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

1

Preliminaries

2

The magnetic coil

3

Global magnitudes

4

Plasma position and shape

Magnetics, A. Alonso, copyleft 2010 2 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Outline

1

Preliminaries

2

The magnetic coil

3

Global magnitudes

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 ǫ0 (1a) ∇ · B = 0 (1b) ∇ × E = −∂B ∂t (1c) ∇ × B = µ0j + µ0ǫ0 ∂E ∂t (1d) The last term in equation (1d) is the displacement current important for EM wave phenomena (c = (µ0ǫ0)−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 ǫ0 (1a) ∇ · B = 0 (1b) ∇ × E = −∂B ∂t (1c) ∇ × B = µ0j + µ0ǫ0 ∂E ∂t (1d) The last term in equation (1d) is the displacement current important for EM wave phenomena (c = (µ0ǫ0)−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

• ∂V

E · dS = 1 ǫ0

• V

ρ dV

• ∂V

B · dS = 0

• ∂S

E · dl = − ∂ ∂t

• S

B · dS

• ∂S

B · dl = µ0

• S

j · dS

Magnetics, A. Alonso, copyleft 2010 6 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Outline

1

Preliminaries

2

The magnetic coil

3

Global magnitudes

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 ≡ −

• ∂S

E · dl = ∂ ∂t

• S

B · dS ≡ d dtΦB where ΦB 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 ≡ −

• ∂S

E · dl = ∂ ∂t

• S

B · dS ≡ d dtΦB 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/m3 and charge q, in equilibrium, the electric field caused by the charge separation balances the j × B force so that jB + nqE = 0 VH = jBL nq = I 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

1

Preliminaries

2

The magnetic coil

3

Global magnitudes

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 ΦB =

N

• i=1
• Si

B · dSi . with dSi = dAui. Assume windings are densely packed N

i →

z

a ndl. Then

ΦB = z

a

ndl

• Si

B · uidA = nA z

a

B · dl = nAµ

• S

j · dS = nAµI. Therefore E = dΦB/dt = nµA˙ 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 Vφ 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 Rp can be calculated by the combination of a Rogowski coil (Iφ) and a Voltage loop (Vφ). 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) IφVφ = P + ∂ ∂t 1 2LI2

φ

• In stationary state P = IφVφ = I2

φRp.

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 Rp can be calculated by the combination of a Rogowski coil (Iφ) and a Voltage loop (Vφ). 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) IφVφ = P + ∂ ∂t 1 2LI2

φ

• In stationary state P = IφVφ = I2

φRp.

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 Rp can be calculated by the combination of a Rogowski coil (Iφ) and a Voltage loop (Vφ). 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) IφVφ = P + ∂ ∂t 1 2LI2

φ

• In stationary state P = IφVφ = I2

φRp.

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: Vθ = d dt

• S

BφdS ≡ d dtBφA 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 Vθ = d dt

• 2π

a Bz(r) rdr

• ≡ d

dtBzπa2 , 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: Vθ = d dt

• S

BφdS ≡ d dtBφA 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 Vθ = d dt

• 2π

a Bz(r) rdr

• ≡ d

dtBzπa2 , 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 (II)

From the equilibrium condition in the radial direction we have ˆ r·(∇p−j×B) ⇒ dp dr = jθBz−jzBθ ≡ − 1 µ0 d dr B2

z

2 + Bθ 1 r d drrBθ

• Multiply by r2 and integrate over the cross section to get

p = 1 2µ0

• B2

θ(a) + B2 z(a) − B2 z

• ,

where we have assumed p(a) = 0. The diamagnetic loops provides an estimate of the βθ βθ ≡ 2µ0p B2

θ(a) = 1 + B2 z(a) − B2 z

B2

θ(a)

gives a measure of the plasma diamagnetism: βθ > 1 - diamagnetic (jθ decreases Bz in the plasma) βθ < 1 - paramagnetic (jθ increases Bz in the plasma)

Magnetics, A. Alonso, copyleft 2010 16 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Diamagnetic loop (II)

From the equilibrium condition in the radial direction we have ˆ r·(∇p−j×B) ⇒ dp dr = jθBz−jzBθ ≡ − 1 µ0 d dr B2

z

2 + Bθ 1 r d drrBθ

• Multiply by r2 and integrate over the cross section to get

p = 1 2µ0

• B2

θ(a) + B2 z(a) − B2 z

• ,

where we have assumed p(a) = 0. The diamagnetic loops provides an estimate of the βθ βθ ≡ 2µ0p B2

