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 wave structures. • Pulsed power semiconductor applications CAS on Small Accelerators

CERN Septa and Kicker examples 4 x MKQAV

• Large Hadron Collider 14 TeV • Super Proton Synchrotron 450 GeV • Proton Synchrotron 26 GeV

4 x KKQAH

LHC 30 x MKD 12 x MKBV 8 x MKBH

4 x MKI 4 x MKP 4 x MKE

SPS 3 x MKDH

MKQH 5 x MKE

2 x MKDV

PS PSComplex

MKQV

2 x TK

4 x MKI

9 x KFA71 4 x KFA45

3 x KFA79

4 x KSW 4 x EK

5 x BI.DIS

Reference [1]

Septum: E ≤ 12 MV/m T = d.c. l= 0.8 – 15m Kicker: V=80kV B = 0.1-0.3 T T = 10 ns - 200μs l=0.2 – 16m RF cavities: High gradients, E ≤ 150MV/m CAS on Small Accelerators

PS septa SEH23

SPS septa ZS

Voltage: 300 kV

CAS on Small Accelerators

spacers

• SPS injection kicker magnets

ferrites

beam gap

30 kV

magnets CAS on Small Accelerators

• SPS extraction kickers

Courtesy: E2V Technologies

Generators Pulse Forming Network 60 kV

Thyratron gas discharge switches Power Semiconductor Diode stack

Magnets 30 kV

72 kV

CAS on Small Accelerators

• Maxwell equations for calculating Electromagnetic fields, voltages, currents – Analytical – Numerical

CAS on Small Accelerators

Breakdown Electrical Fields, Geometry High fields Field enhancement Field steering

Insulation and Breakdown

Medium Gas Liquids Solids Vacuum Charges in fields Ionisation Breakdown CAS on Small Accelerators

NUMERICAL FIELD SIMULATION METHODS -CSM (Charge Simulation Method):

(Coulomb)

Electrode configuration is replaced by a set of discrete charges - FDM (Finite Difference Method): Laplace equation is discretised on a rectangular grid - FEM (Finite Element Method):

Vector Fields (Opera, Tosca), Ansys, Ansoft

Potential distribution corresponds with minimum electric field energy (w=½εE2)

- BEM (Boundary Element Method):

IES (Electro, Oersted)

Potential and its derivative in normal direction on boundary are sufficient CAS on Small Accelerators

Procedure FEM 1. Generate mesh of triangles: 2. Calculate matrix coefficients: [S ]ij = (∇α i ⋅ ∇α j ) A 3. Solve matrix equation:

[S

kf

⎡U f ⎤ S kp ⎢ ⎥ = 0 ⎣U p ⎦

]

4. Determine equipotential lines and/or field lines

Procedure BEM 1. Generate elements along interfaces 2. Generate matrix coefficients:

∂ ln ri H ij = ∫ ds , Gij = ∫ ln ri ds ∂ n Sj n Sj n

∑ (H

3. Solve matrix equation:

j =1

4. Determine potential on abritary position:

1 U ( x0 , y0 ) = 2π

ij

n ⎛ n ⎞ ln r ∂ ⎜ ∑U j ∫ ds − ∑ Q j ∫ ln rds ⎟ ⎜ j =1 S ∂n ⎟ j =1 S ⎝ ⎠ j

j

− πδ ij )U j = ∑ Gij Q j j =1

CAS on Small Accelerators

Generation of High Voltages • AC Sources (50/60 Hz) High voltage transformer Resonance source

(one coil; divided coils; cascade) (series; parallel)

• DC Sources Rectifier circuits Electrostatic generator

(single stage; cascade) (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

Cascaded High voltage transformer Courtesy: Delft Univ.of Techn.

900 kV 400 A

1: primary coil 2: secondary coil 3: tertiary coil

CAS on Small Accelerators

Resonance Source L

cap. deler

L L

ν

C test

Courtesy: Eindhoven Univ.of Techn.

Equivalent Circuit:

L

R

+ Waveform: almost perfect sinusoidal C

+ Power: 1/Q of “normal” transformer + Short circuit: Q→0 results in V→0

H (ω ) = ω0 =

1 LC

Q ω0 ω 1 + Q (ω 0 ω − ω ω 0 ) 2

and Q =

1 L R C

- No resistive load 2

900 kV 100 mA CAS on Small Accelerators

1`

1

D1 C1

(Greinacher; Cockcroft - Walton) Courtesy: Delft Univ.of Techn.

