Magnetics Design 3.1 Important magnetic equations 3.2 Magnetic - - PDF document

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 1]

Magnetics Design

3.1 Important magnetic equations 3.2 Magnetic losses 3.3 Transformer 3.3.1 Ideal transformer (voltages and currents) 3.3.2 Equivalent circuit of transformer (coupling, magnetization current) 3.3.3 Design of transformer 3.4 Inductor design

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 2]

; dt dB nA dt d n V

e

= Φ = H B ∆ ∆ = µ Φ - magnetic flux Weber [ Wb ] B - flux density ] T [ Tesla m Wb

2 =

V - voltage [ V ] Also : Gauss [ G ] 1T = 10,000 G B H BDC HDC Φ Ae

Faraday’s law

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 3]

Ampere’s law

H - magnetic field [ A/m ]

∫

⋅ = I n Hdl

e

H I n l ⋅ = ⋅

e

I n H l ⋅ = [ A/m ] I n

e

l

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 4]

Magnetic losses ~∆B “Good number” = 100mW/cm3 =100KW/m3 P

3

cm mW B ∆

Magnetic losses

B H BDC HDC

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 5]

“Good number”=100mW/cm3=100 kW/m3

Magnetic Losses

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 6]

Magnetic losses ~∆B “Good number” = 100mW/cm3 =100KW/m3 P

3

cm mW B ∆

Magnetic losses

B H BDC HDC

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 7]

Magnetic Losses

• Prof. S. Ben-Yaakov , DC-DC Converters

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• Curves for constant loss: 500mW/cm3
• Figure of merit B*f
• Each material has optimum operating temperature

(minimum loss)

Magnetic Losses

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 9]

Transformer currents

n1 n2 I1 I2 For ideal transformer

2 2 1 1

I n I n =

1 2 2 1

n n I I = At any given moment

2 2 1 1

I n I n = I1,I2 opposite direction. No magnetic energy stored due to useful currents I1, I2 (they cancel each other)

n1 n2 I1 I2

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 10]

Transformer voltages

φ dt d n V

1 1 1

φ = dt d n V

2 2 2

φ =

2 1

g min Assu φ = φ dt d dt d

2 1

φ = φ

2 1 2 1

n n V V =

n1 n2 V1 V2

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• Prof. S. Ben-Yaakov , DC-DC Converters

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Since each winding also represents an inductance, therefore for any winding

= n V

Permissible voltages: AC only on any winding

S1 V S1 S2 S2 S1 V S1 V S2 S2 S1 S2 S1 S2 t t t

A B C

Voltages

• Prof. S. Ben-Yaakov , DC-DC Converters

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Equivalent circuit (preliminary)

2 m m

n L L

1 2 =

∞ → L Ideal transformer

Ideal 1:n Lm1 Llkg1 Ideal 1:n Lm2 Llkg1 Ideal 1:n Lm1 Llkg2

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• Prof. S. Ben-Yaakov , DC-DC Converters

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Leakage inductance

I2 V2 I1 V1 n2 n1

Leakage inductance is the

uncoupled magnetic flux 1:n Lkg1 Lm1 ideal Lkg2

Relationship between Llkg, M and k (coupling coefficient).

2 1 L

L k M ⋅ = ) k 1 ( L L

1 m 1 lkg

− ≅

2 1 lkg 2 lkg

n L L ⋅ ≅

Leakage

• Prof. S. Ben-Yaakov , DC-DC Converters

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1:n Llkg1 Lm1 ideal Llkg2 Vo Llkg1 Lm1 L'lkg2 V'o

2 2 lkg 2 lkg

n L L = ′

2

• n

V V = ′

Leakage

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 15]

Magnetization Current

Vin t Vo t I2 t Im t I1 t Ideal 1:n Lm1 Llkg2 R I2 Vo I1 Im Vin

• Prof. S. Ben-Yaakov , DC-DC Converters

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Transformer

• 1. Bmax ( could be symmetrical or asymmetrical )
• 2. Bmax < Bsat
• 3. In most case ( high frequency ) Bmax limit by magnetic

losses. dt dB A n dt d n V

e 1 1 1

= Φ = I2 V2 I1 V1 n2 n1 H B B sat- Bsat+ B max+ ∆Β Bmax-

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 17]

Symmetrical operation

∫

= Vdt A n 1 B

e 1 − + = max max

B B

e 1

• n

m max

A n t V B 2 B = = ∆

{ }

e max

• n

m 1

A B 2 t , V n = B Vm Ts Bmax- Bmax+ ton V1

s 2 T t

• n

e 1

f 1 ~ t ~ A n

S

• n→
• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 18]

Skin effect

depth skin − δ f 72 ) mm ( = δ Hz in f δ 1 R R

DC AC >

DC Frequency High

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 19]

Skin Effect Solutions

Litz wire Tape

• Prof. S. Ben-Yaakov , DC-DC Converters

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Proximity effect

I I

Current crowding due to magnetic fields

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• Prof. S. Ben-Yaakov , DC-DC Converters

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[ ]

k n w n w A

2 2 A 1 A w

1

⋅ + ⋅ = k - filling factor k<1 J I w

rms 1 A1 =

J - current density A/m2 J ≅ 4.5 A/mm2

1 2 1 2

I n n I = Aw - winding area Aw wA

Aw

• Prof. S. Ben-Yaakov , DC-DC Converters

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2 Jk I n A

rms 1 1 w

⋅ =

rms 1 w 1

I 2 JkA n ⋅ =

{ }

e max

• n

1 1

A B 2 t , V n =

{ }

e max

• n

1 rms 1 w

A B 2 t , V 2 I JkA = ⋅

{ } { }Jk

B 2 I 2 t , V A A A

max rms 1

• n

1 e w p

⋅ = =

{ }

Jk B I 2 t , V A

rms

1

• n

1 p

⋅ ∆ ⋅ =

{ }

Jk B f I 2 D , V A

s 1

• n

1 p

rms

⋅ ∆ ⋅ ⋅ =

Ap

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 23]

