Superconducting TF coil

Superconducting coil geometry

The TF coils are assumed to be supporting each other against the net centering force. This can be described as a vaulted or wedged design. Each coil, illustrated in Figure 1, can be separated in two main sections:

• The winding pack (WP) : section containing the superconducting cables (Blue area in Figure 1). The ground insulation and the insertion gap (the clearance required to allow the winding pack to be inserted, shown as the dark grey area in Figure 1) is considered part of the WP by convention.
• The steel casing: Section holding the WP providing the necessary structural support (light grey area in Figure 1).

The next sub-section describes the different parametrization proposed in PROCESS:

Figure 1: Inboard mid-plane cross-section of the superconducting TF coils considered in PROCESS with associated parametrization. The green variables are calculated by the machine_build module subroutine while the blue ones are specific to the sctfcoil module. The light grey area corresponds to the steel casing containing the Winding pack and providing structural support. The dark grey area corresponds to the winding pack insertion gap, and its ground insulation. Finally the light blue area corresponds to the winding pack containing the conductor cables. More details on the parametrization is discussed in this section.

TF coil inboard radial size

Following the geometry and its parametrization presented in Figure 1, the TF total thickness dr_tf_inboard \( \left( \Delta R_\mathrm{TF} \right) \) is related with the inner and outer case radial thicknesses (dr_tf_nose_case, \( \Delta R_\mathrm{case}^\mathrm{in} \) and dr_tf_plasma_case, \( \Delta R_\mathrm{case}^\mathrm{out} \) respectively) and the WP radial thickness dr_tf_wp_with_insulation \(\Delta R_\mathrm{WP}\) by the following equation :

$$\Delta R_\mathrm{TF} = R_\mathrm{TF}^\mathrm{in} + \Delta R_\mathrm{WP} + \Delta R_\mathrm{case}^\mathrm{out} + \Delta R_\mathrm{case}^\mathrm{in} - R_\mathrm{TF}^\mathrm{in}$$

with \( R_\mathrm{TF}^\mathrm{in} \) (r_tf_inboard_in) the radius of the innermost TF edge, set by the central solenoid coil size and \( N_\mathrm{TF} \) the number of TF coils. Reverted, to provide the WP thickness, the same equation simply becomes:

$$\Delta R_\mathrm{WP} = \left( R_\mathrm{TF}^\mathrm{in} + \Delta R_\mathrm{TF} \right) - R_\mathrm{TF}^\mathrm{in} - \Delta R_\mathrm{case}^\mathrm{out} - \Delta R_\mathrm{case}^\mathrm{in}$$

The TF coil radial thickness (dr_tf_inboard) can parametrized in two ways in PROCESS:

• Direct parametrization: the TF radial inboard thickness width is set as an input variable : dr_tf_inboard (iteration variable 13). The WP radial thickness (dr_tf_wp_with_insulation) is calculated from dr_tf_inboard and the two case radial thicknesses. This parametrization is used by default.

• WP thickness parametrization: the TF inboard radial thickness is calculated from the the case and the WP radial thickness. This option is selected by using the WP thickness (dr_tf_wp_with_insulation, iteration variable 140) as an iteration variable. Doing so, any dr_tf_inboard values will be overwritten and for this reason dr_tf_wp_with_insulation and dr_tf_inboard cannot be used as iteration variables simultaneously. Although not set by default for backward compatibility, this parametrization provides a more stable optimization procedure (negative WP area layer cannot be obtained by construction) and is hence encouraged.

Case geometry

Although not physically divided into pieces, three sections of the case can be considered:

• The nose casing: this section corresponds to the case separating the WP with the machine center. Due to the presence of net electromechanical centering forces, this case has a major structural purpose and is often much larger than the other sides. The nose case dimension is set by its radial thickness that the user can specify using the dr_tf_nose_case input variable (iteration variable 57).

