Magnetics Faradays and Ampers laws Permeability Inductor - PDF document

Mor M. Peretz, Switch-Mode Power Supplies [3-1] Magnetics • Faraday’s and Amper’s laws • Permeability • Inductor • Reluctance model • Air gap • Current crowding • Inductor design • Skin effect, proximity effect • Losses • Transformer • Ideal transformer • Real transformer • Transformer design [3-2] Mor M. Peretz, Switch-Mode Power Supplies Important relationships Faraday � � � � � B t , t � � v t Electrical Core � � � � � � H t , F t i t Ampere 1

Mor M. Peretz, Switch-Mode Power Supplies [3-3] Magnetic quantities Analogies to electrical quantities Magnetic field H Electric field E � � � � � MMF F H � � � Voltage V E M � J I , B , I � J B [3-4] Mor M. Peretz, Switch-Mode Power Supplies Important relationships Faraday’s law � B d � dB � � � � V n nA ; e � H dt dt Amper’s law nI � � � � Hd � n I H � e 2

Mor M. Peretz, Switch-Mode Power Supplies [3-5] Units � - magnetic flux Weber [ Wb ] V - voltage [ V ] Wb 2 � B - flux density Tesla [ T ] m Gauss [ G ] 1T = 10,000 G H - magnetic field [ A/m ] µ - magnetic permeability [ H/m ] [3-6] Mor M. Peretz, Switch-Mode Power Supplies Permeability � B B � � � � r H � 0 H Free space permeability Typical B sat (Vacuum) Ferrite: 0.2 – 0.5 T � � � � � 7 4 10 0 Iron powder: 0.5 – 1 T Typical magnetic material permeability 3 10 � � 10 � 5 r 3

Mor M. Peretz, Switch-Mode Power Supplies [3-7] What is inductance ? � d dI � � V n V L dt dt [3-8] Mor M. Peretz, Switch-Mode Power Supplies Reluctance model � F H � e � � � � � B � H � B � � � A � � � � � � � � � � e 1 R H � � � A 3 1 2 � � F R � � 4

Mor M. Peretz, Switch-Mode Power Supplies [3-9] Inductance with gap � e � R � m A r e � � g R � g A 0 e � � � � � nI R R m g � 2 d n di � � v n dt � R R dt � � m �� � g L [3-10] Mor M. Peretz, Switch-Mode Power Supplies Current crowding � e � R � m A r e � � g R � g A 0 e nI R ���� � g nI � � �� � R R R R g m m g 5

Mor M. Peretz, Switch-Mode Power Supplies [3-11] Inductor design Air gap �� � � g e � � � � � g e e � � � � e � � � � e If � � r re � � � � � � g g [3-12] Mor M. Peretz, Switch-Mode Power Supplies Inductor design Saturation - Ae � d dI � � V n V L dt dt � � � � dB dI � B � I � max pk nA � � dt L � � dt e � � � � dt dt 0 0 � LI nA B max pk e LIpk n � Bmax Ae 6

Mor M. Peretz, Switch-Mode Power Supplies [3-13] Inductor design Aw nW � a A w k I � rms W a J nI � rms A w Jk [3-14] Mor M. Peretz, Switch-Mode Power Supplies Inductor design Ap LI I � � pk rms A A A p e w B Jk max 7

Mor M. Peretz, Switch-Mode Power Supplies [3-15] Inductor design summary • Calculate Ap • Choose core • Calculate n • Calculate lg or adjust gap [3-16] Mor M. Peretz, Switch-Mode Power Supplies Inductor selection Off-shelf product 8

Mor M. Peretz, Switch-Mode Power Supplies [3-17] Inductor selection Off-shelf product [3-18] Mor M. Peretz, Switch-Mode Power Supplies Skin effect High Frequency DC � � skin depth � AC � R 72 1 � � ( mm ) R f DC f in Hz 9

Mor M. Peretz, Switch-Mode Power Supplies [3-19] Skin Effect Solutions Litz wire Foil/tape [3-20] Mor M. Peretz, Switch-Mode Power Supplies Proximity effect I I Current crowding due to magnetic fields 10

Mor M. Peretz, Switch-Mode Power Supplies [3-21] Losses function of � B i f j � B P ~ K i, j – material dependent � � [3-22] Mor M. Peretz, Switch-Mode Power Supplies Losses function of temperature 11

Mor M. Peretz, Switch-Mode Power Supplies [3-23] Losses material selection [3-24] Mor M. Peretz, Switch-Mode Power Supplies Ideal transformer � � � R n I n I m 1 1 2 2 φ � 0 R m � n I n I 1 1 2 2 � � d d � � 1 2 V n V n 1 1 2 2 dt dt 12

Mor M. Peretz, Switch-Mode Power Supplies [3-25] Transformer characteristics • Current flow at the same time at primary and secondary • Each winding represents inductance – Average voltage = 0 • AC only on any winding [3-26] Mor M. Peretz, Switch-Mode Power Supplies Magnetizing inductance � � � R n I n I m 1 1 2 2 � d � V n dt 1 1 1 � � 2 n n d � � 1 � 2 � V i i 1 1 2 R dt � n � � m 1 L m 13

Mor M. Peretz, Switch-Mode Power Supplies [3-27] Leakage inductance � uncoupled magnetic flux L lkg , M (mutual coupling) and k (coupling coefficient) � � M k L L 1 2 � � L L (1 k ) lkg 1 m � � 2 L L n lkg 2 lkg 1 [3-28] Mor M. Peretz, Switch-Mode Power Supplies Magnetizing and leakage inductances V1 V2 I2 Im I1 14

Mor M. Peretz, Switch-Mode Power Supplies [3-29] Transformer design Aw � n W n W � 1 a 1 2 a 2 A w k I � rms W a J � n I n I � 1 1 rms 2 2 rms A w Jk 2 n I � 1 1 rms A w Jk [3-30] Mor M. Peretz, Switch-Mode Power Supplies Transformer design Saturation - Ae 1 � � B Vdt nAe � � � B B max max V T � m on n 2 B A max e 15

Mor M. Peretz, Switch-Mode Power Supplies [3-31] Transformer design Ap � � � V T 2 I � � 1 on 1 rms A A A � p w e 2 B Jk max � � � V T 2 I � 1 on 1 A rms � � p B Jk � � � V D 2 I � 1 on 1 A rms p � � � f B Jk s [3-32] Mor M. Peretz, Switch-Mode Power Supplies Transformer design summary • Calculate Ap • Choose core • Calculate n1 • Calculate n2 16

Mor M. Peretz, Switch-Mode Power Supplies [3-33] Magnetic design Example • Boost converter: P=100W, Vin=10V, Vo=50V, � IL=0.1I Lav • Calculate L • Calculate Ap [3-34] Mor M. Peretz, Switch-Mode Power Supplies 17

Mor M. Peretz, Switch-Mode Power Supplies [3-35] [3-36] Mor M. Peretz, Switch-Mode Power Supplies 18

Mor M. Peretz, Switch-Mode Power Supplies [3-37] [3-38] Mor M. Peretz, Switch-Mode Power Supplies 19

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