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CAS on Small Accelerators: Wave Generators, Transformers, and Breakdown in Gases, Study notes of Engineering

Leeds Beckett University (LBU)Engineering

An in-depth analysis of various aspects of small accelerators, including traveling wave generators (PFL, PFN, transmission line transformers), cascaded high voltage transformers, and breakdown in gases and liquids. Topics covered include voltage and current values, cascaded rectifiers (Greinacher, Cockcroft-Walton), LC-oscillations, Marx Generator, pulse source with transformer, and breakdown field strengths. The document also discusses insulation and breakdown in gases and liquids, as well as applications and disadvantages of vacuum insulation.

Typology: Study notes

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CAS on Small Accelerators

High Voltage Engineering

Enrique Gaxiola

Many thanks to the Electrical Power Systems Group, Eindhoven University of Technology, The Netherlands

& CERN AB-BT Group colleagues

Introductory examples

Theoretical foundation and numerical field

simulation methods

Generation of high voltages

Insulation and Breakdown

Measurement techniques

Discover Study notes of Engineering Leeds Beckett University (LBU)

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CAS on Small Accelerators

High Voltage Engineering

Enrique Gaxiola

Many thanks to the Electrical Power Systems Group, Eindhoven University of Technology, The Netherlands

& CERN AB-BT Group colleagues

Introductory examples

Theoretical foundation and numerical field

simulation methods

Generation of high voltages^ Insulation and Breakdown

Measurement techniques

CAS on Small Accelerators

Introduction E.Gaxiola:

Studied Power Engineering Ph.D. on Dielectric Breakdown in Insulating Gases;

Non-Uniform Fields and Space Charge Effects

Industry R&D on Plasma Physics / Gas Discharges CERN Accelerators & Beam, Beam Transfer, Kicker Innovations: •

Electromagnetism

Beam impedance reduction

Vacuum high voltage breakdown in traveling wavestructures.

Pulsed power semiconductor applications

CAS on Small Accelerators

PS septa SEH

Voltage: 300 kV

SPS septa ZS

CAS on Small Accelerators

SPS injection kicker magnets

30 kV

spacers

beam gap

magnets

ferrites

CAS on Small Accelerators

Maxwell equations for calculating

Electromagnetic fields, voltages, currents

Analytical

Numerical

CAS on Small Accelerators

Breakdown

Electrical

Fields,

Geometry

Medium

Insulation and

Breakdown

High fieldsField enhancementField steering

Charges in fieldsIonisationBreakdown

GasLiquidsSolidsVacuum

CAS on Small Accelerators

Procedure FEM

1. Generate mesh of triangles:2. Calculate matrix coefficients:3. Solve matrix equation:4. Determine equipotential lines and/or field lines

[^

]^

(^

)^

A

S

j

i

ij

α

∇⋅

∇

[^

]^

0

⎤ ⎥ ⎦

⎡ ⎢ ⎣

f p

kp

kf

U U

S

Procedure BEM

1. Generate elements along interfaces2. Generate matrix coefficients:3. Solve matrix equation:4. Determine potential on abritary position:

∫

j

j^

S

i

ij

S

i

ij^

ds

r

G

ds

n

r

H

ln

(^

)^

∑

=

n j

j

ij

n j

j

ij

ij^

Q

G

U

H

1

πδ

∑

∫

∑

∫

=

n j

S j

n j

S j

j

rds

Q

ds

n

r

U

y

x

U

1

0

ln

π

CAS on Small Accelerators

Generation of High Voltages

AC Sources (50/60 Hz)

High voltage transformer

(one coil; divided coils; cascade)

Resonance source

(series; parallel)

DC Sources

Rectifier circuits

(single stage; cascade)

Electrostatic generator

(van de Graaff generator)

Pulse sources

Pulse circuits (single stage; cascade; pulse transformer) Traveling wave generators (PFL; PFN; transmission line transformer)

CAS on Small Accelerators

L

R

C

EquivalentCircuit:

C

L L L

cap.deler

test

ν

Resonance Source

Waveform: almost perfect sinusoidal + Power: 1/Q of “normal” transformer + Short circuit: Q

0 results in V

No resistive load

2 0

0

2

0

Q

H

L C

R

Q

LC

1

and

1

0

=

ω

900 kV100 mA

Courtesy: Eindhoven Univ.of Techn.

CAS on Small Accelerators

Cascaded Rectifier

(Greinacher; Cockcroft - Walton)

max

2

nV

V

DC

=

C

n

C

`n D

n D

`n

C

1

C

` 1 D

1 D

` 1

C

2

C

` 2 D

2 D

` 2

C

n-

C

n-

D `

n-1 D

n-

`

stage 1 stage 2 stage n-1 stage n n

n`

2

2`

1

1`

amplitude: V

max

V

DC

Reduce

δ

V (~n

2 ) and

V (~n

3 ) by:

larger C’s (more energy in cascade)higher f (up to tens of kilohertz)

Voltage: 2 MV

Courtesy: Delft Univ.of Techn.

CAS on Small Accelerators

Cascade Pulse Source

(Marx Generator)

DC

pulse

V

n

V

⋅

=

C``

R

`^2

`

C

` 1

R

`^2 R

`^2

C

` 1 C

` 1

`

stage n stage 3 stage 2 stage 1

V

DC G

1 G

2 G

3 G

n

1

A

1

2

A

2

3

A

3

n

A

n

C`

V

pulse

Total discharge capacity: 1/C1=

∑

1/C1’

Front resistance: R1=R1”+

∑

R1’

Discharge resistance: R2=

∑

R2’

R

: front resistor 1 R

: discharge resistor 2

R

` 1 R

` 2

R`

R

`` 1

C

` 1

R

` 1 R

` 2 R

` 2

C

` 1 C

` 1

RR R`

V

DC

900 kV100 mA

Courtesy: Kema, The Netherlands

CAS on Small Accelerators

G

C

2

R

1

C

1

R

2

U

2

U

1

resistivity neglected

I^1

I^2

d^ dt

t U

1

) (^

Φ

d dt

t

U

2

) (^

Φ

Pulse Source with Transformer

0

:

secondary

0

:

primary

2

(^12) 2

2

(^22) 2

2 2

1

(^22) 2 1

(^12) 2 1 1

=

−

=

−

I

dt

I d

MC

dt

I d

C L

I

dt

I d

MC

dt

I d C L

2 1 2 2 2 2 2 1 1

1 2

2

1 2

2 2

2

1

1 1

⎛^ ⎜ ⎝

C C M C L C L

i i

C

L

MC

C

L

)

//

(

1

,

)

(

1

' 2

1 2 2 1 1 2 2

' 2

1 1 2 2 1 1 1

C C L C L C L k C C L C L C L

eq

= + − ≈ + = + ≈

ω

slow oscillation

fast oscillation

Eigen frequencies from characteristic equation: Approximation: transformer almost ideal: k=M/

(L

L 1

Voltage: 300 kV

CAS on Small Accelerators

Insulation and Breakdown

In Gases

Ionisation and Avalanche Formation Townsend and Streamer Breakdown Paschen Law: Gas Type Breakdown Along Insulator Inhomogeneous Fields, Pulsed Voltages, Corona

Insulating Liquids • Solid Insulation

Breakdown types, Surface tracking, Partial discharges, Polarisation, tan

δ

Vacuum Insulation

Applications, Breakdown, Cathode Triple-Point, Insulator Surface Charging, Conditioning

CAS on Small Accelerators

800kV South Africa

400kV

400kV Geertruidenberg,The Netherlands