• Sidewall casing: this section corresponds to the lateral side of the case, separating the WP with the other vaulted coils. As in the WP geometry is generally squared, the sidewall case thickness may vary with the machine radius. For this reason, the user sets its dimensions though its minimal thickness dx_tf_side_case_min. The user can either directly specify dx_tf_side_case_min or define it as a fraction of the total coil thickness at the inner radius of the WP (r_tf_wp_inboard_inner) with the casths_fraction input. If casths_fraction is set in the input file, the dx_tf_side_case_min value will be overwritten.

• Plasma side casing: this section corresponds to the case section separating the WP with the plasma. As the geometry of this section is rounded, its thickness is set by its minimal value dr_tf_plasma_case (user input). This parameter can also be defined as a fraction of the total TF coil thickness dr_tf_inboard using f_dr_tf_plasma_case. If the f_dr_tf_plasma_case parametrization is used, the dr_tf_plasma_case value will be overwritten.

Winding pack geometry

Several Winding pack geometries can chosen with the i_tf_wp_geom integer switch as shown in Figure 3:

• i_tf_wp_geom = 0 : Rectangular winding pack. It is the only geometry compatible with the integer turn parametrization (i_tf_turns_integer = 1).

• i_tf_wp_geom = 1 : Double rectangle winding pack. The two rectangles are have the same radial thickness and their width in the toroidal direction is defined with the minimal sidewall thickness at their innermost radius.

• i_tf_wp_geom = 2 : Trapezoidal WP. The WP area is defined with a trapezoid, keeping the sidewall case thickness constant. This is however probably not a realistic shape as the turns are generally rectangular. This option has been added mostly to allow comparison with simplified FEA analysis configurations.

Figure 3: Visual illustration of the WP shapes the user can select with the i_tf_wp_geom integer switch. The dx_tf_wp_primary_toroidal and dx_tf_wp_secondary_toroidal parameters, added in option i_tf_wp_geom = 1 are calculated using the minimal sidewall case thickness.

Turns geometry

Figure 4 illustrates the winding pack internal structure and the individual turns structure.

Figure 4: Illustration of the winding pack internal structure. The top right diagram shows the inboard mid-plane cross section of a TF coil with the steel case in light grey and the winding pack in light blue. The dotted lines illustrate the rectangular conductor. The bottom left zoom-in section shows the conductor structure used in the PROCESS module. The red section represent the individual turn electrical insulation, the turquoise the steel jacket/conduit providing structural support, the grey is the area allocated to the superconductor material, copper stabiliser and inter-strand coolant, and the white circle the helium cooling channel. (The superconductor, copper stabiliser and coolant together make up the 'cable'.)

The winding pack is assumed to be made of $N_\mathrm{turn}$ (n_tf_coil_turns) turns. The number of turns can be parametrized in three different ways :

• Current per turn parametrization (default): i_tf_turns_integer = 0 the user sets the value of the current flowing in each turns c_tf_turn. The number of turns necessary to carry the total TF coil current is then deduced from c_tf_turn. There is no guarantee that a realistic turn configuration (with all the turn geometrically fitting in the allocated space) or even have an integer number of turn is used with this parametrization. If the turn thickness dx_tf_turn_general or the cable space thickness dx_tf_turn_cable_space_general is defined by the user, this parametrization is not selected.

• Turn size parametrization: the dimension of the turn dx_tf_turn_general can be set by the user. To do so, the user just have to select the following option: i_tf_turns_integer = 0 and to set a value to the variable dx_tf_turn_general. The area of the corresponding squared turn and the number of turns necessary to fill the WP area is deduced. There is no guarantee that a realistic turn configuration (with all the turn geometrically fitting in the allocated space) or even have an integer number of turns is used with this parametrization. The current per turn c_tf_turn will be overwitten.

• Cable size parametrization: the dimension of the SC cable dx_tf_turn_cable_space_general can be set by the user. To do so, the user just have to select the following option: i_tf_turns_integer = 0 and to set a value to the variable dx_tf_turn_cable_space_general. The area of the corresponding squared turn is deduced adding the steel conduit structure and the turn insulation. The number of turns necessary to fill the WP area is then deduced. There is no guarantee that a realistic turn configuration (with all the turn geometrically fitting in the allocated space) or even have an integer number of turns is used with this parametrization. The current per turn c_tf_turn will be overwitten.

