuzzy decision support system for magnetic component design thesis-d.holt-1996-08

8/10/2019 A Fuzzy Decision Support System for Magnetic Component Design THESIS-D.holt-1996-08

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A Fuzzy Decision Support System forMagnetic Component Design

A96.037(741) Masters Thesis

August 1996 D.Holt


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A Fuzzy Decision Support SystemforMagnetic Component Design

D.Holt

Masters ThesisA96.037(741)August 1996

Supervisors: Prof. Ir. G. HonderdProf. Ir. H.R. van Nauta LemkeDr. Ir. J.B. KlaassensIr. U. Kaymak

Delft University of TechnologyDepartment of Electrical EngineeringControl LaboratoryP.O. Box 50312600 GA Delft

The Netherlands


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To My Parents


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PREFACE

This thesis has been written within the framework of my graduation at the Control Laboratoryat the faculty of Electrical Engineering of Delft University of Technology. It is meant toprovide a concept for easing and enhancing the design of magnetic components by means ofa decision support system. Such a system may also be a help for designers of electricalsystems who have no access to expertise and accessible literature about magnetics.

What made the research especially interesting, was the fact that it enclosed the uncommonmixture of theory on magnetics, magnetic component design and human and fuzzy decisionmaking. During the research, I have been guided by my supervisors with energy and humour.Therefore, I am glad to be in the opportunity to thank Prof. Honderd for his everlasting energyto keep the laboratory prosperous, dr. Klaassens for his positive view on life, ir. Kaymak forhis witty and always supporting presence, and prof. van Nauta Lemke for his comprehensiveview on almost everything. Together with the members of student fraternity VerstelRegel ofthe laboratory, they have made the last period of my student life a very pleasant andinstructive one.

The reader of this report does not need a thorough understanding of magnetics or fuzzydecision making. Necessary information about magnetics can be found in appendix A andabout fuzzy decision making in section 3.4. The decision support system has been written bymeans of the computer language Matlab. Using some basic tools available for processing

fuzzy logic together with the theory described in this thesis, the reader can create such acomputer program him or herself easily.

July 1996Danil Holt


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9Contents

CONTENTS

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2. MAGNETIC COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2. Component elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3. Example: AC-inductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Airgap and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3. DESIGN AND DECISION MAKING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1. Magnetic component design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2. Decision making in design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3. Traditional design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3.2 disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4. Fuzzy Multiple Attribute Decision Making . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 decision matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.2 fuzzy sets theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4.3 fuzzy decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.4 aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4.5 decision tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4. DSS FOR AC-INDUCTOR DESIGN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1. Design information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1.2 additional expert knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2. DSS structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.1 Initial selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2.2 removal of dominated alternatives . . . . . . . . . . . . . . . . . . . . . . . . . 504.2.3 final ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1. Airplane application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2. Locomotive application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3. Mass product application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4. Airplane appl. with L=3.5mH,L=4A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 65


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10 Contents

APPENDIX A:THEORY OF MAGNETIC COMPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.1. Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66A.2. Core characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

A.3. Airgap issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.4. Wire characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75A.5. Power losses and temperature rise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76A.6. Important equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

APPENDIX B: AC-INDUCTOR APPLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.1. Inverter equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83B.2. Design example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

APPENDIX D: SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91


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11Summary

SUMMARY

In many areas of electrical engineering, magnetic components like transformers and inductorsare necessary parts of an electrical circuit. Magnetic components are composed of two corehalves of magnetic material and turns of copper wire. For some applications, an airgap ofarbitrary length can be introduced between the core halves. Designing a magnetic componentmeans finding a suitable combination of core, wire and airgap size, while a great variety ofcriteria have to be satisfied. The criteria consist of physical constraints (e.g. maximum storableenergy, maximum allowable temperature rise), application requirements (e.g. inductance,allowed power loss), and designer preferences (e.g. preferred weight, cost). The criteria dependstrongly on the application the component is designed for.

Many different types and sizes of core and wire are available. Hence, a large number ofdifferent combinations is possible, which complicates the design. Therefore, a designer ofmagnetic components uses a design method that iterates towards some feasible design.Because of the variety of criteria and the mutual dependencies of the component attributes,extensive knowledge and experience of the designer is required to make efficient and effectivedecisions during the iteration steps. Despite the human ability to come to some feasible design,the traditional method is not efficient and may cause a sub-optimal design.

In this thesis, a Decision Support System (DSS) is proposed using fuzzy logic and fuzzyMultiple Attribute Decision Making (MADM). The DSS assist a designer in selecting suitable

combinations of core, wire and airgap size by distinguishing the set of possible alternativesand ranking them corresponding with how much they satisfy the criteria. As an example, asystem is developed that is focused on the design of an AC-inductor. However, the conceptsof the DSS can be used for many kinds of magnetic components.

To create the DSS, two types of information are defined that are necessary to design acomponent. Firstly, the design criteria are set up. Because the criteria can be vague, impreciseor uncertain, the mathematical representation is based on fuzzy set theory. Secondly, necessaryadditional expert knowledge is retrieved, in this case acquired by having an expertparticipating in the realization of the system. The expert knowledge consists of variousinformation such as assumptions and heuristic knowledge.

The decision procedure of the DSS is multi-step to provide both a transparant decisionstructure as well as a faster evaluation of the large number of alternatives. The initial decisionsteps reject the inappropriate combinations using the hard limiting boundaries of the criteria.The final decision step ranks the remaining alternatives by means of fuzzy MADM methods.At the end of the process, the designer can concentrate on the recommended set ofalternatives, and choose the preferred component using his subjective and context sensitivehuman opinion. In this way the quantitative power of computers and qualitative abilities ofhumans are used efficiently. Magnetic component design using the proposed DSS is faster thanthe traditional method and increases the possibility to find a near optimal or optimal inductor

alternative.


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12 Introduction


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13Introduction

1. INTRODUCTION

This thesis is concentrated on the design of magnetic components. Although existing for morethan a hundred years, magnetic components are still essential elements of electrical circuits,serving applications such as transforming power, filtering and resonating. In power electronicsmagnetic components are essential for the operation of converters, in addition to the powersemiconductors and capacitors. In telecommunications they are used for accurate filteringpurposes. The two most commonly used magnetic components are the transformer andinductor, which are basically composed of a core of magnetic material and turns of insulatedcopper wire. The number of available sizes and types of premanufactured cores and wires islarge, so many combinations are possible. For several applications, an airgap of arbitrary widthcan be introduced between two core halves, where energy can be stored. The range of possibleairgap volume results in a respective range of possible numbers of turns of the wire. Thisrange increases the number of alternatives considerably.

Designing a magnetic component means finding the optimal combination of core, wire andnumber of turns, while a great variety of application dependent criteria have to be satisfied.It is important to find this optimal alternative, since a magnetic component has an importantinfluence on weight, volume, efficiency and cost of the overall electrical system it is used for.

In order to manage the large number of component alternatives and the variety of criteria, ahuman designer applies an iterative design procedure to find the optimal alternative. The

decisions will be made while making extensive use of experience, rules-of-thumb andapplication tables in catalogues. Still, the human design method requires a lot of time that isinvolved in iterating, and the method may also result in a sub-optimal final design.

With the developments in artificial intelligence and fuzzy logic, it can be expected thatcomputers not only perform the required calculations, but can also help the designer with thetrade-offs that have to be made. A Decision Support System (DSS) will be able to assist himby selecting a set of feasible alternatives and ranking them depending on how much theysatisfy the design requirements.

This thesis describes several aspects of component design and proposes a Decision Support

System. In chapter 2, the construction of magnetic components is described and an exampleis introduced, namely the AC-inductor in a power electronics application. Chapter 3 explainswhat kind of decisions have to be made in component design and discusses the traditionaldesign method and Fuzzy Multiple Attribute Decision Making, a method that can makedecisions using vague information. In chapter 4 the design information and knowledgenecessary to create the DSS is described, and a system structure is established. Finally, theresults of a computer program are shown and evaluated.Papers on the same subject are reference [9] and [40].