C1` D ` 1 2`

VDC

Cascaded Rectifier

2

D2

stage 1

C2 C2` D ` 2

Dn-1

stage 2

Cn-1

Cn-1` D ` n-1 n`

n

Dn

stage n-1

Cn Cn` D ` n amplitude: Vmax

VDC = 2nVmax stage n

Voltage: 2 MV

Reduce δV (~n2) and ΔV (~n3) by: larger C’s (more energy in cascade) higher f (up to tens of kilohertz) CAS on Small Accelerators

Single-Stage Pulse Source V0

G R1 dU dU + + bU = 0 a 2 dt dt

U(t)

2

C1 a =

1 1 1 1 + + ;b = R1C1 R2C2 R1C2 R1C1 R2C2

U

R2

C2

60 kV 1 kA

spark gaps

τ2 (discharge) τ1 (front) “LC”-oscillations t

Courtesy: Eindhoven Univ.of Techn.

(

U (t ) = V0 e − t / τ 2 − e − t / τ1

)

if C1 >> C2 and R2 >> R1 rise time: τ1=R1C2 discharge time: τ2=R2C1

Standard lightning surge pulse: 1.2 / 50 μs CAS on Small Accelerators

Vpulse R2`

Bn R`

B2 C 1`

G1

C1`

R1`

C``

R2`

R`

C1`

(Marx Generator)

V pulse = n ⋅VDC

R`

C1`

VDC R1: front resistor R2: discharge resistor

R1` R2`

stage 1 R`

VDC

R1`

900 kV 100 mA

R2`

stage 2

A1

C 1`

R2`

stage 3

R2`

B1 R`

R2` A2

G2 C`

R`

R`

R1``

A3 C1`

B3

R2`

stage n

C 1` G3

R`

Cascade Pulse Source

An

Gn

C1`

R1`

Total discharge capacity: 1/C1=∑1/C1’ Front resistance: R1=R1”+∑R1’ Discharge resistance: R2=∑R2’

Courtesy: Kema, Netherlands CAS onThe Small Accelerators

Voltage: 300 kV

Pulse Source with Transformer resistivity neglected

d 2 I1 d 2I2 primary : L1C1 2 − MC1 2 + I1 = 0 U1 dt dt d 2I2 d 2 I1 secondary : L2C2 2 − MC2 2 + I 2 = 0 C1 dt dt

G

R1

U1 (t ) =

U2

I2

I1

C2

R2

dΦ1 dt

U 2 (t ) =

dΦ 2 dt

Eigen frequencies from characteristic equation: ⎛ L1C1 − MC1 ⎞ ⎛ i1 ⎞ 1 ⎛ i1 ⎞ 1 ⎞⎛ 1 ⎞ ⎛ ⎜⎜ ⎟⎟ ⎜⎜ ⎟⎟ = 2 ⎜⎜ ⎟⎟ ⇒ ⎜ L1C1 − 2 ⎟ ⎜ L2C2 − 2 ⎟ − M 2C1C2 = 0 ω ⎠⎝ ω ⎠ ⎝ ⎝ − MC2 L2C2 ⎠ ⎝ i2 ⎠ ω ⎝ i2 ⎠ Approximation: transformer almost ideal: k=M/√(L1L2)→1 ω1 ≈

1 = L1C1 + L2C2

1 L1 (C1 + C2' )

slow oscillation

,

ω2 ≈

1 1− k 2

1 1 + = L1C1 L2C2

1 Leq (C1 // C2' )

fast oscillation CAS on Small Accelerators

Pulse Forming Line / Network charge cable

S

load cable

R

amplitude: ½VDC duration: 2L/v

Z

VDC

80 kV, 10 kA, T=20ns - 10μs Transmission Line Transformer S VDC

Z

cables parallel

cables in series

150 kV 1 kA

Courtesy: Eindhoven Univ.of Techn.