• Integer turn parametrization: i_tf_turns_integer = 1 the user sets the number of layers in the radial direction (n_tf_wp_layers) and the number of turns in the toroidal direction (n_tf_wp_pancakes). The number of turns is integer. The turn cross-section is not necessarily square, giving different averaged structural properties in the radial and toroidal directions. Only a rectangular WP can be used for this parametrization.

The turn internal structure, illustrated in Figure 4, is inspired from the cable-in-conduit-conductor (CICC) design, with the main different being that a rounded squared cable space is used (grey area in Figure 4 ). The rounding curve radius is take as 0.75 of the steel conduit thickness. The turn geometry is set with with the following thicknesses:

• Turn insulation thickness dx_tf_turn_insulation: user input setting the thickness of the inter-turn insulation.

• Steel jacket/conduit thickness dx_tf_turn_steel (iteration variable 58): user input thickness of the turn steel structures. As it is a crucial variable for the TF coil structural properties it is also an iteration variable.

• Helium cooling channel diameter dia_tf_turn_coolant_channel: user input defining the size of the cooling channel.

Cable composition

As the conductor cable composition is only used to correct the area used to compute current density flowing in the superconductor material, to be compared with its critical current density, an average material description is enough for the PROCESSmodels. The composition is set with the following material fractions:

• Cable void fraction (f_a_tf_turn_cable_space_extra_void): user input setting the void fraction between the strands. This fraction does not include the helium cooling pipe at the cable center.

• Copper fraction (f_a_tf_turn_cable_copper): user input setting the copper fraction. This fraction is applied after the void and helium cooling channels areas has been removed from the conductor area. Does not include any copper from REBCO tape if used.

Critical current density for the superconductor

The minimum conductor cross-section is derived from the critical current density for the superconductor in the operating magnetic field and temperature, and is enforced using constraint 33.

Switch i_tf_sc_mat specifies which superconducting material is to be used:

• i_tf_sc_mat == 1 -- Nb$_3$Sn superconductor, ITER critical surface parameterization[^5], standard critical values
• i_tf_sc_mat == 2 -- Bi-2212 high temperature superconductor
• i_tf_sc_mat == 3 -- NbTi superconductor
• i_tf_sc_mat == 4 -- Nb$_3$Sn superconductor, ITER critical surface parameterization[^5], user-defined critical parameters
• i_tf_sc_mat == 5 -- WST Nb$_3$Sn parameterization
• i_tf_sc_mat == 6 -- REBCO HTS tape in CroCo strand
• i_tf_sc_mat == 7 -- Durham Ginzburg-Landau critical surface model for Nb-Ti
• i_tf_sc_mat == 8 -- Durham Ginzburg-Landau critical surface model for REBCO
• i_tf_sc_mat == 9 -- Hazelton experimental data combined with Zhai conceptual model for REBCO

The fraction of copper present in the superconducting filaments is given by f_a_tf_turn_cable_copper (iteration variable number 59). For cases where REBCO tape is used this copper fraction does not include the copper within the tape.

For i_tf_sc_mat = 2, a technology adjustment factor fhts may be used to modify the critical current density fit for the Bi-2212 superconductor, to describe the level of technology assumed (i.e. to account for stress, fatigue, radiation, AC losses, joints or manufacturing variations). The default value for fhts is 0.5 (a value of 1.0 would be very optimistic).

For i_tf_sc_mat = 4, important superconductor properties may be input as follows: - Upper critical field at zero temperature and strain: bcritsc, - Critical temperature at zero field and strain: tcritsc.

The toroidal field falls off at a rate $1/R$, with the peak value occurring at the outer edge of the inboard portion of the TF coil winding pack (radius r_b_tf_inboard_peak).

Three constraints are relevant to the operating current density $J_{\mbox{op}}$ in the TF coils.

