magnet introduction (1)

8/12/2019 MagNet Introduction (1)

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An Introduction to

MAGNETfor

Static 2D Modeling

Infolytica

Corporation

J D Edwards


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Post: Documentation ManagerInfolytica Corporation300 Leo ariseau! "uite ####

Montr$al! %u$&ec '#( )*3Canada

e-mail: info + inf o ly ti c a.com

, #00- Infolytica Corporation

ll rights reser/ed. his document may not &e reproduced! translated into another language!

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he information in this document is su&1ect to change without notice.
mailto:[email protected]:[email protected]:[email protected]


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Contents

Chapter 1 Introduction 1

Overview .......................................................................................................1

Modeling in 2D and 3D .................................................................................3

Magnetic concepts .......................................................................................4

Using MagNet effectively .............................................................................8

Getting help ..................................................................................................

Chapter 2 Tutorial: Ccore Electro!a"net 11

!ntrod"ction ................................................................................................11

Device Model .............................................................................................. 12

#"ilding the $odel ..................................................................................... 14

%olving the $odel....................................................................................... 21

&ost'processing ......................................................................................... 24

Modifying the $odel................................................................................... 28

&ostscript.................................................................................................... 3

Chapter # Case $tudies: Translational Geo!etry %1

!ntrod"ction ................................................................................................41

('core electro$agnet ................................................................................. 41

Magnetic latch with a per$anent $agnet .................................................4)

#"s*ar forces ............................................................................................. 48

+ield in a cylindrical cond"ctor ................................................................. )4

,ylindrical screen in a "nifor$ field .........................................................)-

ransfor$er e/"ivalent circ"it .................................................................. -2

0aria*le el"ctance %tepper Motor ..........................................................-

inear synchrono"s $otor ........................................................................ 2

Chapter % Case $tudies: &otational Geo!etry '(!ntrod"ction ................................................................................................

M"t"al ind"ctance of coaial coils............................................................ 83

0D displace$ent transd"cer ................................................................. 8-

Magnetic p"ll'off force ............................................................................... 8

Moving'coil transd"cer .............................................................................. 3


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Chapter ) $criptin" ('

!ntrod"ction ................................................................................................

(a$ple $odel ...........................................................................................

%cript for the $odel....................................................................................

,reating a new script ............................................................................... 151

6"to$ation with (cel ............................................................................. 152

Appendi* A +ield E,uations and $olution 111

+ield e/"ations ......................................................................................... 111

#o"ndary conditions and sy$$etry....................................................... 113

N"$erical sol"tion ................................................................................... 11)

Appendi* - Ener"y. +orce and Inductance 11(

%tored energy and co'energy .................................................................. 11

+orce calc"lation ...................................................................................... 121

!nd"ctance calc"lation ............................................................................. 122

Appendi* C /pen -oundary I!ple!entation 12)

&eferences 12(


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Chapter 1 1Introduction

2007 Infolytica Corporation

Chapter 2

!ntrod"ction

Overviewhe principal aim of this document is to introduce new users to the power of Maget for sol/ing

#D static magnetic field pro&lems. tutorial with detailed instructions ta4es the first5time

user through the most important features of Maget. his is followed &y a series of case

studiesillustrating modeling techni6ues and introducing further features of the pac4age. he document

concludes with an introduction to ad/anced features that ma4e Maget a uni6uely powerful tool.

7hat is MagNet

Maget is the most ad/anced pac4age currently a/aila&le for modeling electromagnetic de/ices

on a personal computer. It pro/ides a 7/irtual la&oratory8 in which the user can create models

from magnetic materials and coils! /iew displays in the form of field plots and graphs! and get

numerical /alues for 6uantities such as flu9 lin4age and force. Maget user needs only an

elementary 4nowledge of magnetic concepts to model e9isting de/ices! modify designs! and test

new ideas.

Maget is designed as a full 3D5modeling tool for sol/ing static magnetic field and eddy5currentpro&lems. Many de/ices can &e represented /ery well &y #D models! so Maget offers the optionof #D modeling! with a su&stantial sa/ing in computing resources and solution time. With #Dmodels! Maget can also handle pro&lems where currents are induced &y the motion of part ofthe system.

feature of Maget is its use of the latest methods of sol/ing the field e6uations and calculating6uantities such as force and tor6ue. o get relia&le results! the user does not need to &e an e9pertin electromagnetic theory or numerical analysis. e/ertheless the user does need to &e aware ofthe factors that go/ern the accuracy of the solution. ne of the aims of this document is to showhow the user can o&tain accurate results. In #D! pro&lems can &e sol/ed /ery rapidly! so it isusually not necessary to consider the trade5off &etween speed and accuracy. In 3D modeling! onthe other hand! this is an important consideration.

;or the ad/anced user! Maget offers powerful facilities for user5defined ad1ustment of the

model parameters! calculation of further results from the field solution! and control of the

operation of the pac4age with scripts and scripting forms.


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2 Introduction to Ma"Net

i$itations

he information gi/en in this document has &een prepared specifically for the entry5le/el

/ersion of Maget. his /ersion is restricted to static magnetic fields and #D models! without

facilities for parameteri


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Chapter 1 #Introduction

Modeling in 2D and 3D"ome practical pro&lems are essentially three5dimensional e9amples include the rotor of a claw5

pole alternator and the end5winding regions of rotating C machines. ro&lems of this 4ind

re6uire the full 3D modeling capa&ility of Maget. In many cases! howe/er! a #D model will

gi/e useful results. here are two common types of de/ice geometry that allow 3D o&1ects to &e

modeled in two dimensions: translational geometry and rotational geometry.

ranslational geo$etry

ranslational geometry means that the o&1ect has a constant cross5sectional shape generated &y

translation mo/ing the shape in a fi9ed direction. he diagram &elow shows a C5core formed in

this way.

With translational geometry! any slice perpendicular to the a9is has the same shape. otating

electrical machines can often &e represented in this way! and so can many other de/ices such as

transformers and actuators. Ine/ita&ly this #D appro9imation neglects fringing and lea4age fields

in the third dimension! so the model must &e used with caution. he shape is usually drawn in the

x-yplane! with thez-a9is as the a9is of translation.

otational geo$etry

otational geometry means that the o&1ect has a shape formed &y rotation a&out an a9is! li4e

turning on a lathe. he diagram &elow shows an o&1ect formed in this way from the same &asic C

shape used in the diagram a&o/e.

&1ects with rotational geometry are usually descri&ed in cylindrical polar coordinates! with thez5a9is as the a9is of rotation. he rotated shape is then defined in an r-zplane! which ma4es anangle with the 3Dx-a9is. his geometry differs from translational geometry in two important

respects. ;irst! it is a true representation of a real 3D o&1ect! so highly accurate solutions arepossi&le. "econdly! there are different e6uations to &e sol/ed! and different methods re6uired forcalculating 6uantities such as force and inductance. ;or all &uilt5in calculations Maget handlesthese differences automatically! &ut the user needs to &e aware of the difference when using theCalculator facilities.

In Maget! the #D cross5section of a rotationally symmetric model must &e drawn in thex-y

plane! with they-a9is as the a9is of rotation. hex-y coordinates then correspond to the r-z

coordinates of the con/entional cylindrical polar coordinate representation.


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% Introduction to Ma"Net

Magnetic conceptsMaget can &e used to model practical de/ices without 4nowing anything a&out the differential

e6uations of electromagnetism or the numerical methods used to sol/e them. his section re/iews

some &asic magnetic concepts that are re6uired for ma4ing effecti/e use of MagetE more

ad/anced topics are co/ered in appendi9 . he system of units used is the "I or M@" system!

although other systems will &e mentioned in the conte9t of magnetic materials.

Magnetic fl" density #

he fundamental magnetic concept is the magnetic field descri&ed &y the /ector B! which is

termed the magnetic flux density. In two dimensions this field is commonly represented &y cur/ed

lines! 4nown as flu9 lines! which show &oth the direction and the magnitude of B. he direction

of a line gi/es the direction of B! and the spacing of the lines indicates the magnitudeE the closerthe lines! the greater the magnitude. he diagram &elow shows the flu9 plot for a simple

electromagnet where the C5shaped steel core on the left attracts the steel &ar on the right. he

two sides of the magneti


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+lu*densityB

0T1

;rom the point of /iew of the de/ice designer! the magneti


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+lu*density

B

0T1

&er$anent $agnets

ermanent magnets ha/e the property that some magneti


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Using MagNet effectivelyhis section contains a few practical pointers to getting the &est out of Maget. "ome of the

suggestions may not ma4e much sense until the user has had some e9perience of using Maget!

at least to the e9tent of wor4ing through the tutorial in chapter #.

he principle of progressive refine$ent

he time ta4en for Maget to sol/e a pro&lem will depend on the comple9ity of the model andthe desired solution accuracy. ;or this reason alone it is not ad/isa&le to attempt an e9ceedinglydetailed model of a practical de/ice with e/ery geometric feature faithfully copied. here is alsoa practical reason for a/oiding comple9 models initially. he first model is almost certain tocontain mista4esE if it is /ery detailed it will ta4e a long time to sol/e! and an e/en longer time to

re&uild when the solution has re/ealed the mista4es.