θ(a) = 1 + B2 z(a) − B2 z

B2

θ(a)

gives a measure of the plasma diamagnetism: βθ > 1 - diamagnetic (jθ decreases Bz in the plasma) βθ < 1 - paramagnetic (jθ increases Bz in the plasma)

Magnetics, A. Alonso, copyleft 2010 16 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Diamagnetic loop (II)

From the equilibrium condition in the radial direction we have ˆ r·(∇p−j×B) ⇒ dp dr = jθBz−jzBθ ≡ − 1 µ0 d dr B2

z

2 + Bθ 1 r d drrBθ

• Multiply by r2 and integrate over the cross section to get

p = 1 2µ0

• B2

θ(a) + B2 z(a) − B2 z

• ,

where we have assumed p(a) = 0. The diamagnetic loops provides an estimate of the βθ βθ ≡ 2µ0p B2

θ(a) = 1 + B2 z(a) − B2 z

B2

θ(a)

gives a measure of the plasma diamagnetism: βθ > 1 - diamagnetic (jθ decreases Bz in the plasma) βθ < 1 - paramagnetic (jθ increases Bz in the plasma)

Magnetics, A. Alonso, copyleft 2010 16 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Diamagnetic loop (II)

From the equilibrium condition in the radial direction we have ˆ r·(∇p−j×B) ⇒ dp dr = jθBz−jzBθ ≡ − 1 µ0 d dr B2

z

2 + Bθ 1 r d drrBθ

• Multiply by r2 and integrate over the cross section to get

p = 1 2µ0

• B2

θ(a) + B2 z(a) − B2 z

• ,

where we have assumed p(a) = 0. The diamagnetic loops provides an estimate of the βθ βθ ≡ 2µ0p B2

θ(a) = 1 + B2 z(a) − B2 z

B2

θ(a)

gives a measure of the plasma diamagnetism: βθ > 1 - diamagnetic (jθ decreases Bz in the plasma) βθ < 1 - paramagnetic (jθ increases Bz in the plasma)

Magnetics, A. Alonso, copyleft 2010 16 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Diamagnetic loop (II)

From the equilibrium condition in the radial direction we have ˆ r·(∇p−j×B) ⇒ dp dr = jθBz−jzBθ ≡ − 1 µ0 d dr B2

z

2 + Bθ 1 r d drrBθ

• Multiply by r2 and integrate over the cross section to get

p = 1 2µ0

• B2

θ(a) + B2 z(a) − B2 z

• ,

where we have assumed p(a) = 0. The diamagnetic loops provides an estimate of the βθ βθ ≡ 2µ0p B2

θ(a) = 1 + B2 z(a) − B2 z

B2

θ(a)

gives a measure of the plasma diamagnetism: βθ > 1 - diamagnetic (jθ decreases Bz in the plasma) βθ < 1 - paramagnetic (jθ increases Bz in the plasma)

Magnetics, A. Alonso, copyleft 2010 16 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

With the quantities we have estimated in this section it is posible to get an estimate of the energy confinement time τE in

• hmically heated plasmas.

From the definitions of the plasma kinetic energy and βθ W = 3 2

• V

pdV = 3π2a2Rp , βθ = 2µ0p B2

θ(a)

and the relation Bθ(a) = µ0Iz/(2πa), the energy confinement time can be written as τE = W P = 3 8µ0βθ(R/Rp) , where P = I2

z Rp is the ohmic heating.

Magnetics, A. Alonso, copyleft 2010 17 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

With the quantities we have estimated in this section it is posible to get an estimate of the energy confinement time τE in

• hmically heated plasmas.

From the definitions of the plasma kinetic energy and βθ W = 3 2

• V

pdV = 3π2a2Rp , βθ = 2µ0p B2

θ(a)

and the relation Bθ(a) = µ0Iz/(2πa), the energy confinement time can be written as τE = W P = 3 8µ0βθ(R/Rp) , where P = I2

z Rp is the ohmic heating.

