Skin Depth Investigation

Lei Zang

The University of Sheffield

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Introduction: CST STUDIO

• CST STUDIO is a package of tools for designing, simulating and optimizing electromagnetic systems.• CST: Computer Simulation Technology

• It includes various modules dedicated to specific application areas:• Microwave studio, EM Studio, Particle Studio, MPHYSICS Studio, Design Studio, PCB Studio,

CABLE Studio

• CST is based on “MAFIA”--- “solving MAxwell’s equations using the Finite Integration Algorithm*”

*T. Weiland, "A Discretization Method for the Solution of Maxwell’s Equations for Six-Component Fields", Electronics and Communication, (AEÜ), Vol. 31, p.116, 1977

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Some Accelerator Related Example with CST

• CST Microwave Studio: RF Cavity design

• CST EM Studio: Magnet design

• CST Particle Studio: particle and Wakefield

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-20 0 20 40 60 80 1000.0

0.1

0.2

0.3

0.4

0.5

FC2 data Bz

CST Bz

Bz [T

]

Longitudinal Position [mm]

measurement data at low current (1.36 kA)

-20 0 20 40 60 80 100-0.020

-0.015

-0.010

-0.005

0.000

0.005

0.010

FC2 data Bx

CST Bx

Bx [T

]

Longitudinal Position [mm]

Bz

Bx

CST Work Demonstration

Bore

Central Core

Slit

Primary Coil

Primary CurrentInduced Current

FLUX Concentrator

• Pulsed current flows to the

primary coil.

• The induced current flow into

the inner core through the slit.

• Duo to the skin effect, current

has a path near inner surface

Field measurement and simulation results agreed very well!!

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LBNE Horn Inner Conductor

For the first step:

What is the skin depth in the horn neck section?

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Simulation Results Validation

• Software: CST Studio

• Purpose: compare the simulation results with the theoretical calculation

• Geometry: Cylinder, R=15mm

• Material: Aluminium and Aluminium alloy (6061-T6)

Aluminium MaterialAl, Al alloy and Al alloy at 75oC

Current PathAC CurrentFrequency 625Hz

Calculate the current density in the middle position z=100mm

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• The penetration of current in a good conductor is characterized by the skin depth. At this depth, the current density is 1/e~0.37 of that at the conductor surface.

• Where ω is the angular frequency of current, μ is the permeability and σ is the conductivity.

𝛿 = 2

𝜔𝜇𝜎 [𝑚]

CST Simulation Results

Aluminium Skin Depth= 3.46mmTheoretical prediction=3.28mm

6061-T6 Skin Depth=4.17mmTheoretical prediction=4.02mm

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6061-T6 at 75 degree

At 75 degree, 6061-T6’s resistivity=4.7 e-8 ohm-mSkin depth = 4.45mmTheoretical prediction=4.36mm

CST and theory agrees well for a simple cylinder geometry. We proved that the software is capable of doing this calculation.

Now, how about a HORN?

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Horn Neck Skin Depth Approximate Evaluation

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A Tube• Same thickness as Horn neck

• Material 6061-T6 at 75 degree

• Inner radius is 9mm

• Outer radius is 13.5mm

Max=2.3e+11Min=1.8e+11Current density will not drop to 1/e

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Conclusions

• CST Studio could simulate skin effect to get fair accurate results. (Tested with different frequency, material and resistivity)

• Skin depth theoretical equation can only apply to the simple geometry such as a cylinder. In which case, CST agrees well with it.

• A tube geometry has been used as an approximation. The results indicated a current density degradation of 22%.

• Therefore, we can conclude that the current flow in horn neck should be relatively uniform.

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Future Plan

• After obtaining the the skin depth with some approximation, I would like to see the results again with the realistic geometry. I expect they are the same.

• Investigate the magnetic field distortion, which is caused by the input power.

• Generate 3D horn field map to implement it to G4LBNE simulation. A realistic 3D field should return a more accurate neutrino yield.

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