• Criticial current (constraint 33): $J_{\mbox{op}}$ must not exceed the critical value $J_{\mbox{crit}}$. Iteration variable 50 must be active (fiooic). The current density margin can be set using the upper bound of fiooic:

$$J_{\mbox{op}} < \texttt{fiooic} \cdot J_{\mbox{crit}}$$

• Temperature margin (constraint 36) -- The critical current density $J_{\mbox{crit}}$ falls with the temperature of the superconductor. The temperature margin $\Delta T$ is the difference between the current sharing temperature (at which $J_{\mbox{crit}}$ would be equal to $J_{\mbox{op}}$) and the operating temperature. The minimum allowed $\Delta T$ can be set using tmargmin together with constraint equation 36 and iteration variable 54 (ftmargtf). Note that if the temperature margin is positive, $J_{\mbox{op}}$ is guaranteed to be lower than \jcrit, and so constraints 33 and 36 need not both be turned on. It is recommended that only one of these two constraints is activated.

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Quench protection

Superconducting quenches in the TF coil are modelled using a simple 0-D adiabatic heat balance in which the heat rise in the copper during a quench is equal to the heat rise required to increase the material temperature by $dT$. This is known as a ``hotspot criterion'' model, as the maximum temperature during a quench is the termination criterion (the point at which the quench is considered irreparably damaging).

The copper is assumed to carry the full current during a quench, and the materials are assumed to be in thermal equilibrium. We have:

$$P(t)dt = \sum_i c_{P,i}\rho_{i} V_{i} dT$$

Where $P(t)$ is the power deposited in the copper through Joule heating at time t, and the sum on the RHS is over the conductor consitutents (copper, helium, superconductor), where $V_{i}$ is the volume of said constituent. This can be rewritten as:

$$\int J(t)^2 dt = \int_{T_{0}}^{T_{max}} \sum_i f_{i}\dfrac{c_{P,i}\rho_{i}}{\nu_{Cu}} dT$$

The LHS is solved analytically, assuming that the current is at the operational value $J_{op}$ for some quench detection time $t_{detection}$ and then decays exponentially with a time constant $\tau_{discharge}$. The RHS is solved numerically by integration of the temperature-dependent material properties over $dT$.

The resistivity of copper, $\nu_{Cu}$, is an extremely important parameter in this model. The residual-resistance-ratio (RRR) of the copper can be specified, and magneto-resistive and irradiation-induced increases in resistivity are accounted for. The magnetic field at which the copper resisitivity is calculated is kept constant at $B_{TF,peak}$. This is conservative, and to simplify the implementation (keeping the separation of the $t$ and $T$ integrations).

Formally this gives:

$$\begin{align} J_{TF,\mathrm{quench}}^{2} = \frac{1}{\tau_{TF,\mathrm{discharge}}/2 + t_\mathrm{detection}} f_{Cu} \bigg( & f_{He} \int_{T_0}^{T_\mathrm{max}} \frac{\rho_{He} C_{p,He}}{\nu_{Cu}}\, dT \nonumber \\ & + f_{Cu} \int_{T_0}^{T_\mathrm{max}} \frac{\rho_{Cu} C_{p,Cu}}{\nu_{Cu}}\, dT \nonumber \\ & + f_{sc} \int_{T_0}^{T_\mathrm{max}} \frac{\rho_{sc} C_{p,sc}}{\nu_{Cu}}\, dT \bigg) \end{align}$$

• Constraint 35 -- To ensure that $J_{\mbox{op}}$ does not exceed the quench protection current density limit, $J_{TF,\mathrm{quench}}$, constraint equation no. 35 should be turned on with iteration variable 53 ( fjprot).

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Supercoducting TF coil class | SuperconductingTFCoil(TFCoil)

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Winding Pack Geometry | superconducting_tf_wp_geometry()

Depending on the value of i_tf_wp_geom different WP geometries will be configured.