It is generally &est to &egin with a /ery simple model that preser/es the essential features of the

de/ice. "hapes and dimensions can &e simplified. "ome parts do not need to &e modeled at all.

;or e9ample! real coils will ha/e non5magnetic insulation separating the coil from the steel coreof the de/ice. here is no need to model such insulationE the coils can &e drawn touching the steel

without any significant error. he case studies in chapters 3 and ) gi/e some indication of whatcan &e done with simple models.

;or the first solution of a new model! it is desira&le to get a flu9 plot as 6uic4ly as possi&le!&ecause the flu9 plot is an effecti/e tool for re/ealing errors in the structure of the model. tthis stage! there is no need to use the powerful adaption feature of Maget to impro/e thesolution accuracy.

When the new model is producing a sensi&le flu9 plot! and the numerical results for forces!

tor6ues and inductances are plausi&le! the sol/er and adaption options can &e used to impro/e the

accuracy. he case studies gi/e e9amples of the settings that may &e re6uired.


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Getting acc"rate res"lts

Maget uses the finite5element method to sol/e the field e6uations. ;or a #D model! the entire

region is su&di/ided into a mesh of triangular elements! and within each element the true field is

appro9imated &y a polynomial. he accuracy can &e impro/ed &y increasing the order of the

polynomial: this is one of the sol/er options. It can &e further impro/ed &y using smaller elementsin critical regions of the model! which is done automatically when the user sets the adaption

options.

With any numerical method! perfect accuracy is unattaina&le. >/en with full use of the optionsfor impro/ing the accuracy! the solution generated &y Maget will contain errors. In most casesthese errors will &e insignificant! and are li4ely to &e smaller than the changes caused &ymanufacturing tolerances or /ariations in the magnetic properties of the materials.

Calculated /alues for forces and tor6ues are particularly sensiti/e to errors in the field solution! sothese /alues are li4ely to change significantly as the solution accuracy is impro/ed. If these are

the 6uantities of interest in the de/ice! then it is sensi&le to continue refining until the /aluesappear to ha/e con/erged. If you 4now that some tor6ue /alues or force components should &e


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Chapter 2 11Tutorial: Ccore Electro!a"net


Chapter #

"torial: ,'core (lectro$agnet

!ntrod"ctionhis chapter ta4es the user through the complete se6uence of using Maget to model a simple

electromagnetic de/ice: the C5core electromagnet shown &elow. he o&1ecti/es are as

follows:

o e9amine the magnetic field in the /arious parts of the magnetic circuit.

o determine the force on the s6uare armature plate.

o determine the self5inductance of the coil.

o modify the model &y changing the coil current! the core material! the shape of thecore! and the position of the armature.

his is an e9ample of a de/ice that can &e represented 6uite well &y a #D model with translational

geometry! e/en though it is not /ery long in the direction of translation. he important lea4age

field &etween the poles is accurately modeled! and the part of the fringing field that is neglected

in a #D model has only a minor effect on the calculation of force. ;ringing and lea4age are

discussed later! on pages #) and ?#. n important limitation of the model! howe/er! is that thecalculated inductance is /ery inaccurate &ecause the lea4age field in the third dimension has &een

neglected. If an accurate /alue for the inductance is re6uired! a 3D model must &e used. esults

from #D and 3D models are compared on page 3J.


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Ar!ature



Device Model

#rief description

he diagram &elow shows the cross5section of the electromagnet with the dimensions inmillimeters. >ach side of the coil is a s6uare of side #0 mm! and the core is 20 mm thic4throughout. he armature and the core each ha/e a depth of )0 mm! perpendicular to the plane ofthe drawing. he coil has 2000 turns! and the current is initially #.0 .

%5 ) 15

25 25 15

25 15 25 %5

Coil side 1 Coil side 2

Core 15

"ince the electromagnet is surrounded &y air! the magnetic field theoretically e9tends to infinity!

ma4ing it an o!en "oundarypro&lem. In practice the field decays rapidly with distance! and is

insignificant at a distance of a&out 20 times the magnet dimensions.

techni6ue is a/aila&le in Maget for e9act modeling of an open &oundary! and for some typesof pro&lem this is the &est approach: an e9ample is gi/en in chapter 3. 'owe/er! for manyde/ices such as this electromagnet! good results can &e o&tained with the simpler techni6ue ofspecifying an outer &oundary at a distance of = to 20 times the magnet dimensions.

Modeling the electromagnet in/ol/es the following steps:

Draw the cross5section of the core. >9tend it in a straight line to form a solid &ody! and specify the material of the core.

In the same way! construct the armature and the two coil sides.

"pecify the coil.

Define the &ounded region of the pro&lem as an air "ox within which the field will &ecalculated.

Instruct the program to sol/e the e6uations and display the

results. hese steps are descri&ed in detail in the ne9t sections.


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Chapter 2

1#

Getting started

he instructions gi/en &elow assume that you are familiar with Microsoft Windows! that you

ha/e installed Maget! and started the application &y dou&le5clic4ing the Maget icon. he

Maget Main window! shown &elow! should &e /isi&le.

8ro9ect ar ;iew window

1 >9amine the Maget Main window! and identify the parts listed &elow.

he ro1ect &ar displays information a&out the model! with ta&s at the top la&eled&1ect! Material! etc.

he Kiew window is the wor4 area where you construct the model and /iew theresults.

*etween the ro1ect &ar and the Kiew window is a /ertical tool&ar with &uttonsfor drawing and &uilding models.

t the top of the Main window! there is the usual menu &ar! and two hori


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1% Introduction to Ma"Net

#"ilding the $odelIn Maget! the default units of length are meters. It is more con/enient to wor4 with dimensions

in millimeters. roceed as follows to change the default units and grid settings.

!nitial settings

1 n the ;ile menu! clic4 ew. lternati/ely! clic4 the ew &utton.

2 n the ;ile menu! clic4 "a/e. lternati/ely! clic4 the "a/e &utton.

"elect or create a suita&le folder for storing the model.

3 "a/e the model with the name C-core electromagnet.

The model name in the $"%ect !age should change to #-core electromagnet.

4 n the ools menu! clic4 "et nits to display a dialog:

) Clic4 the Length drop5down list.

"elect Millimeters.

- Clic4 @ to close the dialog.

n the Kiew menu! clic4 "et Construction rid to display a dialog:

8 "et the >9tent and "pacing /alues as follows:

Minimum (:60 Ma9imum (: 20

Minimum Y:20 Ma9imum Y: 20

( spacing: 5 Y spacing: 5

Clic4 @.


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25 15

Coil side 1

15

Coil side 2

25

Core 15

Ar!ature


Chapter 2

1)

Displaying the grid

Display the whole of the construction grid as follows.

15 n the Kiew menu! clic4 Construction rid.

&ou should see a grid of a fe' ery small !oints( 'idely s!aced.

11 Clic4 the Dynamic Noom &utton a&o/e the ro1ect &ar.

12 In the Kiew window! drag the pointer downwards to


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3 If you ha/e drawn any lines in the wrong place! correct them as follows.

ress >sc to stop line drawing.

Clic4 the "elect Construction "lice Lines &utton. Clic4 the line that you want to delete.

The selected line should turn red.

ress the Delete 4ey.

Clic4 the dd Line &utton and redraw the line.

4 When you ha/e finished drawing! press >sc to terminate line drawing.

) he finished outline of the core should loo4 li4e this:

,o$pleting the $agnet core

1 Clic4 the "elect Construction "lice "urfaces &utton.

2 Clic4 anywhere inside the core.The interior of the core should fill 'ith a red !attern.

3 Clic4 the Ma4e Component in a Line &utton to display a dialog:

4 In the dialog &o9! change the Distance to 40.


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Ar!ature


Chapter 2

1'

) Clic4 the Material drop5down list.

- "croll down through the list and select the material C20: Cold rolled 2020 steel.

Change the ame from ComponentO2 to Core.

8 Clic4 @.

A com!onent named #ore should "e sho'n in the $"%ect !age of the Pro%ect "ar.

Ma;ing the ar$at"re

1 Clic4 the dd Line &utton! and draw the outline of the armature.

Ma4e sure that that there is a = mm gap &etween the armature and the core.

2 Clic4 the "elect Construction "lice "urfaces &utton! and clic4 inside the armature.

3 Clic4 the Ma4e Component in a Line &utton.

The )istance should "e sho'n as +, mm( and the aterial as #/,: #old olled

/,/, steel.

Change the ame to Armature.

Clic4 @.

4 he &1ect page of the ro1ect &ar should show two components: Core andrmature.

0f the name of a com!onent is 'rong( you can edit the name in the $"%ect !age "y

selecting the name and !ressing 12.

