I. Furno, B. Labitspc.epfl.ch/wp-content/uploads/2018/10/L1_theory_of_magnetics_2012.pdf · - Rogowski coil - Diamagnetic loop • ... • Low drift integration increasingly important

H. Reimerdes!In collaboration with!

I. Furno, B. Labit !with contributions from !

J. Lister, J.-M. Moret and H. Weisen!

“Plasma Diagnostics in Basic Plasma Physics Devices and Tokamaks: From Principles to Practice”!

January 30 – February 3, 2012!

H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012

•  Introduction

•  Magnetic measurements -  Magnetic induction

-  Hall effect

-  Optical measurements

•  Magnetic induction

-  Pick-up coil + Analog integration + Frequency response

-  Saddle loops

-  Rogowski coil

-  Diamagnetic loop

•  Magnetic diagnostics in various devices


•  Introduction


-  Hall effect




-  Saddle loops

-  Rogowski coil




Magnetic Confinement Fusion (MCF)

•  Magnetic field is the basis for confinement/stability

•  Magnetic field directly relates to currents

-  Maxwell’s correction usually negligible in MCF applications

-  Can’t put shunt into a plasma

•  Magnetic field measurements are important for operation as well as physics

-  Measure currents in the plasma and in the vessel

-  Provide estimators for equilibrium and instability control

-  Reconstruct the equilibrium

-  Detect and identify instabilities

€

∇ × B = µ0j Ampère’s law


Magnetic field amplitudes Tokamak toroidal field ~ 1 Tesla (SI) = 1 Vs/m2 = 104 Gauss (cgs) Tokamak poloidal field ~ 10% of toroidal field Asymmetries/instabilities that can affect plasma operation

≥ 10-4 T

Earth magnetic field ~5x10-5 T

Frequencies/time scales Equilibrium field ~1 s Long wave-length instabilities 10-1 – 10-4s Alfven waves 10-3 – 10-5s Turbulence 10-4 – 10-6s


•  Introduction


-  Hall effect




-  Saddle loops

-  Rogowski coil




•  Faraday’s law

-  Integral form

•  Voltage induced in a loop (contouring the surface S)

is a measurement of the component of dB/dt normal to (and integrated over) the plane of the loop

€

U = −dφSdt

€

∇ × E = −∂B∂t

€

E∂S∫ ⋅ dl = −

∂B∂t

⋅ dsS∫ ≡ −

∂φS∂t

-  Use twisted wires as leads to avoid pick-up


•  Advantages

-  Simplest (and highly reliable) way to measure a flux or magnetic field

•  Disadvantages -  Measures only the rate of change of the magnetic field ➜ field

measurements require integration, which is sensitive to small drifts


•  Force on a charge q moving with a velocity v

•  In a current carrying conductor charge separation builds up a Hall field EH until the resulting force cancels the Lorentz force

-  Hall field depends on the the charge carrier density n and their sign €

F = q E + v × B( )

€

EH = −v × B

€

EH = −j × Bnq


•  Advantages -  No need to integrate and hence no drifts

•  Disadvantages -  Sensitive to stray pickup -  Non-linear at high fields -  Transistors prone to radiation damage in a reactor -  Relatively low bandwidth


•  Faraday rotation: Right and left circularly polarized light travelling parallel to a magnetic field in a plasma have different refractive indices

The polarisation angle of linearly polarized light rotates. The rotation angle is proportional to

•  Faraday rotation is usually small ΔθF << π ➜ requires sensitive detection technique

€

ΔθF ∝ λ2 neB||∫ dl

*[D. Véron, Infrared and Millimeter Waves, Vol. 2, chapter D (Faraday rotation)]


•  In typical fusion plasmas Faraday rotation is measured using light in the far-infrared (100 - 400 µm)

•  Simple polarimeter scheme:

-  Linearly polarized light is passed through a plasma

-  Faraday rotation changes the polarisation angle

-  A polarisation sensitive beam splitter separates orthogonal components

-  Ratio of amplitudes at detector 1 and 2 yields the Faraday rotation

•  Various other techniques have been developed


•  Advantages -  Polarimeter can be combined with an interferometer

•  Disadvantages -  Results only in small phase shifts that are difficult to measure