Initial general dimensions are calculated first as follows:

$$\overbrace{R_{\text{TF-inboard,WP-inner}}}^{\texttt{r_tf_wp_inboard_inner}} = \overbrace{R_{\text{TF-inboard,in}}}^{\texttt{r_tf_inboard_in}} + \overbrace{\mathrm{d}R_{\text{TF,nose-case}}}^{\texttt{dr_tf_nose_case}}$$

$$\overbrace{R_{\text{TF-inboard,WP-outer}}}^{\texttt{r_tf_wp_inboard_outer}} = R_{\text{TF-inboard,WP-inner}} + \overbrace{\mathrm{d}R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}$$

$$\overbrace{R_{\text{TF-inboard,WP-centre}}}^{\texttt{r_tf_wp_inboard_centre}} = \frac{R_{\text{TF-inboard,WP-inner}} + R_{\text{TF-inboard,WP-outer}}}{2}$$

Find the straight toroidal width of the TF coil at the inside edge of the winding pack:

$$\mathrm{d}x_{\text{TF,toroidal,WP}} = 2 \times \overbrace{R_{\text{TF,WP-inner}}}^{\texttt{r_tf_wp_inboard_inner}} \times \overbrace{\tan{\left(\phi_{\text{TF,half}}\right)}}^{\texttt{tan_theta_coil}}$$

To find the straight toroidal length of the winding pack we now take off the side case thicknesses:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-min}}}^{\texttt{dx_tf_wp_toroidal_min}} = \mathrm{d}x_{\text{TF,toroidal,WP}} - \left(2 \times \overbrace{\mathrm{d}x_{\text{TF,side case}}}^{\texttt{dx_tf_side_case_min}}\right)$$

We also set a commonly used parameter for the radial thickness of the winding pack without the insulation and insertion gap.

$$\overbrace{\mathrm{d}R_{\text{TF-WP,no-insulation}}}^{\texttt{dr_tf_wp_no_insulation}} = \overbrace{\mathrm{d}R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}} - \\ 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)$$

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Rectangular WP

For a rectangular winding pack (i_tf_wp_geom == 0) of constant shape:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-primary}}}^{\texttt{dx_tf_wp_primary_toroidal}}, \overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-average}}}^{\texttt{dx_tf_wp_toroidal_average}} = \mathrm{d}x_{\text{TF,toroidal,WP-min}}$$

The full winding pack area with insulation is:

$$\overbrace{A_{\text{TF,WP}}}^{\texttt{a_tf_wp_with_insulation}} = \mathrm{d}x_{\text{TF,toroidal,WP-primary}} \times \overbrace{\mathrm{d}R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}$$

The area of the winding pack with no insulation or gap is:

$$\overbrace{A_{\text{TF,WP-no-insulation}}}^{\texttt{a_tf_wp_no_insulation}} = \\ \left(\mathrm{d}R_{\text{TF,WP}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ \times \left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right)$$

The area of the surrounding winding pack insulation is:

$$\overbrace{A_{\text{TF,WP-insulation}}}^{\texttt{a_tf_wp_ground_insulation}} = \\ \left(\left(\mathrm{d}R_{\text{TF,WP}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right) \\ \times \left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ - A_{\text{TF,WP-no-insulation}}$$

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Double rectangular WP

For a double rectangular winding pack (i_tf_wp_geom == 1):

The straight toroidal width of the primary winding pack is:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-primary}}}^{\texttt{dx_tf_wp_primary_toroidal}} = 2 \times \\ \left(\overbrace{R_{\text{TF,WP-centre}}}^{\texttt{r_tf_wp_inboard_centre}} \times \overbrace{\tan{\left(\phi_{\text{TF,half}}\right)}}^{\texttt{tan_theta_coil}} - \overbrace{\mathrm{d}x_{\text{TF,side case}}}^{\texttt{dx_tf_side_case_min}}\right)$$

The straight toroidal width of the secondary winding pack is:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-secondary}}}^{\texttt{dx_tf_wp_secondary_toroidal}} = 2 \times \\ \left(\overbrace{R_{\text{TF,WP-inner}}}^{\texttt{r_tf_wp_inboard_inner}} \times \tan{\left(\phi_{\text{TF,half}}\right)} - \mathrm{d}x_{\text{TF,side case}}\right)$$