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

400kV

800kV South Africa

400kV Geertruidenberg, The Netherlands

CAS on Small Accelerators

1st

free electron

• Cosmic radiation • Shortwave UV • Radio active isotopes

In air: ≈ 2.5 x 1019 molecules/cm3 ≈ 1000 ions/cm3 ≈ 10 electrons/cm3

Free path, effective cross-section

Townsend’s 1st ionisation coefficient α One electron creates α new electrons per unit length A + e → A+ + 2e

ne(x=d) = n0eαd α/p = f (E/P) CAS on Small Accelerators

α-η vs. E/p

• Electro-negative gasses Attachment η of electrons to ions

1.) air 2.) SF6

electrons: ne(x=d) = n0e(α -η)d negative ions: noη (α −η ) d n− ( x = d ) = [e − 1] α −η

Avalanche ≠ Breakdown; creation of secondaries Townsend’s 2nd ionisation coefficient γ

Reference [3]

one ion or photon creates γ new electrons at cathode ne = γn0(eαd-1)

E 1st electron

Breakdown if:

# secondary electrons ≥ n0 αd ≥ ln(1/γ + 1)

steep function of E/p Æ eαd very steep Æ (E/p)critical and Vd well defined Æ γ of weak influence

New 1st electron

+ Avalanche + + ion r o n o + +

Phot

Secondary γ

Paschen law / breakdown field • Townsend breakdown criterion αd = K: Ed B = p ln(Apd/K)

Vd =

with

Bpd ln(Apd/K)

Æ Ed and Vd depend only on p*d

1: SF6 2: air 3: H2 4: Ne

p: pressure

A=σI/kT B=ViσI/kT

d: gap length

Typically practically Ebd= 10 kV/cm at 1 bar in air Reference [2]

Vbd,Paschen min, air ≈ 300 V

• Small p*d, d> λ: collision dominated, small energy build-up, high Vd CAS on Small Accelerators

E0

E0

E0 + Eρ

Streamer breakdown Space charge field Eρ ≈ E0 • Field enhancement extra ionising collisions (α↑) • High excitation ⇒ UV photons when 1 electron grows into ca. 108 then Eρ large enough for streamer breakdown (ne ≈ 2·108 in avalanche head)

Result: • Secondary avalanches, directional effect (channel formation) • Grows out into a breakdown within 1 gap crossing (anode and/or cathode directed)

Characteristic: • Very fast • Independent of electrodes (no need for electrode surface secondaries) • Important at large distances (lightning)

CAS on Small Accelerators

ÆTownsend, unless:

• Townsend: αd ≥ ln(1/γ + 1) ≈ 7...9 (γ ≈ 10-4...10-3) • Streamer: αd ≥ 18...20

• Strong non-uniform field (small electrodes, few secondary electrons) • Pulsed voltages – Townsend – Streamer

slow, ion drift, subsequent gap transitions fast, photons, 1 gap transition

• High pressure – Less diffusion Eρ high – photons absorbed in front of cathode – positive ions slower

Courtesy: Eindhoven Univ.of Techn.

CAS on Small Laser-induced streamer breakdown inAccelerators air.

Breakdown along insulator • • • •

Surface charge (Non-regular) surface conduction Particles / contaminations on surface Non-regularities (scratches, ridges)

⇒ Field enhancement ⇒ Increased breakdown probability Reference [2]

Courtesy: Eindhoven Univ.of Techn.

Prebreakdown along insulatior in air. CAS on Small Accelerators

Breakdown at pulse voltages; time-lag

Reference [2]

ts, wait for first elektron tf, breakdown formation • Townsend or Streamer

Short pulses, high breakdown voltage CAS on Small Accelerators

Non-uniform fields; Corona Breakdown conditions: • Global Æ Full breakdown • Local Æ Streamer breakdown Æ Partial discharge

Reference [2]

Non-uniform field: • Discharge starts in high field region • ....and “extinguishes” in low field region CAS on Small Accelerators

Corona 10 ns

- Power loss; EM noise - Chemical corrosion + Useful applications

Pulsed corona discharges: 20 ns

30 ns

Courtesy: Eindhoven Univ.of Techn.

• Fast, short duration HV pulses • Many streamers, high density • Generation of electrons, radicals, excited molecules, UV • E.g. Flue gas cleaning

40 ns

Transmission line transformer

CAS on Small Accelerators

Solid insulation Breakdown field strength: • Very clean (lab): high • Practical: lower due to imperfections – – – –

Voids Absorbed water Contaminations Structural deformations

Anorganic

Quartz,mica,glas

Natural

Porcelain

Synthetic

Paper

Al O 2 3

Polymerisation

HD,LD,XL – PE Teflon Polystyrene, PVC, polypropene,etc

Epoxy

Spacer

Capacitor

Polyethelene Organic

Feedthrough

Cable

+ Oil

Synthetic

Disc insulator

Hardener Filler

Spec. properties:

Requirements:

• Mechanical strength • Contact with electrodes and semiconducting layers • Resistant to high T, UV, dirt, contamination, rain, ice, desert sand

Problems:

• Surface tracking • Partial discharges

– In voids (in material or at electrodes, often created at production).