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14 Magnetic Components


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15Magnetic Components

2. MAGNETIC COMPONENTS

This chapter explains what magnetic components are and how they are constructed. To clarifysome important facets of a magnetic component, an example is shown of an AC-inductor ina power electronics converter.

2.1. Classification

Magnetic components are those components of an electrical circuit that use a magnetic circuitto fulfil their task. In many areas of electrical engineering they are necessary parts of thecircuit. Various applications are possible, such as transforming power, providing galvanic

isolation, filtering, resonating and other. In power electronics, magnetic components areessential for the operation of power converters, in addition to the power semiconductors andcapacitors. In telecommunications and radio and television, magnetic components are widelyused for filtering applications.

A basic classification of magnetic components is the following:

1. Inductor:- AC inductor,- AC inductor with DC bias;

2. Transformer:- two-winding transformer- multiple winding transformer,- multiple core transformer;

3. Transductor;4. Electrical machines.

The following list gives a short description of several applications:

1. Filter inductors, resonance inductors:controlling energy flow to a load, blocking unwanted frequencies, resonance in LCcircuit;

2. Saturating inductors:regulating voltage or power by saturation of a section of the core;

3. Power transformers:transforming power from one winding to another on a common core with minimumlosses;

4. Current-, voltage-, impedance- or pulse-transformers:

carrying over special current, voltage or impedance levels or waveform shapes asprecisely as possible;


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5. Sensor inductors:Using the airgap properties as position or distance sensor

5. Transductors, magnetic amplifiers:

providing controllable inductance varied by means of a control current;

6. Rotating electrical machines:Converting electrical energy into mechanical energy and vice versa, respectively motorsand generators.

7. Magnetic bearing, magnetic levitation:providing the possibility of frictionless rotation or translation (reference [24])

In this research we will study the design of magnetic components that transform electricalenergy into magnetic energy and vice versa, while satisfying certain characteristics.

Transformers and inductors meet this description. Transductors, however, will be excludedfrom the discussed design methods, because their design is focused on the capability to varycertain component characteristics. Also rotating electrical machines will be excluded, becausetheir design is mainly focused on the optimal transformation of magnetic energy intomechanicalenergy and vice versa.

The transformer as well as the inductor are basically composed of a core of magnetic materialand turns of insulated copper wire. The theoretical backgrounds on magnetics can be foundin appendix A. A simplified explanation on the functioning of the core and the wire is givenin the following lines:One or more of the windings carry a changing electric current, and they induce a changingmagnetic field in the core. The function of the core is to conduct this magnetic field in apredictable way, analogous to the way copper wire conducts electrical current. The energy inthe core carried by the magnetic field can be given to another winding instantly or storedtemporarily in an airgap or in material inserted in the core. The elementary description oftransformers and inductors is the following: a transformer or an inductor consist of a core andone or more windings, utilizing their magnetic circuit to respectively transform or storeenergy.


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2.2. Component elements

Magnetic components are assembled from the following elements:

- magnetic material;- electrical conductive material;- insulation materials (coil former, wire insulation, layer insulation);- thermal parts (heat sink).

To exemplify the construction, Figure 2. shows three basic parts of the component: wire, coilformer (bobbin) and core.

Figure 2. Three basic component elements:a) coil former and wire,

b) core of E-type.

Figure 3. Cross-section of assembled com-ponent.

The component is assembled by winding the wire around the coil former and then joining thecore halves together. Between the core halves, an airgap can be introduced. Figure 3. showsa cross-section of a magnetic component when assembled. The so called winding area orwindow is the area containing the coil former and windings. In the following paragraphsvarious aspects of cores and wires are described.

2.2.1 cores

Cores for magnetic components are made of material that is able to conduct magnetic fluxeasily. Examples are, laminated iron, powdered iron, MPP, Permalloy, Supermalloy andFerrite. The modern high-performance components invariably use a ferrite core. Ferrite is abrittle material consisting of compounds like: MnZn, NiZn, MgZn, LiZn [15] or Fe2O3 withNiO, MnO, ZnO, CoO [28]. Ferrite has a very high electric resistivity, a relatively lowmagnetic resistivity and is easy to manufacture in an arbitrary shape. Some ferrites can beused for very high frequencies up to several MHz.


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More than 15 different ferrite core shapes

Figure 4. RM-core with coil former and clasps.

can be obtained from the manufacturers,such as:

E, EC, Pot, RM, ETD, U, I, X, EP, H, PH,

EFD, ring, rod, tube. The first four shapesare the most commonly used and are applic-able for more general purposes. The relativedimensions of most of these shapes havealready been optimized and are standardizedin IEC norms1. The shapes are available ina range of sizes and in several kinds offerrite. Each shape and ferrite type has somepreferred power or filter application, indica-ted by tables in the catalogues2. However, these tables only give advice to the componentdesigner in his choice of core, but they do not guarantee whether this choice is the best

choice.

The cores are characterized by means of various parameters.The coredimensional parameters are:

Ac : equivalent core leg section areaVc : equivalent core volume (amount of ferrite in mm

3)

V : volume of complete coreAw : available winding area

t : mean length of turn

Acool : component cooling areaA

cool is the outside area of the core, plus the area of the heatsink. In this

thesis, heatsinks are not considered.

The corematerial parameters are:

Bsat : saturation flux density (maximum physically possible )BBbend : bending point of BH-loopc : curie temperature (maximum allowable core temperature)kc,n : parameters that express specific core loss Pc for a certain frequency

: permeability (usually >>1)Kc : cost of core + coil former [Hfl] (without cost of winding and assembling)

By means of these parameters, all calculations necessary for this thesis can be performed.

1) An example of an IEC norm:dimensions of RM-cores made of magnetic oxidesand associated parts,Int. Electrotechnical Commission, Publication 431, Geneva1983.

2) application tables: [15] pp.46-48 or [28] pp.134-138


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2.2.2 wires

The available wire can be solid round, solid flat or

Figure 5. Examples of different wiretypes.

bunched (Litz wire), each available in a range of coppercross-section areas and with different classes of insula-tion. Some examples of wire types and sizes are shownin Figure 5. Each wire type has specific characteristics.For example, flat wire uses space efficiently, but isdifficult to wind and more expensive. Round wire iseasy to wind and cheap, but is not space efficient.

A more sophisticated type of wire is theLitzendrahtorLitz wire (see [12]). Litz wire has conductor strands thatare transposed in such a way that it reduces both

external proximity effects and internal skin effects of the wire. From a designers point ofview it is useful to determine whether it is advantageous to replace solid wire by Litz wirefor a particular application. Litz wire provides a significant improvement over solid wire atlower frequencies, with a continual improvement increase up to a frequency fo. Beyond thisfrequency, local proximity losses begin to increase until the losses of Litz wire are higher thansolid wire at a frequency fmax. Increasing the number of strands will widen the usefulfrequency range, but decreases utilization of the winding area

Wire is characterized by means of the following parameters:N : discrete number of turnsAN : wire section areakw : wire fill factorkac : AC-current resistance factor

Rw,s : specific resistance of wire [m-1]

mw,s : specific weight of wire [kgm-1]

Kw,s : specific cost of wire [Hfl.m-1]

The three latter quantities are not always provided by catalogues, but can easily be calculatedfrom other available quantities in the catalogue.

When choosing the component elements,

Table I. Basic degrees of freedom of core and wire.