The average toroidal straight width is calculated:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-average}}}^{\texttt{dx_tf_wp_toroidal_average}} = \frac{\left(\mathrm{d}x_{\text{TF,toroidal,WP-secondary}} + \mathrm{d}x_{\text{TF,toroidal,WP-primary}}\right)}{2}$$

The total winding pack area is calculated from the average:

$$\overbrace{A_{\text{TF,WP}}}^{\texttt{a_tf_wp_with_insulation}} = \mathrm{d}x_{\text{TF,toroidal,WP-average}} \times \overbrace{\mathrm{d}R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}$$

The area of the winding pack with no insulation or gap is:

$$\overbrace{A_{\text{TF,WP-no-insulation}}}^{\texttt{a_tf_wp_no_insulation}} = 0.5 \times \\ \left(\mathrm{d}R_{\text{TF,WP}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ \times \left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} + \mathrm{d}x_{\text{TF,toroidal,WP-secondary}} - \\ 4\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right)$$

The area of the surrounding winding pack insulation is:

$$\overbrace{A_{\text{TF,WP-insulation}}}^{\texttt{a_tf_wp_ground_insulation}} = 0.5 \times \\ \left(\left(\mathrm{d}R_{\text{TF,WP}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right) \\ \times \left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} + \mathrm{d}x_{\text{TF,toroidal,WP-secondary}} - 4\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ - A_{\text{TF,WP-no-insulation}}$$

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Trapezoidal WP

For a trapezoidal winding pack (i_tf_wp_geom == 2):

The straight toroidal width of the primary winding pack is (longest side of trapezoid):

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-primary}}}^{\texttt{dx_tf_wp_primary_toroidal}} = 2 \times \\ \left(\overbrace{R_{\text{TF,WP-outer}}}^{\texttt{r_tf_wp_inboard_outer}} \times \overbrace{\tan{\left(\phi_{\text{TF,half}}\right)}}^{\texttt{tan_theta_coil}} - \overbrace{\mathrm{d}x_{\text{TF,side case}}}^{\texttt{dx_tf_side_case_min}}\right)$$

The straight toroidal width of the secondary winding pack is (shortest side of trapezoid):

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-secondary}}}^{\texttt{dx_tf_wp_secondary_toroidal}} = 2 \times \\ \left(\overbrace{R_{\text{TF,WP-inner}}}^{\texttt{r_tf_wp_inboard_inner}} \times \tan{\left(\phi_{\text{TF,half}}\right)} - \mathrm{d}x_{\text{TF,side case}}\right)$$

The average toroidal straight width is calculated:

$$\overbrace{\mathrm{d}x_{\text{TF,toroidal,WP-average}}}^{\texttt{dx_tf_wp_toroidal_average}} = \frac{\left(\mathrm{d}x_{\text{TF,toroidal,WP-secondary}} + \mathrm{d}x_{\text{TF,toroidal,WP-primary}}\right)}{2}$$

The total winding pack area is calculated from the average:

$$\overbrace{A_{\text{TF,WP}}}^{\texttt{a_tf_wp_with_insulation}} = \frac{\left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}}+\mathrm{d}x_{\text{TF,toroidal,WP-secondary}}\right)}{2} \\ \times \overbrace{\mathrm{d}R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}$$

The area of the winding pack with no insulation or gap is:

$$\overbrace{A_{\text{TF,WP-no-insulation}}}^{\texttt{a_tf_wp_no_insulation}} = \\ \left(\mathrm{d}R_{\text{TF,WP}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ \times \left(\left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ + \left(\mathrm{d}x_{\text{TF,toroidal,WP-secondary}} - 2\times\left(\overbrace{\mathrm{d}R_{\text{TF,WP-insulation}}}^{\texttt{dx_tf_wp_insulation}} + \overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right)\right) \\ \times 0.5$$

