Electromagnetic Field Simulation of Superconducting Cables

No, they are already in the practical application stage. To give representative examples, the AmpaCity project in Essen, Germany, has been continuously operating a 10kV, 40MVA, approximately 1km HTS cable connected to the commercial grid since 2014. South Korea's KEPCO has also completed a 500m-class demonstration on a 23kV grid. In Japan, Sumitomo Electric has conducted a 66kV, 200m-class demonstration test.

The key point is current density. HTS (High-Temperature Superconducting) wires can carry current at 5 to 10 times the current density of copper. This directly translates to space savings in underground power transmission tunnels in urban areas. For example, to secure the same 40MVA transmission capacity, while conventional 110kV OF cables would require 3 circuits, superconducting cables can do it with a single 10kV circuit. Transformers can also be omitted, making the entire equipment more compact.

Good question. The power required for cooling is typically about 0.5 to 1% of the cable's transmission capacity. On the other hand, the resistive loss of conventional copper cables is 3 to 5%, so even after subtracting the cooling cost, there is a net energy saving. However, this is the core of the design; FEM analysis must precisely evaluate the balance between AC loss in the HTS wire and the cooling load. Let's look at the necessary physical models in order.

"Zero resistance" is an idealized story; actual HTS wires exhibit extremely steep nonlinear resistance near the critical current density $J_c$. This is described by the E-J power law:

Here, $E_0 = 1\,\mu\text{V/cm}$ (the criterion electric field for superconductivity), and $n$ is an exponent representing steepness. A larger $n$ value indicates a material closer to an ideal superconductor. For YBCO-based (REBCO) wires, $n \approx 20\text{--}40$ is typical; for Bi-2223 tapes, $n \approx 10\text{--}25$.

With a large $n$ value, when $J < J_c$, effectively $E \approx 0$ (superconducting state), but the moment $J$ slightly exceeds $J_c$, $E$ increases explosively. Numerically, this means an extremely nonlinear constitutive law. For an FEM solver, the apparent resistivity $\rho = E/J = E_0 J^{n-1}/J_c^n$ changes by many orders of magnitude depending on the current density, making Newton-Raphson method convergence very difficult. In practice, for $n > 30$, convergence often fails without damping or adaptive time-stepping.

No, $J_c$ depends strongly on both temperature $T$ and external magnetic field $B$. This is the biggest constraint in superconducting cable design. A commonly used practical model is the Kim-Anderson type:

$J_{c0}$ is the critical current density under reference conditions (self-field, 77K), $T_c$ is the critical temperature (about 92K for YBCO), $\alpha$ is the power exponent for temperature dependence (typically 1.5 to 2.0), and $B_0$ is a scale parameter for magnetic field dependence. To give a concrete sense of numbers, for REBCO wire at 77K and self-field, $J_c \approx 3 \times 10^{10}\,\text{A/m}^2$, but if cooled to 65K, $J_c$ more than doubles. Conversely, under a 3T external magnetic field, it drops to less than half.

Exactly. The magnetic field distribution within the cable cross-section is determined by the self-field generated by the cable itself, and the temperature distribution is determined by the balance between AC loss and cooling. In other words, determining $J_c$ requires knowing the magnetic field and temperature, determining the magnetic field requires knowing the current (which is constrained by $J_c$), and determining the temperature requires knowing the AC loss (a function of current and magnetic field)—everything is coupled. That's why coupled electromagnetic-thermal FEM analysis is essential.

For direct current, the loss is indeed almost zero. But with alternating current, because the magnetic flux changes over time, three types of loss mechanisms occur. Collectively, these are called AC loss. In superconducting cable design, accurate evaluation of AC loss is directly linked to cooling system design, making it one of the most important calculation items.

Let me explain them in order.

1. Hysteresis Loss $Q_h$ — Loss due to flux pinning within superconducting filaments, the main component of AC loss. Approximated by the Bean model (critical state model):

Here, $d_f$ is the filament width (for REBCO wire, the conductor tape width, typically 4–12mm), $\Delta B$ is the variation amplitude of the external magnetic field. The key point is that it is proportional to $d_f$ — meaning hysteresis loss can be dramatically reduced by subdividing the filaments (striation). This is the reason why REBCO wire striation technology is actively researched.

