**2. Model Description**

#### *2.1. Double-Pancake ECG Model*

The ECG model [16] in Figure 1 was adopted to calculate the distributions of both the spiral and radial currents of a NI–DP coil. The number of total turns in one pancake was defined as *N*t. Each turn was equally divided into *N*e arc elements. The total number of spiral and radial elements for one pancake was defined as *Ni* = *N*<sup>t</sup> × *N*<sup>e</sup> and *Nj* = (*N*<sup>t</sup> − 1) × *N*e, respectively. Each spiral element consists of its own inductance, its mutual inductance with the other elements, and spiral resistance. The *k*-th spiral resistance *Ri,k* is composed of the resistance of the HTS layer *R*sc and that of the other metal layers *R*mt, as illustrated in Figure 2. *Rj,k* represents the *k*-th radial resistance.

**Figure 1.** ECG model of a NI–DP coil.

**Figure 2.** Circuit of a spiral element.

Considering that a DP coil is wound spirally with a single tape, the upper pancake is reflectional rather than translationally symmetric [21] with respect to the lower pancake, which should be considered when numbering spiral elements and calculating the mutual inductance between an arc element of the upper pancake and one of the lower pancakes.

The relationship between the spiral and radial currents of the DP ECG model can be obtained according to Kirchhoff's law at each circuit node. The governing equations are the following Equation (1):

$$\begin{cases} \begin{aligned} I\_k - I\_{k+1} - J\_k &= 0 \\ I\_k - I\_{k+1} + I\_{k-N\_\ell} - I\_k &= 0 \\ I\_k - I\_{k+1} + I\_{k-N\_\ell} &= 0 \\ I\_k + I\_{k-N\_\ell} &= I\_{op} \\ I\_k - I\_{k-1} + I\_{k-N\_\ell} &= 0 \\ I\_k - I\_{k-1} + I\_{k-N\_\ell} - I\_{k-2N\_\ell} &= 0 \\ I\_k - I\_{k-1} + I\_{k-N\_\ell} - I\_{k-2N\_\ell} &= 0 \\ I\_k - I\_{k-1} - I\_{k-2N\_\ell} &= 0 \end{aligned} & \begin{array}{c} \begin{aligned} &k \in [1, N\_\ell] \\ k \in [N\_\ell + 1, N\_j] \end{aligned} \end{cases} \end{cases} \end{cases} \end{cases} \begin{aligned} I\_0 = \begin{cases} I\_k = [1, N\_\ell] \\ k \in [N\_\ell + 1, N\_j - 1] \end{cases} \end{cases} \tag{1}$$

where *Ik*, *Jk*, and *Iop* denote the current in the *k*-th spiral element, radial element, and power supply, respectively.

The governing equations of each circuit loop derived from Kirchhoff's voltage law are the following Equation (2):

$$\begin{cases} \sum\_{p=2}^{N\_c+1} \mathcal{U}\_p - f\_k R\_{j,k} = 0 & ; k = 1 \\ \mathcal{U}\_k - \mathcal{U}\_{k+N\_c} - f\_{k-1} R\_{j,k-1} + f\_k R\_{j,k} = 0 & ; k \in \left[2, 2N\_j - 1\right] \\ \sum\_{p=2N\_i-N\_c}^{2Ni} \mathcal{U}\_p - f\_k R\_{j,k} = 0 & ; k = 2N\_j \end{cases} \tag{2}$$

where *U*<sup>k</sup> denotes the voltage drop along the *k*-th spiral element, consisting of both the inductive and resistive voltages, as shown by the following Equation (3):

$$dL\_k = \sum\_{m=1}^{2N\_i} M\_{k,m} \frac{dI\_m}{dt} + I\_k R\_{i,k} \tag{3}$$

where *Mk,m* represents the self-inductance of the *k*-th spiral element if *k* = *m* and the mutual inductance between the *k*-th and *m*-th spiral elements if *k* = *m*. The self-inductance and mutual inductance are calculated by integrating Neumann's formula [22,23].

Equations (1)–(3) can be expressed in a matrix form (Equation (4)):

$$\begin{cases} \begin{array}{c} A\_1I + A\_2I = b \\ B\_1\frac{dI}{dt} + B\_2I + B\_3I = 0 \end{array} \end{cases} \tag{4}$$

where *I = [I*<sup>1</sup> *I*<sup>2</sup> ... *I*2*Ni] <sup>T</sup>* and *J* = *[J*<sup>1</sup> *J*<sup>2</sup> ... *J*2*Nj] T*.

For the aforementioned ECG model [16], *A*<sup>1</sup> is always a non-singular square matrix, and consequently, unlike the previously proposed method [16], the radial current vector *J* is selected as the state variable, and the spiral current vector *I* can be derived, as shown by Equation (5).

$$I = A\_1^{-1}(b - A\_2I) \tag{5}$$

To solve the system of ordinary differential Equation (4), iterative methods including the Runge–Kutta fourth-order method were adopted, and the calculation and postprocessing were conducted in MATLAB R2021b. The geometry of the coil in profiles of current distribution [24,25] in the radial direction was enlarged for better illustration.
