Introduction to the Diagnosis of Magnetically Confined Thermonuclear - PowerPoint PPT Presentation

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Introduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma Magnetics J. Arturo Alonso Laboratorio Nacional de Fusión EURATOM-CIEMAT E6 P2.10 arturo.alonso@ciemat.es version 0.1 (September 24, 2011) Magnetics, A. Alonso, copyleft 2010 1 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Outline Preliminaries 1 2 The magnetic coil Global magnitudes 3 4 Plasma position and shape Magnetics, A. Alonso, copyleft 2010 2 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Outline Preliminaries 1 2 The magnetic coil Global magnitudes 3 4 Plasma position and shape Magnetics, A. Alonso, copyleft 2010 3 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary MHD equilibrium The MHD equilibrium equation reads j × B = ∇ p ⇒ B · ∇ p = 0 , j · ∇ p = 0 . Magnetics, A. Alonso, copyleft 2010 4 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Maxwell Equations (I) The physical basis of nearly all the magnetic measurements ∇ · E = ρ (1a) ǫ 0 ∇ · B = 0 (1b) ∇ × E = − ∂ B (1c) ∂ t ∂ E ∇ × B = µ 0 j + µ 0 ǫ 0 (1d) ∂ t The last term in equation (1d) is the displacement current important for EM wave phenomena ( c = ( µ 0 ǫ 0 ) − 1 / 2 ). The dynamics we are interested here have typical frequencies ω and wavelengths k such that ω/ k ≪ c so that the displacement current can be ignored. Magnetics, A. Alonso, copyleft 2010 5 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Maxwell Equations (I) The physical basis of nearly all the magnetic measurements ∇ · E = ρ (1a) ǫ 0 ∇ · B = 0 (1b) ∇ × E = − ∂ B (1c) ∂ t ∂ E ∇ × B = µ 0 j + µ 0 ǫ 0 (1d) ∂ t The last term in equation (1d) is the displacement current important for EM wave phenomena ( c = ( µ 0 ǫ 0 ) − 1 / 2 ). The dynamics we are interested here have typical frequencies ω and wavelengths k such that ω/ k ≪ c so that the displacement current can be ignored. Magnetics, A. Alonso, copyleft 2010 5 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Maxwell Equations (II) By using the integral theorems (see [Jackson(1999)]) we can write equations (1) in their integral form � E · d S = 1 � ρ dV ǫ 0 ∂ V V � B · d S = 0 ∂ V � E · d l = − ∂ � B · d S ∂ t ∂ S S � � B · d l = µ 0 j · d S ∂ S S Magnetics, A. Alonso, copyleft 2010 6 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Outline Preliminaries 1 2 The magnetic coil Global magnitudes 3 4 Plasma position and shape Magnetics, A. Alonso, copyleft 2010 7 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary The magnetic coil The simplest measuring device used to measure magnetic fields is a simple coil. From the MEs we can compute the electromotive force E (Volts): E · d l = ∂ � � B · d S ≡ d E ≡ − dt Φ B ∂ t ∂ S S where Φ B is magnetic flux through the sur- face encircled by the circuit. There is an electric current running on the circuit (or a potential difference for an open circuit) whenever there exist a time variation of the magnetic flux through the circuit-limited surface. Magnetics, A. Alonso, copyleft 2010 8 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary The magnetic coil The simplest measuring device used to measure magnetic fields is a simple coil. From the MEs we can compute the electromotive force E (Volts): E · d l = ∂ � � B · d S ≡ d E ≡ − dt Φ B ∂ t ∂ S S Assume B ( x , t ) ≈ B ( t ) in the surface of the coil. Then E = ˙ BA . Increase the effective area with N windings so that E = ˙ BNA . We can recover B ( t ) from the time integral of ˙ B ( t ) (electronics). Magnetics, A. Alonso, copyleft 2010 8 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Hall effect detectors Measure static magnetic fields by use of the solid-state Hall effect : A current flowing along a slab of a semiconductor material in the presence of a magnetic field creates a potential difference accross the slab n carriers/m 3 and charge q , in equilibrium, the electric field caused by the charge separation balances