Transformer design stages

• 1. Calculate Ap
• 2. Look for core
• 3. Calculate n1 by:
• 4. Calculate n2

{ }

Jk B f I 2 D , V A

s 1

• n

1 p

rms

⋅ ∆ ⋅ ⋅ = In symmetrical operation ∆B = Bmax

+ - Bmax

• In asymmetrical operation

∆B = Bmax - 0

{ }

e max

• n

m 1

A B 2 t , V n =

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 24]

Inductor design

Need to store energy ( in transformer n1·I1= n2·I2 )

r

• µ

µ = µ µo - air (vacuum) permeability µr - relative permeability I L B H µ

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• Prof. S. Ben-Yaakov , DC-DC Converters

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µo = 1.26·10-6 m Henry If µ is high B will reach quickly Bsat Need to slower µ B H Ho <

2 r

µ

2 r

µ

1 r

µ

1 r

µ Bo µr of ferrites ∼ 2000 - 4000 B = µH

Permeability

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 26]

Same Φ magnetic lines in ferromagnetic material and in air.

e g

l l <<

e e g

l l l ≅ + Discrete air gap Φ µo µr

e

l

g

l Distributed air gap

Gaps

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• Prof. S. Ben-Yaakov , DC-DC Converters

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Current crowding due to magnetic fields RAC high around gap

Current Crowding

• Prof. S. Ben-Yaakov , DC-DC Converters

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Φ = constant B ≅ constant

• g

B H µ =

m m

B H µ =

g g e m e

H H H nI l l l + = =

• g

B m e B e H µ + µ = l l l

e g

l l <<

e e g

l l l ≅ +

e

l

g

l

Inductance with Gap

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• Prof. S. Ben-Yaakov , DC-DC Converters

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g e

• a

m m e

B B B H l l µ + µ = µ =

• g

m e e

B B H µ + µ = l l l

Dividing out le and defining

H B

e =

µ

Inductance with Gap

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 30]

        µ + µ = µ

g e

• m

e

1 1 1 l l         µ + µ µ = µ µ

g e

• rm
• re

1 1 1 l l         + µ = µ

g e rm re

1 1 1 l l         µ µ +         = µ

g e rm rm g e re

1 l l l l

rm g e g e rm re

µ +                 µ = µ l l l l

rm g e

If µ < l l         ≈ µ

g e re

l l

Gap Calculation

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• Prof. S. Ben-Yaakov , DC-DC Converters

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dt dI L V = dt d n V Φ = dt dI n nA dt dH nA dt dB nA dt d n

e e e e

l µ = µ = = Φ dt d n dt dI L Φ = L-? dt dI A n dt dI L

e e 2

l µ =

e e 2A

n L l µ =

Inductance

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 32]

2 2 1 2 1

n n L L       = Inductor design B H Bmax

Two windings on same core

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 33]

dt d n dt dI L Φ =

∫ ∫

      =      

max pk

B e I

dt dt dB nA dt dt dI L L Ipk = nAeBmax

max e pk

B A LI n =

max pk e

nB LI A =

rms w

I JkA n = quick design and check

Saturation Limits

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 34]

Jk B I LI A A A

max rms pk w e p

= =

2 rms pk

LI I LI ≈ 2 LI stored Energy

2

= Air gapped core Design

• 1. Calculate Ap
• 2. Choose a core
• 3. Iterate
• 4. Calculate ( or increase gap until L is as required )

g

l

Ap

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• Prof. S. Ben-Yaakov , DC-DC Converters

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Cores

Transformer core Inductor core

air gap

• Prof. S. Ben-Yaakov , DC-DC Converters

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• 1. E - core
• 2. TOROID
• 3. ARENCO
• 4. POT

Cores

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• Prof. S. Ben-Yaakov , DC-DC Converters

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Commercial cores

• Prof. S. Ben-Yaakov , DC-DC Converters

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Distributed gap core

turn H A

y L =

) turns 1000 H sometime (

y

L for n turns:

L 2 A

n L ⋅ =

The concept of AL

Distributed air gap

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• Prof. S. Ben-Yaakov , DC-DC Converters

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AL

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 40]

Toroid Data

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 41]

1 Amp/m =79.5 Oe

L decreases with DC current !

Permeability change

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 42]

These curves are measured by feeding ac signals. If the current is composed of DC + ripple, core loss is due

• nly to ripple component !

DC bias tend to increase loss “Good number”=100mW/cm3 Misleading notations ! ∆B NOT B

Losses

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• Prof. S. Ben-Yaakov , DC-DC Converters

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“Hot Spot” - Critical parameter

• Temp. Rize
• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 44]

Hanna Curve

• r

H max B e V 2 LI max HB max B 1 e V 2 LI H µ µ = µ = = = max B e A e l LI nI Hn max B e A pk LI H Hn max B e A pk LI n = = =

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 45]

Hanna Curve

• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 46]

Core Size Selection

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• Prof. S. Ben-Yaakov , DC-DC Converters

[3- 47]

Basic Design of Distributed Gap Core

) nI ( f

e

l = µ

• 1. Calculate LI2
• 2. Look up manufacturer data
• 3. Select Core
• 4. Calculate
• 5. Check Lmin
• 6. Calculate losses. Temp rise and and
• 7. Iterate

) ( L A L n 1000 1000 =