CORE WIRE

material type

shape cross section area

size fill factor

several discrete degrees of freedom for thecore and wire are available. Table I. gives anoverview of their basic degrees of freedom.Data sheets with some samples of the differentcores and the wires are shown in Table III. andTable IV. in appendix C. It is obvious thatmany different combinations of cores and wiresare possible. The number of alternatives even

increases if we allow a variable number ofturns.


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2.3. Example: AC-inductor

In magnetic component design there is no such thing as one design procedure for all applica-tions or one design procedure for one component type: different components in differentapplications entail differences in the design procedure and criteria. The concepts described inthis thesis can be used for all components. In order to come to a detailed insight in the aspectsof magnetic component design, this thesis is focused on the design of an AC-inductor.

The use of an AC-inductor as example has two advantages. The first advantage is that theapplied equations are less complicated because the core is symmetrically excitated, like intransformer design. The second advantage is, that the number number of componentalternatives is very high because the airgap can take variable discrete lengths. Deciding aboutmore alternatives makes the decision process more comprehensive and clarifying.

2.3.1 Application

Figure 6. Inductor used in output filter.

Because inductors have the ability to store energy, they are typically used for filtering purp-oses like smoothing an electrical signal, or gaining (resonance). In Figure 6., the inductor L

has been used as low pass filter applied in inverters that convert DC voltage into AC power.The purpose of the LC-filter is to remove the high frequency components from voltage es

*.Voltagees

* may be generated by some switch-bridge. Appendix B gives relevant equations forthis application. dc to AC inverters are used for many applications, like in locomotives,airplanes, cars or other applications that use batteries.

Besides electrical specifications like current wave shape, maximum current, inductance or fre-quency, the AC-inductor also has to satisfy several specifications and preferences that dependstrongly on the application the inverter is used for. Examples are minimum and maximumvalues for weight, volume, power loss, cost or temperature rise.


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Possible examples of criteria imposed on an AC-inductor in for example aircraft applicationsare:

- low weight- low size

- low power losses- cost unimportant- reliable- component temperature low during operation

It will not be possible to satisfy all these requirements on the AC-inductor ideally. Severetrade-offs have to be made. The AC-inductor we want to design consists of a core with anairgap and a winding with a certain number of turns. Different alternative combinations ofcore, wire and number of turns might satisfy the requirements, some maybe more than others.

To obtain a better insight in the characteristics of the inductor alternatives, it might be useful

to give some physical relationships of the AC-inductor in the next section. Appendix A andB provide a more detailed explanation.

2.3.2 Airgap and Equations

An AC-inductor has the property to store energy in its magnetic circuit by means of a smallairgap or a piece of low permeability material introduced between two core halves.Manufacturers offer cores that have a gap that is cut beforehand or dispensed in the corematerial. However, many component designers create their own gap by inserting a piece ofsynthetic material with a certain width between the core halves (Figure 3.). In this waydesigners have much more freedom in choice of core size, geometry and material. Becausethis method is widely used and also very effective, we will incorporate the method in thisresearch. The typical airgap length g is 0.1mm...2mm. In this thesis, calculations are mainlyperformed by means of airgap volumeVg, and not airgap length g. The reason is, thatVg can

easily be calculated into magnetic flux density or number of turnsNand vice versa.B

The most important relationships in inductor design are the relationships between:L : inductance,L : peak inductor current,N : number of turns (discrete),

: excitation flux density in the core,

BVg : airgap volume.

Appendix A, equation (A.19) shows that if relative permeability r 1 (for ferrite), then almostall energy is stored in the airgap volume Vg. Assuming that L and L are specified, then

equation (1) (equivalent to (A.46) of appendix A) shows that the maximum energy storedWcin the airgapVg is:

with o the magnetic constant.

(1)Wc1

2LL

2B

2Vg

2o


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From this equation can be found, that product 2Vg of equation (1) determines the energyB

stored in the airgap: when airgap volume Vg is chosen 4x smaller, then excitation has toBbe 2x larger in order to store the same amount of energy Wc. This equation is not dependenton the applied core or wire.

Also the number of turnsN is inversely proportional to , shown by equation (2) (equivalentB

to (A.45)). The ratio betweenNand depends on the core section areaAcand the electricalBspecifications L and L:

Vg can take any value, resulting in a continuous range of and thereforeN. However,Ncan

(2)B 1

N

LLAc

B

only be discrete, so each alternative core/wire/Nwill take only discrete values for andVg.B

In this chapter we have described what magnetic components are, of which elements theyconsist, and we have given some application example and equations. But, how is the magneticcomponent designed? What is a good component and how do we come to such a design? Howdo we make the necessary decisions? The next chapter will answer these questions.


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3. DESIGN AND DECISION MAKING

This chapter describes the role of decision making in magnetic component design and givessome examples of traditional design methods. Then, a decision making method is introducedthat handles vague information. With this method, vagueness and imprecision as used inhuman reasoning can be implemented in a computer program. In that way the fuzzy aspectsof magnetic component design are dealt with, like fuzzy criteria and latent aggregation3.

3.1. Magnetic component design

To assemble a magnetic component, a set of cores of different shapes, sizes and ferrite gradesis available. Furthermore, a set of different wire types is available, each type with a range ofcopper cross section areas. Because the size of the airgap can be chosen arbitrarily, also thenumbers of turns can vary. In this thesis the general goal of inductor design is to find the core,the wire and the number of turns, so that the combination satisfies the imposed criteria as wellas possible. Figure 7. clarifies how an inductor alternative is composed.

Figure 7. The construction of inductor alternatives from the elementscore, wire and number of turns.

Why do we want to find a so called best inductor in this discrete optimization problem? Thereason for this is that an inductor, like a transformer, has an important influence on the costand on the overall performance of the system for which it is applied.

3) Latent aggregation is the hidden way in which humans combine judgments upondifferent aspects of a subject into one overall judgment upon the subject.


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The influence on the overall performance is caused by three troublesome characteristics ofmagnetic components:

1) they are often relatively large and heavy items of a circuit,2) they produce losses and heat,

3) they are not easy to wind and assemble automatically.

In practice, it will be difficult to find a design that has only optimal parameter values, becauseof the interaction and interdependence of the parameters. To achieve the most desirable design,component parameters affecting others have to be traded off as necessary. For example, if asmall volume and a low weight are of great significance, then reduction in weight and volumemay be possible by selecting a more efficient core material. However, this has a severepenalty of increased cost. Hence, to find the best possible component alternative, careful trade-offs must be effected to achieve the design goal.

To find the appropriate combination of elements, decisions have to be made about which core,

wire and number of turns to take. This decision process can be performed in several ways, likechoosing each component element separately, or choosing between complete alternatives, anddeciding in many steps or in one step. It can be stated that the design of an inductor is thedecision process that in the end satisfies the design goal. The goal is to find the combinationof elements that form the best achievable inductor. What actually is a decision process? Theanswer is given in the next section.

3.2. Decision making in design

The three basic ingredients of the decision making process are the decision maker, thealternatives and the criteria, defined as follows:

1) The decision maker:The decision maker is the person or the computer performing some algorithm that willproduce an ordering on the set of inductor alternatives as the result of a decision makingprocess. The most significant difference between computers and humans as decision makeris, that computers have strong quantitative power, while humans are able to deal withqualitative aspects of decision problems. Therefore, a computer method to make a decisionwill be different from the method of a human decider

2) The inductor alternatives:The alternatives are the discrete elements we have to choose from. They can be regarded asdiscrete elements in a continuous alternative space. In our design problem, the set of inductoralternatives consists of all physically possible combinations of core, wire and number of turns.Each alternative has its own specific parameter values.


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An example of an alternative and its parameter values is the following:

Example of an AC-inductor alternative:

- Core type : E-coresize : EC70material : 3C80

- Wire type : Solid Round Enamelled Wirecopper area : 1.22mm2

fill factor :kw = 0.6

- Number of turns : 185.

Attributes of this alternative (application: L = 4mH, L= 4A):

Vg = 837mm3 : airgap volume (airgap length 3mm)= 310mT : flux density excitationB

m = 460g : weightV = 81.1cm3 : sizeK = .31,- : costPc = 3W : core lossPw = 10W : wire lossPtot = 13W : total power losses = 46C : temperature rise

Some of the attributes are directly used as performance parameters in the decision makingprocess, such as weight, volume, cost, total power loss or temperature rise. Other parameters

play an indirect role, such as the airgap volume Vg or flux density .B

3) The objectives and constraints (criteria):In order to accomplish the general design goal, the component aimed at must satisfy theprescribed objectives and the imposed constraints as well as possible. As will be described in3.4.3, the objectives and the constraints can be used in an identical way in the decisionproblem, because they can be modelled in an identical way. Together they are also called the

decision criteria and they are the standards of judgment to test the acceptability of analternative. The criteria imposed in inductor design may consist of physical constraints,application requirements (as described in section 2.3) or designer preferences. These criteriacan be hard, which means that the criterion has a strict boundary, or soft, meaning that itsboundary is vague, imprecise or uncertain.


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Some examples of criteria are given below. We can see that the vague boundaries areexpressed by human expressions like about or near to.

arbitrary examples of criteria:

physical constraints:- the airgap width must not be larger than about 1.5mm, because of the fringing field,- the flux density has decreasing preference from 300mT up to 375mT,- max = 125C.

application requirements:- the total inductor weight has to be as small as possible,-m < 400g,- the inductor must be able to operate with a maximum ambient temperature of 60C.

designer preferences:- the total inductor weight must be near to 400g,

- the costs of the component are not important at all,- an inductor that costs more than $20 is possible, but not preferred.

We have described the general design goal and the decision making aspects of inductor design.The decision problem is a Multiple Attribute Decision Making problem (MADM). MADMrefers to finding the best component alternative out of all discrete alternatives in the presenceof multiple, usually conflicting, criteria upon the attributes. Human designers and computerseach have their own characteristic way of coping with these conflicting criteria, as describedin 3.3 and 3.4.

3.3. Traditional design procedure

In this section the human design method is described. Finding an appropriate alternative in thetraditional way means a continuous sequence of calculating and deciding about componentelement. To come to a final design, a human designer will not assess all possible alternatives,but will cautiously iterate towards an acceptable component.

3.3.1 methods

Traditionally, designing a magnetic component meant calculating the optimal core geometry,

while making trade-offs between parameters and criteria. In this period, a computer could beused to solve the large amount of tedious equations [11]. After this design, the component hadto be manufactured custom-made. Nowadays, we make use of premanufactured cores that havepredetermined and optimized core geometries and sizes. However, the large number ofavailable sizes and types of cores and wires complicates the design. Only experienceddesigners in this field are capable of determining a limited collection of parts to be able todesign the optimal component within the range of requirements. To help a designer with thedecisions, manufacturers provide their catalogues with tables, graphs and advices. However,because the criteria can be vague, imprecise or uncertain, a human decider has to make useof his human ability to deal with vague information.


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We start with a short description of an expert designing a magnetic component. In general,a human designer starts with the selection of a core from the catalogue or stock as initial step.This selection is performed by using experience, physical and heuristic knowledge, rules-of-thumb and advices from the catalogue. After this initial selection, the design is in essence an

iterative trial and error process split in the following aspects:

1. Magnetic design:- calculation of the magnetic field:- stored energy,- magnetizing current,- air gap,- permeability.

resulting in:- number of turns N,- losses magnetic circuit.

2. electric design:- type of conductor (mainly litz wire),- mass of winding,- increase of dc resistance fac,- copper fill factor fw.

resulting in:- winding losses.

3. thermal design:- total losses,- cooling surface,- maximum temperature rise of the surface,

resulting in:- maximum temperature rise of the hot spot,

4. economic design:costs of materials,costs of production.

The outcome of each step determines whether the designer has to choose another core or wire.

Three examples of inductor design are shown in the following sections. Two textualprocedures come from [29] and [15], and Figure 8. comes from [14] (pp. 766). The bold andunderlined words in the textual versions show the places where a human decision has to beperformed, or where human knowledge has to be used.


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From reference [29]:

Figure 8. Example of iterative design procedure forinductor design ([14] pp.766).

1) choose the wire size nearest that specified byequation ...

2) selecta core with dimensions reasonably closeto the optimum specified by equation ...,

3) calculate turns per layer and number of layers,and determine the total winding build

4) check for fit. The wire size may need to beadjustedup or down.

5) calculate the inductance of the actual design toensure that no numerical errors have beenmade, using Equation

From reference [15]:Design procedures:1) On the basis of the operating characteristics and

design limitations,selectthe core size, materialgrade, inductance factor and conductor typeusing the information given in the data sheets.

2) Using the adjustment curve, check that therange of adjustment is sufficient to cover thetolerance on ALor eand that of the resonatingcapacitor. Make anallowance of about 1% forcircuit strays.

3) Calculate the number of turns required from theALor value for the core.4) Selecta conductor size to fill the coil former.5) From the voltage across the inductor, ERMS,

determine peak flux density Bpeak. If this is in excess of 1 mT,checkthat hysteresis loss and distortion areac-ceptable(by reference to the a.c. signal-level characteristics in the core data).

3.3.2 disadvantages

The human procedure has two drawbacks. The first drawback is that the designer has limitedinsight in the consequences of a decision. Because it is not clear whether a certain choice ofcore or wire leads to a better inductor or not, many iterations may be necessary. Even if a

computer is used to perform the tedious calculations, then still the method is inefficient Thesecond drawback of the human procedure is, that the designer may reject many alternativestoo early in the decision process, because he does not assess all possible configurations ofcores and wires. Finding the optimal design may not be possible in this way. Despite thedesigners ability to make fuzzy and intelligent decisions, the human design method isinefficient an ineffective.

To get around the drawbacks of this method, designers are supported with tables in books orcatalogues that simplify the initial choice of the core . The first computer algorithms wereimplemented in the 60s, taking over calculations. However, the critical decisions have always

been taken by the designer himself, because only human knowledge and experience couldhandle the fuzzy criteria and complex trade-offs.


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With the developments in artificial intelligence and fuzzy logic, new doors have been openedtowards automated decision making in magnetic component design. The next section describesa method to rank a set of feasible component alternatives, dependent on how much theysatisfy the design constraints and preferences. To model the imprecise and uncertain criteria,

the concepts of fuzzy sets and fuzzy decision making are introduced.

3.4. Fuzzy Multiple Attribute Decision Making

In order to create a Decision Support System for a human designer, we will introduce a meansthat allows a computer to find some good component alternatives in the presence of multiple,conflicting criteria. Section 3.2 shows that imposed criteria can be vague or imprecise. In this,the deficiency of a computer is that it cannot process these qualitative concepts. To overcomethis problem, we will introduce fuzzy sets in order to provide a mathematical tool that modelsvague and imprecise criteria. We enter the area of Fuzzy Multiple Attribute Decision Making.A lot have been written about Fuzzy MADM. Reference [3] provides an extensive

bibliography on the subject. Other references are [2], [17], [25] and [31].

3.4.1 decision matrix

To illustrate the interaction of alternatives, criteria, and decision, in Figure 9. a decision matrixis shown:

Figure 9. General decision matrix of a decision problem

attr./crit.weights

c1w1

cmwm D

a1 r11 r1m d1

r21 r2m

alternatives ratings final rating

an rn1 rnm di

The definition of the decision elements is:

{a1,a2, ,an}- the discretealternatives aiform a very large set of possible combinations of

core, wire and number of turns.

{c1,c2, ,cm}- the criteria ci are the set of objectives and constraints that the alternatives

should satisfy (e.g. criterion cj: inductor weight m should be smaller than400g). By means of these criteria the performance or appraisal of thealternative attributes is measured.


