LECTURE 8 INTRODUCTION TO MAGNETICS (2)

Transcription

1 ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 8 INTRODUCTION TO MAGNETICS (2) Aknowledment-These handouts and leture notes iven in lass are based on material from Prof. Peter Sauer s ECE 330 leture notes. Some slides are taken from Ali Bazi s presentations Dislaimer- These handouts only provide hihlihts and should not be used to replae the ourse textbook. 9/14/2017

2 MAGNETIC CIRCUITS Maxwell s Equations: Ampere s Law: The maneti field in any losed iruit is proportional to the eletri urrent flowin throuh the loop. Faraday s Law: C C H. dl J.nda The line interal of the eletri field around a losed loop is S db E. dl. n da dt equal to the neative of the rate of hane of the maneti flux S throuh the area enlosed by the loop.. 9/14/2017 2

3 Conservation of Chare: Gauss s Law: MAGNETIC CIRCUITS S J. n da 0 The net maneti flux out of any losed surfae is zero. (for a maneti dipole,in any losed surfae the maneti flux inward toward the south pole will equal the flux outward from the north pole). S B.nda 0 9/14/2017 Soure:study.om 3

4 MAGNETIC CIRCUITS What do these symbols mean? 9/14/2017 Interal over a losed ontour C: Surfae S define by C: S Interal over a losed surfae S: Lenth of the ontour C: H is the maneti field intensity (A.turns/m) B is the maneti flux density (Tesla or Wb/m 2 ) E is the eletri field (V/m) J is the urrent density (A/m 2 ) n is the normal vetor to S. dl C C S 4

5 STATIC MAGNETIC CIRCUIT In stati maneti iruits, there are no movin members. The most important devie in this ateory is the transformer. We use Ampere s urrent law (ACL) and Gauss s law (GL) for maneti fields to derive useful flux-urrent terminal relations. The analysis is helpful in the desin of indutors, and study of transformers. 9/14/2017 5

6 MAGNETIC CIRCUIT - TOROID Current flowin in a ondutor produes a maneti field. Voltae produes an eletri field. Common example: Soure: eletronishub.or i(t) v(t) 9/14/ A r o r N 1 H + v(t) - H, B i(t) i(t) in i(t) out

7 MAGNETIC CIRCUIT - TOROID Applyin Ampere s law where is the mean lenth of the ore. H. d we et H an be approximated as 2π(r o +r 1 )/2. Assumin a linear relationship between B and H where B H and μ is the permeability (H/m). μ = μ r μ o where μ r is the relative permeability and μ o is the permeability of free spae. μ o = 4π x10-7 H/m. 9/14/2017 7

8 MAGNETIC CIRCUIT - TOROID The flux density in the ore is Sine B is the flux density (Wb/m 2 ), then the flux is A A B Define the manetomotive fore (mmf) as Define the relutane to be B 9/14/ N i. mmf ampere turns / weber A H. N i

9 Maneti Ciruit Equivalent Then, The permeane is mmf 1/ Wb / ampere turn It has similarity to a resistive iruit with the followin equivalenes Eletri Ciruit Maneti Ciruit Voltae mmf Current Flux Resistane Relutane Condutane Permeane 9/14/2017 9

10 MAGNETIC VS. ELECTRIC CIRCUITS The followin analoies hold: Differenes: Leakae. μ is not perfetly onstant. Saturation KVL and KCL analoous to total MMFs aross a loop add up to zero, and total flux enterin or leavin a node is zero. 9/14/

11 INDUCTANCE Faraday s law an be written as: C d E. d B. n da. dt S E is (V/m) and is (m) => left side is voltae. Define the flux linkae as N B A. Then, d d NBA d 2 A di () t v ( t ) N N. dt dt dt dt Define the indutane 2 A L N l, then di() t v( t) L. dt 9/14/

12 MAGNETIC CIRCUIT TOROID WITH AIR GAP Some manetis have air aps that store enery. Bak to the Toroid example but with air ap. The ap lenth is and its permeability is 7 4x10 H / m A H. d H H i(t) v(t) N r o r 1 H Soure: softsolder.om 9/14/

13 MAGNETIC CIRCUIT TOROID WITH AIR GAP Therefore, B H and B H o o. Assumin all flux passes throuh the air ap, A B A B. Then, ( ) A A 9/14/

14 FRINGING Frinin ours when the flux jumps around the air ap to join the other side of the ore. b a Frinin an be simply modeled by havin A >A Two methods to aount for this: - Empirial approximation - A is iven as perentae times A ab A ( a )( b ) A, 1 ka k 9/14/ A

15 EXAMPLE 400 At, 6 m, 0.1 m, A 1 m 2 A A 4 1.1, r 10 H/ m, Find the flux. H H i(t) v(t) N r o A 1 A x 10 Wb o A r 1 H 9/14/

16 READING MATERIAL Readin material: Setion 3.2. Reommended readin for next time: Setion /14/