Chapter 3 Magnetic Fields

Transcription

1 Chapte 3 Magnetic Fields Chapte 3 Magnetic Fields Intoduction Electic Fields Electostatics Relationships Pemittivity Magnetic Fields Relationships Pemeability Magnetization cuve Intensity & Density Moto & Geneato Hysteesis and eddy cuents Mechanical motion Relationships Gavity Linea v. otational Mathematical manipulation Maxwell Exemplas Applications... 10

2 Electic Powe Concepts Duham 3.1 Intoduction Cicuits ae based on the elements constained to a solid stuctue. Fields ae analogous to a gas whee the pefomance is distibuted though space. 3. Electic Fields Electostatics 3..1 Relationships Electic fields ae the esult of a chage. When the chage is measued fom a point o node it is called statics. When the chage is dispesed in space like a gas the enegy is called a dynamic field. A capacito is simply two electically chaged conductos that ae sepaated by a dielectic. Enegy elationships ae used to convet between electic, magnetic, and mechanical systems. The fundamental elationships fo an electic field ae listed. W(enegy) Fs Vq NI enegy convesion s(closed loop) F V E (electicintensity) V/mete q s D (electic density) q E A q = chage 1 electon = 1.601x10-19 Coulomb Coulomb m 3.. Pemittivity Pemittivity is the dielectic o chage insulation mateial popety. o Faad/m o Foce occus on chage due to chage 1. qq F 4 1 Electic field intensity at point is due to point chage at point 1. E F q q1 4 A adial electic field comes fom a line chage on the z-axis. E L

3 Chapte 3 Magnetics 3 L coulomb line chage density - m A plane electic field aises fom a sheet chage in x-y plane. E s coulomb s sheet chage density - m Enegy is dependent on the electic field, which can be conveted to voltage. 1 W qe s D 1 W qv CV 3.3 Magnetic Fields Relationships Magnetic fields ae the esult of a pole o flux. When the pole stength is measued fom a point o node it is called statics. When the magnetics ae dispesed in space like a gas the enegy is called a dynamic field. An inducto is simply is simply a coil of wie that ceates a magnetic field. Enegy elationships ae used to convet between electic, magnetic, and mechanical systems. The fundamental elationships fo magnetic devices ae listed. W(enegy) Fs Vq NI enegy convesion s(closed loop) F (magneto-motive foce) NI R H dl mmf = Amp-tuns F H (magneticintensity) NI Amp/m s i B (field density) H A l N R (eluctance) A L 3.3. Pemeability Webe m l N Ai gap l g A coss sectional aea Pemeability is the magnetic popety of mateial. o o Heny/m PAGE 3

4 4 Electic Powe Concepts Duham (coppe) (amophous steel) 000 (laminated steel) 6000 Foce occus on pole due to pole 1. F 4 1 (fluxlinkage) n LI EXAMPLES 3.1 Given: Laminated ion coe depth of coe.05 m width of coe.05 m length of ai gap.0007 m effective length of coe 0.3 m elative pemeability of Fe 000 ai gap flux density 0.5 T numbe of tuns 00 Find: a. total flux (include flaing at ai gap) B A BA AIR Wb b. flux density in the ion B B Fe Fe B AIR Wb/m c. magnetic intensity in the ion B H B B H 0 B H 08.8 A/m AIR Wb d. total mmf equied on the coe mmf NI Hl mmf H A-tuns e. cuent equied on the coe mmf NI mmf I N 6.66 A 00