Moisture content high T losses bonding

Moulding in mold

Types of solid insulation materials

Surface tracking Remedies: • • • •

Geometry Creeping distance (IEC-norm) Surface treatment (emaillate, coating) Washing CAS on Small Accelerators

Vacuum insulation Applications:

Advantages:

What is vacuum?

Disadvantages:

• • • • •

Vacuum circuit breaker Cathode Ray Tubes / accelerators Elektron microscope X-ray tube Transceiver tube

• “Pressure at which no collisions for Brownian “temperature” movements of electrons” • λ >> characteristic distances • E.g. p = 10-6 bar, λ = 400 m

• • • • •

“Self healing” No dielectric losses High breakdown fieldstrength Non flammable Non toxic, non contaminating

• Requires hermetic containment and mechanical support • Quality determind by: – electrodes and insulators – Material choice, machining – Contaminations, conditioning

Characteristics of vacuum breakdown No 1st electron from “gas” • Cathode emission – primary: photoemission, thermic emission, field emission, Schottkyemission – secondary: e.g. e- bombarded anode → +ion collides at cathode → e-

No breakdown medium • No multiplication through collision ionisation • Medium in which the breakdown occurs has to be created (“evaporated” from electrodes, insulators)

Important: prevent field emission • Keep field at cathode and “cathode triple point” as low as possible • Insulator surface charging, conditioning CAS on Small Accelerators

14

Breakdown vs.field cathode-triple-point

Courtesy: ESTEC / ESA

Insulator surface charging

Conditioning effect lost when charge compensated

01

Conditioning # Conditioning breakdowns needed to reach 50kV hold voltage Ref [12]

CAS on Small Accelerators

Insulating liquids Requirements: Applications: • Transformers • Cables • Capacitors • Bushings

• Pure, dry and free of gases • εr (high for C’s, low for trafo) (demi water εr,d.c. = 80) • Stable (T), non-flammable, non toxic (pcb’s), ageing, viscosity

• No interface problems • Combined cooling/insulation • “Cheap” (no mould) • Liquid tight housing Courtesy: Sandia labs, U.S.A.

Breakdown fieldstrength: • • • •

Very clean (lab): high 1 - 4 MV/cm (In practice much lower) Important at production:outgassing, filtering, drying Mineral oil (“old” time application, cheap, flammable) Synthetic oil (purer, specifically made, more expensive) – Silicon oil (very stable up to high T, non-toxic, expensive)

• Liquid H2, N2, Ar, He (supra-conductors) • Demi-water (incidental applications, pulsed power) • Limitation Vbd: – Inclusions: Partial discharges Æ Oil decomposition → Breakdown – Growth (pressure increase) – “extension” in field direction”

• Particles drift to region with highest E → bridge formation → breakdown +

+ +

-

-

CAS on Small Accelerators

Transformer: • Mineral oil: Insulation and cooling • Paper: Barrier for charge carriers and chain formation – Mechanical strength

• Ageing

– Thermical and electrical (partial discharges) – Lifetime: 30 years, strongly dependent on temperature, short-circuits, overloading , over-voltages – Breakage of oil moleculs, Creation of gasses, Concentration of various gas components indication for exceeded temperature (as specified in IEC599)

• Lifetime – Time in which paper looses 50 % of its mechanical strenght – Strongly dependent on: • Moisture (from 0.2 % to 2 % accelerated ageing factor 20) • Oxygen (presence accelerates ageing by a factor 2)

Measurement techniques Partial discharges • UV, fast electrons, ions, heat • Deterioration void:

– Oxidation, degradation through ion-impact – “Pitting”, followed by treeing

• Eventually breakdown

Acceptable lifetime? Preferably no partial discharges.