the j × B force so that jB + nqE = 0 V H = jBL I nq = nqdB Caveats: affected by stray pickups, non-linear for large B and T ( � 140 ◦ C). Magnetics, A. Alonso, copyleft 2010 9 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Outline Preliminaries 1 2 The magnetic coil Global magnitudes 3 4 Plasma position and shape Magnetics, A. Alonso, copyleft 2010 10 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary The Rogowski coil Rogowski coils are used to measure the current flowing in a conducting medium by sensing the induced magnetic field. The magnetic flux through a Rogowski coil is N � � Φ B = B · d S i . S i i = 1 with d S i = dA u i . � z Assume windings are densely packed � N i → a ndl . Then � z � z � � Φ B = ndl B · u i dA = nA B · d l = nA µ j · d S = nA µ I . a S i a S Therefore E = d Φ B / dt = n µ A ˙ I . Magnetics, A. Alonso, copyleft 2010 11 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary The Rogowski coil From the Torpex tokamak in EPL-CRPP , Laussane, Switzerland. Magnetics, A. Alonso, copyleft 2010 12 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary The Voltage loop The Voltage loop measures the inductive electric potential V φ that drives the plasma current in a tokamak. Magnetics, A. Alonso, copyleft 2010 13 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Ohmic power and plasma resistance The heating (Ohmic) power P and plasma resistance R p can be calculated by the combination of a Rogowski coil ( I φ ) and a Voltage loop ( V φ ). Poynting’s theorem state the conservation of EM energy (see notes) ∂ u ∂ t + ∇ · S = − j · E , For the case of interest in a tokamak discharge this simplyfies to (see notes) I φ V φ = P + ∂ � 1 � 2 LI 2 φ ∂ t In stationary state P = I φ V φ = I 2 φ R p . Magnetics, A. Alonso, copyleft 2010 14 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Ohmic power and plasma resistance The heating (Ohmic) power P and plasma resistance R p can be calculated by the combination of a Rogowski coil ( I φ ) and a Voltage loop ( V φ ). Poynting’s theorem state the conservation of EM energy (see notes) ∂ u ∂ t + ∇ · S = − j · E , For the case of interest in a tokamak discharge this simplyfies to (see notes) I φ V φ = P + ∂ � 1 � 2 LI 2 φ ∂ t In stationary state P = I φ V φ = I 2 φ R p . Magnetics, A. Alonso, copyleft 2010 14 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Ohmic power and plasma resistance The heating (Ohmic) power P and plasma resistance R p can be calculated by the combination of a Rogowski coil ( I φ ) and a Voltage loop ( V φ ). Poynting’s theorem state the conservation of EM energy (see notes) ∂ u ∂ t + ∇ · S = − j · E , For the case of interest in a tokamak discharge this simplyfies to (see notes) I φ V φ = P + ∂ � 1 � 2 LI 2 φ ∂ t In stationary state P = I φ V φ = I 2 φ R p . Magnetics, A. Alonso, copyleft 2010 14 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Diamagnetic loop (I) It should now be evident that a diamagnetic loop like the one in the figure gives a measure of the cross -section-averaged toroidal B -field: V θ = d � B φ dS ≡ d dt � B φ � A dt S In what follows we will show how to relate this measurement to the plasma pressure profile. For ilustration we will work in cylindrical geometry. In this approximation � a � � V θ = d ≡ d dt � B z � π a 2 , 2 π B z ( r ) rdr dt 0 where z stands for the axial ( ∼ φ toroidal) direction. Magnetics, A. Alonso, copyleft 2010 15 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary Diamagnetic loop (I) It should now be evident that a diamagnetic loop like the one in the figure gives a measure of the cross -section-averaged toroidal B -field: V θ = d � B φ dS ≡ d dt � B φ � A dt S In what follows we will show how to relate this measurement to the plasma pressure profile. For ilustration we will work in cylindrical geometry. In this approximation � a � � V θ = d ≡ d dt � B z � π a 2 , 2 π B z ( r ) rdr dt 0 where z stands for the axial ( ∼ φ toroidal) direction. Magnetics, A. Alonso, copyleft 2010 15 / 26