Semi-analytical modeling AC loss of a flat stack of Y-Ba-Cu-O tapes

Panasyuk, George Y.; Ebbing, Charles R.; Murphy, John P.; Gheorghiu, Nadina; Sumption, Mike D.; Haugan, Timothy J.

doi:10.1063/5.0229155

We propose semi-analytical models to compute alternating current (AC) power loss in a stack of N high-temperature superconductor YBa₂Cu₃O_7−x (or Y–Ba–Cu–O) tapes subjected to a time-varying magnetic field perpendicular to the tapes with zero transport current. The models take into account screening of the interior superconducting tapes of the stack from the external magnetic field. We validate the results by experiments carried out at temperature $T = 77.2 K$ under an applied magnetic field with the amplitude of its induction $B_{ext} = 0.57 T$ and frequencies up to 110 Hz. As follows from our models, the AC loss per tape in stacks of N tapes decreases with N in agreement with experiments. The approach is extended to compute the AC loss for lower temperatures, larger magnetic fields strengths, and for frequencies up to several kHz. These studies are important for understanding and predicting the AC loss for contemporary motors and generators.

I. INTRODUCTION

High-temperature superconductors (HTSs), in particular, the last generation (Y, Re)BCO tapes, have numerous applications that include power transmission, rotating machinery, medical imaging, and many others.¹ In order to achieve high power for HTS devices, one needs to stack the HTS tapes, as a single tape may not have enough current carrying capacity. However, the properties of a stack of superconductive tapes, in particular, their AC losses, are quite different from the properties of a single tape. This is because the internal tapes in the stack are, at least partially, screened from the external magnetic field. Thus, the problem of determining the AC losses occurs to be both experimentally and computationally challenging, and a number of theoretical and experimental studies related to stacks of HTSs were recently performed. The authors in Refs. 2–5 demonstrated decrease in the eddy loss in metal layers adjacent to a (Y, Re) BCO tape as compared to stand-alone layers. This effect was attributed to the shielding of magnetic field by the neighboring (Y,Re)BCO layer/layers. Measurements of AC loss stacks with $N \leq 4$ in Ref. 6 and with $N \leq 3$ in Refs. 7 and 8 revealed decreasing AC loss per tape in stacks with $N > 1$ compared to AC loss in the single tape. Using a vibrating sample magnetometer equipped with 12 Tesla SC magnet, the authors in Ref. 9 have determined the AC loss in a stack of ReBCO tapes and found a small decrease of AC loss per tape on N at T = 77 K. A numerical model, also developed in Ref. 9, demonstrates independence of the AC loss per tape on N. Pure numerical studies^10,11 show also independence of the AC loss per tape on N. In Refs. 12 and 13, analytical approaches to model infinite (in vertical and horizontal directions) stacks were developed. Recently, a numerical study of AC losses in a single tape and a four-tape stack carrying AC current with or without a DC offset was performed.¹⁴ Experimental and numerical studies of superconductive coils and shielding of AC magnetic fields^15–22 are closely related research directions. In Ref. 15, the critical current and the AC loss in a stack of pancake coils was measured and also simulated. The result of that work enabled the authors to predict the AC loss of a stack of coated conductor pancake coils and to reduce the AC loss by optimizing the coil design. In spite of experiments revealing higher AC losses in HTS tapes with magnetic substrates than in HTS tapes with non-magnetic substrates, it was shown that two identical coils with magnetic and non-magnetic substrates have about the same amount of transport loss.¹⁶ More accurate techniques to measure AC losses in HTS coils as well as a new and simple sub-cooling technique with an open liquid nitrogen bath were developed in Ref. 17. In Ref. 18, the authors measured and simulated the AC loss in an HTS racetrack coil under both AC applied magnetic field and DC current. Simulation revealed that the transient state can expand over a few cycles, even when the superconductor presents no flux creep. A 2D numerical model was developed and experimentally validated in Ref. 19, enabling accurate prediction of the AC losses in the racetrack using prolonged coils used as components in superconductor linear motor systems. While results of numerical modeling are in good agreement with the measurements, results of application of the analytical infinite stack model from Ref. 12 give an order of magnitude smaller AC loss. It was experimentally shown and numerically confirmed that the dynamic resistance and total loss in a double pancake coil are smaller than in a double racetrack coil.²⁰ This can be attributed to the large perpendicular magnetic field component in the straight section of the racetrack coils. In Refs. 21 and 22, shielding characteristics of 4–4.6 cm wide HTS tapes were measured as functions of the magnitudes of DC or AC magnetic field, temperature, and the frequency of AC magnetic field. As found, the shielding fraction increases with the addition of more SC layers, while it does not change with the increase in the interlayer separation. Nowadays, intensified research is being devoted to studying next-generation motors and generators, with magnetic fields up to 5 T, frequencies up to 2 kHz, and temperatures down to 20 K.^23,24 As the number of stacked tapes in the field coil windings can reach 50–80,^23,24 new modeling approaches should consider stacks of N ∼ 10–100 or more tapes. To our knowledge, experimental data are not yet available for cryogenic machines, thus analytical and computational modeling are needed in order to understand and predict the AC loss for such parameter ranges.