Magnetics, A. Alonso, copyleft 2010 17 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

Rogoski → Iz plasma current Voltage loop → Vz transformer-induced voltage Diamagnetic loop → Bz → βθ polidal β Rogowski + Voltage loop → Rp the plasma resistivity Rogowski + Voltage loop + Diamagnetic loop → τE Energy confinement time

Magnetics, A. Alonso, copyleft 2010 18 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

Rogoski → Iz plasma current Voltage loop → Vz transformer-induced voltage Diamagnetic loop → Bz → βθ polidal β Rogowski + Voltage loop → Rp the plasma resistivity Rogowski + Voltage loop + Diamagnetic loop → τE Energy confinement time

Magnetics, A. Alonso, copyleft 2010 18 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

Rogoski → Iz plasma current Voltage loop → Vz transformer-induced voltage Diamagnetic loop → Bz → βθ polidal β Rogowski + Voltage loop → Rp the plasma resistivity Rogowski + Voltage loop + Diamagnetic loop → τE Energy confinement time

Magnetics, A. Alonso, copyleft 2010 18 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

Rogoski → Iz plasma current Voltage loop → Vz transformer-induced voltage Diamagnetic loop → Bz → βθ polidal β Rogowski + Voltage loop → Rp the plasma resistivity Rogowski + Voltage loop + Diamagnetic loop → τE Energy confinement time

Magnetics, A. Alonso, copyleft 2010 18 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Energy confinement time

Rogoski → Iz plasma current Voltage loop → Vz transformer-induced voltage Diamagnetic loop → Bz → βθ polidal β Rogowski + Voltage loop → Rp the plasma resistivity Rogowski + Voltage loop + Diamagnetic loop → τE Energy confinement time

Magnetics, A. Alonso, copyleft 2010 18 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Outline

1

Preliminaries

2

The magnetic coil

3

Global magnitudes

4

Plasma position and shape

Magnetics, A. Alonso, copyleft 2010 19 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Plasma position

If the plasma shifts slightly off the central position, the poloidal B-field caused by the plasma current is no longer uniform:

plasma coil

Bθ(θ) = µ0Iz 2πa

• sin2 θ + (cos θ − ∆/a)2−1/2

≈ µ0Iz 2πa

• 1 + ∆

a cos θ

• ,

The poloidal cosine (m = 1) Fourier component gives a measure of the plasma shift. To measure this component one can use a Rogowski coil with a winding density varying like cos θ, or. . .

Magnetics, A. Alonso, copyleft 2010 20 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Plasma position

If the plasma shifts slightly off the central position, the poloidal B-field caused by the plasma current is no longer uniform:

plasma coil

Bθ(θ) = µ0Iz 2πa

• sin2 θ + (cos θ − ∆/a)2−1/2

≈ µ0Iz 2πa

• 1 + ∆

a cos θ

• ,

The poloidal cosine (m = 1) Fourier component gives a measure of the plasma shift. To measure this component one can use a Rogowski coil with a winding density varying like cos θ, or. . .

Magnetics, A. Alonso, copyleft 2010 20 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Mirnov coils

. . . one can use an array of discrete pick-up coils known as Mirnov coils. In general, any B-field (or other physical magnitude) can be decomposed into poloidal and toroidal modes by Fourier-transforming in the θ and φ angles. B(r, θ, φ) =

• (n,m)

ˆ Bnm(r)ei(mθ+nφ) Toroidally symmetric equilibrium implies ˆ Bnm = 0 for n = 0. m = 1 relates to plasma position, m = 2, 3 . . . to plasma elipticity, triangularity, . . . Plasma instablities can have ˆ Bnm = 0 for any (m, n).

Magnetics, A. Alonso, copyleft 2010 21 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Mirnov coils

. . . one can use an array of discrete pick-up coils known as Mirnov coils. In general, any B-field (or other physical magnitude) can be decomposed into poloidal and toroidal modes by Fourier-transforming in the θ and φ angles. B(r, θ, φ) =

• (n,m)

ˆ Bnm(r)ei(mθ+nφ) Toroidally symmetric equilibrium implies ˆ Bnm = 0 for n = 0. m = 1 relates to plasma position, m = 2, 3 . . . to plasma elipticity, triangularity, . . . Plasma instablities can have ˆ Bnm = 0 for any (m, n).

Magnetics, A. Alonso, copyleft 2010 21 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Mirnov coils

. . . one can use an array of discrete pick-up coils known as Mirnov coils. In general, any B-field (or other physical magnitude) can be decomposed into poloidal and toroidal modes by Fourier-transforming in the θ and φ angles. B(r, θ, φ) =

• (n,m)

ˆ Bnm(r)ei(mθ+nφ) Toroidally symmetric equilibrium implies ˆ Bnm = 0 for n = 0. m = 1 relates to plasma position, m = 2, 3 . . . to plasma elipticity, triangularity, . . . Plasma instablities can have ˆ Bnm = 0 for any (m, n).

