I. Furno, B. Labitspc.epfl.ch/wp-content/uploads/2018/10/L1_theory_of_magnetics_2012.pdf · -...
Transcript of I. Furno, B. Labitspc.epfl.ch/wp-content/uploads/2018/10/L1_theory_of_magnetics_2012.pdf · -...
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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!
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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
![Page 3: 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](https://reader030.fdocuments.us/reader030/viewer/2022040716/5e1fdb091c1c25096f5a5e7b/html5/thumbnails/3.jpg)
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
![Page 4: 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](https://reader030.fdocuments.us/reader030/viewer/2022040716/5e1fdb091c1c25096f5a5e7b/html5/thumbnails/4.jpg)
H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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
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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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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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
![Page 17: 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](https://reader030.fdocuments.us/reader030/viewer/2022040716/5e1fdb091c1c25096f5a5e7b/html5/thumbnails/17.jpg)
H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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( )
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• Advantages
- Requires no circuit contact with the current, which is measured
- Measurement is independent of the current distribution
• Disadvantages - ?
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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∑
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
• 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)]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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)]
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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)
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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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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
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H. Reimerdes, Theory of Magnetic Diagnostics, Jan 30 – Feb 3, 2012
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