-  Line integrated measurement

-  Requires electron density


•  Stark effect: Splitting of spectral lines of atoms and molecules due to the presence of an external electric field (electric analogue to the Zeeman effect)

➜  Measure the stark splitting due to Lorentz field EL = v x B experienced by fast moving atoms in a neutral beam (diagnostic or heating)

-  “Motional” Stark effect typically much stronger than Zeeman effect -  Spatial resolution obtained from intersection between viewing

optic and beam

[Courtesy of C. Holcomb, LLNL]

*[D. Wroblewski, L.L. Lao, Rev. Sci. Instr. 63 (1992) 5140]


•  Advantages -  Local measurement of the magnetic field -  Multiple views also reveal radial electric field Er of the plasma

•  Disadvantages -  Polarisation measurement requires an extremely accurate

knowledge of the optical properties of the diagnostic -  Optical properties can change with time due to coating of PFCs -  Requires multiple views to separate Er from EL

Neutral beam particles (typically H or D) are excited

E field splits line emission (typically Balmer α) into π and σ components: σ component polarised ⊥ to E π component polarised || to E

Measure polarisation angle of one Stark component or Measure the splitting or intensity ratio of two Stark components


•  Introduction


-  Hall effect




-  Saddle loops

-  Rogowski coil




•  Magnetic pick-up coils (Mirnov probes) ➜ Equilibrium (axisymmetric field) & perturbations (non-axisymmetric field)

•  Flux loops ➜ Equilibrium

•  Saddle loops ➜ Perturbations

•  Rogowski coil ➜ Plasma current •  Diamagnetic loop ➜ Stored energy


•  In practice A is not sufficiently well known ➜ characterize a probe by its effective area Aeff

•  Applications: Detection of fast growing or rotating instabilities

•  Induced voltage

•  For a small rigid probe with a cross-sectional area A and N turns

-  Probe measures the magnetic field component perpendicular to the probe surface B⊥

€

B ∇B( )⊥

>> L

€

Uprobe = −∂B∂t

⋅ dsS∫

€

Uprobe = −NA dB⊥dt

€

Uprobe = −AeffdB⊥dt

⇔ dB⊥dt

= −Uprobe

Aeff

➜ Practicum II (Tuesday)


•  Designed to withstand 400º C

•  Mounted inside the vacuum vessel behind graphite protective tiles

-  Fit in 12 mm gap

1 mm mineral insulated coaxial wire

Ceramic body

23mm

*[J.-M. Moret, et al., Rev. Sci. Instr. 69 (1998) 2333]


•  Assume a magnetic field oscillating with a frequency w:

➜ High frequency signals are strongly amplified -  Fast events (e.g. vertical displacement events) induce large

voltages an can require signal attenuation at high frequencies (i.e. high pass filters)

•  Typical values in TCV

-  Equilibrium changes: dB/dt ~ 0.1-1.0 T/s (1-10 Hz) -  MHD modes: dB/dt ~ 102 T/s (1-10 kHz)

-  Disruptions: dB/dt > 103 T/s (1-10 kHz)

➜ In TCV signals above 100 Hz are attenuated (less amplified)

€

U = −iωAeff B

€

B t( ) = B e iωt


•  Assume a magnetic field oscillating with a frequency w:

➜  High frequency signals are strongly amplified -  Fast events (e.g. vertical displacement events) induce large

voltages an can require signal attenuation at high frequencies (i.e. high pass filters)

•  Typical values in TCV

-  Equilibrium changes: dB/dt ~ 0.1-1.0 T/s (1-10 Hz) -  MHD modes: dB/dt ~ 102 T/s (1-10 kHz)

-  Disruptions: dB/dt > 103 T/s (1-10 kHz)

➜  In TCV signals above 100 Hz are attenuated (less amplified)

€

U = −iωAeff B

€

B t( ) = B e iωt


•  Analog integration circuit

-  Current Usensor/R charges the capacitors C

-  Differential input avoids ground loops between sensors and integrators

-  Integrators require an offset compensation, which holds Uout to zero before an experiment