2. Coupling Loss $Q_c$ — Loss due to coupling currents flowing through the matrix metal (silver sheath or Hastelloy substrate) between filaments:

$\tau_c$ is the coupling time constant, $\tau_c = \mu_0 l_p^2 / (8\pi^2 \rho_m)$. $l_p$ is the twist pitch length, $\rho_m$ is the resistivity of the matrix. Shorter twist pitch reduces $\tau_c$ and thus coupling loss. For NbTi wires, twist pitch is 10–20mm; for REBCO wires, an equivalent short pitch is achieved through transposed (displaced) structures.

3. Eddy Current Loss $Q_e$ — Eddy current loss induced in normal-conducting metal parts like the stabilizing copper layer of HTS wires or the cable former (central conductor):

$d$ is the conductor thickness, $\rho_{cu}$ is the resistivity of copper. Copper at 77K has about 7 times higher conductivity than at room temperature, so eddy current loss can be larger than expected. Especially if the cable former is made of copper, this loss can become non-negligible.

For an AmpaCity-class 10kV, 2.3kA HTS cable, AC loss is about 1–3 W/m. The required power for the refrigerator to remove this at 77K, considering Carnot efficiency, is 10–15 times the loss (COP ≈ 0.07–0.1), so in terms of electrical input, it's 10–45 W/m. For a 1km cable, the entire cooling system would be equivalent to 10–45 kW. This is sufficiently smaller than the transmission loss of copper cables (100–200 W/m for the same capacity), making it economically viable.

Exactly. That is the essence of quench (irreversible loss of the superconducting state). Writing the governing equation for electromagnetic-thermal coupling:

The second term on the right, $\mathbf{E} \cdot \mathbf{J}$, is the AC loss (+ Joule heating during transition to normal state), and the third term, $q_{cool}$, is cooling by liquid nitrogen. From the E-J power law, $E \cdot J = E_0 J^{n+1}/J_c^n$, so when $J$ approaches $J_c$, heat generation increases sharply → temperature rises → $J_c$ decreases → heat generation increases further, entering a positive feedback loop. This is quench.

If a local quench occurs, that region transitions to the normal state, generating significant Joule heat. HTS wires have poor thermal conductivity (amorphous-like ceramic layers), so quench propagation speed is slow (a few mm/s to a few cm/s, extremely slow compared to NbTi/Nb₃Sn's tens of m/s). This is problematic because delayed detection can lead to local temperatures reaching several hundred degrees, burning out the wire. Therefore, quench detection and propagation analysis is the most critical issue for safety design. Predicting quench initiation conditions and propagation speed through coupled FEM analysis is essential for designing protection circuits.

Solving Maxwell's equations is the same, but the extremely nonlinear E-J characteristics of superconductors create numerical difficulties. The choice of formulation determines the success of the calculation. There are mainly two approaches.

The A-V method (Magnetic Vector Potential method) is the standard method for electromagnetic field analysis, using the magnetic vector potential $\mathbf{A}$ and electric scalar potential $V$ as unknowns, with $\mathbf{B} = \nabla \times \mathbf{A}$, $\mathbf{E} = -\partial\mathbf{A}/\partial t - \nabla V$:

However, in the superconductor region, the nonlinear resistivity $\rho_{sc}(J)$ based on the E-J power law is used instead of $\sigma$. The problem is that this $\rho_{sc}$ changes by more than 10 orders of magnitude depending on $J$, causing the condition number of the Jacobian matrix to deteriorate extremely.

In contrast, the H-formulation, which has become mainstream in recent superconducting analysis, is:

H-formulation has three advantages. First, current density is naturally obtained as $\mathbf{J} = \nabla \times \mathbf{H}$, making evaluation of E-J characteristics direct. Second, magnetic field penetration can be handled naturally by simply setting a very large resistivity ($\rho_{air} \sim 1\,\Omega\cdot\text{m}$) in the region outside the superconductor (air region). Third, in 2D cross-sectional analysis, it becomes a single scalar variable ($H_z$), resulting in fewer degrees of freedom and better computational efficiency.

The HTS modeling guide for COMSOL also recommends H-formulation, and the majority of academic papers adopt this approach.

Sharp observation. REBCO wire is typically 4–12mm wide, with a total thickness of 0.1mm, and the superconducting YBCO layer within it is only 1–2μm. If the cable outer diameter is 100mm, the scale...