5 Chapte 3 Magnetics Magnetization cuve The magnetic cicuit fo a machine includes the cuent and tuns, the feomagnetic metal, and ai gaps. The magnetization cuve shown below is a non-linea elationship fo the magnetic cicuit, and is typical of all magnetic cicuits. It is used to show the convesion between epesentations of magnetic enegy The fist potion of the cuve has a physical anomaly nea zeo. The potion less than 5000 is in the unsatuated egion. Thee is an appoximate popotional change in the vetical axis as the hoizontal changes. About 5000 is called the knee. That is the tansition egion. The top potion of the cuve above about 5000 is the satuated egion. Thee is vey little change in the vetical paamete as the hoizontal is inceased. The values ae stictly epesentative. Diffeent mateial alloys will yield othe ange of values. Nevetheless, the geneal shape and fom can be used fo a vaiety of poblems. The cuve then can epesent a numbe of diffeent elationships. Some of the moe common ae shown in the table below. X-axis name unit Y-axis name unit Cuve name unit F mmf A-tuns flux Webe R eluctance H intensity A-tuns/m B density Wb/m µ pemeability H/m I field I Amps V teminal Volts synchonous F field mmf A-tuns E a Intenal gen Volts dc machine EXAMPLES 3.1 Given: Magnetization cuve shown above Find: a. Fo an intensity H of 5000, what is the density B? Read chat along bottom to Read vetical to cuve. Read esult on left column of 50 b. Fo a density B of 50 what is the elative pemeability? PAGE 5

6 6 Electic Powe Concepts Duham B H B H 5000 c. Fo a flux of 50, what is the mmf? Read left column to 50 Read ight to cuve Read esult down to Intensity & Density The location and diection of a magnetic field ae detemined by the configuation of the conducto. Note that the intensity and density ae elated by the pemeability μ. Theefoe, eithe can be detemined fom these equations. B H Magnetic flux lines ae continuous about a souce, and pependicula at all points to souce, in a paallel plane. An aow shows the diection of a field. When pojected on a plane, a X is the tail of the aow with the field going into the plane. A dot is the point of the aow with the field coming out of the page. The diection is detemined by the ight hand ule. Cul the ight hand with fou finges in the diection of the magnetic field. The thumb points in the diection of the cuent. H I A magnetic field is poduced by staight conducto caying cuent I. I H 4 sin sin 1 Enegy is dependent on the magnetic field, which can be conveted to cuent. 1 W Hs B 1 W I LI Powe density in watts pe cubic mete is the Poynting vecto, the poduct of the density coss with the intensity. W B H V I θ 1 θ I H 3.5 Moto & Geneato Motos and geneatos combine electic, magnetic, and mechanical enegy. The thee-dimensions fo mechanical foce, electic cuent, and magnetic field ae elated by thee finges of the ight hand. The field diection follows the ight hand ule. F i( l B ) Fist Finge, x-axis F B I F

7 Chapte 3 Magnetics 7 Middle Finge, y-axis I, length, velocity Thumb, z-axis B In anothe application of the ight hand ule, with the finges culed in the diection of cuent, the thumb will point in the diection of the magnetic flux, φ. The moto elationship comes fom an electic cuent flow though a magnetic field. This ceates a mechanical foce. F i( l B ) F q( vb ) The geneato elationship comes fom a wie moving though a field. This ceates an electical voltage. e(voltage) ( vb ) l Since voltage and cuent ae not taditionally consideed vectos, the length of the line caying the cuent is used fo the diection. The coss-poduct vecto diections ae detemined by the cul of the ight hand. Stat with the finges of the hand pointing in the diection of the fist vecto. With the hand culing fom one vecto towad the second vecto diection, the coss-poduct esult is in the diection of the thumb. The dot poduct is in the same diection as the two vectos Hysteesis and eddy cuents The non-linea natue of magnetic devices ceates some athe inteesting phenomenon. The non-linea chaacteistic of a magnet causes it to follow diffeent path when the magnet is being enegized and when it is de-enegized. This altenative tace is called hysteesis. Hysteesis is the lag between cause and effect. The system does not instantly espond to the enegy applied to it. Rathe the system eacts slowly, o does not etun completely to its oiginal state. The system s state depends on the immediate histoy. Fo example, when pessed putty will assume a new shape, and when the pessue is emoved it will not etun to its oiginal shape immediately and entiely. It is hysteesis o esidual magnetism that causes data to be stoed on a magnetic tape and had disks. The esidual magnetism is also used to get a geneato woking. In motos, which epeatedly ae enegized and de-enegized, hysteesis causes loss and educes efficiency. Anothe phenomenon of magnetic systems is eddy cuent. Like the eddy pools in a flowing steam, these ae small swils of cuent that oppose change in enegy. Eddy cuents develop on magnetic mateials, such as steel, because of anomalies in the molecula stuctue. To keep the eddy cuents fom becoming lage, the steel coe is made in thins layes. Altenatively in some systems, slits may be cut in the steel coe. Eddy cuents ae used to ceate dynamic baking. In motos which ae continually enegized and de-enegized, eddy cuents causes loss and educes efficiency. PAGE 7