%5 ) 15

25 25 15

25 15 25 %5

Coil side 1 Coil side 2

Core 15


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Ma;ing the two coil sides

1 Draw the outline of coil side 2.

&ou do not need to dra' a com!lete square. 3ust dra' the additional lines required

for this sha!e.

2 "elect the coil side! and ma4e the component with the following entries in the dialog&o9:

he Distance should &e )0 mm.

;or the material! select Copper: =.--e- "iemensFmeter.

>nter the ame as Coil side 1.

3 In a similar way! ma4e Coil side #.

4 he &1ect page should show two new components for these coil sides.

Defining the coil

he ne9t step is to lin4 the two sides to form a coil! and to specify the num&er of turns and thecurrent.

1 In the &1ect page! clic4 Coil side 2.

The com!onent name should "e highlighted( and the left-hand coil side mar4ed in the

5ie' 'indo'.

2 'old down the Ctrl 4ey! and clic4 Coil side #.

Both com!onent names should "e highlighted( and the coil sides mar4ed in the 5ie'

'indo'.

3 n the Model menu! clic4 Ma4e "imple Coil.

#oil6/ should a!!ear in the $"%ect !age.

4 "elect the Coil page of the ro1ect &ar! &y clic4ing the Coil ta&.

#oil details should "e dis!layed.

) Clic4 on 2 urn Athe si9th item in the list for CoilO2B.

ress ;#.

The dis!lay should change to an edit "ox dis!laying the num"er /.

Change 2 to 1000 and press >nter.

- Clic4 on 0 rms.

ress ;#.

Change 0 to 2 and press >nter.Although the "ox sho's * current( in this static !ro"lem it is actually the

alue of the constant coil current.

"elect the &1ect page of the ro1ect *ar! &y clic4ing the &1ect ta&.

This dis!lays the names of the model com!onents again.


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Chapter 2

1(

6ir *o

n outer &oundary is added to the model &y creating a new component called an air "ox( whichencloses all the other components. he default &oundary condition for the air &o9 is ;lu9angential Asee appendi9 B! which means that the outer &oundary is a flu9 line. his is a

reasona&le appro9imation if the &oundary is sufficiently far away from the de/ice.

he air &o9 is much larger than the electromagnet! so it is not con/enient to draw it with the

mouse. Instead! coordinates are entered with the 4ey&oard.

If the air &o9 is made in the same way as the other components! it will contain holescorresponding to the shapes of those components. his is undesira&le! &ecause it will causepro&lems later when the model is modified. o pre/ent holes &eing formed! an option must &eselected in the Ma4e Component in a Line dialog! as descri&ed in step 22 &elow.

1 n the ools menu! clic4 @ey&oard Input *ar if there is no chec4 mar4 &eside it.

The 7ey"oard 0n!ut "ar should "e dis!layed at the "ottom of the ain 'indo'(a"oe the *tatus "ar( 'ith a text "ox for entering coordinates.

2 Clic4 the dd Circle &utton.

The *tatus "ar at the "ottom of the 'indo' should sho':"pecify the center point and a point on the radius of the circleP

3 Clic4 in the te9t &o9 of the @ey&oard Input &ar.

If the co5ordinates are not shown as A0! 0B! edit the te9t.

4 ress >nter! or clic4 the >nter &utton.

The status text should change to:

CenterG A0! 0! 0B! adiusG A0! 0! 0B

If the status te9t has not changed! press >nter again.

) Change the co5ordinates in the te9t &o9 to A400! 0B. ress >nter! or clic4 the >nter&utton.

This should create a circle of radius +,, mm( 'hich is too large to dis!lay. The

status text should change "ac4 to: "pecify the center point and a point on theP

If the status te9t has not changed! press >nter again.

- ress >sc to terminate circle drawing.

Clic4 the Kiew ll &utton.This dis!lays the large circle for the air "ox.

8 In the Kiew menu! clic4 Construction rid.

This turns off the dis!lay of grid !oints( 'hich is no longer required.

Clic4 the "elect Construction "lice "urfaces &utton.

15 Clic4 anywhere inside the circle.

#ontinued8


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11 Clic4 the Ma4e Component in a Line &uttonto ma4e the air &o9.

Ignore 'oles must &e acti/e. If the chec4 &o9 is empty! clic4 the &o9.

This is essential9 other'ise later !arts of the tutorial 'ill fail.

he Distance should &e )0 mm.

;or the material! select I.

A0 is at the to! of the list of o' 1requency aterials. )o not select 5irtual Air(

'hich is a s!ecial material in agNet.

>nter the ame as Air o!.

Clic4 @.

12 ;rom the ;ile menu! select "a/e! or clic4 the "a/e &utton.

0t is good !ractice to sae often( in case the !rogram crashes.

0iewing the $odel

o /iew the electromagnet in more detail again! use the


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Chapter 2

21

%olving the $odelMaget uses the finite5element method of sol/ing the electromagnetic field e6uations. his

su&di/ides a #D model into small triangular elements! forming a mesh that co/ers the entire

region. he true field within each element is appro9imated &y a polynomial in terms of the field

/alues at a small num&er of points! and Maget sol/es for the un4nown field /alues at these

points for all the elements. ;or e9ample! a first5order polynomial 1ust gi/es a linear interpolation&etween the field /alues at the /ertices of the triangles.

he accuracy will &e higher with a fine mesh or a high5order polynomial. *y default! a first5order

polynomial is used! which is fast &ut not /ery accurate.

!nitial sol"tion

1 n the Kiew menu! clic4 Initial #D Mesh.

This should sho' the default mesh that agNet uses to sole the field equations.

2 n the "ol/e menu! clic4 "tatic #D.

The *oler Progress dialog should a!!ear "riefly. ;hen the solution is com!lete( thePost Processing "ar should "e dis!layed at the "ottom of the ain 'indo'.

3 Close the ost rocessing &ar as follows:

n the ools menu! clic4 ost rocessing *ar.

lternati/ely! clic4 the 'ide &o9 on the left5hand side of the &ar.

0iewing the fl" plot

1 "elect the ;ield page of the ro1ect &ar &y clic4ing the ;ield ta&.

The 1ield !age has ta"s at the "ottom for #ontour( *haded and Arro'. The #ontour

!age is actie "y default.

2 Clic4 ;lu9 ;unction! then clic4 pdate Kiew.

A contour !lot of the flux function is the ordinary flux !lot. 0t should "e similar to the!lot "elo'. The !lot is irregular "ecause the solution is not ery accurate at this

stage.


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!$proving the sol"tion acc"racy

o impro/e the solution accuracy! the polynomial order of the elements can &e increased as

follows.

1 n the "ol/e menu! clic4 "et "ol/er ptions to display a dialog:

Increase the olynomial rder from 2 to # &y clic4ing the up arrow.

Clic4 @.


;hen the soler finishes( the flux lines should "e smoother( indicating a moreaccurate solution.

3 "a/e the model again.

efining the $esh

Increasing the polynomial order has made some impro/ement! &ut the real pro&lem is that the

mesh is too coarse in parts of the model. Maget can refine the mesh automatically a process

termed ada!tion.

1 n the "ol/e menu! clic4 "et daption ptions to display a dialog:

Clic4 se h5adaption.

In the ercentage of >lements to efine &o9! enter 25

In the olerance &o9! enter 0.5

At each ste!( agNet 'ill select the 'orst 2


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Chapter 2

2#


The *oler Progress dialog sho's the ada!tion ste!s. The !rocess continues until the

change in the calculated alue of stored energy is less than the s!ecified tolerance of

,.


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2% Introduction to Ma"Net

&ost'processingfter a field solution has &een o&tained! other 6uantities can &e calculated and displayed. his is

termed!ost-!rocessing. Maget has a ost rocessing &ar that displays glo&al 6uantities such as

force and flu9 lin4age! and also gi/es access to the Calculator! which is descri&ed in chapter ?. Inaddition! color maps of the field can &e displayed! and the mouse used to display field /alues at

any point in the model.

Getting fl" density val"es

1 In the ;ield page of the ro1ect &ar:

Clic4 the "haded ta&

Clic4 Q*Q "moothed

Clic4 pdate Kiew.

This should sho' a color ma! of the flux density magnitude( su!erim!osed on the flux

!lot.2 n the /ertical tool&ar! clic4 the ;ield ro&e &utton.

Mo/e the mouse pointer anywhere in the model region! ut do not clic".

he "tatus &ar should show two /alues at the left5hand side: the flu9 function!and Q*Q "moothed.

0gnore the flux function. The other num"er is the flux density magnitude at the

!osition of the !ointer.

Mo/e the pointer without clic4ing! and o&ser/e the change on the "tatus &ar.

3 'old the mouse still! with the pointer anywhere in the model region! and clic4 once.

new area should appear &elow the Kiew window! called the e9t utput &ar.

his displays the coordinates of the point! and the flu9 density magnitude. >/ery clic4 in the model region displays a new set of /alues.

4 Close the e9t utput &ar &y clic4ing this item on the ools menu! or clic4ing the'ide &o9 on the left5hand side of the &ar.

otice two features of the magnetic field in this de/ice. ;irst! the magnetic field in the airgap

&etween the poles and the armature is not confined to the pole region! &ut spreads into the

surrounding airE this is termedfringing. "econdly! some flu9 ta4es a short cut across the space

&etween the poles! instead of crossing the airgap and passing through the armatureE this is termed

lea4age.


29/137


Chapter 2

2)

Graphs of fl" density

Maget pro/ides four items 55 three on the Tools menu and one on theExtensions menu 55 forplotting graphs of field /alues in the model. he first three of these items ena&le the user to definea path as a line! arc or circle and then display graphs of field /alues along the path. he fourth is a

simple1ield *am!ler! which displays graphs of /alues along lines parallel to thex ory a9es.

Field Sampler

It is instructi/e to display graphs of the flu9 density componentsBx andBy along a line passingthrough the airgap &etween the armature and the poles! with end5point coordinates A#.=! 30B andA#.=! 30B. his will show the /ariation of the flu9 density in each airgap! and the rapid decay&eyond the edges of the poles.

1 n the >9tensions menu! clic41ield *am!ler to display a dialog:

2 In the Data >ntry panel! clic4 >nd oint.

3 >nter data /alues as follows for the ( and Y coordinates:

( Y

"tart #.= 30

>nd #.= 30

Iterations 2 ?2

*ince the y-coordinate range is >, mm( setting the iteration alue to >/ 'ill sam!le

the field at increments of / mm.

4 In the raph ptions! select Y Kalues.

) Clic4 %uery nd Create raph.

A gra!h 'indo' should "e dis!layed.


30/137



- Clic4 the Ma9iminergy page. If this is not /isi&le! clic4 the

lo&al %uantities ta& on the left! and then clic4 the >nergy ta&. If you cannot see two /alues

displayed! do the following.

Mo/e the pointer to find the split &ar &etween the ost rocessing &ar and the

@ey&oard Input &ar.The !ointer changes from an arro' to a s!lit sym"ol.

Drag the split &ar upward a short distance and release it. epeat until the /alues are/isi&le.

You may also need to drag the te9t panels in the ost rocessing &ar.


31/137


Chapter 2

2'

he displayed /alues should &e similar to the following:

"tored Magnetic >nergy 0.3#02 R

Co5energy 0.3#2= R

"ee appendi9 * for a discussion of energy and co5energy. he difference &etween these two

/alues is an indication of the le/el of saturation in the steel parts of the model.

Flux linkage and inductance

In the ost rocessing &ar! clic4 the ;lu9 Lin4age ta&. he displayed /alue of the flu9 lin4agewith the coil should &e similar to the following:

CoilO2 0.3#0S W&

he self5inductance of the coil may &e calculated from the flu9 lin4age:

=

=0.3#0S

= 0.2?0) ' =2?0.) m'i #.0

lternati/ely! the self5inductance may &e calculated from the stored magnetic energy:

=#;

=# 0.3#02

= 0.2?00 ' =2?0.0 m'i#

A#.0B#

"ee appendi9 * for details of these methods of calculating inductance.

Forces

Maget automatically calculates forces on all &odies in the model. &ody is defined as one of:

a set of connected components surrounded &y air!

a current5carrying component.

In this case! there are two &odies: the armature! and the set comprising the core and the two coil

sides.

1 n the ost rocessing &ar! clic4 the ;orce ta&.

2 If you cannot see the force /alues displayed for the two &odies! do the following.

Mo/e the pointer to find the split &ar &etween the ost rocessing &ar and the@ey&oard Input &ar.

The !ointer changes from an arro' to a s!lit sym"ol.

Drag the split &ar upward a short distance and release it. epeat until the force/alues are /isi&le.

You may also need to drag the te9t panels in the ost rocessing &ar.


32/137



he force components and magnitudes! in newtons! should &e similar to the following:

Bod# fx fy fz Qf Q

Core T

Coil side 2 T Coil side #

T#).#3 0.03?) 0 #).#3

rmature #).#0 0.02)0 0 #).#0

"ince this is a #D model! thez components are


33/137

2 Clic4 the arameters ta&.

$"sere that there is a alue 2 in the Ex!ression field for the #urrent !arameter:

3 >dit the >9pression field for the Current parameter so that it contains the followinglist of /alues:

2, 4, 6, $, 104 ress >nter to accept the list.

0f the field turns red( there is an error in the field( 'hich must "e corrected.

) Clic4 @ to close the roperties dialog.

- In the &1ect page of the ro1ect *ar! clic4 the name of the modelAC5core electromagnetB.

This cancels the selection of the #oil6/ o"%ect. 0f it is left selected( it interferes'ith the field !lots descri"ed "elo'.

"elect the ro&lem page of the ro1ect *ar.

&ser/e that fi/e pro&lems ha/e &een created! each with a different coil current. If Maget is licensed for parameteri


34/137

Post-processing 1

he instructions &elow are applica&le when Maget is licensed for parameteri


35/137

Clic4 the Magnetic ermea&ility ta& and resi


36/137

4 Display a graph of force /alues as follows.

;or the rmature! clic4 in the te9t &o9 for the ;orce Kector! where the forcecomponents are shown.

The ?ra!h *election "utton should "e ena"led.

Clic4 the raph "election &utton to display a graph:

The ta"s at the "ottom of the data area select the x( y or z com!onent of force.

$nly the x-com!onent is of interest in this model. Note that the current alues are

not dis!layed in this gra!h( "ecause agNet uses this method to sho' theariation of force 'ith any !arameter. The current alues corres!onding to the

!ro"lem num"ers / to < are 2( +( >( @ and /, A res!ectiely.

Ma9imi


37/137

6ltering the core $aterial

he procedure &elow will change the core material from ordinary low5car&on steel AC20B to a

high 6uality magnet iron Aem4oB. similar procedure can &e used to change the material of any

other component in the model.

1 Display the model again:

n the Kiew menu! clic4 "olid Model.

2 "elect the &1ect page of the ro1ect &ar.

3 ight5clic4 the Core component! and select roperties to display a dialog:

4 Clic4 the Material ta&! and from the drop5down list select em4o: "oft pure iron.

) Clic4 @ to apply this change and close the Core roperties dialog.

The color of the core should hae changed to re!resent the ne' material.

- n the "ol/e menu! clic4 "tatic #D.

Kiew the solution results as &efore.

em4o has a higher saturation flu9 density than C20! so a greater proportion of the core is

unsaturated! resulting in a larger force on the armature. he results should &e similar to the

following! which were o&tained with Maget /ersion ?.#2.2.

Material %orce at 2 A %orce at 6 A %orce at 10 A

C20 #).#0 2)?.J #0J.)

em4o #).-- 2=2.) ##0.=

his change of material has made a small impro/ement. ;or significant impro/ement! the

shape of the core must &e changed.


38/137

,hanging the geo$etry

With large /alues of coil current! part of the core is saturated magnetically! so the electromagnetis less effecti/e. he diagram &elow shows a change to the shape of the core that will put morematerial in the critical areas! there&y reducing the /alue of the flu9 density in these parts of thecore.

25

25 25

) 15

15

25 25

25

25 %5

15

o ma4e this change! the original model will &e modified with theshift and distort facilities inMaget. *efore doing this! the original drawing lines will &e deleted as descri&ed &elow.

1 "a/e the model as C-core electromagnet mod.

2 n the Kiew menu! clic4 Construction rid.

This dis!lays the construction grid again.

3 Clic4 the "elect Construction "lice Lines &utton.

4 "elect all the lines as follows:

osition the pointer outside the model.

'old down the left mouse &utton! and drag out a ru&&er5&and &o9 enclosing the

entire model. elease the &utton.

All the lines should "e mar4ed in red. A line 'ill not "e mar4ed if the ru""er-"and

"ox fails to enclose it com!letely.

If some lines are not mar4ed! select all the lines again.

) n the 4ey&oard! press the Delete 4ey.

All the mar4ed lines should "e deleted.


39/137

- Mo/e the left5hand coil side as follows:

In the &1ect page! clic4 Coil side 2.

n the Model menu! clic4 "hift Components to display a dialog:

In the "hift Kector te9t &o9! change the te9t to &10, 0, 0'.

Clic4 @ to apply this shift and close the dialog.

The left-hand coil side should hae moed /, mm to the left.

Change the shape of the core as follows:

n the Model menu! clic4 Distort Kertices.

The model ertices are mar4ed in the 5ie' 'indo':

;1

;2

Clic4 the /erte9 la&eled K2 in the diagram.

The selected ertex is colored red. Mo/e the pointer to the re6uired position! displaced from the original position &y

20 mm in &othx andy. Do not press the mouse &utton while doing this.

u""er "and lines follo' the !ointer.

Clic4 again to mo/e the /erte9 to the new position:

;1

;2

epeat for the /erte9 K#.


40/137

8 his has only changed part of the component. Complete the change as follows:

Clic4 the "how (YANB &utton a&o/e the Kiew window.