Magnetics, A. Alonso, copyleft 2010 21 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Mirnov coils

. . . one can use an array of discrete pick-up coils known as Mirnov coils. In general, any B-field (or other physical magnitude) can be decomposed into poloidal and toroidal modes by Fourier-transforming in the θ and φ angles. B(r, θ, φ) =

• (n,m)

ˆ Bnm(r)ei(mθ+nφ) Toroidally symmetric equilibrium implies ˆ Bnm = 0 for n = 0. m = 1 relates to plasma position, m = 2, 3 . . . to plasma elipticity, triangularity, . . . Plasma instablities can have ˆ Bnm = 0 for any (m, n).

Magnetics, A. Alonso, copyleft 2010 21 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Mirnov coils at JET

Magnetics, A. Alonso, copyleft 2010 22 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Instabilities and real-time control

[Hutchinson(2002)] Plasma instabilities can terminate in a disruption. Magnetics in a real-time control loop Modify the external knobs (currents and heating power) to restore the plasma equilibrium.

Magnetics, A. Alonso, copyleft 2010 23 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Instabilities and real-time control

[Hutchinson(2002)] Plasma instabilities can terminate in a disruption. Magnetics in a real-time control loop Modify the external knobs (currents and heating power) to restore the plasma equilibrium.

Magnetics, A. Alonso, copyleft 2010 23 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Instabilities and real-time control

[Hutchinson(2002)] Plasma instabilities can terminate in a disruption. Magnetics in a real-time control loop Modify the external knobs (currents and heating power) to restore the plasma equilibrium.

Magnetics, A. Alonso, copyleft 2010 23 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Magnetic surfaces are constant flux surfaces

Poloidal flux χ (either through a ribbon χr or through a disk χd) is constant on magnetic surfaces.

Magnetics, A. Alonso, copyleft 2010 24 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Poloidal Flux loops for Equilibrium

Numerical codes are used to solve the tokamak equilibrium Grad-Shafranov equation that best agrees with the magnetic measurements A commonly used such code is called EFIT (Equilibrium Fitting). The Grad-Shafranov equation is formulated in terms of the poloidal magnetic flux. To aid equilibrium reconstructions with boundary contitions → poloidal flux loops (differences cancel the transformer core contribution)

Poloidal Flux Loops Mirnov coils

Magnetics, A. Alonso, copyleft 2010 25 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Poloidal Flux loops for Equilibrium

Numerical codes are used to solve the tokamak equilibrium Grad-Shafranov equation that best agrees with the magnetic measurements A commonly used such code is called EFIT (Equilibrium Fitting). The Grad-Shafranov equation is formulated in terms of the poloidal magnetic flux. To aid equilibrium reconstructions with boundary contitions → poloidal flux loops (differences cancel the transformer core contribution)

Poloidal Flux Loops Mirnov coils

Magnetics, A. Alonso, copyleft 2010 25 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

Poloidal Flux loops for Equilibrium

Numerical codes are used to solve the tokamak equilibrium Grad-Shafranov equation that best agrees with the magnetic measurements A commonly used such code is called EFIT (Equilibrium Fitting). The Grad-Shafranov equation is formulated in terms of the poloidal magnetic flux. To aid equilibrium reconstructions with boundary contitions → poloidal flux loops (differences cancel the transformer core contribution)

Poloidal Flux Loops Mirnov coils

Magnetics, A. Alonso, copyleft 2010 25 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

After this lecture you must. . .

understand, by means of MEqs, the measurement of a simple coil know what is a

1 Rogoski coil 2 diamagnetic and a voltage loop 3 Mirnov coil 4 poloidal flux loop

understand how they measure and know what information (direct and derived) they provide.

Magnetics, A. Alonso, copyleft 2010 26 / 26

Preliminaries The magnetic coil Global magnitudes Plasma position and shape Summary

• I. H. Hutchinson.

Principles of plasma diagnostics. Cambridge and New York, Cambridge University Press, 2002, 2002. URL http://www3.cambridge.org/uk/catalogue/ catalogue.asp?isbn=9780521803892. John David Jackson. Classical electrodynamics; 3rd ed. Wiley, New York, NY, 1999.

Magnetics, A. Alonso, copyleft 2010 26 / 26