8 8 Electic Powe Concepts Duham 3.6 Mechanical motion Relationships Mechanical motion is the esult of mass. When the mass is measued fom a point o node it is called statics. When the mass is in motion it is call dynamics. Enegy elationships ae used to convet between electic, magnetic, and mechanical systems. The fundamental elationships fo thee types of field ae listed. W(Enegy) Fs Vq NI W P powe t 3.6. Gavity Gavity constant is the mass popety of mateial. o N m / kg 10 Foce occus on mass due to mass 1. F mm 4 1 Enegy is dependent on the mass field. F ma W Fs mas W mgh W 1 W mv W mc mv potential enegy aveage kinetic enegy instanteous kinetic enegy total enegy convesion enegy convesion Linea v. otational The motion of mass has both linea and otational popeties. Because of the otation of machines, the angula calculations ae moe common. Name Linea Units Angula Units displacement s mete θ adian velocity v=s/t m/sec ω= θ/t ad/sec acceleation a=v/t m/sec α= ω/t ad/sec mass m kilogam moment of inetia J kg-m foce F=ma Newton toque via angula T=J α Nt-m toque via linea T=s x F Nt-m enegy W=s F Joule W= T θ Joule powe P=Fv Watt P= T ω Watt

9 Chapte 3 Magnetics Mathematical manipulation Numeous equations and elationships have been boached in this chapte. These ae the basis fo most electical, magnetic, and mechanical machine analysis. The study equies manipulation of elationships in a vaiety of ways and combinations. Thee is a simple technique fo systematic manipulation: Begin with a known quantity. Daw a hoizontal line so the units can be manipulated until the coect fom is obtained. Then the equation will be coect, except fo convesion constants. As an example, convet evolutions pe minute (RPM) to adians pe second. ev min ad ad/sec min 60sec ev Maxwell Maxwell s Equations ae a summay of all that is electical and magnetic in a Calculus fom. Although they ae not easily used, they do povide a mathematically cleve summay. The integal fom is tedious, but the point fom supplies definitions. E B t H J D t D E 0 The del is the thee-dimensional patial deivative with espect to the x, y, and z axes. The electic-magnetic enegy equation contains all the infomation in one equation[1]. Note that all the definitions ae in the point equation. In addition, all the field infomation is in the distibuted equation. W is enegy and V is volume in these equations. zq W q b d s W tv s b d y t z y ys t y axial distance y adial distance otational distance Realizing the othogonal natue of the fields, the equations inheently contain the diections and equie only vecto algeba. [1] "A Composite Appoach to Electical Engineeing," Macus O. Duham, Institute of Electical and Electonic Enginees Region V, 88CH5617-6/ , Coloado Spings, CO, Mach 1988, pp Applications Engineeing Appoach to Maxwell and Othe Mathematically Intense Poblems, Macus O. Duham, Robet A. Duham, and Kaen D. Duham, Institute of Electical and Electonics Enginees PCIC, Septembe 00. "Applications Enginees Don't Do Haiy Math", Macus O. Duham, Robet A. Duham, and Kaen D. Duham, Poceedings of 35th Annual Fonties in Powe Confeence, OSU, Stillwate, OK, Octobe 00. PAGE 9

10 10 Electic Powe Concepts Duham "Electomagnetics in One Equation Without Maxwell", Macus O. Duham, Ameican Association fo Advancement of Science - SWARM, Tulsa, OK, Apil Exemplas An exempla is typical o epesentative of a system. These examples ae epesentative of eal wold situations Applications Applications ae an oppotunity to demonstate familiaity, comfot, and compehension of the topics. Application 5-1 SITUATION: