Superconducting Coils

In fusion reactors, magnetic fields of 5-10 T or higher are required to confine plasma at temperatures exceeding 100 million degrees. Generating such strong magnetic fields with conventional copper coils (normal conducting coils) would result in enormous power losses due to Joule heating, exceeding the energy produced by fusion. Therefore, the use of superconducting coils, which have zero electrical resistance, is essential.

Superconducting technology is a foundational technology that determines the feasibility of fusion reactors, representing an interdisciplinary field that integrates materials science, cryogenic engineering, electromagnetism, and structural mechanics. This article comprehensively covers superconductivity from fundamental physics to the design, manufacturing, and operation of practical coils.

Necessity of Superconducting Coils

In normal conducting coils, passing current generates Joule heat $P = I^2 R$. When generating fusion reactor-scale magnetic fields (millions of ampere-turns), this Joule heating reaches several hundred MW, making it unviable as an energy generation device.

Quantitatively, if ITER-scale toroidal field coils were made of copper, the required excitation power would be approximately:

$$P_{\text{Cu}} = \rho_{\text{Cu}} \frac{L}{A} I^2 \approx 200-500 \text{ MW}$$

where $\rho_{\text{Cu}} \approx 1.7 \times 10^{-8}$ Ωm is the resistivity of copper, $L$ is the conductor length, $A$ is the conductor cross-sectional area, and $I$ is the current. Meanwhile, fusion output is expected to be around 500 MW, so magnetic field generation alone would consume most of the power output.

Superconducting coils offer the following advantages:

• Zero DC electrical resistance (no Joule losses during steady-state operation)
• High current density achievable (enables coil miniaturization)
• Capable of generating high magnetic fields (up to the critical field)
• Power consumption limited to cooling systems only (typically several MW)

However, maintaining the superconducting state requires cooling to cryogenic temperatures (around 4 K), and there is energy consumption from the refrigeration system. Considering the Carnot efficiency ratio to actual efficiency:

$$\eta_{\text{ref}} = \frac{T_{\text{cold}}}{T_{\text{hot}} - T_{\text{cold}}} \times \eta_{\text{Carnot}} \approx \frac{4}{300 - 4} \times 0.3 \approx 0.004$$

This means approximately 250-300 W of electrical power is needed to remove 1 W of heat at 4 K. Nevertheless, power consumption during operation is 2-3 orders of magnitude smaller compared to normal conducting coils.

Historical Development of Superconducting Technology

The history of superconductivity spans over a century, from the discovery of the phenomenon to its application in fusion coils.

Discovery of Superconductivity (1911)

In 1911, at Leiden University in the Netherlands, Heike Kamerlingh Onnes discovered that electrical resistance suddenly disappeared in mercury cooled with liquid helium at around 4.2 K. This discovery earned Onnes the Nobel Prize in Physics in 1913.

Initially, superconductivity was primarily a subject of scientific curiosity, but the property of zero electrical resistance suggested potential applications for powerful electromagnets.

Advancement of Theoretical Understanding (1930s-1960s)

In 1933, Meissner and Ochsenfeld discovered that superconductors expel magnetic fields from their interior (Meissner effect). This demonstrated that superconductivity is not merely zero resistance but a new quantum state.

In 1935, the London brothers proposed the London equations describing the electromagnetic behavior of superconductors:

$$\nabla^2 \mathbf{B} = \frac{\mathbf{B}}{\lambda_L^2}$$

where $\lambda_L$ is the London penetration depth.

In 1950, Ginzburg and Landau developed a phenomenological theory of superconductivity, introducing the order parameter $\psi$. This theory became the foundation for explaining the behavior of Type II superconductors.

In 1957, Bardeen, Cooper, and Schrieffer published BCS theory, providing the first complete microscopic explanation of superconductivity. All three received the Nobel Prize in Physics in 1972.

Development of Practical Materials (1960s-1980s)

In 1961, Kunzler et al. demonstrated that niobium-tin (Nb₃Sn) maintained a current density of about 10⁵ A/cm² at 8.8 T and 4.2 K. This discovery opened the path to practical high-field superconducting magnets.

From the late 1960s through the 1970s, manufacturing technologies for NbTi alloys and Nb₃Sn compounds were established, and applications to MRI devices and particle accelerators began.

In fusion research, testing of superconducting coils in small experimental devices began in the 1970s, and superconducting tokamaks such as T-7 (Soviet Union) and TRIAM-1M (Japan) began operation in the 1980s.

Discovery of High-Temperature Superconductivity (1986)

In 1986, Bednorz and Müller at IBM Zurich Research Laboratory discovered that a copper oxide ceramic (La-Ba-Cu-O) exhibited superconductivity at about 35 K. This high critical temperature, inexplicable by conventional BCS theory, sent shockwaves through physics and materials science. Both researchers received the Nobel Prize in Physics the following year in 1987.

In 1987, the Y-Ba-Cu-O (YBCO) system reached a critical temperature of 92 K, exceeding the liquid nitrogen temperature (77 K). This opened the possibility of superconducting applications without expensive liquid helium.

Full-Scale Application to Fusion (1990s-Present)

In the 1990s, with the materialization of the ITER project, large-scale conductor development programs were launched. Cable-in-conduit conductors (CICC) using Nb₃Sn and NbTi were developed in Japan, Europe, the United States, and Russia, and model coil tests demonstrated performance at fusion reactor scales.

In the 2000s, manufacturing technology for REBCO (RE-Ba-Cu-O) tapes advanced, making long-length, high-quality tapes available. In the 2020s, the MIT/Commonwealth Fusion Systems SPARC project decided to fully adopt REBCO, bringing high-temperature superconducting applications to fusion into reality.

Fundamental Theory of Superconductivity

Discovery and Basic Properties of Superconductivity

In 1911, Dutch physicist Kamerlingh Onnes discovered that when mercury was cooled to cryogenic temperatures, its electrical resistance suddenly became zero at about 4.2 K. This phenomenon was named superconductivity, and similar phenomena were subsequently observed in many metals and alloys.

The basic properties of superconductors are as follows:

1. Zero resistance: DC electrical resistance becomes strictly zero below the critical temperature
2. Meissner effect: Perfect diamagnetism where magnetic fields are expelled from the superconductor interior
3. Flux quantization: Magnetic flux in a superconducting ring is quantized in integer multiples of $\Phi_0 = h/2e \approx 2.07 \times 10^{-15}$ Wb
4. Josephson effect: Superconducting current flows between two superconductors even with an insulating barrier between them

BCS Theory

In 1957, Bardeen, Cooper, and Schrieffer proposed a theory (BCS theory) explaining the microscopic mechanism of superconductivity. According to this theory, superconductivity is based on the formation of electron pairs (Cooper pairs).

The mechanism of Cooper pair formation is explained as follows: When an electron passes through the lattice, it attracts surrounding positive ions, creating a local concentration of positive charge. This positive charge region attracts another electron, resulting in two electrons having an indirect attractive interaction.

The binding energy of Cooper pairs (superconducting gap) $\Delta$ is:

$$\Delta(T=0) = 1.76 k_B T_c$$

where $k_B$ is the Boltzmann constant and $T_c$ is the critical temperature. Because this energy gap is maintained at finite temperatures, energy loss through scattering does not occur, and zero resistance is achieved.

The critical temperature derived from BCS theory is:

$$T_c = 1.13 \theta_D \exp\left(-\frac{1}{N(0)V}\right)$$

where $\theta_D$ is the Debye temperature, $N(0)$ is the density of states at the Fermi level, and $V$ is the strength of electron-lattice interaction. This equation suggests that conventional superconductors have a theoretical upper limit for critical temperature (about 30-40 K).

Type I and Type II Superconductors

Superconductors are classified into two types based on their response to magnetic fields:

Type I superconductors (most pure metals): They transition abruptly to the normal conducting state when the critical field $H_c$ is exceeded. The thermodynamic critical field is:

$$H_c(T) = H_c(0) \left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

Type II superconductors (alloys, compounds): They have two critical fields: lower critical field $H_{c1}$ and upper critical field $H_{c2}$. In the range $H_{c1} < H < H_{c2}$, magnetic flux penetrates the superconductor as quantized vortices (fluxons), creating a mixed state.

$$H_{c1} = \frac{\Phi_0}{4\pi\mu_0\lambda^2} \ln\kappa$$ $$H_{c2} = \frac{\Phi_0}{2\pi\mu_0\xi^2}$$

where $\lambda$ is the London penetration depth, $\xi$ is the coherence length, and $\kappa = \lambda/\xi$ is the Ginzburg-Landau parameter. When $\kappa > 1/\sqrt{2}$, the material is a Type II superconductor.

All superconducting materials used in fusion are Type II superconductors, utilizing high $H_{c2}$ to generate strong magnetic fields.

Critical Surface and Operating Margin

The superconducting state is limited by a critical surface characterized by three parameters:

• Critical temperature $T_c$: Transition to superconducting state below this temperature
• Upper critical field $B_{c2}$: Transition to normal conducting state above this field
• Critical current density $J_c$: Transition to normal conducting state above this current density

These are interdependent and can be approximated empirically by:

$$J_c(B, T) = J_{c0} \left(1 - \frac{T}{T_c}\right)^n \left(1 - \frac{B}{B_{c2}(T)}\right)^m \frac{B_{c2}(T)^p}{B^q}$$

where $J_{c0}$, $n$, $m$, $p$, and $q$ are material-specific parameters.

In actual operation, sufficient margin from the critical surface must be ensured. The temperature margin $\Delta T_{\text{margin}}$ is:

$$\Delta T_{\text{margin}} = T_{\text{cs}} - T_{\text{op}}$$

where $T_{\text{cs}}$ is the current sharing temperature at the operating field and current, and $T_{\text{op}}$ is the operating temperature. Fusion coils typically ensure a margin of 1-2 K or more.

Superconducting Materials

Superconducting materials used in fusion reactors must satisfy many requirements including high-field characteristics, mechanical strength, radiation resistance, and manufacturability.

NbTi (Niobium-Titanium Alloy)

NbTi is the most widely used practical superconducting material.

Basic properties:

• Critical temperature: $T_c = 9.6$ K
• Upper critical field: $B_{c2}(4.2 \text{ K}) = 11.5$ T
• Critical current density: $J_c(5 \text{ T}, 4.2 \text{ K}) \approx 3000$ A/mm²

Temperature dependence:

$$B_{c2}(T) = B_{c2}(0) \left[1 - \left(\frac{T}{T_c}\right)^{1.7}\right]$$

Advantages:

• Excellent ductility, easy to process
• Easily manufactured as copper composite
• Low cost, high reliability
• No heat treatment required

Disadvantages:

• Relatively low critical field (difficult to use above 12 T)
• Low critical temperature

Fusion applications:

• ITER PF coils (maximum field 6.4 T)
• External coils for many experimental devices

Nb₃Sn (Niobium-3-Tin)

Nb₃Sn is an A15-type intermetallic compound used for high-field applications.

Basic properties:

• Critical temperature: $T_c = 18.3$ K
• Upper critical field: $B_{c2}(4.2 \text{ K}) = 23-28$ T (depending on composition)
• Critical current density: $J_c(12 \text{ T}, 4.2 \text{ K}) \approx 1000$ A/mm²

Field dependence (Kramer model):

$$J_c(B) \propto B^{-1/2} (1 - B/B_{c2})^2$$

More precisely, including the effect of strain $\varepsilon$:

$$J_c(B, T, \varepsilon) = \frac{C_0}{B} (B_{c2}^*(T, \varepsilon))^n \left(1 - \frac{B}{B_{c2}^*(T, \varepsilon)}\right)^2 \left(\frac{T_c^*(\varepsilon)}{T}\right)^{n-3} \left(1 - \frac{T^{1.52}}{T_c^*(\varepsilon)^{1.52}}\right)^2$$

Strain sensitivity: Nb₃Sn is sensitive to strain, with critical current dropping sharply beyond 0.2-0.3% strain:

$$\frac{J_c(\varepsilon)}{J_c(0)} = 1 - a|\varepsilon - \varepsilon_0|^{1.7}$$

where $\varepsilon_0$ is residual strain and $a$ is a material constant.

Advantages:

• High critical field (usable above 20 T)
• Higher critical temperature than NbTi

Disadvantages:

• Brittle and mechanically delicate
• Requires high-temperature heat treatment (approximately 650°C for hundreds of hours)
• Strain sensitive
• Higher cost than NbTi

Fusion applications:

• ITER TF coils (maximum field 11.8 T)
• ITER CS coils (maximum field 13 T)
• High-field experimental devices

Nb₃Al (Niobium-3-Aluminum)

Nb₃Al is an A15-type compound with superior strain tolerance compared to Nb₃Sn.

Basic properties:

• Critical temperature: $T_c = 18.9$ K
• Upper critical field: $B_{c2}(4.2 \text{ K}) = 29-33$ T
• Critical current density: $J_c(12 \text{ T}, 4.2 \text{ K}) \approx 600-900$ A/mm²

Strain characteristics: Compared to Nb₃Sn, degradation with strain is more gradual, maintaining practical properties up to about 0.5% strain.

Disadvantages:

• Difficult to manufacture (requires special processes such as rapid quenching)
• Long-length wire manufacturing technology still developing

REBCO (Rare Earth-Based High-Temperature Superconductors)

REBCO (Rare Earth Barium Copper Oxide) is a copper oxide high-temperature superconductor with the composition $\text{REBa}_2\text{Cu}_3\text{O}_{7-\delta}$ (RE = Y, Gd, Sm, etc.).

Basic properties:

• Critical temperature: $T_c = 90-93$ K
• Irreversibility field: $B_{\text{irr}}(4.2 \text{ K}) > 100$ T
• Critical current density: $J_c(20 \text{ T}, 4.2 \text{ K}) > 1000$ A/mm²

Anisotropy: REBCO has strong crystalline anisotropy, with critical current depending on field direction:

$$J_c(B, \theta) = \frac{J_c(B, 0°)}{\sqrt{\cos^2\theta + \gamma^{-2}\sin^2\theta}}$$

where $\theta$ is the angle between the field and c-axis, and $\gamma \approx 5-7$ is the anisotropy parameter.

Tape form: REBCO is manufactured in thin film form, with typical dimensions:

• Width: 4-12 mm
• Thickness: 0.05-0.1 mm (superconducting layer approximately 1-2 μm)
• Engineering current density: $J_e \approx 500-1000$ A/mm²

Advantages:

• Very high critical field (superior above 20 T)
• Higher operating temperatures possible (cooling is easier at 20-30 K)
• Excellent mechanical strength

Disadvantages:

• High cost (10-100 times NbTi)
• Limited manufacturing length (currently several hundred m/piece)
• Joint technology still developing
• Can have large AC losses

Fusion applications:

• All coils for SPARC (MIT/CFS)
• Future compact reactors

Bi-2212 (Bismuth-Based High-Temperature Superconductor)

Bi₂Sr₂CaCu₂O₈₊ₓ (Bi-2212) is a promising high-temperature superconducting material after REBCO.

Basic properties:

• Critical temperature: $T_c = 85$ K
• Irreversibility field: $B_{\text{irr}}(4.2 \text{ K}) \approx 100$ T
• Critical current density: $J_c(20 \text{ T}, 4.2 \text{ K}) \approx 600-1000$ A/mm²

Manufacturing method: Bi-2212 is manufactured by the PIT (Powder-In-Tube) method, where oxide precursor powder is filled into a silver sheath, processed, and then subjected to partial melt heat treatment. Heat treatment conditions are:

$$T_{\text{max}} \approx 888°\text{C} \text{ (oxygen partial pressure 1 atm)}$$

The Bi-2212 phase crystallizes during cooling, and superconducting properties emerge.

Advantages:

• Can be manufactured in round wire form (easy winding)
• Isotropic current transport properties
• Relatively easy to apply to CICC

Disadvantages:

• Void formation due to gas generation during heat treatment (solvable with overpressure heat treatment)
• Lower irreversibility field than REBCO
• Cost of silver sheath

Fusion applications: Due to its round wire characteristic, Bi-2212 has high compatibility with conventional LTS conductor designs and is being researched for cable-in-conduit conductor applications.

Bi-2223 (Bismuth-Based First Generation HTS)

Bi₂Sr₂Ca₂Cu₃O₁₀₊ₓ (Bi-2223) was the first commercialized high-temperature superconducting material.

Basic properties:

• Critical temperature: $T_c = 110$ K
• Irreversibility field: $B_{\text{irr}}(77 \text{ K}) \approx 0.2$ T, $B_{\text{irr}}(20 \text{ K}) \approx 20$ T

Bi-2223 is not suitable for fusion coils due to its weak field properties at 77 K, but it has a track record in applications such as power cables and fault current limiters.

MgB₂ (Magnesium Diboride)

MgB₂, whose superconductivity was discovered in 2001, has attracted attention as a metallic superconductor.

Basic properties:

• Critical temperature: $T_c = 39$ K
• Upper critical field: $B_{c2}(4.2 \text{ K}) \approx 15-20$ T
• Critical current density: $J_c(4 \text{ T}, 4.2 \text{ K}) \approx 1000$ A/mm²

Advantages:

• Inexpensive and abundant raw materials
• BCS-type superconductor with well-understood theory
• Can operate at around 20 K

Disadvantages:

• Insufficient high-field properties (performance degradation above 10 T)
• High anisotropy

MgB₂ has insufficient field properties for main fusion reactor coils, but applications to low-field correction coils and current leads are being considered.

Comprehensive Comparison of Material Properties

Property	NbTi	Nb₃Sn	REBCO	Bi-2212	MgB₂
Critical temperature $T_c$	9.6 K	18.3 K	92 K	85 K	39 K
$B_{c2}$(4.2 K)	11.5 T	25 T	> 100 T	~100 T	18 T
$J_c$(12 T, 4.2 K)	-	1000 A/mm²	2000 A/mm²	800 A/mm²	-
Conductor form	Round wire	Round wire	Tape	Round wire	Round wire
Strain tolerance	> 1%	0.2-0.3%	> 0.5%	0.3%	0.4%
Workability	Excellent	Fair (brittle after heat treatment)	Tape form	Fair (requires heat treatment)	Fair
Relative cost	1	3-5	50-100	20-40	2-5
Radiation resistance	Good	Good	Needs verification	Needs verification	Good
Maturity	High	High	Medium	Medium	Low

Toroidal Field Coils (TF Coils)

TF coils generate the toroidal magnetic field that confines plasma in the toroidal direction. They are the largest and most electromagnetically demanding coils in a tokamak.

Basic Principle and Field Distribution

From Ampère’s law, the toroidal field strength is inversely proportional to the major radius:

$$B_\phi(R) = \frac{\mu_0 N I}{2\pi R} = B_0 \frac{R_0}{R}$$

where $N$ is the number of coil turns, $I$ is the coil current, $R_0$ is the plasma major radius, and $B_0$ is the field strength at $R_0$.

The total magnetomotive force required to obtain the specified field at the plasma center is:

$$NI = \frac{2\pi R_0 B_0}{\mu_0}$$

For ITER with $R_0 = 6.2$ m and $B_0 = 5.3$ T, $NI \approx 1.66 \times 10^8$ A (166 MA).

Theory of D-Shaped Geometry

TF coils typically have a D-shape (or Princeton-D) geometry. This shape is designed to minimize pure bending moments and support the coil with in-plane tension only.

In steady state, the following equilibrium condition holds for the coil conductor:

$$T = \int B \cdot J \, dA$$

where $T$ is tension, $B$ is local field, and $J$ is current density.

The ideal pure-tension shape is designed considering the 1/R dependence of the field and satisfies the differential equation:

$$\frac{d^2 r}{dz^2} = -\frac{r}{r_0^2} \left(1 + \left(\frac{dr}{dz}\right)^2\right)^{3/2}$$

Electromagnetic Forces

Multiple electromagnetic forces act on TF coils.

1. In-plane forces (bursting force): The interaction between the coil current and its own magnetic field generates a force that tries to expand the coil radially.

$$F_{\text{burst}} = \frac{1}{2\mu_0} B^2 \cdot A_{\text{coil}}$$

2. Centering force: Due to the 1/R dependence of the toroidal field, a difference in magnetic pressure occurs between the inner (high-field) and outer (low-field) sides of the coil, generating a force that pulls the entire coil toward the torus center.

$$F_{\text{centering}} = \frac{B_{\text{in}}^2 - B_{\text{out}}^2}{2\mu_0} \cdot A$$

The centering force per ITER TF coil is approximately 6,600 metric tons (65 MN). For all 18 coils together, this reaches approximately 1,200 MN.

3. Out-of-plane forces (overturning forces): The interaction between TF coil current and poloidal field (from plasma current and PF coils) generates moments that try to tilt the coil.

$$\tau_{\text{OOP}} = \int (I \times B_p) \cdot r \, dl$$

Structural Design

Radial plate structure: ITER’s TF coils adopt a structure where grooves are machined into D-shaped steel radial plates and conductors are fitted into them. This provides:

• Direct support of conductor electromagnetic forces by the plate
• High-precision winding geometry
• Easy manufacturing and assembly

Wedge configuration: The 18 TF coils contact each other in a wedge shape at the center, supporting the centering force. Contact pressure between coils is:

$$p_{\text{wedge}} = \frac{F_{\text{centering}}}{A_{\text{contact}}}$$

Coil case: A stainless steel case housing the conductors and radial plates provides overall rigidity and strength.

ITER TF Coil Specifications

Parameter	Value
Number of coils	18
Superconducting material	Nb₃Sn
Rated current	68 kA
Maximum field	11.8 T
Stored energy (total)	41 GJ
Coil outer dimensions	Height 14 m × Width 9 m
Coil weight	Approximately 300 tons/coil
Conductor length (per coil)	Approximately 6 km

Poloidal Field Coils (PF Coils)

PF coils generate poloidal field components to control plasma position and shape.

Functions and Roles

1. Plasma equilibrium by vertical field: Toroidal plasma experiences an outward expansion force due to field gradient drift and toroidal drift. To compensate for this, a vertical field (downward or upward when viewed from outside) is required.

The required vertical field depends on plasma pressure and beta value, approximately:

$$B_v = \frac{\mu_0 I_p}{4\pi R_0} \left(\ln\frac{8R_0}{a} + \beta_p + \frac{l_i}{2} - \frac{3}{2}\right)$$

where $I_p$ is plasma current, $a$ is plasma minor radius, $\beta_p$ is poloidal beta, and $l_i$ is internal inductance.

2. Shape control: Controls plasma cross-sectional shape (elongation $\kappa$, triangularity $\delta$).

Plasma with elongation has vertical instability, requiring control faster than the damping time to suppress it:

$$\gamma = \frac{1}{\tau_w} \frac{n - 1}{n_{\text{crit}} - n}$$

where $\tau_w$ is the vacuum vessel resistive decay time and $n$ is the field decay index.

3. Divertor configuration: Creates X-points (magnetic null points) at the plasma boundary, forming a divertor configuration.

Layout and Design

PF coils are typically placed outside the TF coils for the following reasons:

• Avoiding interference with TF coils
• Difficulty in disassembling and connecting superconducting coils
• Ensuring large control range

However, the farther the coils, the greater the magnetomotive force required for the same field.

Equilibrium coil design: The required current distribution is calculated by plasma equilibrium codes. Typically, multiple pairs of coils are arranged to independently control vertical field components, curvature components, and higher-order multipole components.

ITER PF Coil Specifications

Coil Name	Major Radius	Superconducting Material	Rated Current	Maximum Field
PF1	3.9 m	NbTi	48 kA	5.1 T
PF2	8.3 m	NbTi	48 kA	4.2 T
PF3	11.9 m	NbTi	48 kA	4.8 T
PF4	11.9 m	NbTi	48 kA	5.8 T
PF5	8.3 m	NbTi	52 kA	6.4 T
PF6	3.9 m	NbTi	48 kA	6.0 T

PF coils are arranged symmetrically top and bottom, consisting of a total of 6 coils (3 pairs). Since the maximum field is below 10 T, NbTi can be used.

Central Solenoid (CS Coil)

The CS coil is a solenoid coil placed at the central axis of the torus, functioning as the primary winding of a transformer to inductively drive plasma current.

Operating Principle

Using the transformer principle, flux changes in the CS coil induce loop voltage in the plasma:

$$V_{\text{loop}} = -\frac{d\Phi_{\text{CS}}}{dt}$$

By Ohm’s law, this loop voltage drives plasma current:

$$V_{\text{loop}} = I_p R_p + L_p \frac{dI_p}{dt}$$

where $R_p$ is plasma resistance and $L_p$ is plasma self-inductance.

In steady state, $V_{\text{loop}} = I_p R_p$, and a constant loop voltage is needed to maintain plasma current. Using Spitzer resistivity:

$$\eta = \frac{m_e \nu_{ei}}{n_e e^2} \propto \frac{Z_{\text{eff}}}{T_e^{3/2}}$$

In high-temperature plasma, resistivity becomes very small, so the required loop voltage also becomes small.

Flux Supply Capability

By varying the CS coil current from $+I_{\text{max}}$ to $-I_{\text{max}}$, maximum flux swing is obtained.

The maximum flux that the CS coil can supply is determined by coil geometry and maximum field:

$$\Phi_{\text{CS}} = \int B \cdot dA = \mu_0 n I \cdot \pi R_{\text{CS}}^2$$

More precisely, for a finite-thickness solenoid:

$$\Phi_{\text{max}} = 2 B_{\text{max}} \int_0^{R_{\text{CS}}} \frac{r}{R_{\text{CS}}} 2\pi r \, dr = \frac{4\pi}{3} R_{\text{CS}}^2 B_{\text{max}}$$

Considering the swing range, available flux is:

$$\Delta\Phi = 2\Phi_{\text{max}} = \frac{8\pi}{3} R_{\text{CS}}^2 B_{\text{max}}$$

Pulse Duration Limitations

The inductive pulse duration of a fusion reactor is limited by the flux supply capability of the CS coil:

$$\tau_{\text{pulse}} = \frac{\Delta\Phi}{V_{\text{loop}}} = \frac{\Delta\Phi}{I_p R_p}$$

ITER aims for pulse operation of approximately 400 seconds. Longer operation (steady-state operation) requires non-inductive current drive such as neutral beam injection and radio-frequency current drive.

Electromagnetic Forces and Structure

Large electromagnetic forces act on the CS coil:

1. Radial compression force: The coil current and its own axial field generate a force that compresses the conductor inward.

$$\sigma_r = \frac{B^2}{2\mu_0}$$

At a maximum field of 13 T, this corresponds to a magnetic pressure of approximately 67 MPa.

2. Axial forces: Interaction with the upper and lower PF coils and plasma current generates axial forces.

3. Induction forces during rapid current changes: During rapid CS coil current changes, transient induced currents occur, generating additional electromagnetic forces.

ITER CS Coil Specifications

Parameter	Value
Number of modules	6
Superconducting material	Nb₃Sn
Rated current	45 kA
Maximum field	13 T
Outer diameter	4.1 m
Inner diameter	2.1 m
Height	12.4 m (total)
Stored energy	6.4 GJ

The CS coil consists of 6 independent modules, each individually controllable. This enables flexible flux control during plasma startup.

Cable-In-Conduit Conductor (CICC)

CICC (Cable-In-Conduit Conductor) is the most widely adopted conductor format for large superconducting magnets.

Basic Structure

CICC consists of the following elements:

1. Superconducting strands: Fine superconducting wires (strands) about 0.5-1 mm in diameter
2. Copper strands: Pure copper wires for stabilization
3. Cable: Multi-stage twisted structure of strands
4. Conduit: Outer jacket made of stainless steel or Inconel
5. Coolant channel: Supercritical helium flowing through gaps inside the conduit

Twisted Structure

Strands are twisted in multiple stages to form a cable. Typical structures are:

$$3 \times 3 \times 3 \times 3 \times 6 = 486 \text{ strands}$$

or:

$$(2 \text{SC} + 1 \text{Cu}) \times 3 \times 4 \times 4 \times 6 = 864 \text{ strands}$$

The twist pitch at each stage is designed considering the balance between AC losses and mechanical properties.

The self-field cancellation effect of twisting is characterized by the ratio of twist pitch $p$ to strand radius $r$:

$$\frac{B_{\text{self}}}{B_{\text{external}}} \propto \frac{r}{p}$$

Void Fraction

The gaps (voids) inside the cable serve as coolant flow paths. The void fraction $f_v$ is:

$$f_v = 1 - \frac{A_{\text{cable}}}{A_{\text{conduit}}}$$

ITER conductors adopt $f_v \approx 0.29-0.35$. A larger void fraction improves cooling but reduces mechanical rigidity.

Current Sharing

Ideally, current should be equally distributed among all strands in a CICC. However, non-uniformity actually occurs due to the following factors:

• Variation in contact resistance between strands
• Differences in strand length (due to twisting)
• Differences in local temperature and field distribution

Current non-uniformity is evaluated by:

$$\Delta I = \frac{V}{\sum_i R_i^{-1}} \left(\frac{1}{R_i} - \frac{1}{N}\sum_j \frac{1}{R_j}\right)$$

Cooling Characteristics

Supercritical helium (SHe: approximately 4.5 K, 0.6 MPa) circulates inside the conduit to cool the conductor.

Heat transfer coefficient is:

$$h = \frac{Nu \cdot k_{\text{He}}}{D_h}$$

where $Nu$ is the Nusselt number, $k_{\text{He}}$ is the thermal conductivity of helium, and $D_h$ is the hydraulic diameter.

Pressure drop is:

$$\Delta p = f \frac{L}{D_h} \frac{\rho v^2}{2}$$

where $f$ is the friction coefficient, $L$ is conductor length, $\rho$ is helium density, and $v$ is flow velocity.

ITER Conductor Specifications

Parameter	TF Conductor	CS Conductor	PF Conductor
Superconducting material	Nb₃Sn	Nb₃Sn	NbTi
Number of strands	900	576	1440
Strand diameter	0.82 mm	0.83 mm	0.73 mm
Cu/SC ratio	1.0	1.0	1.6
Rated current	68 kA	45 kA	45-52 kA
Conduit outer diameter	43.7 mm	49 mm	Approximately 50 mm
Void fraction	0.29	0.33	0.35

Quench and Protection Systems

Quench is a phenomenon where part of a superconductor transitions to the normal conducting state by exceeding critical conditions, and the resulting heat propagates in a chain reaction. Without proper protection, it can lead to coil damage and fire.

Physics of Quench

Initiation mechanism: When part of a conductor exceeds the critical surface due to local temperature rise, mechanical disturbance, or field variation, normal transition begins.

In the normal region, Joule heating occurs from the current:

$$\dot{Q} = \rho J^2 = \frac{I^2 \rho}{A^2}$$

When this heat is transmitted to the surroundings and adjacent parts also exceed the critical temperature, the normal region propagates.

Propagation velocity: Assuming thermal equilibrium, quench propagation velocity is:

$$v_q = \frac{J \sqrt{\rho k}}{C (T_c - T_{\text{op}})}$$

where $k$ is thermal conductivity and $C$ is volumetric heat capacity. Typical values are 1-100 m/s.

In Nb₃Sn coils, propagation velocity is relatively slow, making local hot spots more likely and protection particularly important.

Energy Balance and Maximum Temperature

The maximum temperature during quench is determined by stored energy and conductor cross-section.

Temperature rise under adiabatic conditions is:

$$\int_{T_0}^{T_{\text{max}}} C(T) \, dT = \int_0^{t_d} \rho(T) J^2 \, dt$$

Rearranging, the integral quantity (U function) that determines maximum temperature is obtained:

$$U(T_{\text{max}}) = \int_{T_0}^{T_{\text{max}}} \frac{C(T)}{\rho(T)} dT = J^2 t_d$$

The protection goal is to keep $T_{\text{max}} < 150-200$ K. Above this, thermal stress and insulation degradation become problematic.

Detection Systems

Early quench detection is the first step in protection.

Voltage detection method: Normal transition generates resistive voltage:

$$V_R = I \cdot R_n = I \int \rho \frac{dl}{A}$$

To distinguish from inductive voltage, the coil is segmented and differential voltage is measured:

$$V_{\text{diff}} = V_1 - V_2 \cdot \frac{L_1}{L_2}$$

Detection threshold is typically 0.1-1 V with detection time of tens of milliseconds.

Other detection methods:

• Temperature measurement (local but direct)
• Flow rate change (due to helium vaporization)
• Optical fiber distributed temperature measurement

Protection Methods

Energy extraction: A method of releasing energy to external resistors.

$$\frac{d(LI)}{dt} = -IR_{\text{ext}} - V_{\text{magnet}}$$

Time constant is:

$$\tau = \frac{L}{R_{\text{ext}}}$$

Most energy is dissipated in external resistors, but discharge voltage becomes high, so insulation design is important.

Coupled coil method: Multiple coils are magnetically coupled so that when one quenches, induced currents flow in the other coils to distribute energy.

Heater-induced quench: After quench detection, heaters apply heat to the entire coil, intentionally quenching the whole thing. This:

• Distributes energy throughout the coil
• Prevents local overheating
• Suppresses maximum temperature

Heaters are placed near conductors and heat them above the critical temperature in tens of milliseconds.

ITER Protection System

ITER adopts the following protection strategy:

1. Quench detection: Differential voltage method (threshold 0.5 V, judgment time 1 second)
2. Energy extraction: Discharge resistors (TF: 74 mΩ, CS/PF: variable)
3. Heaters: For quench propagation promotion
4. Bypass diodes: Inter-coil voltage limiting

The TF coil system discharge time constant is approximately 11 seconds, with maximum discharge voltage designed at 5 kV.

Quench Characteristics of HTS Coils

High-temperature superconducting coils exhibit different quench characteristics from low-temperature superconducting coils, presenting new challenges for protection design.

Slow propagation velocity: HTS quench propagation velocity is significantly slower than LTS, typically 1-10 mm/s (LTS is 1-100 m/s).

$$v_{\text{quench}}^{\text{HTS}} \ll v_{\text{quench}}^{\text{LTS}}$$

This slow propagation is due to the following factors:

• High critical temperature (large temperature margin)
• Good thermal conduction (copper/silver stabilizers)
• High heat capacity

Risk of local overheating: Slow propagation velocity means energy concentrates locally. Local temperature rises sharply before the normal region spreads:

$$\Delta T_{\text{local}} = \frac{E_{\text{stored}}}{C_p \cdot V_{\text{quench}}}$$

where $V_{\text{quench}}$ is the quench region volume. Local temperatures 1-2 orders of magnitude higher than LTS are possible, increasing conductor damage risk.

Difficulty of detection: Due to slow propagation and large temperature margin, it takes time before detectable voltage develops. Detection delay becomes a problem with conventional voltage detection.

HTS detection technologies:

• Acoustic emission (AE) detection: Detects elastic waves from mechanical deformation
• Optical fiber distributed temperature measurement: Continuous temperature distribution monitoring using OFDR/OTDR technology
• High-density voltage tap placement
• Fast current distribution measurement

Protection strategy: For HTS coils, quench prevention is more important than post-quench protection.

1. Large operating margin: Operate below 70-80% of critical current
2. Excellent cooling design: Quickly remove local heat generation
3. High-density monitoring: Early warning system
4. NI/PI technology: Utilize self-protection function

In No-Insulation coils, current bypasses between turns during normal transition, making them inherently safer:

$$I_{\text{bypass}} = \frac{V_{\text{quench}}}{R_{\text{turn-to-turn}}}$$

Cryogenic Systems

Large-scale cryogenic systems are essential for superconducting coil operation. The refrigeration system for cooling to liquid helium temperature (4.2 K) or supercritical helium temperature (4.5 K) is an important component that directly affects fusion reactor energy balance.

Physical Properties of Helium

Helium has the lowest boiling point of all elements (4.2 K at atmospheric pressure) and is an essential coolant for superconducting coil cooling.

Phase diagram and states: Helium’s phase diagram differs significantly from other substances.

• Boiling point at atmospheric pressure: 4.22 K
• Critical point: 5.2 K, 0.227 MPa
• Lambda point: 2.17 K (superfluid transition)
• Solidification requires pressures above about 2.5 MPa (does not solidify even at absolute zero at atmospheric pressure)

Supercritical helium (SHe): Fusion coils are operated in a supercritical state above the critical point pressure (approximately 0.3-0.6 MPa) and at about 4.5 K. In this state:

• No distinction between liquid and gas, avoiding two-phase flow problems
• Continuous density changes, stable heat transfer
• Large flow rates possible while suppressing pressure drop

Supercritical helium density is:

$$\rho_{\text{SHe}}(4.5 \text{ K}, 0.6 \text{ MPa}) \approx 130 \text{ kg/m}^3$$

Specific heat capacity is:

$$c_p \approx 3-5 \text{ kJ/(kg·K)}$$

Superfluid helium (He II): Below 2.17 K, helium transitions to the superfluid state (He II). Superfluid helium has effectively infinite thermal conductivity, with almost no temperature gradient.

Superfluid helium cooling is used in some accelerator magnets (such as LHC), but is not typically used in fusion coils due to complexity and cost considerations.

Isotopes: Natural helium consists of ⁴He (99.99986%) and ³He (0.00014%). ³He has an extremely low superfluid transition temperature of 2.6 mK and is used in special cryogenic experiments, but only ⁴He is used for fusion coils.

Refrigeration Cycle

Helium refrigerators use the Claude cycle or its variants.

Ideal Carnot efficiency:

$$\text{COP}_{\text{Carnot}} = \frac{T_c}{T_h - T_c}$$

For 4.5 K and 300 K, $\text{COP} = 0.015$ (meaning 67 W is needed for 1 W of cooling).

Actual efficiency: Actual refrigerators are about 20-30% of Carnot efficiency, so:

$$\eta_{\text{actual}} \approx 0.25 \times 0.015 \approx 0.004$$

That is, approximately 250-300 W of electrical power is needed to cool 1 W at 4.5 K.

Types of Heat Loads

1. Static heat loads:

• Radiation heat: Radiation from thermal shields
• Conduction heat: Heat penetration through support structures
• Heat penetration from piping and leads

Radiation heat is:

$$\dot{Q}_{\text{rad}} = \varepsilon \sigma A (T_h^4 - T_c^4)$$

This can be significantly reduced with multi-layer insulation (MLI).

2. Dynamic heat loads:

• AC losses: Eddy current losses and hysteresis losses from field variations
• Nuclear heating: Heating from neutron irradiation (specific to fusion reactors)
• Mechanical losses: Heating from friction and vibration

AC losses: AC losses during field variations are:

$$P_{\text{AC}} = P_{\text{hysteresis}} + P_{\text{coupling}} + P_{\text{eddy}}$$

Hysteresis loss is:

$$P_{\text{hys}} = \frac{2}{3\pi} \mu_0 J_c d_f \Delta B \cdot f$$

where $d_f$ is filament diameter, $\Delta B$ is field variation amplitude, and $f$ is frequency.

ITER Cryogenic System

Parameter	Value
4.5 K equivalent cooling capacity	75 kW
80 K shield cooling capacity	1.3 MW
Liquid helium storage	25,000 liters
Helium circulation flow rate	3.4 kg/s
Required power	Approximately 25 MW

Cooling Loops

1. Coil cooling loop: Circulates supercritical helium (4.5 K, 0.6 MPa).

Flow rate is determined from heat load and allowable temperature rise:

$$\dot{m} = \frac{\dot{Q}}{c_p \Delta T}$$

2. Structure cooling loop: Cools coil cases and support structures to 4.5-10 K.

3. Shield cooling loop: Cools thermal shields to 80 K, reducing radiation heat penetration to 4.5 K.

4. Current lead cooling: Current leads from room temperature to cryogenic temperature are a major heat load source. Using high-temperature superconducting current leads can significantly reduce heat penetration.

Mechanical Support Structure and Electromagnetic Forces

Superconducting coil systems require mechanical structures that can withstand enormous electromagnetic forces.

Evaluation of Electromagnetic Forces

Lorentz force: The body force from current-field interaction is:

$$\mathbf{f} = \mathbf{J} \times \mathbf{B}$$

The force acting on a conductor is:

$$\mathbf{F} = \int_V \mathbf{J} \times \mathbf{B} \, dV = I \int_L d\mathbf{l} \times \mathbf{B}$$

Magnetic pressure: The pressure at the field boundary surface is:

$$p_{\text{mag}} = \frac{B^2}{2\mu_0}$$

At 12 T, this corresponds to $p \approx 57$ MPa (570 atmospheres).

Structural Materials

Structural materials used at cryogenic temperatures have the following requirements:

• High strength and toughness (no low-temperature embrittlement)
• Low thermal shrinkage (small dimensional change during cooling)
• Good weldability
• Non-magnetic (avoid effects of stray fields)

Main materials:

Material	Yield Strength (4 K)	Application
316LN stainless steel	1200 MPa	Coil case
JK2LB (cryogenic steel)	1400 MPa	Conduit
Inconel 718	1600 MPa	High-stress parts
Titanium alloy	1500 MPa	Support structure

Thermal Contraction

Materials contract when cooled from 300 K to 4.5 K:

$$\Delta L = L \int_{T_2}^{T_1} \alpha(T) \, dT$$

Material	Thermal Contraction (300→4 K)
Stainless steel	0.30%
Nb₃Sn conductor	0.26%
Copper	0.33%
Epoxy resin	1.0%

Differential thermal contraction between different materials causes stress and delamination, requiring consideration during design.

TF Coil Support Structure

Inter-coil structure (ICS): Connects adjacent TF coils and supports out-of-plane forces.

Gravity support: Flexible support is needed that bears the TF coil’s own weight (approximately 300 tons/coil) while allowing thermal contraction.

$$F_g = Mg = 300 \times 10^3 \times 9.8 \approx 3 \text{ MN}$$

Preload: Compressive force is applied in advance to prevent gaps between coils.

Relative Motion with Vacuum Vessel

Superconducting coils are at 4 K, while the vacuum vessel is at 200°C (during baking) and 100°C (during operation), creating a temperature difference.

Differential thermal contraction from this temperature difference is:

$$\Delta = (T_{\text{VV}} - T_{\text{coil}}) \times \alpha_{\text{avg}} \times L$$

Flexible support structures absorb this relative motion.

Details of ITER Magnet System

ITER uses the world’s largest superconducting magnet system, and its technology forms the foundation for future fusion reactors.

Overall Configuration

Coil System	Main Function	Superconductor
TF Coils (18 units)	Toroidal field generation	Nb₃Sn
PF Coils (6 units)	Plasma shape and position control	NbTi
CS Coils (6 modules)	Plasma current drive	Nb₃Sn
Correction Coils (18 units)	Error field correction	NbTi

Stored Energy

Total stored energy of the magnet system is:

$$E = \frac{1}{2}LI^2$$

System	Stored Energy
TF Coils	41 GJ
CS Coils	6.4 GJ
PF Coils	4 GJ
Total	51 GJ

This is equivalent to approximately 12 tons of TNT, making safe management extremely important.

Manufacturing and Assembly

TF coil manufacturing process:

1. Nb₃Sn strand manufacturing (bronze method, internal tin method)
2. Cable stranding
3. Jacket insertion
4. Heat treatment (650°C, approximately 200 hours)
5. Winding onto radial plates
6. Insulation treatment (epoxy impregnation)
7. Assembly into coil case

International sharing:

• TF Coils: Japan (9 units), Europe (9 units)
• CS Coils: United States
• PF Coils: Europe, China, Russia

Operating Scenarios

Plasma startup:

1. TF coil excitation (approximately 30 minutes)
2. CS coil pre-magnetization
3. Initial field configuration formation by PF coils
4. Plasma ignition (CS current decrease)
5. Plasma current ramp-up

Steady-state operation:

• TF Coils: 68 kA steady
• CS Coils: Stepwise change for flux supply
• PF Coils: Plasma equilibrium maintenance

Shutdown:

• Plasma termination
• Gradual demagnetization (thermal stress reduction)

High-Temperature Superconducting Applications for Fusion

High-temperature superconducting (HTS) materials, particularly REBCO, have the potential to revolutionize fusion coils.

Advantages of HTS

1. High-field operation: REBCO maintains high critical current even at fields above 20 T.

$$J_c(\text{REBCO}, 20\text{ T}) > J_c(\text{Nb}_3\text{Sn}, 12\text{ T})$$

2. High-temperature operation: Operation at 20-30 K is possible, leading to simplification and improved reliability of refrigeration systems.

$$\text{COP}(20\text{ K}) \approx 10 \times \text{COP}(4\text{ K})$$

3. Temperature margin: The high critical temperature allows for large temperature margins.

Application to Compact Reactors

Higher fields can increase fusion power output per plasma volume.

Fusion power density depends on the product of $\beta$ and field:

$$P_{\text{fus}} \propto \langle p \rangle^2 \propto (\beta B^2)^2 = \beta^2 B^4$$

Doubling the field increases power density by 16 times at the same $\beta$, enabling significant device miniaturization.

SPARC Project

SPARC, under development by MIT and Commonwealth Fusion Systems, is the first fusion experimental device to fully adopt REBCO.

Parameter	ITER	SPARC
Major radius	6.2 m	1.85 m
Toroidal field	5.3 T	12.2 T
Plasma current	15 MA	8.7 MA
Fusion power	500 MW	140 MW (target)
Q value	10 (target)	> 2 (target)
Superconductor	Nb₃Sn/NbTi	REBCO

Technical Challenges

1. Conductor development: Technology is needed to assemble REBCO tapes into high-current conductors.

• CORC (Conductor on Round Core)
• STAR (Symmetric Tape Round)
• Stacked tape conductors

2. Joint technology: Long coils require many joints, with each joint resistance required to be below 1 nΩ.

$$P_{\text{joint}} = I^2 R_{\text{joint}}$$

3. Radiation resistance: Neutron irradiation degradation of REBCO is not fully understood. Demonstration of long-term stability at fluences above $10^{18}$ n/cm² is needed.

4. Cost reduction: Current REBCO costs 50-100 times more than Nb₃Sn. Cost reduction to about 1/10 is expected over the next 10-20 years through mass production effects.

Future Outlook

With HTS technology maturation, the following are expected:

• Realization of compact fusion reactors (major radius 2-3 m)
• Significant construction cost reduction
• Faster commercialization

Meanwhile, Nb₃Sn technology is mature and, once demonstrated at ITER, could be applied to early commercial reactors. The two technologies are not competing but are expected to be used according to application requirements.

Superconducting Joint Technology

Many conductor joints are essential for manufacturing large superconducting coils. Joints are sources of electrical resistance and cause heating and energy loss, requiring extremely low joint resistance.

Joint Resistance Requirements

The resistance allowable at superconducting coil joints is determined by the balance between heat generation and cooling capacity. Heat generation at joints is:

$$P_{\text{joint}} = I^2 R_{\text{joint}}$$

For example, with ITER TF coils carrying 68 kA, if joint resistance is 1 nΩ:

$$P = (68 \times 10^3)^2 \times 10^{-9} = 4.6 \text{ W}$$

Since approximately 300 W of power is needed to cool 1 W at 4.5 K, this heat generation cannot be ignored as a load on the power system. ITER conductors require joint resistance of 1-2 nΩ or less.

LTS Conductor Joint Technology

Lap joint (wrap joint): For NbTi conductors, cables are overlapped for a certain length, and current is transferred through copper stabilizers.

Joint resistance is determined by joint length $L_j$ and contact area $A_c$:

$$R_{\text{joint}} = \frac{\rho_{\text{Cu}}}{A_c} L_j + R_{\text{contact}}$$

The residual resistivity of copper at low temperatures depends on purity, with high-purity copper (RRR > 100) having $\rho \approx 2 \times 10^{-10}$ Ωm.

Solder joint: Conductors are joined by crimping or molten solder. The joint does not need to be maintained in a superconducting state; current transfer occurs through the copper matrix.

Solder materials include Sn-Pb, Sn-Ag, In-Ag, etc., with low-temperature creep properties being important.

Nb₃Sn Conductor Joint Challenges

Nb₃Sn is a brittle material, making mechanical handling after reaction heat treatment difficult. Therefore, two joining methods are adopted:

Wind-and-React method: Heat treatment is performed after winding. Since joints are also heat treated, joining itself is relatively easy, but the overall manufacturing process becomes complex.

React-and-Wind method: Winding is performed after heat treatment. Joints require separate processing, with the following methods used:

• Additional heat treatment of joints only
• Lap joint with copper sleeve
• Mechanical crimp joint

HTS Conductor Joint Technology

REBCO tape joining is a key technology challenge for fusion coil realization.

Resistive joints: Method of joining REBCO tapes through intermediate materials of copper or silver. It has a track record and high reliability, but finite joint resistance remains.

Typical resistance values:

• Solder joint: 10-100 nΩ·cm²
• Pressure joint: 1-10 nΩ·cm²

Superconducting joints: “Zero-resistance joints” where superconducting current flows at the joint are essential for realizing persistent current mode.

REBCO superconducting joint methods:

1. Melt diffusion bonding: Partial melting and recrystallization of REBCO layer at high temperature
2. Epitaxial bonding: Additional growth of REBCO thin film at the joint
3. REBCO adhesive method: Joining using REBCO nanoparticle paste

The achievement criterion for superconducting joints is maintaining critical current density at 50% or more of the parent material:

$$J_c^{\text{joint}} > 0.5 \times J_c^{\text{tape}}$$

Joint length and current capacity: For finite-resistance joints, effective current capacity is ensured by increasing joint length. Current transfer length $\lambda_T$ is:

$$\lambda_T = \sqrt{\frac{\rho_t \cdot t}{R_{ct}}}$$

where $\rho_t$ is tape transverse resistivity, $t$ is tape thickness, and $R_{ct}$ is contact resistance.

Joint length of 3-5 times the current transfer length is recommended:

$$L_{\text{joint}} \geq 5 \lambda_T$$

No-Insulation (NI) Coil Technology

As a new approach specific to HTS, there are NI coils that omit inter-conductor insulation.

Principle: Normally, coil conductors are insulated, but NI coils allow direct contact between conductors. In steady state, current flows through the superconducting layer, and during quench, bypass current flows through inter-conductor contact resistance.

Advantages:

• Self-protection function during quench (current bypasses)
• Simplification of manufacturing
• Improved thermal conduction

Disadvantages:

• Complex current distribution during transients
• Long charge/discharge time constants
• Not suitable for AC applications

NI coil time constant is:

$$\tau = \frac{L}{R_{\text{turn-to-turn}}} = \frac{L \cdot N^2}{R_{ct} \cdot A_{\text{overlap}}}$$

where $N$ is number of turns, $R_{ct}$ is contact resistance per unit area, and $A_{\text{overlap}}$ is contact area.

Since transient response is important in fusion reactors, careful consideration is needed for NI technology application, but high self-protection is attractive. Compromise approaches such as Metal-Insulation (MI) and Partial-Insulation (PI) are also being researched.

HTS Conductor Configurations

Multiple approaches have been developed to assemble REBCO tapes into high-current conductors.

CORC (Conductor on Round Core)

Developed by Advanced Conductor Technologies, REBCO tapes are spirally wound around a round core.

Structure:

• Central copper or steel core
• Multiple layers of REBCO tape wound helically in alternating directions
• Outer copper sheath

Characteristics:

• Round cross-section compatible with conventional CICC design
• Strong against bending radius (core curvature limits tape strain)
• 100 kA-class high-current conductors possible

Current capacity is:

$$I_c = N_{\text{tapes}} \times I_{c,\text{tape}} \times \eta$$

where $\eta \approx 0.8-0.9$ is a reduction factor due to winding angle, etc.

STAR (Symmetric Tape Round)

Developed by Bruker, REBCO tapes are arranged symmetrically.

Structure:

• Central spiral-grooved former
• Tapes placed along grooves
• Mechanically stable structure

Advantages:

• Self-balancing against electromagnetic forces
• High current density
• Good cooling characteristics

Roebel Cable

A flat conductor made by punching and weaving REBCO tapes.

Manufacturing process:

1. Meander-shaped cuts in REBCO tape
2. Weaving multiple tapes
3. Crimping and soldering

Characteristics:

• Low AC losses (current averaging)
• Flat form for easy winding
• High manufacturing cost

AC losses depend on twist pitch $p$:

$$P_{\text{AC}} \propto \frac{1}{p^2}$$

Twisted-Stack Cable

A conductor made by stacking multiple REBCO tapes and adding twist. An approach applying ITER conductor design philosophy to HTS.

Structure:

• Stack of several to tens of tapes
• Entire stack twisted
• Covered with copper sheath

Challenges:

• Non-uniform current sharing between tapes
• Joint complexity
• Manufacturing technology maturity

Comparison of Conductor Configurations

Type	Cross-section	Current Density	Manufacturability	Maturity
CORC	Round	High	Good	Medium
STAR	Round	High	Good	Low
Roebel	Flat	Medium	Difficult	Medium
Twisted-Stack	Flat/Round	Medium	Good	Low

The SPARC project has adopted a design based on CORC conductors.

Radiation Environment and Neutron Irradiation Effects

Fusion reactor coils are exposed to high-energy neutrons from the plasma. Material damage from 14.1 MeV D-T fusion neutrons is an important constraint in superconducting coil design.

Neutron Fluence and Shielding

The fluence of neutrons emitted from fusion plasma that reach the coils after penetrating shields depends on blanket/shield thickness and composition.

Neutron fluence attenuation versus shield thickness $d$:

$$\Phi(d) = \Phi_0 \exp\left(-\frac{d}{\lambda}\right)$$

where $\lambda$ is mean free path (approximately 10 cm for lithium hydride, approximately 15 cm for steel).

In ITER’s design, neutron fluence at the coil surface is kept below:

$$\Phi_{\text{total}} < 10^{22} \text{ n/m}^2$$

throughout the operating lifetime.

Irradiation Effects on LTS Materials

NbTi: NbTi shows relatively good radiation resistance. At fluences around $10^{22}$ n/m², critical current density reduction is about 10-20%.

$J_c$ change due to irradiation:

$$\frac{\Delta J_c}{J_c} = -k \cdot \Phi^{0.5}$$

where $k$ is a material constant.

Nb₃Sn: Nb₃Sn is more sensitive to irradiation. In particular, combined effects with strain can accelerate degradation.

Critical temperature decrease:

$$\Delta T_c = -\alpha \cdot (dpa)$$

where $dpa$ (displacement per atom) is the number of atomic displacements per atom, with $10^{22}$ n/m² corresponding to about $10^{-3}$ dpa. $\alpha \approx 3-5$ K/dpa.

Irradiation Effects on HTS Materials

Radiation resistance of REBCO and Bi-based HTS is less studied compared to LTS.

REBCO: Preliminary studies show no significant $J_c$ degradation up to about $10^{18}$ n/cm². However, long-term stability at higher fluences is undemonstrated.

In copper oxide superconductors, irradiation may generate oxygen vacancies and change carrier density:

$$\delta_{O} \rightarrow \delta_{O} + \Delta\delta$$

This affects both $T_c$ and $J_c$.

Challenges:

• Property evaluation at fluences above $10^{22}$ n/m²
• Irradiation testing at operating temperature (many tests at room temperature)
• Annealing effects of irradiation damage

Radiation Degradation of Organic Insulation Materials

Organic insulation materials such as epoxy resins are vulnerable to radiation.

Mechanical strength decrease versus absorbed dose $D$ (Gy):

$$\sigma(D) = \sigma_0 \exp\left(-\frac{D}{D_0}\right)$$

General epoxy resins have $D_0$ around $10^7$ Gy and degrade rapidly above this.

ITER uses radiation-resistant TGPAP/DDS epoxy resin systems.

Nuclear Heating

Neutron absorption causes direct heating inside coils.

Nuclear heating density is:

$$\dot{Q}_n = \Phi \cdot \sigma_a \cdot E_a \cdot N$$

where $\sigma_a$ is absorption cross-section, $E_a$ is energy per absorption reaction, and $N$ is atomic number density.

Nuclear heating in ITER TF coils is estimated at approximately 14 kW (total), becoming a heat load on the 4.5 K cooling system.

Helium Generation and Embrittlement

Neutron irradiation generates helium through (n,α) reactions. Boron (¹⁰B) has a particularly large (n,α) cross-section:

$$^{10}\text{B} + n \rightarrow ^{7}\text{Li} + ^{4}\text{He}$$

Generated helium accumulates at grain boundaries and can cause high-temperature embrittlement. This effect potentially manifesting during Nb₃Sn heat treatment is a concern.

Impact on Shield Design

Radiation resistance is an important constraint in shield optimization.

Thicker shielding:

• Reduces neutron fluence (extends coil lifetime)
• Decreases toroidal field utilization efficiency (increases major radius)
• Increases construction cost

$$R_0 = R_{\text{plasma}} + a + \Delta_{\text{blanket}} + \Delta_{\text{shield}} + \Delta_{\text{gap}} + \Delta_{\text{coil}}$$

High-field coils (HTS) can achieve the same field with smaller coils, enabling operation with thinner shielding and contributing to overall reactor compactness.

Future Challenges and Outlook

Superconducting coil technology is steadily progressing toward fusion power realization, but further technology development is needed for commercial reactor deployment.

Materials and Conductor Challenges

HTS cost reduction: REBCO tapes currently cost 50-100 times more than Nb₃Sn, becoming a barrier to large-scale fusion reactor application.

Cost reduction directions:

• Manufacturing process efficiency (continuous deposition, yield improvement)
• Tape width expansion (increased current capacity per piece)
• Substrate/buffer layer simplification
• Mass production effects

Reduction to 1/5-1/10 of current costs is expected by the 2030s.

Length extension: The length per REBCO tape is currently limited to several hundred meters, leading to increased joint numbers.

$$N_{\text{joints}} = \frac{L_{\text{conductor}}}{L_{\text{tape}}}$$

Development of continuous long-length tapes exceeding 1 km is underway.

Critical current density improvement: Further $J_c$ improvement is possible through optimization of pinning centers.

Introduction of artificial pinning centers (BaZrO₃, BaHfO₃, etc.):

$$J_c^{\text{enhanced}} = J_c^{\text{intrinsic}} + \Delta J_c^{\text{pinning}}$$

Research is ongoing to more than double $J_c$ at high fields (above 20 T).

Design and Manufacturing Challenges

HTS quench protection: Local overheating due to slow propagation velocity is one of the biggest challenges in HTS coil design.

Required technology development:

• Fast, high-precision quench detection systems
• Active thermal propagation technology (heater-induced)
• Self-protecting conductor design (NI/PI)
• Predictive maintenance technology

Large coil manufacturing: Manufacturing coils beyond ITER scale requires further technological innovation.

• Large vacuum furnaces (for heat treatment)
• Precision winding technology
• Non-destructive inspection technology
• Quality control systems

Modularization and sectorization: Future commercial reactors require coil designs that consider maintainability.

• Divisible joint technology
• Sector-by-sector replacement
• Remote joining/disconnection technology

Operation and Maintenance Challenges

Long-term reliability: Commercial reactors require 30-40 year operating lifetimes.

Degradation factors:

• Radiation damage accumulation
• Thermal cycle fatigue
• Repeated stress from electromagnetic forces
• Aging of insulation materials

Development of accelerated testing and lifetime prediction models is needed.

Remote maintenance: Direct human maintenance is difficult in activated environments.

• Robot inspection and repair
• Optimization of coil replacement procedures
• Design for Maintenance (maintainability consideration at design stage)

Deployment to DEMO Reactor

The DEMO (demonstration reactor) planned after ITER is expected to have the following superconducting coil requirements:

Parameter	ITER	DEMO (Expected)
Toroidal field	5.3 T	5-6 T
Major radius	6.2 m	8-9 m
Neutron fluence	$10^{22}$ n/m²	$10^{23}$ n/m²
Pulse duration	400 seconds	Steady-state (several hours)
Operating lifetime	20 years	30-40 years

DEMO will have one order of magnitude higher neutron fluence, making shield design optimization and material radiation resistance demonstration essential.

Compact Reactor Approach

Compact reactors utilizing HTS (SPARC, ARC, etc.) aim for commercialization through a different approach.

Advantages:

• Significant major radius reduction through high fields ($R \propto 1/B^{0.5}$)
• Construction cost reduction
• Shortened development period

Challenges:

• HTS coil technology maturity
• Response to high thermal and neutron loads
• Difficulty of coil maintenance and replacement

Technology Roadmap

The development roadmap for superconducting coil technology is as follows:

2020s:

• ITER magnet manufacturing completion and testing
• SPARC construction and demonstration
• HTS conductor performance improvement

2030s:

• Full ITER operation and data accumulation
• DEMO coil design finalization
• HTS cost reduction

2040s:

• DEMO construction
• Commercial reactor design
• Standardization and mass production technology establishment

Superconducting coils are the heart of fusion reactors, and their technological advancement holds the key to fusion energy realization. Through large-scale demonstration at ITER, optimal designs leveraging the characteristics of both LTS and HTS will advance, and the technological foundation for commercialization in the 2050s is expected to be established.

Related Topics

• Tokamak Confinement - Magnetic field configuration and components of tokamak
• ITER Project - The world’s largest tokamak experimental reactor
• SPARC - Compact tokamak using high-temperature superconductors
• JT-60SA - Japan’s superconducting tokamak
• Charged Particle Motion - Particle motion in magnetic fields
• MHD - Fundamentals of magnetohydrodynamics