Exercises in the lectures on Exercises in the lectures on - - PowerPoint PPT Presentation

Exercises in the lectures on Exercises in the lectures on “ Superconducting RF - I and - II ” p g

Kenji Saito, KEK

Exercise I.

Using the Abrikosov’s theory:

2 2

( / 2 ) ,

c

c hc e H φ κ κ λ λ λξ = = = h

2 2

2

c

H φ φ λ ξ ξ = =

2 2

, 2 * 2 2 2 2

c

e λ λ π π λξ

2 2

2 2 2

c

ξ πξ π λξ

7 2 15 2

/ 2 2.0678 10 hc e Gauss cm φ

−

= = × ⋅ 1) write down ξ, λ λ by HC and HC2, ξ

RF

• 15

2

=2.0678 10 T m × ⋅ 2) get the T-dependences of ξ, HC2, κ, HC

RF, from the given T-dependences

• f λ

λ and HC:

2

(0) ( ) (0) 1 ( / ) ( ) H T H T T T λ λ

⎡ ⎤ ⎣ ⎦

= =

4

RF RF C C C

,

( ) ( ) (0) 1 ( / ) ( ) 1 ( / )

H ,here H is given as H 2 .

C C C C

H T H T T T T T λ

κ

⎡ ⎤ ⎣ ⎦

= − = −

= ⋅

C C

κ

Exercise II.

1) Get the following formula for the surface resistance Rs for good electric conductor.

1 1 µω µσω

2) Calculate the δ and RS for a 1300MHz copper cavity, when the σ is given as 1/σ =1.72E-8 [Ωm] at 20OC.

S

1 1 R 2 2 µω µσω σ σ σδ = = =

1/σ 1.72E 8 [Ωm] at 20 C. 3) If the RRR of the copper material is 40, calculate the Rs at 4.2K.

Exercise III Exercise III.

By the two fluid model, electric conductivity is given as the bellow:

2 2 n

= , = v

n s s s

n e n q i E i m m σ σ σ σ ω ⎛ ⎞ − = − ⎜ ⎟ ⎝ ⎠ J E

Put this complex electric conductivity into the formula of surface impedance: Z=RS+iXS, show the surface resistance and admittance for superconductor are:

2 2 2 3

1 n e λ λ ⋅

v

e s

m m ω ⎝ ⎠

2 2 3

1 , and 2 v

n S n L S L n S

n e R X m σ ω µ λ ωµλ σ = = = ⋅

nn is the number of unpaired electrons (quatsi particle), then it could be written by Boltzman statistics as: then it could be written by Boltzman statistics as:

2

(0) v

B

k T n s

e n e m σ

Δ −

= ⋅

Show the formula of surface resistance in case of superconductor as: Show the formula of surface resistance in case of superconductor as:

2

( , ) ( , , , ) exp( )

S L C B

R T f A T f k T λ ξ Δ = ⋅ ⋅ − l

Exercise IV.

Get the formulas in lecture note p.65

2 2

1 ,

z t t z t z

B i E z c k ω εµ ω ⎡ ∂ ⎤ ⎛ ⎞ = ∇ + ×∇ ⎜ ⎟ ⎢ ⎥ ∂ ⎛ ⎞ ⎝ ⎠ ⎣ ⎦ ⎜ ⎟ B e

2 2 2

1

z t t z t z

k c E i B z c εµ ω ω ⎝ ⎠ ⎣ ⎦ − ⎜ ⎟ ⎝ ⎠ ⎡ ∂ ⎤ ⎛ ⎞ = ∇ − ×∇ ⎜ ⎟ ⎢ ⎥ ∂ ⎛ ⎞ ⎝ ⎠ ⎣ ⎦ E e

2 2

z c k c ω εµ ∂ ⎛ ⎞ ⎝ ⎠ ⎣ ⎦ − ⎜ ⎟ ⎝ ⎠

Exercise V. e c se V.

Make design a 1300MHz TM010 – mode single cell Pill Box cavity 1.What is the diameter of the cell? 2 What is the cell length?

• 2. What is the cell length?

Exercise VI. Superfish outputs

f0=1293.77430MHz Ploss=118.1551W RsQ=265 171 Ω RsQ=265.171 Ω Qo=28257.6 (Rsh/Q)=109.24 Ω Hp=1753.44 A/m

Calculate the following cavity RF parameters from above Superfish outputs.

Hp 1753.44 A/m Ep=0.946176 MV/m

Calculate the following cavity RF parameters from above Superfish outputs. Rsh [Ω] = Accelerating Voltage V [MV]= RF wave length λ[ λ[m] ] = Gradient Eacc = V/Leff [MV/m]= ,defined as Leff =λ/2

eff [

] ,

eff

Hp/Eacc[Oe/(MV/m)] = , use 1A/m= 4π10-3 Oe Ep/Eacc = E [MV/ ] Z Eacc [MV/m] = Z= Geometrical factor Γ Γ [Ω] =

Exercise VII.

Calculate the cable correction factors: Cin, Cr and Ct, when measurement results are: when measurement results are: pin=55.5μW, po=50.0mW, pr=10.72μW, pt=3.04mW and

’ 39 0

W

’ 22 6

W

’ 27 9

W po

’=39.0mW, pin ’=22.6mW, pt ’=27.9mW

Exercise VIII. Calculate βin∗, β

, βin, β , βt, Ploss[W]’, QL, Qin, Qo, Qt, Rs[Ω], Eacc[MV/m], Ep[MV/m], and Hp[Oe],

h lt when measure results are : f0=1303.590529MHz, τ1/2=23.6 msec, τ1/2 23.6 msec, coupling over, pin=3.11mW, pr=192nW, pt=0.142mW. For the cable correction factors, use the results of the exercise VII. RF cavity parameters are given as following: Γ 269Ω E /E 1 83 H /E 45 2 O /[MV/ ] Γ=269Ω, Ep/Eacc=1.83, Hp/Eacc=45.2 Oe/[MV/m], and

[ ]

[ / ] 86.94

t t

W

Eacc MV m P Q = ⋅