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{w1,w2, ,wm}- the relative importance wj of the criteria cj (e.g. wj expresses that criterion

on the cost of an inductor is 2 times more important than the criterion onpower loss). The term importance is used instead of the common term

weight, because the latter term is used in the sense of mass in inductordesign.

rij =r(ai,cj)- theratings(scores)rijexpress the performance of alternativeaiwith respect

to criterioncj(e.g.rij=0.7 means thatai is fairly compatible with criterioncj).

di =D(ri1,ri2, ,rim,w1,w2, ,wm)- Final rating di is the overall rating of alternative ai formed by combining

ratings {ri1, ,rim}, if necessary using importances {w1,w2, ,wm}. The operator

that performs the aggregation is goal function D.

Latent aggregation is the hidden way in which humans combine judgments upon differentaspects of a subject into one overall judgment upon the subject.

An inductor alternative ai is optimal with respect to the general design goal when it obtainsthe highest final rating di. di reflects how much the design goal is satisfied. Its value isdetermined by the following three elements of the decision problem:

- The implicit or explicit definition of the criteria cj(the objectives and the constraints).

- The allotment of the importance wj- The choice ofgoal function D that combines rij and wj intodi.

For many decision problems only fuzzy information is available, like vague specifications orpreferences imposed by human opinion. To implement this fuzziness, the information can bemodelled by means of fuzzy sets, as shown in the following section. Several elements of thedecision matrix can be fuzzy, like the criteria, the importance, or the ratings. We assume thatin our problem only the criteria are fuzzy, and the ratings are not fuzzy (crisp).

Because ratings in our decision problem rijare crisp values, the decision making method lookslike traditional non-fuzzy decision making with someutility function Uas goal function. The

most important difference is that the concept of this utility function depends heavily onprobability theory for the axioms that characterize it. Our goal function D, is induced fromhuman fuzzy values and from empirical determination of operators that express the hiddenway in which humans combine judgments on different aspects of an alternative into oneoverall judgment.


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3.4.2 fuzzy sets theory

Fuzzy logic has been proposed by Zadeh in the 1960s (reference [34]) as a means to modeluncertainty of non-probabilistic nature. It is an extension of conventional (Boolean) logic. A

good example of this uncertainty is vague, imprecise or uncertain information as used byhuman reasoning. Fuzzy logic introduces the concept of partial truth values that lie between

completely trueandcompletely false. There is a strong relation between fuzzy logic and fuzzysets theory, similarly to the relationship between Boolean logic and conventional set theory.Fuzzy sets theory is introduced as an extension of conventional sets theory, allowing partialmembership in the set. A fuzzy set C is defined in the universe of discourse X via itscharacteristic function, usually called membership function, fc(x) :X [0,1] that is definedas follows:4

fc(x) = 1 x belongs completely to Cfc(x) (0,1) x belongs partially to Cfc(x) = 0 x does not belong to C

The terms membership function and fuzzy set are sometimes used interchangeably. Thevalue of the membership functionfc(x) is calledmembership degree. In Figure 10. an exampleis shown of the fuzzy set A that describes the fuzzy setfc(m) of the human expression about300grams.

Figure 10. fuzzy set of m is about 300g. Figure 11. Fuzzy set with trapezoidal shape.

The valuem=275g belongs with a degree off(x)=0.4 to the set weight of about 300g. The

value m=340g has a degree of membership of f(x)=0.1. The slopes of a fuzzy set can beregarded as fuzzy boundaries. A non-fuzzy parameter value or range that only carriesmembership values 0 or 1 is called crisp. The fuzzy set can also be represented as atrapezoidal figure constructed of four points a., b., c. and d. as shown in Figure 11. This shapedoes hardly differ from the smooth version, while calculations are easier to perform. Pointsb. and c. will be called the shoulder values of the fuzzy set, indicated with subscript s (e.g.mmaxs).

4) Literature on fuzzy sets normally uses parameter (x) to express the membershipfunction. However, because the theory on magnetics uses to express thepermeability of magnetic material, we have chosen f(x) as membership function.


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The formal fuzzy set theory states that the imprecision expressed by a fuzzy set is meant inthe sense of human vagueness rather than the lack of knowledge about a parameter value asin tolerance analysis. However, because fuzzy sets theory is such a powerful modellinglanguage, in this thesis some context dependent modifications have been allowed. Lack of

precise knowledge and other uncertain situations will be modeled by the fuzzy set.

The next section explains how the fuzzy criteria in our decision problem are represented byfuzzy sets.

3.4.3 fuzzy decision criteria

In a by now classical paper, Bellman and Zadeh [1] suggested fuzzy set theory as a suitableconceptual framework for decision making under various objectives and constraints. Theobjectives and constraints, or criteriaci, that have a vague, imprecise or uncertain nature canbe modeled by the fuzzy setfc(x). This fuzzy set represents to what degree an attribute value

of an alternative satisfies the criteria. Figure 12. shows some arbitrarily chosen fuzzy sets fc(x)of criteriacjthat can be imposed on the design of an inductor. Section 4.1.1 explains how theshoulder values are determined:

Figure 12. Examples of membership functions expressing criteria.

Explanation on some criteria of Figure 12.:In Figure 12.a, fuzzy setsfc(B) andfc(Vg) entail the physical constraints imposed on magneticflux densityBand airgap volumeVg. They express to which degree a value ofBorVgbelongsto the set of appropriate values.- The upper boundary offc(B) is saturation flux density Bsat.Bsatis a hard limit imposed

by the physical properties of magnetic material. However, it is uncertain whether theflux density right below Bsat (the near saturation area) is appropriate, so a fuzzyboundary is used.

- The upper boundary of the maximum allowable airgap volume Vg of a core is fuzzybecause of imprecise knowledge of the expert.

These examples of physical constraints are rather independent of the application.


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Figure 12.b shows the fuzzy setsfc(),fc(m) andfc(Ploss) that express designer preferences andapplication requirements.- The maximum temperature rise is determined by a combination of a fuzzy ambient

temperature amb and the crisp maximum allowable temperaturemax of the inductor.- The crisp maximum weightmmaxis prescribed by the application requirements. However,the fuzzy preference of the designer is that a lower weight is more preferred.

Figure 13. clarifies that some criterion on inductor weight can be expressed by means of afuzzy set. An inductor with a weight ofm = 550g, has a judgment or rating of 0.9.

Figure 13. Criterion, fuzzy set and rating.

The examples have only described how the criteria are set up qualitatively. However, rating

rijwill be numerically expressed on a scale from zero to one. The actual judgement expressedby such a value can be explained by for example a scale of five linguistic expressions(reference [27]):

Figure 14. Linguistic scale for the evaluation of the ratings.

VerbalRating Appreciation:

1 Very Good

0.75 Good0.5 Fairly Good0.25 Poor0 Very Bad


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Two examples of judgment are compared, as shown in Figure 15.. An inductor alternative hasa weight ofm=750g and a temperature rise of=50C. These attribute values are assessedby means of criteria fc(m) andfc() imposed in for example an aircraft application:

Figure 15. Assignment of two attribute ratings to some inductor alternative.

In Figure 16., the respective judgments are written down numerically and verbal. If we wouldjudge the inductor on only weight and temperature rise, what final ratingdiwould the inductorhave?

Figure 16. Judgments upon two attributes weight and temp.rise. Inwhat final rating do they result?.

ratings verbal expr.

m = 750g r= 0.25 Poor

= 50C r= 1 Very Good

final rating: d= ... ...


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In this section has been described how fuzzy criteria cjare modeled as fuzzy setsfc(x). In thisway, each single attribute valuexof an alternative is judged, receiving a rating of performancerij. Now, goal function D has to combine all single ratings into a final rating di. The finalrating di has to approximate the human opinion on alternativeai. Therefore, we derive D by

investigating how a designer comes to a final judgment. Figure 17. gives an overview of thedecision matrix and used parameters:

Figure 17. Overview of decision matrix.