€

Uout = −1RC

Usensor dt + const.t0

t∫

•  Low drift integration increasingly important with pulse length

•  Applications: Plasma position and shape control, equilibrium reconstruction, detection of non-rotating instabilities


•  Main effects are

-  Conductor with capacitance CP and resistance RP

-  Coupling to a shielding with inductance LS and resistance RS

-  Amplifying chain

Shielding Probe

➜ Shown circuit diagram introduces two poles


•  Transfer function of a pick-up coil (s ≡ iw)

-  Ideal probe

-  Real probe

+ “zero-pole-gain” representation of the transfer function

➜ Practicum II (Tuesday)

€

UB

= −sAeff

€

UB

= −sAeff

-Pprobes − Pprobe( )


•  TCV uses an amplifier with a pole at ~100Hz and a zero at ~3.5kHz

-  f < 100Hz: gain x30

-  f>3.5 KHz: gain x1


•  Toroidal pick-up: If the probe axis has a toroidal component, the much larger toroidal field can pollute poloidal field measurements

➜  Determine coupling by pulsing toroidal field without a plasma

-  Probe signals can be corrected in real time


•  Measure three components of the magnetic field at three locations

Bz

Br Bt

€

µ0 jt = ∇ × B( )t

➜  Yields tangential component of the local current density


•  Applications: Loop voltage, plasma position and shape control, equilibrium reconstruction

•  Poloidal flux loops measure the poloidal flux

-  Dominated by flux from Ohmic transformer ➜ measurement of loop voltage Uloop

-  Integrate signal to obtain poloidal flux ψp

-  Difference between flux loops related to poloidal field

-  Large contribution from poloidal field coils ➜ requires high precision to retrieve information about the plasma

€

ψp ≡ 2π ʹ′ R BZ d ʹ′ R 0

R∫

€

Ufl = −dψp

dt


•  Applications: Fast measurements of the stored energy, equilibrium reconstruction

•  Diamagnetic loops measure the toroidal flux

•  Diamagnetic flux is the contribution from the plasma

•  In cylindrical geometry

€

Φp ≡ Bt dsS∫

€

Udia = −dΦt

dt

€

Φdia = Bt - Bt,vac( )dsS∫

€

Φdia =µ02IP2

8πBt

1 − βp( )-  For βp<1 (>1) the plasma increases (decreases) the absolute value

of the toroidal field, i.e. the plasma is paramagnetic (diamagnetic)


•  Applications: Fast measurements of the stored energy, equilibrium reconstruction

•  Diamagnetic loops measure the toroidal flux

•  Diamagnetic flux is the contribution from the plasma

•  In cylindrical geometry

€

Φp ≡ Bt dsS∫

€

Udia = −dΦt

dt

€

Φdia = Bt - Bt,vac( )dsS∫

€

Φdia =µ02IP2

8πBt

1 − βp( )-  For βp<1 (>1) the plasma increases (decreases) the absolute value

of the toroidal field, i.e. the plasma is paramagnetic (diamagnetic)


•  Multiple-turn uniformly wound solenoid with n turns per unit length and a loop area A

-  Replace sum with integral

€

URC = − A dB⊥dt

dNdl

dlL∫ = −nA d

dtB⊥L∫ dl

•  Current measurement based on the integral form of Ampere‘s law

€

B ⋅ dlL∫ = µ0I

•  Completely enclose current to be measured

-  Integrate voltage URC to deduce the current I •  Note, that a Rogowski coil measurement outside the vessel includes

toroidal vessel currents in addition to the plasma current

€

URC = − A dB⊥dtturns

∑

€

URC = −nAµ0dIdt


•  Advantages

-  Requires no circuit contact with the current, which is measured

-  Measurement is independent of the current distribution

•  Disadvantages -  ?


•  TCV does not have a Rogowski coil ➜ use poloidal field measurements Bm instead

-  Sensitive to currents close to the probes, including currents outside the integration contour €

IP =1µ0

B dl∫ = cmBmm∑


•  Typically single loop coils with an axis in the radial direction and a large toroidal extend

-  Often mounted on the vacuum vessel ➜ measures the magnetic field normal to the wall

+ Eddy currents in the wall attenuate high frequency signals

-  Small equilibrium component ➜ large fraction of signal from perturbations

-  Averaging of the field over a large area suppresses short wavelength perturbations

•  Applications: Detection of long-wavelength non-rotating perturbations (locked modes, resistive wall modes)


•  Magnetic probes can be be used in an active diagnostic to measure stability properties of the plasma (e.g. frequencies, damping rates)

•  Excite weakly damped Alfvén eigenmodes1: fext > 104 Hz –  Actuator: Short-wavelength internal antenna –  Detector: Poloidal field probes

•  Drive weakly damped resistive wall mode2: fext ≈ 0-100 Hz –  Actuator: In- or external saddle coils –  Detector: Radial or poloidal field probes

1[A. Fasoli, et al., Phys. Rev. Lett. 75 (1995) 645] 2[H. Reimerdes, et al., Phys. Rev. Lett. 93 (2004) 135002]


•  Probe high beta plasmas with slowly rotating n=1, 2 fields

•  Measure plasma responds (same frequency, same n) with poloidal field probes

➜  Plasma responds yields measurement of the ideal MHD no-wall limit

*[H. Reimerdes, et al., APS (2007)]


TCV (in/ex-vessel) TORPEX ITER (in/ex-vessel)* Pol. field probes 203/0 - 186/360 (fast) “ 9 (cluster) >200/- Flux loops (n=0) 0/61 - 124/5 Diamagnetic loops 0/2 - 24 Saddle loops 0/24 - 72/0 Rogowski - 1 2-9 Fibre-optic IP sensor - - 4

*[J. Lister, et al., “The Magnetic Diagnostic Set for ITER”, 25th SOFT, Rostock (2008)]


Reactor environment

•  Neutron radiation damaging to electronics

Interpretation of measurements

•  Integrator drifts in long-pulses •  Thermal and radiation induced EMF

•  Greater distance between plasma and diagnostics (behind blanket modules) -  Shielding due to conducting structures in blanket modules

Maintenance and repair

•  Limited or no accessibility once machine has been assembled and started to operate

➜  All these challenges are considered as solved (or solvable)

H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012 *[J. Lister, et al., “The Magnetic Diagnostic Set for ITER”, 25th SOFT, Rostock (2008)]

Equilibrium coils (Consorzio RFX) ~5cm

Equilibrium coils (CEA)

25 cm 9 mm


Plasma diagnostics

•  Equipe TFR, “Tokamak plasma diagnostics”, Nucl. Fusion 18 (1978) 647

•  I.A. Hutchinson, “Principles of plasma diagnostics”, Cambridge University Press.

Magnetic diagnostics

•  J.-M. Moret, et al., “Magnetic measurements on the TCV tokamak”, Rev. Sci. Instrum. 69 (1998) 2333.

•  E.J. Strait, “Magnetic diagnostic system of the DIII-D tokamak”, Rev. Sci. Instrum. 77 (2006) 023502


TCV TORPEX

Toroidal field (R0=0.88m) BT ≤ 1.4 T ≤ 0.1 T

Density ne 1019-1020 m-3 1016-1018 m-3

Electron temperature Te < 10 keV 1-20 eV

Ion temperature Ti < 1 keV <1eV

Plasma frequency ωp/(2π)=(nee2/ε0me)0.5 /(2π) 28-120 GHz 1-10 GHz

Electron cyclotron frequency ωce/(2π)=eB/me /(2π) ~ 41 GHz ~ 3 GHz

Ion cyclotron frequency ωci/(2π)=ZieB/mi /(2π) ~ 11 MHz ~ 1.5 MHz

Documents

I. Furno, B. Labitspc.epfl.ch/wp-content/uploads/2018/10/L1_theory_of_magnetics_2012.pdf · - Rogowski coil - Diamagnetic loop • ... • Low drift integration increasingly important