• High sensitivity measurements on often large objects • Qapp ≠ Qreal, still useful, because measure for dissipated energy, thereby for induced damage • relative measurement

AC voltage phase resolved discharge pattern detection Æ Type of defect CAS on Small Accelerators

Partial discharges Before

V a

c

c Qtot = Q

a >> b Cobject α=1/(2RC) β=[1/(LC) - α2]-1/2 • Calibration through injecting known charge

Loss angle, tan(δ) Sources:

CHV

• Conduction σ (for DC or LF) • Partial discharges • Polarisation jωCV

Schering bridge: • i=0, RC=R4C4 • Gives: tan(δ) – parallel: – serie:

1/ωRC ωRC

δ

test object

i

i R4

1/R.V

Tan δ:

Shielding

• “Bulk” parameter • No difference between phases

PD: • Detection of weakest spot • Largest activity and asymmetry in “blue” phase (ridge discharges)

C4

R3

Summary Seen many basic high voltage engineering technology aspects here: – High voltage generation – Field calculations – Discharge phenomena

The above to be applied in your practical accelerator environments as needed: – Vacuum feed through: Triple points – Breakdown field strength in air 10kV/cm – Challenging calculations for real practical geometries. CAS on Small Accelerators

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

M.Benedikt, P.Collier, V.Mertens, J.Poole, K.Schindl (Eds.), ”LHC Design Report”, Vol. III, The LHC Injector Chain CERN-2004-003, 15 December 2004. E. Kuffel, W.S. Zaengl, J. Kuffel: “High Voltage Engineering: Fundamentals”, second edition, ButterworthHeinemann, 2000. A.J. Schwab, “Hochspannungsmesstechnik”, Zweite Auflage, Springer, 1981. L.L. Alston: “High-Voltage Technology”, Oxford University Press, 1968. K.J. Binns, P.J. Lawrenson, C.W. Trowbridge: “The Analytical and Numerical Solution of Electric and Magnetic Fields”, Wiley, 1992. R.P. Feynman, R.B. Leighton, M. Sands: “The Feynman Lectures on Physics”, Addison-Wesley Publishing Company, 1977. E. Kreyszig: “Advanced Engineering Mathematics”, Wiley, 1979. L.V. Bewley: “Two-dimensional Fields in Electrical Engineering”, Dover, 1963. Energy Information Administration: http://www.eia.doe.gov. R.F. Harrington, “Field Computation by Moment Methods”, The Macmillan Company, New York, 1968, pp.1 -35. P.P. Silvester, R.L. Ferrari: “Finite Elements for Electrical Engineers”, Cambridge University Press, 1983. J. Wetzer et al., “Final Report of the Study on Optimization of Insulators for Bridged Vacuum Gaps”, EHC/PW/PW/RAP93027, Rider to ESTEC Contract 7186/87/NL/JG(SC), 1993.

CAS on Small Accelerators

Appendix I

Maxwell equations in integral form

∫∫ D ⋅ dA = ∫∫∫ ρ dV = Q

Electrostatic ∫∫ D ⋅ dA = Qomsl.

(1)

∫ E ⋅ dl = −

∫ E ⋅ dl = 0

(2)

Magnetostatic ∫∫ B ⋅ dA = 0

(3)

∫ H ⋅ dl = I

(4)

omsl .

dφ omsl. d ⋅ = − B dA dt dt ∫∫

∫∫ B ⋅ dA = 0 ∫ H ⋅ dl = ∫∫ ( J +

∂D ) ⋅ dA ∂t

omsl .

Maxwell equations in differential form ∇⋅D = ρ ∇× E = −

∂B ∂t

∇⋅B = 0

∇× H = J +

∂D ∂t

Electrostatic ∇⋅D = ρ

No space charge ∇⋅D = 0

(5)

∇× E = 0

∇× E = 0

(6)

Magnetostatic ∇⋅B = 0

In area without source ∇⋅B = 0

(7)

∇× H = J

∇× H = 0

(8) CAS on Small Accelerators

Appendix III Finite

Element Method (FEM)

Field energy minimal inside each closed region G: 2

W = ∫ ε E dV = ∫ ε ∇U dV 1 2

2

1 2

G

G

Assume U satisfies Laplace equation, but U' does not, then

(

WU' - WU ≥ 0:

)

WU ' − WU = 12 ε ∫∫∫ ∇U ' − ∇U dV = L = 12 ε ∫∫∫ ∇U '−∇U dV ≥ 0 2

2

2

G

G

Field energy for one element (2-dim) Potential is linear inside element: U = a + b x + c y = (1 x

on corners:

⎛ U1 ⎞ ⎛1 x1 ⎜ ⎟ ⎜ ⎜ U 2 ⎟ = ⎜1 x2 ⎜ U ⎟ ⎜1 x ⎝ 3⎠ ⎝ 3

y1 ⎞ ⎛ a ⎞ ⎟⎜ ⎟ y2 ⎟ ⎜ b ⎟ y3 ⎠⎟ ⎜⎝ c ⎟⎠

Potential can be written as: U = (1 x

⎛1 x1 ⎜ y ) ⎜1 x 2 ⎜1 x ⎝ 3

y1 ⎞ ⎟ y2 ⎟ y3 ⎟⎠

−1

⎛ U1 ⎞ ⎜ ⎟ 3 ⎜U 2 ⎟ = ∑ U iα i ( x, y ) ⎜ ⎟ i =1 ⎝U 3 ⎠

y ⎛a⎞ ⎜ ⎟ y )⎜ b ⎟ ⎜c⎟ ⎝ ⎠

1 2

e 3 x

Field energy in element (e):

[ ]

W ( e ) = 12 εU T S(e) U with Sij( e ) = ∫∫ (∇α i ⋅ ∇α j )dxdy CAS on Small Accelerators α’s are linear in x and y ⇒ ∇α is constant: Sij=(∇αi·∇αj) A(e)

All elements together:

Total field energy of n elements U T = (U1 L U m U m +1 L U n ) ≡ (U f U p ) free

prescribed

free: potential values to be determined prescribed: potential according to boundary conditions

[

W = εU [S ]U = ε U 1 2

1 2

T

Partial derivatives of W to Uk are zero for 1≤k≤m (m equations):

∂W =0 ⇒ ∂U k

[S

kf

S kp

]

T f'

U

T p'

]

⎡S f ' f ⎢S ⎣ p' f

S f ' p ⎤ ⎡U f ⎤ S p ' p ⎥⎦ ⎢⎣U p ⎥⎦

⎡U f ⎤ ⎢U ⎥ = 0 ⎣ p⎦

Boundary Element Method (BEM) Boundaries uniquely prescribe potential distribution

Laplace equation : Δu = 0 border Γ1 : u ( x , y )

border Γ2 : q ( x , y ) ≡

(Dirichlet ) ∂u ∂n



n

r = ( x − x0 ) 2 + ( y − y0 ) 2 6

P (x0,y0) ) c

P r

(Neumann)

(x,y)

∂ ln r 1 ⎛ ⎞ P0=(x0,y0) inside Γ: u ( x0 , y0 ) = − q x y r u ( x , y ) ( , ) ln ⎜ ⎟ds ∫ ∂n 2π Γ ⎝ ⎠ (0,0)

Problem: u(x,y) and q(x,y) not both known at the same time CAS on Small Accelerators

∫ (u∇v − v∇u )⋅ nˆds = ∫∫ (uΔv − vΔu )dxdy

Green II (2-dim):

Δv(x,y)=0 for P≠P0(x0,y0)

Choose v(x,y)=ln(1/r)

Exclude region σ around P0 by means of circle c

∂ ln r −1 −1 ∂u −1 −1 ∫Γ+c (u ∂n − ln r ∂n )ds = Ω∫∫−σ(u Δ ln r − ln r Δu )dxdy = 0 ⎛ ∂ ln r −1 ∂u ⎞ ∂u ⎞ ⎛ ∂ ln r − ∫ ⎜u − ln r ⎟ ds + ∫ ⎜⎜ u − ln r −1 ⎟⎟ ds = 0 ∂n ∂n ⎠ ∂n ∂n ⎠ c⎝ Γ⎝

lim 2π ⎛ 1 ∂u ⎞ + u ln ε ⎜ ⎟ ε dϑ = 2π u ( x0 , y0 ) ε ↓ 0 ∫0 ⎝ ε ∂n ⎠

u ( xi , yi ) =

Pi=(xi,yi) on border Γ: Discretisation:

1

π

∫ (u ( x, y) Γ

n

πU ( xi , yi ) = ∑ U j ∫ j =1

In matrix notation:

∑ (H n

j =1

− πδ ij )U j = ∑ Gij Q j n

ij

j =1

∂ ln r − q ( x, y ) ln r )ds ∂n

∂ ln rij

Sj

∂n

n

ds − ∑ Q j ∫ ln rij ds j =1

Sj

H

G

ij

ij

2 1

n node

S

element

6 j+1 j

Generates missing information

CAS on Small Accelerators