Measurement of magnetic permeability of steel laminations of - - PowerPoint PPT Presentation

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Measurement of magnetic permeability of steel laminations

• f Booster gradient magnets

Yury Tokpanov supervisors: Valery Lebedev, Bill Pellico

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Problem

Calculation of impedance of Booster gradient

magnets

Unknown magnetic permeability of the steel

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Idea of measurement

Electromagnetic wave propagation in strip lines depends upon properties of materials, including magnetic permeability

Microstrip line Strip line Network analyzer

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1D transmission line

U I IR L x t I U C x t ∂ ∂  = − −   ∂ ∂  ∂ ∂  = −  ∂ ∂  exp( ) exp( ) U A i t ikz B i ikz ω ω = − + + exp( ) exp( ) A B I i t ikz i t ikz R i L R i L γ γ ω ω ω ω = − − + + +

U R i L I i C ω ρ ω + = =

2

( ) k i C R i L ω ω = − + Basic element of transmission line: Telegrapher’s equations: Harmonic solutions:

Wave impedance:

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Microstrip line parameters

W C H εε =

H L W µ =

The simplest formulae (valid if W>>H) for parameters per unit length:

(1 ) 2 2

strip ground

i R sqrt sqrt W ωµ ωµ σ σ   +     = +              

More complicated formulae exist, which take into account edge effects. If resistive losses are negligible (for example, in the case of copper), then

0 1

2 L i C δ ρ ρ   ≈ ≈ +    

1 2 kl l LC i δ ω ωτ   ≈ ≈ −    

' '' i ε ε ε = −

Loss tangent:

'' tan ' ε δ ε =

( ) '( ) ''( ) i µ ω µ ω µ ω = −

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S-parameters

Z ρ κ =

ɶ

4 exp(

) U i t ikz ω −

Definition: Our case (symmetric):

2 exp(

) U i t ikz ω −

3 exp(

) U i t ikz ω + ɶ

0 exp(

) U i t ikz ω −

ɶ

1 exp(

) U i t ikz ω + S-parametes are measured by network analyzer

21 2

2 2 cos ( 1)sin S kl i kl κ κ κ = + +

2 11 2

( 1) tan 2 ( 1) tan i kl S i kl κ κ κ + = + +

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Experimental setup

Copper strip line Steel microstrip line Copper microstrip line Tapering Network analyzer

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Copper microstrip line

17.4 ρ = Ω

0.02 δ =

9 sec

1.91 10 rad τ

−

= ⋅

17.4 ρ = Ω

1.4 H mm =

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S21 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S21 phase S21m S21m fit S21p S21p fit

12 W mm =

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S 11 m ag n itud e

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S11 p h ase S11m S11m fit S11p S11p fit

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Tapered copper microstrip line

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S 21 m ag n itud e

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S 21 p h ase S21m S21m fit S21p S21p fit

0.02 δ =

17.3 ρ = Ω

9 sec

1.84 10 rad τ

−

= ⋅ 12 W mm = 1.4 H mm =

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S 11 m ag n itu de

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S11 p h ase S11m S11m fit S11p S11p fit

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Strip transmission line

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08 frequency, Hz S21 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S21 phase S21m S21m fit S21p S21p fit

0.02 δ =

10.1 ρ = Ω

9 sec

2 10 rad τ

−

= ⋅ 12 W mm =

1.4 H mm =

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08 frequency, Hz S11 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S11 phase S11m S11m fit S11p S11p fit

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Weakly-linked resonator

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08 frequency, Hz S21 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S21 phase S21m S21p

9 sec

1.88 10 rad τ

−

= ⋅

instead of

9 sec

1.85 10 rad τ

−

= ⋅ 12 W mm =

0.8 H mm =

12

Steel

1

s c

R i L ρ ρ ω = +

1

s c

R i L τ τ ω = +

(1 ) 2

s

i R sqrt W ωµµ σ   + =    

c c

L l ρ τ ⋅ =

2

1 1

s a r

f f i f f µ µ = +   + −   

How to take into account resistive losses:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S21 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S21 phase S21m copper S21m steel S21p copper S21p steel

Landau-Lifshitz ferromagnetic resonance model:

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2.3 10

s

S m σ = ⋅

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.00E+05 1.00E+08 2.00E+08 3.00E+08 4.00E+08 5.00E+08 6.00E+08 7.00E+08 8.00E+08 9.00E+08

frequency, Hz

S11 magnitude

• 180
• 150
• 120
• 90
• 60
• 30

30 60 90 120 150 180 S11 phase S11m copper S11m steel S11p copper S11p steel

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Results

Technique for determining necessary parameters is

developed

Experimental investigation of the problem is carried

• ut

Rough estimation of magnetic permeability is obtained

Plans

Solve problem of additional phase shift Carry out experiments in strong dc magnetic field

(~1-2 T)

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Thank you!