This sho's the "ac4 ie' of the model:

;1

;#

;%

;2

Mo/e /ertices K3 and K) to the same positions as K2 and K#.

Clic4 the "how (YATNB &utton.

This restores the original ie'.

n the "ol/e menu! clic4 "tatic #D.

15 Kiew the solution results as &efore.

With a core material of em4o! the results should &e similar to the following! which were

o&tained with Maget /ersion ?.#2.2.

Model %orce at 2 A %orce at 6 A %orce at 10 A

riginal #).J? 2=2.) ##0.=

Modified #=.2J ##).0 )3J.=

In the a&sence of saturation! the force would increase &y J times when the current increases from# to ? ! and &y #= times when the current increases from # to 20 . With the modified

model! the force increase is S.SJ times at ? ! and 2-.= times at 20 . hus! the change hasalmost eliminated saturation for currents of up to ?. here is considera&le impro/ement in the

performance at 20 ! &ut saturation is still significant at this current.

Moving the ar$at"re

If the armature is displaced from its position of alignment with the magnet poles! there will &e arestoring force that /aries with the displacement. o e9amine this effect! another form ofparameteri


41/137

defined parameter will &e used to /ary they5coordinate of ashift ector that determines the

displacement of the armature.

It is necessary to remo/e the list of /alues from the coil current parameter! otherwise Maget will

generate a new pro&lem for each com&ination of /alues for the current parameter and the shift

parameter.

Creating a user-defined parameter

1 In the &1ect page! right5clic4 the model name C5core electromagnet mod! and selectroperties to display a dialog:

2 "elect the arameters page and scroll down to the end of the list of parameters:

3 In the first /acant arameter field! enter ()ift.

4 In the ype drop5down list for this parameter! select um&er.

) In the >9pression field! enter the following list of /alues:0, 1, 2, *, 4, 5

- ress >nter to accept the list.


Clic4 @ to close the Model roperties dialog.


42/137

Parameterizing the model

1 In the &1ect page! right5clic4 CoilO2 and select roperties.

2 In the Coil roperties dialog! open the arameters page.

3 >dit the list of Current /alues so that only the /alue # remains.

4 Close the Coil roperties dialog.

) In the &1ect page! right5clic4 rmature and select roperties.

- In the rmature roperties dialog! open the arameters page.

"croll down the list of parameters to find "hiftKector:

8 In the >9pression field! enter the following array for the /ector! including the s6uare&rac4ets:

+0, ()iftmm, 0

The name of the user-defined !arameter *hift must "e !receded "y a = sym"ol. The

suffix =mm conerts alues from millimeters to the "asic units of meters.

ress >nter to accept the array.


15 Clic4 @ to close the roperties dialog.

11 "elect the ro&lem page of the ro1ect &ar.

&ser/e that fi/e pro&lems ha/e &een created! each with a different armatureshift /ector.


43/137

"hift AmmB 0 2 # 3 ) =

;9 AB #=.2J #=.03 #).?= #3.J3 ##.J2 #2.SS

;y AB 0.03 0.SS 2.-= #.=2 3.2J 3.S#

Soling and post-processing

1 "ol/e as "tatic #D.

;ith the "asic ersion of agNet( only the first !ro"lem 'ill "e soled automatically.

*olutions of the other !ro"lems can "e o"tained "y the method descri"ed on !age 2C.

2 Kiew the solution and the glo&al 6uantities in the same way as &efore.

3 Display graphs of thex andy components of force on the armature.

The results should "e similar to the follo'ing( 'hich 'ere o"tained 'ith agNet

ersion >.2/./.

&ostscripthe ta&le &elow compares the #D results for the original C5core electromagnet with the results

from a 3D model! using Maget /ersion ?.#2.2.

2 *

"tored magnetic energy 0.3#02 R 0.)SJ# R

Co5energy 0.3#2= R 0.)J-0 R

;lu9 lin4age 0.3#0S W& 0.)SSS W&

"elf5inductance 2?0.) m' #)).) m'

;orce on the armature #).#0 #).-?

hese results show that the #D model predicts the force of attraction with good accuracy! &ut it

seriously under5estimates the flu9 lin4age and hence the inductance of the coil. he 3D solution

includes all the fringing and lea4age field components! some of which are ignored in the #D

model. In this de/ice the fringing field ma4es only a small difference to the force on the armature!

&ut the lea4age field has a significant effect on the flu9 lin4age. here is a corresponding increasein the stored magnetic energy and co5energy.

he 3D solution gi/es an inductance /alue that is =#U higher than the #D result! whereas the

force /alue is only #.3U higher.


44/137


45/137

Chapter #

%1


Chapter 3

,ase %t"dies:

ranslational Geo$etry

!ntrod"ctionhe case studies in this chapter co/er a range of modeling pro&lems for de/ices with translational

geometry. De/ices with rotational geometry are discussed in chapter ). hese case studies are

arranged in order of increasing difficulty! progressi/ely introducing further features of Maget!

so it is ad/isa&le to wor4 through them in se6uence. he detailed descriptions of &asic Maget

operations gi/en in chapter # will not &e repeated! &ut any new operations will &e fully e9plained.

In some of the case studies! a #D model does not always gi/e accurate results &ecause the de/ice

is not /ery long in the translational direction. "ince the user needs to &e aware of the limitations

of #D modeling! these case studies also include 3D results for comparison. ;or all of the case

studies in chapter 3 and chapter )! the instructions assume that a new model is &eing started! as

descri&ed in the tutorial in chapter #. o a/oid tedious repetition! this instruction is gi/en in

a&&re/iated form in the case studies.

"ome of the case studies re6uire shapes to &e drawn from arcs and straight lines. Drawing an arcre6uires the user to specify the coordinates of the center! the start point and the end point. rcsare always drawn counter5cloc4wise from the start point to the end point. If the arc

drawing tool is selected from the Draw tool&ar! the order of the points is center! start! end.he Draw menu has other arc tools with the points in different orders.

('core electro$agnethe diagram &elow shows an >5core electromagnet. It is similar in principle to the C5coreelectromagnet of chapter #! &ut it is a &etter magnetic design &ecause the coil is nearer to theairgap! and &oth sides of the coil are acti/e. he o&1ecti/es are to determine the self5inductance ofthe coil and the force on the armature! and to e9plore the magnetic field distri&ution in the de/ice.


46/137

#* #*

8

#5 #5


%2 Introduction to Ma"Net

Modeling the device

he diagram &elow shows the #D model of the de/ice! with the dimensions as multiples of a

&asic unitxE the airgap lengthg is independent ofx. he shaded areas are the two coil regions!

representing the two sides of the coil shown in the 3D /iew a&o/e.

4*

*

" * * 2* * *

%*

he dimensions in this diagram are chosen so that the armature and the core can &e made fromlaminations punched from sheet steel without any waste! as shown in the diagram &elow.

he &asic dimensionx is 20 mm! the airgap lengthg is = mm! and the depth of the electromagnetis ?0 mm. he coil has 2000 turns! carrying a current of #.0 .

45

15

) 15 15 25 15 15

%5

s with the C5core electromagnet! an air &o9 must &e drawn around the model with a radius of

a&out 20 times the model dimensions.


47/137

Creating the model

1 "tart a new model and sa/e it as /-core electromagnet.

2 "et the model length units to millimeters.

3 "et and display the construction grid.

4 Let the coordinate origin &e the point in the center of the top edge of the middlepole.

) Construct components for the core and the armature:

"weep distance: ?0 mm.

Material: C20: Cold rolled 2020 steel.

- Construct components for the two coil sides:


Material: Copper: =.--e- "iemensFmeter.

Ma4e a single coil from the two components:

um&er of turns: 2000.

Current: #.0 .

8 Construct an air &o9 from a circle centered at the origin! with a radius of )00 mm:

Ignore 'oles must &e acti/e in Ma4e Component In Line.



he following sol/er settings should gi/e an accurate #D solution without e9cessi/e computing

time. he user is in/ited to try the effect of different settings.

1 "et the options for sol/ing as follows in the "ol/e menu:

"ol/er ptions: Material ype Default!ewton tolerance 2U! Ma9imum ewton Iterations #0!C tolerance 0.02U.

olynomial order #.

daption ptions: se h5adaption!ercentage of >lements to efine #=U!olerance 0.#U!Ma9imum um&er of "teps 20.


3 Inspect the contour plot of the flu9 function and the shaded plot of the smoothed QBQ/alues.

4 Inspect the computed glo&al 6uantities! and calculate self5inductance /alues asfollows Asee appendi9 * for detailsB:

;rom the flu9 lin4age: G F i( where is the flu9 lin4age for the coil! and iis the coil current.

;rom the stored energy: G #; F i#! where ; is the stored magnetic energy.


48/137

%a$ple res"lts

he results &elow were o&tained with Maget /ersion ?.#2.2. ;or comparison! results from a 3D

solution are also gi/en.