The next section gives an explanation on criteria aggregation and multi step decision making.

3.4.4 aggregation

In the previous section is shown that the decision criteria can be modeled by means of fuzzysets, taking into account subjective factors like human opinion or imprecise information. Thesecriteria assign an individual rate of suitability or preference to the attributes. To come to a

useful final judgment of an alternative, its attribute ratings have to be combined in such a waythat final rating di will be equal to the judgement of a human decision maker. This is theactual decision making, performed by means of goal function D.


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Goal function D aggregates the fuzzy criteria into a so called fuzzy decision in themultidimensional space of attributes (see [1]). In this continuous space, a discrete alternativeresults in one crisp final ratingdi. Hence, criteria aggregationD can be viewed as a matter ofaggregating fuzzy sets by means of fuzzy set-theoretic operations. Figure 18. illustrates criteria

fc(m) and fc() that are aggregated by means of the example goal functionD = min[fc(m),fc()]. An alternative with weightm=450g and =50C obtains a final ratingdi=min[0.5, 1]=0.5.

Figure 18. Example of final ratingdiof alternativeaiby aggregation of criteriaf(m)andf().

In this example we have chosen the minimum function as goal function. However, theproblem of which aggregation operator to use has not yet been considered. Since long, manydifferent descriptions have been defined of operators that might correspond with latent humanaggregation. Some overviews and ideas can be found in references [35] (pp.23-43), [7] (pp.73-98), [22], [18], [20], [21] or [19]. Why would we do so much effort to try to model a humandecision? The advantage of finding the right operator is that the resulting final ratings willapproximate a realistic judgment of the designer. This makes the artificial decision processmore reliable and efficient.

Three basic classes of aggregation operators can be distinguished: operators expressing theintersection of fuzzy sets (conjunction, t-norms), the union of fuzzy sets (disjunction, t-conorms) and the class of averaging operators, which operatebetweenintersection and union.An infinite number of possible definitions for union, intersection or average can be chosen.

To illustrate this, Figure 19. shows the respective areas where the outcomes of the aggregationbetween f() and f(m) are located. Note that, because alternative ai is crisp, an aggegationoperator combines crisp ratingsrij. The concept, however, is one of fuzzy latent aggregation.


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Figure 19. Mapping of t-norms, t-conorms and averaging operators thataggregate two criteria.

Table II. shows somenumerical results of twoinductor alternatives a1and

a2 that are assessed onweight and cost. Weassume that these twoattributes have equal imp-ortance.

The operator might correspond with a human connective if the values of the final ratingsr{0 1.0} equal the human judgment:

Table II. Example of the aggregation with t-norms, t-conorms and compensatory operators.

final ratingwith t-norm

final ratingwith t-conorm

final ratingwith average

ratingweight ratingcost min prod-uct max algebr.sum geom.mean arith.mean

inductora1

0.3 0.7 0.30 0.21 0.70 0.79 0.46 0.50

inductora2

0.5 0.5 0.50 0.25 0.50 0.75 0.50 0.50

Using a single t-norm or t-conorm as decision operator implies that there is respectively fullcompensation or no compensation when ratings are compared. This is shown in Table II.where the final ratings resulting from t-norms are very low (bold), while those from t-conormsare very high (underlined). In human decision making, however, there is usually a trade-offbetween criteria. In that case, aggregation performed by the human mind shows compensatorybehaviour [36]. Compensatory operators can be based on a combination of t-norms and t-conorms or can be not based on these norms, like averaging operators. The result of theaveraging operator is shown in Table II. on the right hand side. Criteria can have unequalimportance. The aggregation operator should take these importance into account by means ofsome parameterwj.


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A way to observe a whole range of operators is to introduce an extra parameter in theoperator. Two examples of parametric operators with unequally weighted criteria are:

From Zimmerman and Zysno [36]:

=0 : product operator

(3)

D1() (

x

i 1

rw

j

ij )(1 ) (

x

i 1

(1 rij)wj)

with [0,1] andx

j 1wj x

=1 : algebraic sum

rij and wj are the ratings and the respective importances. x is the number of attributes.can

be interpreted as a grade of compensation. As increases from zero to one, the operatorchanges from non-compensatory to full compensatory.

Another parametric operator is from van Nauta Lemke [22] [17] [18]:

(4)

D2(s)

x

j 1

wjr sij

1/s

with s /{0} andx

j 1

wj 1.

Also D2(0)

x

j 1 rwj

ij

This averaging operator unifies many commonly used operators:

s- : minimum operator,s=-1 : harmonic mean,s0 : geometric mean,s=1 : arithmetic mean,s=2 : quadratic mean,s : maximum operator.

Parameter s can be interpreted as an index of optimism. As s increases from minus infinityto plus infinity, larger membership values will have more and more effect on the final ratingof the aggregation operator. Neutral operators are the geometric and arithmetic mean.

3.4.5 decision tree

A means to get ideas about which operators and importances to choose, is to draw up adecision tree, that shows the decision hierarchies. Attributes that play a similar role in thedecision process are grouped. When necessary, it is allowed to put the same attribute in morethan one group. The attributes within groups are mutually traded-off against each other,

resulting in a sub-decision that must correspond to the judgment a human designer would passon that group. Then, the ratings of the various groups are assigned. Because a human decision


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maker often decides with the help of groups and hierarchies, the aggregation operators andimportances can be found in an intuitive way from this decision tree. An example of adecision tree for an inductor in an airplane application is given in Figure 20..

Figure 20. example of a decision tree: tree for an AC-inductor in an aircraftapplication.

Such a hierarchical structure provides a better insight in the trade-offs between attributes,because of the following features:

- comparable attributes can be put together,- different operators can be used for different steps,- the importances and aggregation operators can be found more easily, because the

tree resembles the way humans decide

In this chapter we have seen what magnetic component design is, how it is done traditionally,and we have introduced a way to make decisions by judging the alternatives by means offuzzy criteria and aggregation operators. In the next chapter, a system is set up that performsthe necessary calculations and fuzzy ranking by means of fuzzy MADM in order to determinea preferred set of component alternatives.


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40 DSS for AC-inductor Design


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41DSS for AC-inductor Design

4. DSS FOR AC-INDUCTOR DESIGN

During the last three decades, computers have been used to an increasing degree to supporthuman decision making in different ways. Nowadays, these Decision Support Systems arewidely used throughout many different industries. Some examples of DSSs in variousapplications are [37], [38] and [39]. The advantage of a Decision Support System (DSS) isthat it couples the computers number-crunching and memory capability with human commonsense and subjective judgment. Therefore, it is expected that a DSS makes the search for theoptimal component more efficient and effective than with the traditional human designmethods.

The DSS proposed in this thesis will assist a designer of magnetic components by selectinga set of appropriate component alternatives and ranking them depending on how well theysatisfy the criteria. The DSS cannot replace the human designer, because the latter usesirreplaceable subjective and context sensitive decision criteria. The final decision will thereforebe a trade-off between computer suggestions and human subjective judgment. To improve thequality of this final decision, we can also implement some user interaction in the DSS duringthe selection and ranking process.

To create the DSS, firstly the information that is necessary to design a component is deter-mined. Then a DSS structure is proposed for the example of an AC-inductor.

4.1. Design information

To construct the DSS, several kinds of information have to be retrieved, such as the designcriteria and expert knowledge like assumptions, equations and rules. A lot of this informationcan be retrieved by interviewing an expert. However, because the approach of the computerdesign procedure is very different from the experts way, not all information can be obtainedat once. Therefore, there is a need for a continuing personal working relationship with theexpert during the construction of the DSS.

