Magnetics Faradays and Ampers laws Permeability Inductor - - PDF document

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Magnetics Faradays and Ampers laws Permeability Inductor - - PDF document

Feb 03, 2023 239 likes •450 views

Mor M. Peretz, Switch-Mode Power Supplies [3-1] Magnetics Faradays and Ampers laws Permeability Inductor Reluctance model Air gap Current crowding Inductor design Skin effect, proximity effect

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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

Mor M. Peretz, Switch-Mode Power Supplies

[3-2]

Important relationships

v t

  • ,

B t t , H t F t

i t

Faraday Ampere Core Electrical

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Mor M. Peretz, Switch-Mode Power Supplies

[3-3]

Magnetic quantities

Analogies to electrical quantities

  • ,

B

  • M

F H Magnetic field H

  • Electric field E
  • V

E , J I

  • B

I J

MMF Voltage

Mor M. Peretz, Switch-Mode Power Supplies

[3-4]

Important relationships

; dt dB nA dt d n V

e

  • H

B Faraday’s law Amper’s law

e

nI H

  • I

n Hd

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Mor M. Peretz, Switch-Mode Power Supplies

[3-5]

Units

  • magnetic flux Weber [ Wb ]

B - flux density ] T [ Tesla m Wb

2

V - voltage [ V ] Gauss [ G ] 1T = 10,000 G H - magnetic field [ A/m ] µ - magnetic permeability [ H/m ]

Mor M. Peretz, Switch-Mode Power Supplies

[3-6]

Permeability

  • B

H

Free space permeability (Vacuum)

7

10 4

  • Typical magnetic material permeability

5 3 10

10

  • r
  • Typical Bsat

Ferrite: 0.2 – 0.5 T Iron powder: 0.5 – 1 T

  • B

H

r

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Mor M. Peretz, Switch-Mode Power Supplies

[3-7]

What is inductance ?

  • d

dI V n V L dt dt

Mor M. Peretz, Switch-Mode Power Supplies

[3-8]

Reluctance model

  • e

F H

  • B

H

  • B

A

  • 1

e

R H A

  • F

R

  • 3

1 2

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Mor M. Peretz, Switch-Mode Power Supplies

[3-9]

Inductance with gap

  • e

m r e

R A

  • g

g e

R A

  • m

g

nI R R

  • 2

m g L

d n di v n dt R R dt

Mor M. Peretz, Switch-Mode Power Supplies

[3-10]

Current crowding

  • e

m r e

R A

  • g

g e

R A

  • g

m

g R R m g

nI R nI R R

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Mor M. Peretz, Switch-Mode Power Supplies

[3-11]

Inductor design Air gap

e g

  • e

e g

  • r

g e

  • If
  • g

e re

  • Mor M. Peretz, Switch-Mode Power Supplies

[3-12]

Inductor design Saturation - Ae

  • d

dI V n V L dt dt

Ae Bmax LIpk n

  • max

pk

B I e

dB dI nA dt L dt dt dt

  • max

pk e

LI nA B

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Mor M. Peretz, Switch-Mode Power Supplies

[3-13]

Inductor design Aw

  • a

w rms a rms w

nW A k I W J nI A Jk

Mor M. Peretz, Switch-Mode Power Supplies

[3-14]

Inductor design Ap

  • max

pk rms p e w

LI I A A A B Jk

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Mor M. Peretz, Switch-Mode Power Supplies

[3-15]

Inductor design summary

  • Calculate Ap
  • Choose core
  • Calculate n
  • Calculate lg or adjust gap

Mor M. Peretz, Switch-Mode Power Supplies

[3-16]

Inductor selection Off-shelf product

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Mor M. Peretz, Switch-Mode Power Supplies

[3-17]

Inductor selection Off-shelf product

Mor M. Peretz, Switch-Mode Power Supplies

[3-18]

Skin effect

depth skin

  • f

72 ) mm (

  • Hz

in f

  • 1

R R

DC AC

DC Frequency High

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Mor M. Peretz, Switch-Mode Power Supplies

[3-19]

Skin Effect Solutions

Litz wire Foil/tape

Mor M. Peretz, Switch-Mode Power Supplies

[3-20]

Proximity effect

I I

Current crowding due to magnetic fields

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Mor M. Peretz, Switch-Mode Power Supplies

[3-21]

Losses function of B

j if

B K ~ P

i, j – material dependent

  • Mor M. Peretz, Switch-Mode Power Supplies

[3-22]

Losses function of temperature

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Mor M. Peretz, Switch-Mode Power Supplies

[3-23]

Losses material selection

Mor M. Peretz, Switch-Mode Power Supplies

[3-24]

Ideal transformer

  • 1

1 2 2

n I n I

  • 1

2 1 1 2 2

d d V n V n dt dt φ

  • 1

1 2 2 m

R n I n I

m

R

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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

Mor M. Peretz, Switch-Mode Power Supplies

[3-26]

Magnetizing inductance

  • 1

1 2 2 m

R n I n I

  • 1

1 1

d V n dt

  • 2

1 2 1 1 2 1

m

m L

n n d V i i R dt n

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Mor M. Peretz, Switch-Mode Power Supplies

[3-27]

uncoupled magnetic flux

Llkg, M (mutual coupling) and k (coupling coefficient)

  • 1

2

M k L L

  • 1

(1 )

lkg m

L L k

  • 2

2 1 lkg lkg

L L n

Leakage inductance

Mor M. Peretz, Switch-Mode Power Supplies

[3-28]

Magnetizing and leakage inductances

V1 V2 I2 Im I1

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Mor M. Peretz, Switch-Mode Power Supplies

[3-29]

Transformer design Aw

  • 1

1 2 2 1 1 2 2 1 1

2

a a w rms a rms rms w rms w

n W n W A k I W J n I n I A Jk n I A Jk

Mor M. Peretz, Switch-Mode Power Supplies

[3-30]

Transformer design Saturation - Ae

  • Vdt

nAe 1 B

  • max

2

m

  • n

e

V T n B A

  • max

max

B B

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SLIDE 16

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Mor M. Peretz, Switch-Mode Power Supplies

[3-31]

Transformer design Ap

  • 1

1 max

2 2

  • n

rms p w e

V T I A A A B Jk

  • 1

1

2

rms

  • n

p

V T I A B Jk

  • 1

1

2

rms

  • n

p s

V D I A f B Jk

Mor M. Peretz, Switch-Mode Power Supplies

[3-32]

Transformer design summary

  • Calculate Ap
  • Choose core
  • Calculate n1
  • Calculate n2
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Mor M. Peretz, Switch-Mode Power Supplies

[3-33]

Magnetic design Example

  • Boost converter: P=100W, Vin=10V, Vo=50V, IL=0.1ILav
  • Calculate L
  • Calculate Ap

Mor M. Peretz, Switch-Mode Power Supplies

[3-34]

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Mor M. Peretz, Switch-Mode Power Supplies

[3-35]

Mor M. Peretz, Switch-Mode Power Supplies

[3-36]

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Mor M. Peretz, Switch-Mode Power Supplies

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Mor M. Peretz, Switch-Mode Power Supplies

[3-38]

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Mor M. Peretz, Switch-Mode Power Supplies

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