2 *

"tored magnetic energy: 0.-=3# R 0.SS-? R

Co5energy: 0.-=?2 R 0.SJ0= R

;orce on core and coils: T?).2) T??.)0

;orce on armature: ?3.J3 ??.#2

;lu9 lin4age: 0.-=)? W& 0.SSS# W&

Inductance from flu9 lin4age: 0.3--3 ' 0.)))2 '

Inductance from stored energy: 0.3-?? ' 0.))3S '

Disc"ssion

2! model

"ince the airgap length and the coil ampere5turns are the same as for the C5core electromagnet!

the airgap flu9 density /alues are e9pected to &e similar. he total pole5face area is three times as

great! so the force of attraction and the self5inductance are also e9pected to &e a&out three times

as great! which is the case. If the coil ends are neglected! the coil resistance will &e only twice thatof the C5core electromagnet! so the >5core electromagnet appears to &e a &etter electromagnetic

de/ice.

"! model

s with the C5core electromagnet! the 3D solution gi/es a significantly higher /alue for the self5inductance: in this case 2=U higher than the #D result. "imilarly! the 3D solution gi/es a higher/alue for the force: 3.?U higher than the #D result for the force on the armature. ;or comparison!with the C5core electromagnet the 3D solution gi/es an inductance /alue that is =#U higher thanthe #D result! and a force /alue that is #.3U higher.


49/137

8

25

25

Magnetic latch with a per$anent $agnethe diagram &elow shows a magnetic latch! which ta4es the form of a C5core permanent magnet

with steel poles and a steel armature. his is similar to the C5core electromagnet of chapter #!

e9cept that the e9citation for the magnetic circuit is pro/ided &y a permanent magnet. hedimensions of the poles! armature and airgap are the same as for the C5core electromagnet! andthe thic4ness of the permanent magnet is #0 mm.

Modeling the device

he diagram &elow shows the cross5section of the de/ice with the dimensions in mm. ll parts

ha/e a depth of )0 mm. he poles and the armature are made from 2020 cold5rolled steel! and the

magnet from lnico5J*.

%5 ) 15

15

25

15

s with the C5core electromagnet! an air &o9 must &e drawn around the model with a radius of

a&out 20 times the model dimensions.


50/137

Creating the model

1 "tart a new model and sa/e it as M latc).



4 Let the coordinate origin &e the point mid5way &etween the poles.

) Construct components for the poles and the armature:

"weep distance: )0 mm.

Material: C20: Cold rolled 2020 steel.

- Construct the component for the magnet:


Material: LJ: lnico5J*.

ype: uniform.

Direction: A0! 2! 0B.

This s!ecifies that the "loc4 is uniformly magnetized( 'ith the magnetization ector in

the !ositie y direction.

Construct an air &o9 from a circle centered at the origin! with a radius of )00 mm:

Ignore 'oles must &e acti/e in Ma4e Component in a Line.



he following sol/er settings should gi/e an accurate #D solution without e9cessi/e computing

time. he user is in/ited to try the effect of different settings.

1 "et the options for sol/ing as follows in the "ol/e menu: "ol/er ptions: Material ype Default!

ewton tolerance 2U! Ma9imum ewton Iterations #0!

C tolerance 0.02U.

olynomial order #.

daption ptions: se h5adaption!ercentage of >lements to efine #=U!

olerance 0.2U!Ma9imum um&er of "teps 20.


3 Inspect the contour plot of the flu9 function and the shaded plot of the smoothed QBQ/alues.

4 Inspect the computed glo&al 6uantities.


51/137

%a$ple res"lts

he results &elow were o&tained with Maget /ersion ?.#2.2. ;or comparison! results from a 3D

solution are also gi/en.

2 *"tored magnetic energy: 0.?0=0 R 0.J22# R

Co5energy: 0.?0=0 R 0.J2)? R

;orce on poles and magnet: T3-.#0 T32.-=

;orce on armature: 3-.2- 32.?=

Disc"ssion

Energy

he stored magnetic energy /alues are negati/e in this model &ecause B and H are in oppositedirections in the permanent5magnet material! so the energy density calculated from e6uation *52

Appendi9 *B is negati/e in this part of the model. When the total energy is calculated &y

integrating the energy density o/er the /olume of the model! the negati/e energy in the

permanent5magnet material e9ceeds the positi/e energy in the rest of the model. Kersions of

Maget earlier than ?.22.# reported a positi/e /alue for the stored energy! calculated &y a

different method. "ee the Maget help for further information.

2! model

he airgap flu9 density is somewhat higher than in the C5core electromagnet! gi/ing an increasedforce of attraction! and the ma9imum flu9 density in the steel is also higher. *ecause the

permanent magnet has a low recoil permea&ility! there is much more lea4age flu9 at the &ac4 ofthe magnet than with the coil in the C5core electromagnet.

"! model

In this 3D model the force on the armature is 2=U lower than the /alue from the #D model. In

contrast! with the C5core electromagnet! the force from a 3D model is #.3U larger. he reduction

in force can &e e9plained as follows. here is significant flu9 lea4age &etween the poles at the

ends! which is ignored in the #D model. his is reflected in the larger /alue for stored magnetic

energy in the 3D model. s a result! less flu9 crosses the airgap! so there is a reduction in the

force of attraction. his effect is greater if the width of the permanent magnet is reduced.


52/137

=f

#"s*ar forceshe diagram &elow shows two long non5magnetic &us&ars! where the force due to the currents is

to &e calculated in two cases: currents in the same direction! and currents in opposite directions.

;or this case study the &ars are 0.# m apart! 0.# m high! and 0.2 m wide. he current density is= Fmm

#! so the current in each &ar is 200 4.

here is a simple analytical e9pression for the force per unit length &etween two infinitely long&us&ars in free space:

# 20-

4 i2i#d

VFm A352B

where i2 and i# are the currents in the &ars! 4 is a constant that depends on the shape of theconductors and their separation! and d is the distance &etween centers of the &us&ars. his is&ased on the formula gi/en &y *ewley V2 and "teele V#. he /alue of 4 in this case study can &ecalculated from the analytical formula to &e 0.J)JS0 when d G 0.3 m! so the theoretical force is?.33#0 4FmE this ser/es as a chec4 on the results from Maget.


53/137

Modeling the device

he diagram &elow shows the #D model of the &us&ars! with dimensions in meters.

51 52 51

52 52

-ar


54/137

$dding the open #oundary

1 n the *oundary menu! select pen *oundary to display the dialog:

2 Change radius of the inner &oundary to 1.5! and the scale factor to 0.1.

3 Clic4 @.

This should create t'o cylindrical air "oxes( one named airspace enclosing the

"us"ars( and a much smaller "ox named e9terior. Each circular edge is su"diidedinto +, !oints( and there is an een !eriodic "oundary condition lin4ing the

corres!onding edges of the t'o "oxes. This ma4es the interior of the small "ox

re!resent the s!ace outside the large "ox. The screen dis!lay should resem"le the

follo'ing:



"ol/er ptions: Material ype Default!ewton tolerance 2U! Ma9imum ewton Iterations #0!

C tolerance 0.002U.

olynomial order #.




3 Inspect the resulting solution mesh as well as the force /alues.

4 Compare the force with the theoretical /alue of ?.33# 4.


55/137

Chapter #

)1

,"rrents in the sa$e direction

t sufficiently large distances from the conductors! the field lines will appro9imate to circles. circular air &o9 is therefore a suita&le choice for the closed &oundary. he default ;lu9 angential&oundary condition is appropriate in this case! &ut the pro&lem is to choose a suita&le si


56/137

)2 Introduction to Ma"Net

13 In the /ertical tool&ar! clic4 the "elect Component Kertices &utton.

The air "ox has four ertices( sho'n as square dots at the ends of the lines in the

diagram a"oe.

14 Clic4 a /erte9 of the air &o9! then right5clic4 and select roperties.

The 5ertex Pro!erties dialog is dis!layed.

1) In the Kerte9 page! note the Local osition parameter.

1or exam!le( D,.+( ,F

1- pen the arameters page.

1 In the >9pression field for the osition parameter! enter an e9pression such as:+Bo!adius, 0 where the sign corresponds to the sign in the Local osition

parameter.

18 epeat the procedure for the other three /ertices.

&ie'ing the parameterized model

1 In the ro1ect &ar! clic4 the ro&lem ta&.

2 Clic4 pro&lem #! then clic4 pdate Kiew.

$"sere the increased size of the air "ox in relation to the "us"ars.

3 "imilarly! /iew pro&lems 3! )! and = in succession.

$"sere the !rogressie increase in the radius of the air "ox. *ince this is a 2)

model( the de!th of the air "ox has no significance.



"ol/er ptions: Material ypeDefault!

ewton tolerance 2U! Ma9imum ewton Iterations #0!

C tolerance 0.002U.

olynomial order #.

daption ptions: se h5adaption!ercentage of >lements to efine #=U!olerance 0.02U!Ma9imum um&er of "teps 20.


;ith the "asic ersion of agNet( only the first !ro"lem 'ill "e soled automatically.

*olutions of the other !ro"lems can "e o"tained "y the method descri"ed in cha!ter 2

D!age 2CF.

3 ote the /alues of force o&tained with different si


57/137

Chapter #

)#

%a$ple res"lts

he results &elow were o&tained with Maget /ersion ?.#2.2.

Currents in opposite directions

With the @el/in open &oundary! the computed force /alues are as follows:;orce on *arO2 ?33#.3

;orce on *arO# ?33#.)

Mean force magnitude ?33#.)

Currents in the same direction

Kalues for the mean force magnitude on the two &ars are as follows! for different air &o9 si


58/137

)% Introduction to Ma"Net

+ield in a cylindrical cond"ctorhe magnetic field of a long solid cylindrical conductor is well 4nownE the flu9 lines are circles

centered on the a9is. utside the conductor! the magnitude of the flu9 density /aries as 2Fr! where

r is the radial distance from the a9is. Inside the conductor the magnitude /aries directly with r! so

the field /anishes on the a9is.

If a cylindrical hole is &ored in the conductor from one end to the other! there will &e no magnetic

field in this hole pro/ided the hole is coa9ial with the conductor. In other words! there is no field

inside a hollow cylindrical conductor.

remar4a&le result! which is not widely 4nown! is the nature of the field in the hole when it is

&ored off5center! so that its a9is is displaced from the a9is of the cylinder! as shown in the

diagram &elow. his field is perfectly uniform the flu9 lines are parallel straight lines in a

direction normal to the plane containing the two a9es V3.

Modeling the device

lthough the dimensions of the cylinders are not critical! the si


59/137

Chapter #

))




C tolerance 0.0002U.

olynomial order #. daption ptions: se h5adaption!

ercentage of >lements to efine #=U!olerance 0.002U!Ma9imum um&er of "teps 20.


3 Inspect the contour plot of the flu9 function and the shaded plot of QBQ smoothed.

$"sere the nature of the flux lines in the hole. Gse the 1ield Pro"e to get alues of

the flux density at seeral !oints in the hole.

4 se the ;ield "ampler on the >9tensions menu to o&tain /alues forBy o/er a grid ofnine e6ually spaced points in the hole as follows.

In the Data >ntry panel! clic4 "pacing.

>nter data /alues as follows for the ( and Y coordinates:

( Y

"tart #0 #0

"pacing #0 #0

Iterations 3 3

;rom the ;ield ame drop5down list! select *y "moothed.

Clic4 %uery.

A set of nine alues of By should "e dis!layed in the text area at the to! of the 1ield

*am!ler dialog.

%a$ple res"lts

he ta&le &elow shows the de/iation ofBy from the mean /alue By0 of 0.200002 o/er a grid ofnine e6ually spaced points in the hole. he 6uantity displayed is the percentage de/iation By FBy0. hese results were o&tained with Maget /ersion ?.#2.2.

x AmmBy AmmB

#0 )0 ?0

#0 T0.002-U 0.000#U 0.0022U

0 T0.0020U 0.000#U 0.002)U

T#0 T0.002?U 0.0003U 0.0022U


60/137

)4 Introduction to Ma"Net

,ylindrical screen in a "nifor$ field

Description of the pro*le$

he diagram &elow shows the flu9 plot of a hollow iron cylinder placed in a trans/erse magnetic

field. In the a&sence of the cylinder the field is uniform! so the diagram demonstrates the

screening effect of the cylinder.

his pro&lem ma4es an interesting case study for Maget! firstly &ecause the results can &ecompared with an analytical solution if the magnetic material is linear V3! and secondly &ecauseit illustrates a techni6ue for producing a uniform magnetic field. ;or this case study! the innerradius of the cylinder is #) mm! the outer radius is 30 mm! the relati/e permea&ility of the iron is2000! and the magnitude of the applied field is 0.2 .

Modeling the device

o produce a uniform magnetic field in Maget! we use a 7coil8 formed from two sla&s! asshown in the diagram &elow. he arrows show the direction of current flow! and the coil is

assumed to &e /ery long in this direction! gi/ing a #D field.


61/137

Chapter #

)'

If this coil is surrounded &y air! the field ta4es the form shown &elow.

s this flu9 plot shows! the field in the middle of the coil is appro9imately uniform. It can &emade e9actly uniform &y a simple modification: the coil is enclosed in a close5fitting air &o9 with

a ;ield ormal &oundary condition on four surfaces. he diagram &elow shows the resulting fieldpattern.

*oundary conditions are discussed in appendi9 . his use of the ;ield ormal condition

effecti/ely em&eds the de/ice in a material of infinite permea&ility. ;lu9 lines ha/e to enter thetop and &ottom &ounding surface at right angles! and they find return paths round the sides Anot

/isi&le in MagetB of


62/137

-

;or the screening pro&lem! we need to generate a uniform field o/er a region that is significantly

larger than the cylinder. region #)0 mm s6uare should &e satisfactory for a cylinder with an

outer diameter of ?0 mm. he re6uired current is therefore:

i =B 0l =

0.20.#)=2J.20

4

A353B

0 ) 20

Creating the model

1 "tart a new model and sa/e it as C#lindrical screen 1.


3 "et and display the construction grid! using a spacing of ? mm.

4 Let the coordinate origin &e the center of the hollow cylinder! which is also the centerof the coil.

) Construct the cylinder from two concentric circles! of radii #) mm and 30 mm.

se a sweep distance of 200 mm and the material M3: elati/e permea&ility2000.

- Construct the left5hand coil side from a rectangle #)0 mm long and ? mm wide:

Left5hand edge: 2#0 mm from the origin.

"weep distance: 200 mm.


ame of the component: Coil side 1.

Ma4e the right5hand coil side &y copying! as follows.

"elect the left5hand coil side &y clic4ing Coil side 2 in the &1ect page.

In the Model menu! clic4 "hift Components.

Clic4 the chec4 &o9 for ma4ing a copy.

>nter the following /alue for the shift /ector:A2*4, 0, 0B

Clic4 @.

ename the new component Coil side 2.

8 Ma4e a coil from the two coil sides:

um&er of turns: 2.

Current: 2J200 .

$ir #ox and #oundary conditions

1 Construct a s6uare air &o9 that encloses the coil! with a side length of #)0 mm:



2 In the &1ect page! clic4 the T sym&ol &eside the air &o9 component to display itstree directory.


63/137

3 "elect four faces as follows:

"elect ;aceO3 &y clic4ing.

'old down the "hift 4ey and clic4 ;aceO?.

1our faces should "e highlighted.

4 In the *oundary menu! clic4 ;ield ormal.$ne "oundary condition should a!!ear in the $"%ect !age.

) se the Dynamic otation tool to inspect the faces of the air &o9.

$"sere the !attern for the 1ield Normal "oundary condition on four faces.


- "et the options for sol/ing as follows in the "ol/e menu:


C tolerance 0.02U.

olynomial order #.

daption ptions: se h5adaption!ercentage of >lements to efine #=U!olerance 0.02U!Ma9imum um&er of "teps 20.

"ol/e as "tatic #D.

8 Inspect the flu9 plot and the shaded plot of Q*Q smoothed.

se the field pro&e to e9plore the flu9 density /alues inside and outside thecylinder! and in the material of the cylinder.

Confirm that the /alue inside the cylindrical ca/ity is a&out 2U of the /alue far

away from the cylinder.

"how that the field is uniform inside the cylindrical ca/ity &y increasing the num&erof flu9 lines as follows:

In the ro1ect &ar! select the Kiew page.

ight5clic4 Contour lot and select roperties

Change the um&er of Inter/als from #2 to 2001.

Clic4 pply.

The region outside the caity should turn "lac4 "ecause there are so many flux lines(

"ut in the caity the lines should "e straight and !arallel.


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(ploiting sy$$etry

It is possi&le to e9ploit the symmetry of this pro&lem &y modeling only one 6uarter of the de/ice!

as shown &elow. he coil current is half of the /alue for the full model! and the re6uired &oundary

conditions are ;ield ormal on faces ! * and C! and ;lu9 angential on face D. his is a usefultechni6ue for more comple9 models. C

D -

Creating the model A

1 "tart a new model and sa/e it as C#lindrical screen 2.


3 "et and display the construction grid! using a spacing of ? mm.

4 Let the coordinate origin &e the center of the hollow cylinder! which is also the centerof the coil.

) Construct the 6uarter cylinder from two concentric arcs! of radii #) mm and 30 mm!

as follows: Clic4 the dd rc &utton. Clic4 near the origin for the center of the arc.

Clic4 near the point A#)! 0B for the first point on the arc.

Clic4 near the point A0! #)B for the second point on the arc.

Draw the second arc of radius 30 mm in a similar way.

dd two lines to complete the outline.

Ma4e the component! using a sweep distance of 200 mm and the material M3:elati/e permea&ility 2000.

- Construct the coil side from a rectangle 2#0 mm long and ? mm wide: ight5hand edge: 2#0 mm from the origin.



ame of the component: Coil side.

Ma4e a coil from the single coil side: um&er of turns: 2.

Current: -J?0 .


65/137

$ir #ox and #oundary conditions

1 "elect and delete all the construction5slice lines and circles.

2 Construct a s6uare air &o9 that encloses the coil:

"ide 2#0 mm.


3 In the &1ect page! clic4 the T sym&ol &eside the air &o9 component to display itstree directory.

4 "elect three faces corresponding to ! * and C in the diagram as follows:

"elect ;aceO3 &y clic4ing.

'old down the "hift 4ey and clic4 ;aceO=.

Three faces should "e highlighted.

) In the *oundary menu! clic4 ;ield ormal.

- In a similar way! apply the ;lu9 angential &oundary condition to ;aceO?.

se the Dynamic otation tool to inspect the faces of the air &o9! to chec4 that the&oundary conditions ha/e &een applied correctly.




C tolerance 0.02U.

olynomial order #.




3 Inspect the flu9 plot and the shaded plot of Q*Q smoothed.

4 Chec4 that the flu9 density /alues are similar to those o&tained with the full model ofthe cylinder and coil.

Disc"ssion

he theoretical /alue V3 for the flu9 density inside the cylindrical ca/ity is 2.202 m for thecylinder used in this case study! when it is placed in a uniform field of 200 m! of infinite e9tent.

When a/eraged o/er a grid of nine points in the hole! the /alue in the model is 2.2?3 m. hedifference of =.?U can &e attri&uted to the appro9imate representation of an infinite field. Inpractice this error is unimportant! &ecause the screening effect is strongly dependent on therelati/e permea&ility of the screen! which is li4ely to &e highly /aria&le in practice. o see thiseffect! try replacing the linear material M3 with a non5linear material such as C20.


66/137

ransfor$er e/"ivalent circ"ithe diagram &elow shows a con/entional shell5type transformer with the secondary wound o/er

the top of the primary. ;or simplicity! the windings ha/e e6ual num&ers of turns in this casestudy.

he full e6ui/alent circuit for the transformer is shown &elow! wherex2 andx# are the primary

and secondary lea4age reactances respecti/ely! andm is the magneti


67/137

the flu9 lin4age with winding 2 when winding # carries a current i#.


68/137

i

Maget calculates flu9 lin4ages for each coil! so it is possi&le in principle to determine2!# and

! and hence to find the /alues for the reactances in the e6ui/alent circuit. potential difficulty

with this approach! howe/er! is that the lea4age inductances in e6uation 35= are the small

differences &etween large 6uantities. ny errors in the calculation of2!# and will &e

magnified enormously in the resulting /alues of l2 and l#. similar pro&lem occurs in

e9perimental wor4! where it is not possi&le to measure2!# and with sufficient accuracy to

get good results for l2 and l#&y su&traction.

>9perimentally! the reactance parameters are determined from open5circuit and short5circuit tests.

In the open5circuit test! the secondary is open5circuited so that0# G 0! and the normal /oltage is

applied to the primary winding. Measurements at the primary terminals gi/e the /alue of the total

primary reactancex2 Tm.

In the short5circuit test! the secondary terminals are short5circuited and a low /oltage is applied

to the primary! sufficient to circulate the normal full5load current. he secondary current is then

close to its normal full5load /alue. nder these conditions the current flowing in the magnetiach coil has 2000 turns. nder no5load conditions! when only one coil carries current! the

current is #.0 . ;or the simulated short5circuit condition where the coils carry e6ual and opposite

currents! the current is #00 ! gi/ing a current density of = MFm#

Aor = Fmm#B.


69/137

;or this pro&lem it is necessary to use a linear material! &ecause e6uations 35= to 35- are /alid for

a linear system only. suita&le choice is M3! with a constant relati/e permea&ility of 2000!

which is reasona&ly representati/e of transformer steel under normal operating conditions.

*ecause there is /irtually no e9ternal field! the air &o9 can &e 6uite close to the core of the

transformer.

Creating the model

1 "tart a new model and sa/e it as 3ransformer.

2 "et the model length units to meters.


4 Let the coordinate origin &e in the center of the transformer.

) Construct a component for the core:

Ignore 'oles must not e actie in Ma4e Component In Line.

"weep distance: 0.? m.

Material: M3: elati/e permea&ility 2000.

- Construct components for the four coil sides:



;or the primary winding! ma4e a single coil from two components &y selecting the"tart ;ace of side 2 and the >nd ;ace of side 2X.

um&er of turns: 2000.

Current: #.0 .

8 ;or the secondary winding! ma4e a single coil from two components &y selecting theend face of side # and the start face of side #X. his is the opposite of the primary

winding. um&er of turns: 2000.

Current: 0 .

Construct an air &o9 from a rectangle with /ertices at the following points:A0.=! 0.)B! A0.=! 0.)B! A0.=! 0.)B! A0.=! 0.)B.






C tolerance 0.02U.

olynomial order #.


70/137

daption ptions: se h5adaption!ercentage of >lements to efine #=U!olerance 0.0=U!Ma9imum um&er of "teps 20.


This solution is for the no-load condition 'ith the !rimary energized.

3 Inspect the contour plot of the flu9 function and the shaded plot of the smoothed Q*Q/alues.

4 Inspect the computed glo&al 6uantities and calculate inductances as follows Aseeappendi9 * for detailsB:

"elf5inductance:2 G 11 F i2( where 11 is the flu9 lin4age with the primary! andi2 is the coil current.

Mutual inductance: G 21 F i2! where 21 is the flu9 lin4age with the secondary.

) Change the primary current to 0 and the secondary current to #.0 . "ol/e againand calculate the inductances:

"elf5inductance:# G 22 F i#( where 22 is the flu9 lin4age with thesecondary! and i# is the coil current.

Mutual inductance: G 12 F i#! where 12 is the flu9 lin4age with the primary.

- Change the primary current to #00 and the secondary current to #00 . "ol/e againand calculate the total lea4age inductance from e6uation 35-

%a$ple res"lts

he results &elow were o&tained with Maget /ersion ?.#2.2. ;or comparison! results are also

gi/en for a 3D model of the same de/ice.

i2 G #.0 ! i# G 0 2 * "tored

magnetic energy: #2#.2J#3 R #2#.J#30 R ;lu9lin4age 11: #2#.2J#3 W& #2#.J##J W& ;lu9

lin4age 21: #22.J)2) W& #2#.=0?S W&

"elf5inductance2: 20?.0J?# ' 20?.)?2= '

Mutual inductance: 20=.J-0- ' 20?.#=3) '

Lea4age inductance l2: 0.2#== ' 0.#0S2 '

i2 G 0! i# G #.0

"tored magnetic energy: #2#.2J#2 R #2#.JS0S R

;lu9 lin4age 22: #2#.2J#2 W& #2#.JS0S W&

;lu9 lin4age 12: #22.J)20 W& #2#.=0-2 W&

"elf5inductance#: 20?.0J?2 ' 20?.)J0) '

Mutual inductance: 20=.J-0= ' 20?.#=3? '

Lea4age inductance l#: 0.2#=? ' 0.#3?J '

i2 G #00 ! i# G #00

"tored magnetic energy: =0#0 R SS-? R

Lea4age inductance l2 T l#: 0.#=20 ' 0.))3S '


71/137

Disc"ssion

2! results

he first two solutions gi/e /ery similar /alues for self5inductance! mutual inductance and

lea4age inductance. t first sight this is surprising &ecause the primary winding occupies the

space &etween the secondary and the center lim& of the core! so the two are not o&/iouslye6ui/alent. *ut in a #D model! if the core were infinitely permea&le the current and flu9 patterns

would ha/e a symmetry that results in e6ual inductance /alues. >/idently a permea&ility of 2000

is sufficient to gi/e /ery similar results.

he result from the energy calculation is identical to the sum of the two lea4age reactancescomputed from the separate flu9 lin4ages. his is useful confirmation of the accuracy of the flu9calculation! and shows that it is sometimes possi&le to o&tain a good /alue for the lea4agereactance &y su&traction.

"! results

hese results are significantly different from the #D solution. *oth of the lea4age inductances arelarger than their #D counterparts &ecause of the end5winding field! and as e9pected the secondary

/alue is considera&ly larger than the primary /alue.

he lea4age inductance from the energy calculation is slightly smaller than the sum of the two

lea4age reactances computed from the separate flu9 lin4ages! which is an indication of numerical

error in the 3D result.


72/137

0aria*le el"ctance %tepper Motorhe diagrams &elow show 3D and #D models of a simple /aria&le5reluctance stepper motor.

he motor has a 35phase stator! with two coils in each phase shown &y the colors in the #D

model. 'ere! the red phase is energi


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(uilding the stator core and poles

he stator core and poles will &e made &y drawing intersecting lines and circles! and deleting the

unwanted parts.

1 "tart a new model and sa/e it as (teer motor.

2 "et the model length units to millimeters.3 "et and display the construction grid with a spacing of = mm.

4 Let the coordinate orig

magnet introduction (1)

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