This section defines the criteria and expert knowledge necessary to construct the DSS. Thetypical example of the design of an AC-inductor used in an aircraft is continued.

4.1.1 design criteria

Most of the criteria (objectives and constraints) contain vague, imprecise or uncertainboundaries. For simplicity, these fuzzy criteria will be modeled by means of membershipdegrees fc(x). The trapezoidal fuzzy sets expressing the criteria have outer boundaries f(cmax)and f(xmin) dictated by the hard limiting constraints and crisp specifications. As alreadymentioned, the fuzziness near these outer boundaries may be caused by various types ofuncertainty. The meaning of the shoulder values f(xmaxs) and f(xmins) of the trapezoidal fuzzy

set is to represent some point between where the information is supposed to be preciselyknown or correct and where the information is uncertain or vague.


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As we have seen in section 3.2, the criteria (objectives and constraints) can be split in physicalconstraints, application specifications and designer preferences. We will describe the fuzzycriteria respectively.

1. Physical constraints

The physical constraints are the physical limits of the alternatives. In our example of the AC-

inductor, the limits are the maximum of airgap volume Vg, the maximum of flux density Band the maximum of temperature rise . Some numerical examples are given that are validfor the core E55/28/21 with ferrite grade 3C80. Also, L=4mH andL=4A.

a. Airgap volume fc(Vg)

The upper boundary of the airgap volumeVgdepends on the geometry of the core. Therefore,each core has its own specific upper limit on airgap volume.- The maximum airgap volumeVgmaxis caused by cross-over of magnetic flux as explained

in A.3.3. The core parameters Ac, Awi and wi, explained in appendix A, Figure 43.,describe the upper limit in equation (A.24):

- The shoulder valueVgmaxs is determined by a rule-of-thumb of the expert (section A.3.3,

VgmaxwiA

2

c

5Awi Ac

Figure 42.):Vgmaxs=0.2Acw

- The physical lower boundariesVgmin and Vgmins are determined by the maximum flux

density max.B

The resulting criterion f(Vg) for core E55/28/21 is shown in Figure 22.a

b. Flux density excitation fc( )B

- The upper boundary maxof the flux density

Figure 21. Saturation flux densityBsatand bend-ing point.

Bin the core material is the saturation flux

densityBsat. A flux density higher than thismaximum is not possible.

- Flux densities in a near saturation state arenot very appropriate and involve strong non-linear behaviour as can be seen in Figure 21.As shoulder value Bmaxs of the fuzzy cri-terion will chosen the shoulder valueBbendofthe BH-loop. Below Bbend the core materialis in normal operational state. The fuzzy

upper boundary of fc( ) is shown inBFigure 22.c


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- The lower limits mins and min are not determined by material properties, but by theB Bamount of energy that can be stored in the airgapVg. Because there exists a fuzzy upper

airgap boundaryVgmaxsandVgmax, the lower fuzzy boundary off( ) can be calculated andBis shown in Figure 22.b. for core material 3C80. Equation i. is equation (A.46)):

Bmin

oLL2

Vgmax, and Bmins

oLL

2

Vgmaxs

The physical constraints f(Vg) and f( ) that have just been described are fuzzy boundariesBdepending on respectively core geometry and core material. These boundaries can be com-bined into one unique membership function f(N). f(N) expresses the appropriateness of thephysical state of the component when a certain number of turns is used on a certain core. Theadvantage of this function is that N is a visible attribute of a component alternative, and

directly depends on the hidden attributesf(Vg) andf( ). Figure 22. shows howf(N) is derived.B

Figure 22. Fuzzy criteriaf(Vg) andf(B) combined into membership function of physical appropriateness f(N).

f(Vg) andf( ) are aggregated intof(B) by means of the minimum operator (Figure 22.d.). ThisBmeans that the judgement on physical appropriateness is always based on the attribute thatscores worst. The rating of the worst attribute cannot be compensated by a better rating of theother. Also, the aggregation presupposes that the criteria have equal importance. After the

aggregation, the resulting f( ) is calculated into f(N) by means of equation ii., which isB(A.45). Note that Ncan only adopt a discrete number of turns.


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c. Temperature riseAn exact estimation of the temperature rise of a component needs extensive calculations.For practical situations, more simple equations like equation (A.39) can be used:

In this case, is the temperature rise at the surface of the component.

PtotcoolAcool

- The maximum allowable temperature risemax of the component is found using themaximum temperature rise at the hot spot: (A.40)

max : crisp maximum allowable temperature of the component. maxis determined

max max amb hot

by the component materials:curiefor ferrite, or somemaxof wire insulationor coil former (typical value: 125C).

amb : ambient/environment temperature.hot : extra temperature rise at the hot spot of the component compared to the

temperature rise at the surface of the core

However, more a lower temperature than themaximum is appreciated.

- To determinemaxs

, we use the uncertainty of thetemperature rise at the hot spot hot inside thecomponent. We take for ambient temperatureambits crisp worst case value. Ifhotmin


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2) Specifications

The component specifications are the constraints imposed by the application. Some exampleshave already been mentioned in section 2.3.

Thefilter shown in appendix B has the following specifications:uo = 100sin(t)V : output voltageio = 4sin(t)A : output current*s = 100V : input pulse peak voltage

fp = 20kHz : pulse frequencyR = 20 ohm : load resistance

Appendix B describes that these values may result in the following electrical specifications:L = 4mH : inductance

L = 4A : inductor peak current(C = 40F : capacitance)

These specifications are provided as exact (crisp) values, hereby forming strict limiting bound-aries. However, this assumption does not always make sense, because in practice specificationshave a range of tolerance. For example, as we can see in appendix B, there exists a substantialrange of tolerance around inductanceL=4mH of the AC-inductor, becauseL is achosenvaluewe can control. If such a range ofL would be allowed, then the set of suitable alternativesmay be larger than the case when only a single value for L is allowed. This means that the

unjustly crisp specification ofL might obstruct the finding of a real optimal component. Thisis also the case for other crisp specifications, like the inductor current. Nevertheless,L andLare assumed crisp values in this thesis for simplicity, but fuzziness of these parameters is arecommendation for further research.

If the DC-AC inverter will be used for some airplane application, the following crisp upperboundaries specifications that determine fc(Cmax) can chosen:

m < 600g : total weightV < 140cm3 : total sizeK < .80,- : total costPtot < 20W : total power losses

amb = 50C : ambient temperature during operation

3) Designer preferencesAlthough the designer has to take the above-mentioned limitations into account, he usually hassome personal preferences about the above mentioned properties of the component. The prefer-ences of the designer depend on aspects such as knowledge and experience, applicationcontext and other human insights.


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Figure 25. shows typical fuzzy sets that reflect some abitrary preferences of designer of anAC-inductor:

Figure 25. Designer preferences, the specifications taken into account.

We conclude that the trade-offs that are made between the physical constraints, specificationsand preferences are usually trade offs-between soft criteria.

4.1.2 additional expert knowledge

Besides the preferences and knowledge included in the decision criteria, additional expertinformation is retrieved in order to realize a more efficient decision procedure. The expertinformation may consist of rules, procedures, assumptions or any other kind of applicableknowledge.

For the design of an ac-inductor, the following information is retrieved during the constructionof the decision procedure:

a. The equations and assumptions expressing the relations between the component para-meters.

Example: We assume that the energy stored in the inductor is negligible, except for theenergy stored in the airgap. We assume that the cooling factor coolis equal for allcomponent alternatives.

Many approximations have been made in the equations to reduce the complexity of the design

problem. Hence, there is no guarantee that the attributes of the component alternatives willtake the calculated values if constructed in practice. Human design experts cope with this


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problem, too. Because the DSS proposes a whole set of preferred alternatives, the designerwill have numerous possibilities available for assessment. Without the DSS, he would haveto pass the iterative design procedure several times to find more than one possible alternative.

b. Heuristic information and the rules of thumb.

Example:

- Typical heuristic information is the knowledge about the general multi step character ofthe decision structure and the sequence of decision steps. The first decision steps dis-tinguish the set of possible alternatives using the limiting boundaries. The most efficientsequence of the steps is found to be:

1. core selection2. calculation of the range of number of turns for each core

3. wire selection for each core4. determination of the inappropriate numbers of turns.

- As described in section 4.1.1, a shoulder value for a membership function f(Vgmaxs)isdefined by means of a rule-of-thumb of the expert.

- More applicable knowledge is thinkable, such as the information that round wires areeasier to wind and therefor simplify the assembly of an alternative, while they are alsoless costly.

Not all expert knowledge is useful for implementation in the DSS. Some rules-of-thumb areused by a human designer to reduce the number of calculations. For computer implementation,these simplifications only reduce the set of possible alternatives unjustly. Examples are:- The expert states that temperature rise should not be too low because a low temperature

during operation is caused by a large core. This relationship between temperature riseand volume is taken into account in MADM, because all attributes are traded off againsteach other anyway.

- The expert also states that thewinding area of the core should be completely filled withwire in order to get the most efficient inductor. Because a computer has muchcomputation power, we can calculate the efficiency for all alternatives. It might be the

case that some alternative does not have a filled window area, but is still better thansome other alternative.

We now have defined the information necessary to rank alternatives by means of MADM andthe information for how to perform the ranking. In the following section, section 4.2, thestructure of the DSS will be set up.


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4.2. DSS structure

In this section, a structure for a DSS is proposed that selects and ranks the alternatives for anAC-inductor. The DSS is structured in such a way that the calculation effort is reasonablysmall, while no alternatives are rejected unjustly. Therefore, the decisions are taken in ahierarchical manner. At each level of the hierarchy, a number of alternatives is rejected, whichare not within the set of feasible alternatives. The initial set of alternatives is the set of allpossible combinations of cores, wires and number of turns. Figure 26. gives a basic illustrationof the steps: steps 1. and 2. distinguish the (groups of) feasible alternatives. Step 3. is aranking step.

Figure 26. Structure of Decision Support System that leads to efficientand effective decision making

A detailed explanation of the steps is given in the following sections:


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4.2.1 Initial selection

The initial selection determines the set of useful inductor alternatives ai. The totally

inappropriate alternatives are rejected. This means that the component elements or thecomplete inductor alternatives that have attributes that lie outside the (crisp) outer boundariesof the criteria will be rejected. The outcome of the initial selection is a set of inductoralternatives that satisfies the criteria to some degree. Figure 27. illustrates the selections andrejections:

Figure 27. Initial selection cores, wires and number of turns. The set of possible alternatives results,but no rating have been assigned yet.

As first step, not shown in Figure 27., physical constraints f( ) and f(N) are calculated forB

each core by means ofLand L. The respective maxima and minima of maxand Nmaxare usedB

as outer boundaries. The following three rejection steps are performed, resulting in a set ofpossible alternatives:

1) core preliminary rejected if:

- the core cannot store the required maximum energy when the maximum values maxandBVgmax are used,

- the minimum core losses caused by minproduce a temperature rise that is too high,B- the volume, weight, or cost or minimum core losses of only the core are already too

high.


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2) applied for a certain core of the remaining set, a wire is rejected if:- the minimum wire losses caused byNmin added to the minimum core losses caused by

min produce a temperature rise that is too high,B- the wire does not fit in the former when the minimum number of turnsNminis applied.

3) A provisional range ofNis determined for each remaining core/wire combination {xi,yj}.- Rejected are the combinations of {xi,yj,N} (i.e. the inductor alternatives) that do not

satisfy one or more of the outer limits of the criteria on weight, cost, core losses andtemperature rise.

The possible inductor alternativesairemain. Up to now, only the hard limiting boundaries ofthe criteria are used. To come to a final ranking based on attribute ratings, the fuzzyboundaries of the criteria have to be known.

4.2.2 removal of dominated alternatives

If the criteriacj are known by means of membership functions fc(x), then all ratings rij of allpossible alternatives ai are known. The second decision step will reject the alternatives thatare dominated by any other alternative. A dominated alternative has all ratings lower than theratings of some other alternative. A dominated alternative is irrelevant, because there willalways be another alternative that is better. In each kind of ranking process, these alternativesalways have the lowest ranking places.

If many attributes are used in the decision process, than the computational effort to distinguishthe dominated alternatives might be very high. In that case, it is possible to allow the

dominated alternatives to the final ranking part. If a ranking algorithm is applied, then the setof dominated alternatives will always be the set of least preferred alternatives.

4.2.3 final ranking

The final ranking determines the order of preference of the remaining alternatives with respectto the design goal. In section 3.4 we have introduced Fuzzy Multiple Attribute DecisionMaking as a means to assign final ratings to the alternatives. The required fuzzy criteria cjhave been set up in section 4.1.1. They result in ratings rij for each attribute value of analternative. The attributes are:

f(N) : physical appropriateness

m : weightV : sizeK : costPtot : power loss : temperature rise

The ratings are combined into a final rating diby some goal functionD. The goal function isdefined by one or more aggregation operators and importances wj and can be found byobserving the way a human designer judges an alternative. The alternatives with the highestratings form the set of most preferred alternatives.


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In chapter 5 the results of an AC-inductor in three applications will be described:1) airplane application,2) locomotive application,3) mass product (e.g. car)

In this section, the determination of the final rating is extensively described by means of thefirst example of an AC-inductor in an airplane application.

CRITERIA AND IMPORTANCES:

Figure 28. Criteria imposed on inductor in airplane application.

The preferences are chosenas in Figure 28.

These criteria do not have equal importance. The designer has knowledge about theimportance, expressed by the following statements:1) The most important aspects are the weightm and the sizeVof the inductor.m is more

important thanV, because especiallym is an extremely expensive aspect in an aircraft.2) Reliability is also an important aspect. Reliability is connected with the temperature

rise and with physical appropriateness f(N). However, these attributes are not very

strong indicators of reliability and are therefore less important than m and V.3) The operational cost caused by the power loss Ptot is much more important than thematerial and production cost Kof the component

4) The operational cost of the component (Ptot) and the cost of material and production (K)are less important than the other attributes.


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DECISION TREE:

The importances can be assigned intuitively from the statements described before. Later in thissection, some other method is used to verify these importances. The decision tree of

Figure 29. can be set up:

Figure 29. Decision tree of AC-inductor in aircraft application.

CHOICE OF OPERATOR:

It is very difficult to retrieve the operators I...IV of Figure 29. from a human decision maker,because he often cannot explain the way he aggregates two or more attributes. Also, a veryprecise identification may be considered of little use, because we only need to realize certainproperties of an operator. Therefore, operators that emperically give adequate final ratings (i.e.ratings that are comparable with human judgment) will suffice for this thesis.

Operators I, II and III judge the attributes that play similar roles in the decision problem.

When considering one group, a decider in magnetic component design wants to make someneutral compensatory trade-off between attributes. In that case, an averaging operator likethe weighted mean seems a natural choice. The difference between harmonic, geometric orarithmetic mean can be explained by the interpretation of parameter s of equation (4) (section3.4.4). Parameter s expresses the grade of optimism.


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Some arbitrary numerical results of the different averaging operators are shown to determinewhich is the most appropriate one:

ratings s=-1 s0 s=1r

1

r2

harm. geom. arithm. mean

0 1.0 0 0 0.50.1 0.9 0.18 0.30 0.50.2 0.8 0.32 0.40 0.50.3 0.7 0.42 0.46 0.5

0.2 0.7 0.31 0.37 0.450.3 0.6 0

a fuzzy decision support system for magnetic component design thesis-d.holt-1996-08

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