The area of the surrounding winding pack insulation is:

$$\overbrace{A_{\text{TF,WP-insulation}}}^{\texttt{a_tf_wp_ground_insulation}} = \\ \left(\left(\mathrm{d}R_{\text{TF,WP}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right) \\ \times \left(\left(\mathrm{d}x_{\text{TF,toroidal,WP-primary}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ + \left(\mathrm{d}x_{\text{TF,toroidal,WP-secondary}} - 2\times\overbrace{\mathrm{d}R_{\text{TF,WP-gap}}}^{\texttt{dx_tf_wp_insertion_gap}}\right)\right) \\ \times 0.5 \\ - A_{\text{TF,WP-no-insulation}}$$

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Case geometry | superconducting_tf_case_geometry()

The areas of the total casing surrounding the winding packs on the inboard and outboard leg are calculated:

$$\overbrace{A_{\text{TF,case-inboard}}}^{\texttt{a_tf_coil_inboard_case}} = \frac{\overbrace{A_{\text{TF,inboard-total}}}^{\texttt{a_tf_inboard_total}}}{\underbrace{N_{\text{TF,coils}}}_{\texttt{n_tf_coils}}} - \overbrace{A_{\text{TF,WP-no-insulation}}}^{\texttt{a_tf_wp_with_insulation}}$$

$$\overbrace{A_{\text{TF,case-outboard}}}^{\texttt{a_tf_coil_outboard_case}} = \overbrace{A_{\text{TF,outboard}}}^{\texttt{a_tf_leg_outboard}} - A_{\text{TF,WP-no-insulation}}$$

The plasma facing front case area is calculated:

If i_tf_case_geom == 0 then the front case is circular so:

$$\overbrace{A_{\text{TF,plasma-case}}}^{\texttt{a_tf_plasma_case}} = \left(\overbrace{\phi_{\text{TF,half}}}^{\texttt{rad_tf_coil_inboard_toroidal_half}} \times \overbrace{R_{\text{TF,inboard-out}}^2}^{\texttt{r_tf_inboard_out}}\right) \\ - \left(\overbrace{\tan{(\phi_{\text{TF,half}})}}^{\texttt{tan_theta_coil}} \times \overbrace{R_{\text{TF,WP-outer}}^2}^{\texttt{r_tf_wp_inboard_outer}}\right)$$

The first term is equal to the area of an arc segment of radius $R_{\text{TF,inboard-out}}$. Since the value of rad_tf_coil_inboard_toroidal_half is a fraction of $\pi$ for each TF coil it can be substituted as the fraction of a full circle.

If i_tf_case_geom == 1 then the front case is straight so:

$$\overbrace{A_{\text{TF,plasma-case}}}^{\texttt{a_tf_plasma_case}} = \left(\left(\overbrace{R_{\text{TF,WP-outer}}}^{\texttt{r_tf_wp_inboard_outer}} \times \overbrace{\Delta R_{\text{TF,plasma-case}}}^{\texttt{dr_tf_plasma_case}}\right)^2- R_{\text{TF,WP-outer}}^2\right) \\ \times \tan{(\phi_{\text{TF,half}})}$$

Next the nose case area is calculated:

$$\overbrace{A_{\text{TF,nose-case}}}^{\texttt{a_tf_coil_nose_case}} = \left(\tan{(\phi_{\text{TF,half}})} \times \overbrace{R_{\text{TF,WP-inner}}^2}^{\texttt{r_tf_wp_inboard_inner}}\right) \\ - \left(\phi_{\text{TF,half}} \times \overbrace{R_{\text{TF,inboard-in}}^2}^{\texttt{r_tf_inboard_in}}\right)$$

Finally the average side case thickness is calculated:

If i_tf_wp_geom == 0 then a rectangular casing is:

$$\overbrace{\Delta x_{\text{TF,side-case-average}}}^{\texttt{dx_tf_side_case_average}} = \overbrace{\Delta x_{\text{TF,side-case-min}}}^{\texttt{dx_tf_side_case_min}} + \frac{1}{2}\left(\tan{(\phi_{\text{TF,half}})} \times \overbrace{\Delta R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}\right)$$

This is equal to the sidewall casing thickness at the very centre of the winding pack.

If i_tf_wp_geom == 1 then a double rectangular casing is:

$$\overbrace{\Delta x_{\text{TF,side-case-average}}}^{\texttt{dx_tf_side_case_average}} = \overbrace{\Delta x_{\text{TF,side-case-min}}}^{\texttt{dx_tf_side_case_min}} + \frac{1}{4}\left(\tan{(\phi_{\text{TF,half}})} \times \overbrace{\Delta R_{\text{TF,WP}}}^{\texttt{dr_tf_wp_with_insulation}}\right)$$

Finally, if i_tf_wp_geom == 2 then a trapezoidal casing is:

$$\overbrace{\Delta x_{\text{TF,side-case-average}}}^{\texttt{dx_tf_side_case_average}} = \overbrace{\Delta x_{\text{TF,side-case-min}}}^{\texttt{dx_tf_side_case_min}}$$

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On coil ripple | peak_b_tf_inboard_with_ripple()

The ratio of TF coil magnetic field increase with respect to the axisymmetric formula has been defined as :

$$f_\mathrm{rip}^\mathrm{coil} = \frac{B_\mathrm{rip}}{B_\mathrm{nom}}$$

with $B_\mathrm{rip}$ being the maximum field measured at the middle of the plasma facing sides of the winding pack and $B_\mathrm{nom}$ the nominal maximum field obtained with the axisymmetric formula (see section TF coils current). The same FIESTA runs have been used to estimate the on-coil ripple. This peaking factor has been fitted separately for 16, 18 and 20 coils using the following formula:

$$f_\mathrm{rip}^\mathrm{coil} = A_0 + A_1 e^{-t} + A_2 z + A_3 zt$$

with the $A_n$ the fitted coefficients and $t$ the relative winding pack lateral thickness defined as:

$$t = \frac{\Delta R_\mathrm{tWP}^\mathrm{out}}{\Delta R_\mathrm{tWP}^\mathrm{out\ max}}$$

with $\Delta R_\mathrm{tWP}^\mathrm{out}$ defined in Figure 1 and 3 and $\Delta R_\mathrm{tWP}^\mathrm{out\ max}$ the same value calculated without sidewall case as illustrated in Figure 12.

Figure 5: Inboard midplane cross section showing the filaments used for the on-coil ripple calculations with `FIESTA` with 16 coils. A radial and toroidal winding pack thickness of 0.5 m and 0.8 m for the left graph, while a radial and toroidal thickness of 1.2 m and 1.9 m, respectively, is used in the right figure. The right graph provides a visual illustration of the case where the parameter $t$ is close to unity.

And the relative winding pack radial thickness $z$ given by

$$z = \frac{\Delta R_\mathrm{WP}}{\Delta R_\mathrm{tWP}^\mathrm{out\ max}}$$

The three fits (for 16, 18 and 20 coils) are valid for:

• relative toroidal thickness: $t\in[0.35-0.99]$

• relative radial thickness: $z\in[0.2-0.7]$

• Number of TF coils: Individual fits has been made for 16, 18 and 20. For any other number of coils, the FIESTA calculations are not used and a default ripple increase of 9% is taken ($f_\mathrm{rip}^\mathrm{coil} = 1.09$). This default value is also used for 17 and 19 coils.

Figure 6 shows a contour plot of the on-coil ripple peaking factor as a function of the winding pack sizing parameters for 16 coil.

Figure 6: Contour plots of the on-coil peaking factor \(f_\mathrm{rip}^\mathrm{coil}\) obtained with FIESTA and used in the PROCESS scaling for 16 coils. The horizontal and vertical axis corresponds to the relative transverse \(\mathit{t}\) and radial thickness \(\mathit{z}\), respectively.

These ripple calculations are out of the spherical tokamak design range, which generally have fewer coils (between 10 and 14) and more radially thick winding packs. It is also worth mentioning that the the ripple must be evaluated layer-by-layer for graded coil designs, to get the genuine B field of each layer used to quantify the SC cross-section area per layer. Finally, resistive coils do not suffer from on-coil ripple as there is no radial case present.