Our experiments, accomplished in the AFRL laboratory, demonstrate a noticeable decrease in the AC loss per unit volume (or per tape) of the stack with N, which is in agreement with Refs. 6–8, though in contradiction with Refs. 10 and 11. Thus, the goal of this research is to resolve this issue by constructing models that properly consider screening of magnetic field by the outer SC layers. In Sec. II, we briefly describe our experimental setup and the parameters for the stack samples. Section III describes the proposed semi-analytical models that are used to compute the AC losses. Section IV presents results of the models' application and their comparison to our experimental results and to the results of experimental study from Ref. 9. Results for the AC loss for lower temperatures, larger magnetic fields strengths, and for frequencies up to several kHz are also described. Finally, Sec. V concludes the article.

II. EXPERIMENT

AC losses in samples with a single tape or in stacks of Y–Ba–Cu–O tapes are measured using our spinning magnet calorimeter (SMC) that was described in Refs. 25–27. This in-house built AC loss measurement system has a spinning rotor consisting of a set of permanent magnets arranged in a Halbach array with the sample exposed to a rotating AC magnetic field. The sample to be measured on is placed at a small radial distance from the spinning rotor containing magnets with pole orientations that alternate along the circumference. Thus, the sample experiences both radial and tangential fields that are approximately sinusoidal in character. Details of the waveform shape and harmonics are given in Ref. 26. The resulting magnetic field has a maximum radial component $B_{0 r} = 0.57 T$ and a resulting maximum rate $d B_{0 r} / d t = 272 T / s$ ⁠, while the maximum tangential component $B_{0 t} = 0.24 T$ and a maximum rate $d B_{0 t} / d t = 125 T / s$ ⁠. The AC loss at $T = 77.2 K$ and frequencies for the AC magnetic field up to 110 Hz is measured using nitrogen boiloff from a double wall calorimeter feeding a gas flow meter. The sample is immersed in liquid nitrogen such that the gas flow from the inner calorimeter is used for AC loss measurement.^25–27 For all measurements in this research, the tapes are oriented perpendicular to the radial field of the SMC, hence the tangential fields can be ignored. The system is calibrated using the power input from a known resistor, as discussed in Ref. 26. Background loss (due to heat leak into the calorimeter at zero field) is accounted for. There are two kinds of conductor used for samples. Both samples are SuperPower tapes of thickness 0.1 mm. One has width w = 12 mm, having a layer of silver stabilizer and no copper stabilizer and the other one has w = 4 mm mostly a copper stabilizer. In order to eliminate coupling losses, 25 μm thick Kapton tapes were inserted between the HTC tapes. The samples' specifications are shown in Table I.

TABLE I.

Tape (wire) specifications.

Type	Label	Hastelloy thickness (μm)	Total Cu thickness (μm)	YBCO thickness (μm)	Total Ag thickness (μm)	I_c @77.2 K (A)
SCS4050-AP	M4-305	109	53	1.6	3	126
SF12100	M4-244-3	111	0	1.6	8	451

Type	Label	Hastelloy thickness (μm)	Total Cu thickness (μm)	YBCO thickness (μm)	Total Ag thickness (μm)	I_c @77.2 K (A)
SCS4050-AP	M4-305	109	53	1.6	3	126
SF12100	M4-244-3	111	0	1.6	8	451

Using these basic coated conductors, two stack series were made and used for experiments: one from tape type M4-244-3 (with N = 1, 5, 7, 10, 20, 30, and 40) and the other one from tape type M4-305-10 (with N = 1, 3, 5, 10, and 25).

III. MODELING

A cross section of a stack of long tapes is shown in Fig. 1. The width of a tape in the stack is

w = 2 a

⁠, the SC layer thickness is d, and the thickness of each tape (in which we include also the Kapton layer) is D. In accordance to Ampère's law,

A (r) = \frac{μ_{0}}{4 π} \int \frac{j (r^{'})}{| r - r^{'} |} d^{3} r^{'} = A (r) z,

(1)

where

j (r) = j (r) z

is the current density flowing along the z axis (perpendicular to the figure's plane). A system of equations for

j (r)

can be derived in a way similar to the approach adopted in Refs. 28–30. However, due to the screening of the magnetic field by the SC layers, one needs to consider the variation of the magnetic field induction

B

across each SC layer. Thus, each of the SC layers is divided in M intervals along the x direction with the width

d x = 2 a / M

and by

q_{\max}

intervals (sublayers) along the y direction with the width

d y = d / q_{\max}

⁠. One can assume that j varies insignificantly within each of these intervals, provided that M and

q_{\max}

are large enough integers. For the case here of zero transport current, one can express the vector potential

A_{s i}

at the ith interval of the sth sublayer as

\begin{aligned} A_{s i} & = - \frac{μ_{0} a^{2}}{4 π} \int_{- 1}^{1} d {\hat{x}}^{'} \int_{0}^{(N - 1) \hat{D} + N \hat{d}} d {\hat{y}}^{'} \ln [{({\hat{x}}^{'} - x_{i})}^{2} \\ + {({\hat{y}}^{'} - y_{s})}^{2}] j ({\hat{x}}^{'}, {\hat{y}}^{'}) = \sum_{k, j} K_{s i, k j} j_{k j}, \end{aligned}

(2)

where

j_{k j}

is the elemental current density flowing through the jth interval of the kth sublayer. It is also assumed that

N \geq 1

⁠;

s, k = 1, 2, \dots N_{tot} = N q_{\max}

⁠;

i, j = 1, 2, \dots M

⁠,

\hat{D} = D / a

⁠. Thus, the matrix element is

K_{s i, k j} = - \frac{μ_{0} a^{2}}{2 π} {\hat{K}}_{s i, k j} with {\hat{K}}_{s i, k j} = \frac{1}{2} \int_{{\hat{x}}_{j} - {\hat{x}}_{i} - d \hat{x} / 2}^{{\hat{x}}_{j} - {\hat{x}}_{i} + d \hat{x} / 2} d u \int_{{\hat{y}}_{k} - {\hat{y}}_{s} - d \hat{y} / 2}^{{\hat{y}}_{k} - {\hat{y}}_{s} + d \hat{y} / 2} d v \ln (u^{2} + v^{2}) .

(3)

FIG. 1.

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(a) Sketch of a stack of Y–Ba–Cu–O tapes. The SC layers of thickness d are shown as blue strips. (b) Illustration of the division of each SC layer in $q_{\max}$ sublayers (⁠ $q_{\max} = 5$ in this figure) and substitution of SC layers from two adjacent tapes by one with double thickness.

In , the dimensionless spatial parameters

{\hat{x}}_{i} = (a_{i} + b_{i}) / 2

⁠,

a_{i} = (i - M / 2 - 1) d \hat{x}

⁠,

b_{i} = a_{i} + d \hat{x}

⁠, and

d \hat{x} = d x / a = 2 / M

along the x direction are introduced. Taking into account that

j ({\hat{x}}^{'}, {\hat{y}}^{'}) \neq 0

only if y belongs to a SC layer, it is convenient to present

{\hat{y}}_{k, s}

as

{\hat{y}}_{k} = (l^{'} - 1) \hat{D} + (q^{'} - 1 / 2) d \hat{y}

and

{\hat{y}}_{s} = (l - 1) \hat{D} + (q - 1 / 2) d \hat{y}

⁠, where

d \hat{y} = d y / a

along the y direction, and

l^{'}

⁠, l = 1, 2, …, N. Thus, the sum over k in is transformed to a sum over

l^{'}

and

q^{'}

⁠: for each next value of

l^{'}

⁠, the index

q^{'}

varies from

q^{'} = 1

until

q^{'} = q_{\max}

⁠, where

q_{\max} \geq 10

in our case. Clearly, each value of the index k unequivocally determines the values of both

l^{'}

and

q^{'}

⁠, and vice versa. The integrations in give

{\hat{K}}_{s i, k j} = \frac{1}{2} {{[f (u, v)]}_{{\hat{y}}_{k} - {\hat{y}}_{s} - d \hat{y} / 2}^{{\hat{y}}_{k} - {\hat{y}}_{s} + d \hat{y} / 2}}_{{\hat{x}}_{j} - {\hat{x}}_{i} - d \hat{x} / 2}^{{\hat{x}}_{j} - {\hat{x}}_{i} + d \hat{x} / 2},

(4)

with³⁰

f (u, v) = u v \ln (u^{2} + v^{2}) - 3 u v + u^{2} arctg (v / u) + v^{2} arctg (u / v)

⁠. As in Refs. 28–30, we accept the strongly nonlinear current–voltage characteristic

E = E_{c} | j / j_{c} |^{n} j / | j | z \equiv E z

with

E_{c} = 10^{- 4} V / m

⁠. Assuming also sinusoidal time variation with the frequency

f = 2 π ω

of the external vector potential

A_{ext, i} (t) = - a {\hat{x}}_{i} B_{0} \sin (ω t) z

that must be added to , one arrives at the following system of equations for the elements of the current density

j_{k j}

⁠:

\sum_{j = 1}^{M} \sum_{k = 1}^{N} {\hat{K}}_{s i, k j} \frac{d j_{k j}}{d t} = \frac{2 π}{μ_{0} a^{2}} [a {\hat{x}}_{i} ω B_{0} \cos (ω t) - E_{c} sign (j_{s i}) {| \frac{j_{s i}}{j_{c}} |}^{n}] .

(5)

Here,

B_{0}

is the amplitude of the magnetic field. Due to the screening of the magnetic field by currents within the SC layers,

B_{0} \to B_{0} (y)

⁠, which brings also a y dependence in other quantities. These dependences will be determined later. Using definition for

{\hat{K}}_{s i, k j}

⁠, one finds that

{\hat{K}}_{s i, k j} = {\hat{K}}_{s j, k i} = {\hat{K}}_{k i, s j} = {\hat{K}}_{k j, s i},

(6)

or, introducing double indices

m = (s, i)

and

m^{'} = (k, j)

⁠, one can write that

{\hat{K}}_{m, m^{'}} = {\hat{K}}_{m^{'}, m}

(7)

and rewrite as

\sum_{m^{'}} {\hat{K}}_{m, m^{'}} \frac{d j_{m^{'}}}{d t} = \frac{2 π}{μ_{0} a^{2}} [a {\hat{x}}_{m} ω B_{0 m} \cos (ω t) - E_{c} sign (j_{m}) {| \frac{j_{m}}{j_{c}} |}^{n}],

(8)

where

B_{0} \to B_{0 m}

in order to reflect its dependence on the sublayer's number and

{\hat{x}}_{m} \equiv {\hat{x}}_{i}

for any s. Multiplying by

j_{m}

⁠, summing over m, integrating over a period 1/f, and using , one finds that

\sum_{m, m^{'}} {\hat{K}}_{m, m^{'}} \oint d t j_{m} \frac{d j_{m^{'}}}{d t} = \frac{1}{2} \sum_{m, m^{'}} {\hat{K}}_{m, m^{'}} \oint d t \frac{d}{d t} (j_{m} j_{m^{'}}) = 0,

(9)

for the established periodic regime. Thus, we have two different expressions for the hysteretic AC power losses per cycle and per unit length of the stack,

\begin{aligned} Q = d x d y \sum_{m} \oint d t j_{m} E_{m} and \\ Q_{1} = d x d y a ω \sum_{m} \oint d t j_{m} B_{0 m} {\hat{x}}_{m} \cos (ω t), \end{aligned}

(10)

where

E_{m} = E_{c} | j_{m} / j_{c} |^{n} j_{m} / | j_{m} |

⁠. A correct solution of must satisfy relation

Q = Q_{1}

⁠, and this relation can be used to check the solution for .

Solving with the initial conditions

j_{k j} (0) = 0

allows one to find

j_{k j} (t)

⁠. As found in Ref. 30, the periodic regime for

j_{k j} (t)

is already established for

t \geq 1 / f

⁠. Thus, it is enough to consider a solution

j_{k j} (t)

only for the interval

(1 / f, 2 / f)

⁠. Moreover, due to a symmetry relation,

j_{k j} (t) = j_{k j} (3 / 2 f - t)

for

3 / 2 f \leq t \leq 2 / f

⁠, it is enough to find

j_{k j} (t)

within even a smaller time interval

1 / f \leq t \leq 3 / 2 f

⁠. A possible approximate analytical solution to the system can be obtained by equating the expression in the square brackets in to zero, resulting in

\begin{aligned} {\hat{J}}_{m} = C_{m}^{1 / n} | {\hat{x}}_{m} |^{1 / n} | \cos (τ) |^{1 / n} sign [\cos (τ)] sign ({\hat{x}}_{m}), \\ {\hat{j}}_{m} \equiv j_{m} / j_{c} . \end{aligned}

(11)

To check for the applicability of this solution and find its limits, we rewrite as

Λ \equiv r \sum_{m^{'}} {\hat{K}}_{m, m^{'}} \frac{d {\hat{\dot{j}}}_{m^{'}}}{d τ} = {\hat{x}}_{m} \cos (ω t) - C_{m}^{- 1} sign ({\hat{j}}_{m}) | {\hat{j}}_{m} |^{n},

(12)

where

r = μ_{0} j_{c} a / (2 π B_{0 m}) = 2 10^{- 7} j_{c} a / B_{0 m}

and

C_{m} = a B_{0 m} ω / E_{c}

⁠. Thus, solution is applicable if

| Λ | ≪ 1

⁠. As one finds, solution breaks if

| τ |

is too close to

π / 2.

Indeed, after substituting into

Λ

and introducing

K_{\max} = max_{m} | \sum_{m^{'}} {\hat{K}}_{m, m^{'}} C_{m^{'}}^{1 / n} {| {\hat{x}}_{m^{'}} |}^{1 / n} sign ({\hat{x}}_{m^{'}}) |,

(13)

one can represent the applicability condition for solution as

| π / 2 - τ |^{1 - 1 / n} ≫ \frac{r}{n} K_{\max} \equiv δ τ_{\min} = δ τ_{\min} (T, B_{0}),

(14)

where we also approximated

\sin τ | \cos τ |^{1 / n - 1} \approx | π / 2 - τ |^{1 / n - 1}

⁠. As found by numerically solving , the reason for why becomes inaccurate in the vicinity of

τ_{0} = π / 2,

where our approximate solution vanishes, is because the numerical (accurate) solution reaches zero at

τ

slightly less than

π / 2

by a value of the order of

δ τ_{\min}

⁠. Using our solution in Q from , one can find the corresponding energy loss from each SC sublayer

Q_{s}

as

\begin{aligned} 6 Q_{s} \equiv Q_{appr, s} = 8 a^{2} d \hat{x} B_{0 s} {\tilde{J}}_{c} C_{s}^{\frac{1}{n}} S I_{τ}, where \\ I_{τ} = \int_{0}^{π / 2} d τ {[\cos (τ)]}^{2 p} = \frac{\sqrt{π}}{2} Γ (p + 1), and \\ S = \sum_{i = M / 2 + 1}^{M} {\hat{x}}_{i}^{2 p}, \end{aligned}

(15)

where

Γ (z) is the gamma function and

2 p = 1 + 1 / n \leq 2

⁠. Exactly the same result can be found from using expression

Q_{1}

for the hysteretic AC loss in . In order to estimate an error associated with our approximate solution , we introduce a stricter condition, demanding that

| π / 2 - τ |^{1 - 1 / n} ≫ 10 δ τ_{\min} (T, B_{0}) \equiv δ τ_{\min 1}

Assuming that is still accurate at

| π / 2 - τ | \sim δ τ_{\min 1}

⁠, one can find that the relative error in for each SC sublayer and for the total hysteretic AC loss

\sum_{s} Q_{appr, s}

is of the order of

\begin{aligned} r & \equiv δ Q_{appr, s} / Q_{appr, s} \\ \approx \int_{π / 2 - δ τ_{\min 1}}^{π / 2} d τ {[\cos (τ)]}^{2 p} / \int_{0}^{π / 2} d τ {[\cos (τ)]}^{2 p} \\ \approx \frac{2}{π} {(δ τ_{\min 1})}^{2 + 1 / n}, \end{aligned}

(16)

where we use

n ≳ 10 ≫ 1

and approximate

Γ (3 / 2 + 1 / 2 n)

\approx Γ (3 / 2) = \sqrt{π} / 2.

As shown below, is applicable for the whole region of T and

B_{0}

considered here. Particular to our experimental study described below,

r < 4 10^{- 4} ≪ 1

⁠. We also compared our result to the result found in Ref. 30 at

N = 1

using parameters of the HTC tape provided in Ref. 30 and found coincidence with their solution within several percentages.

The dependence of the magnetic field on the y coordinate is determined from screening of B by the currents flowing inside the SC layers. This dependence can be found by solving the London equation.³¹

λ^{2} \nabla^{2} B (x, y) \approx λ^{2} B^{''} (y) = B (y)

(17)

inside SC layers. In , one can neglect the x dependence, because the characteristic scale of variation of B along the x direction is smaller than along the

y

direction by the factor

a / d

⁠. The magnetic field penetration depth λ is used as a fitting parameter here, with its temperature and field dependences described in Sec. IV. We also assume that magnetic field penetrates the spaces between SC layers only along the y direction. In the case of just one SC layer, solution to with the boundary condition

B (0) = B (d) = B_{0}

is

B_{0} (y) = B_{0} \frac{\cos h [(y - d / 2) / λ]}{\cos h (d / 2 λ)} .

(18)

Unlike its strong variation inside the SC layers, the magnetic field varies insignificantly between the SC layers. Indeed, the skin depth

δ_{skin} = {(\frac{ρ}{π μ_{0} f})}^{- 1 / 2} \sim 1 mm

at the considered experimental temperature

T \approx 77 K

and the corresponding values for the resistivities

ρ

in copper and silver even at

f \sim 1 kHz

⁠, which is much larger than the thicknesses of Cu and Ag sublayers. Hence, one can assume that the magnetic field at the entrance of a SC layer is approximately equal to its value at the exit from the previous SC layer. In this case, one can reduce the problem of finding

B_{0} (y)

in the stack of several SC layers separated by the distance

D - d

by a system when all these layers are attached to each other. For example, as illustrated in Fig. 1(b), the magnetic field distribution in the two-layer stack [Fig. 1(b), top] is substituted by just one SC layer having double thickness [Fig. 1(b), bottom]. The magnetic field in this case can be determined from by substitution

d \to 2 d

⁠:

B_{0} (y) = B_{0} \frac{\cos h [(y - d) / λ]}{\cos h (d / λ)},

(19)

and in our general case of N SC layers,

B_{0} (y) = B_{0} \frac{\cos h [(y - N d / 2) / λ]}{\cos h (Nd / 2 λ)} .

(20)

Here

y \equiv y_{k}

with

k = 1, 2, \dots, q_{\max}

corresponding to the first SC layer,

k = q_{\max} + 1, q_{\max} + 2, \dots, 2 q_{\max}

corresponding to the second SC layer, and, finally,

k = (N - 1) q_{\max}

+ 1, (N - 1) q_{\max} + 2, \dots, N q_{\max}

corresponding to the last

(N^{t h})

SC layer. This construction can be applied solely to computing the hysteresis part of the AC losses

Q \equiv Q_{N}

in the stack. In order to compute

Q_{N}

⁠, one needs to take into account that

J_{c}

and n depend on B. These dependencies were determined experimentally and one can find them in Ref. 32. Thus, if

B_{k} \leq B_{0} (y) \leq B_{k + 1},

the values for

X (y) = J_{c} (y)

and

n (y)

can be determined using the linear interpolation

X (y) = \frac{B_{0} (y) - B_{k + 1}}{B_{k} - B_{k + 1}} X_{k} + \frac{B_{0} (y) - B_{k}}{B_{k + 1} - B_{k}} X_{k + 1},

(21)

which provides a good enough accuracy taking into account the rather smooth

J_{c} (B)

and

n (B)

dependencies. Here

J_{c} (B)

is the critical sheet current density flowing through the SC layer when it is under

B_{0} (y)

⁠. We denote by B₀ the amplitude of the external field and “B₀(y)” the (decreased) amplitude of the magnetic field inside the SC layers. The quantities

J_{c k}

⁠,

n_{k}

⁠, and

B_{k}

can be found in Ref. 32 for the whole interval

0 \leq B_{0} (y) \leq B_{0}

and for any type of a Y–Ba–Cu–O tape. Knowing

B_{0} (y), J_{c} (y)

⁠, and

n (y)

⁠, one can determine the y dependencies for the current density and electric field inside the SC layers using our approximation ,

{\hat{J}}_{i} (y) = C {(y)}^{1 / n (y)} | {\hat{x}}_{i} |^{1 / n (y)} | \cos (τ) |^{1 / n (y)} sign [\cos (τ)] sign ({\hat{x}}_{i})

(22)

and

E_{i} (y) = E_{c} {[{\hat{J}}_{i} (y)]}^{n (y)} sign ({\hat{J}}_{i} (y)) .

(23)

Substituting and into , one finds eventually the following expression for the hysteresis part of the AC losses in our stack of N Y–Ba–Cu–O tapes,

P_{N}^{hys} = \frac{1}{d} \int_{0}^{N \hat{d} / 2} d \hat{y} p (y),

(24)

where

\hat{d} = d / a

⁠,

\begin{aligned} p (y) & = 16 a^{3} d \hat{x} f B_{0} (y) J_{c} (y) {[\frac{a B_{0} (y) ω}{E_{c}}]}^{1 / n (y)} \\ \times \sum_{i = M / 2 + 1}^{M} y_{i}^{1 + 1 / n (y)} \int_{0}^{π / 2} d τ {[\cos (τ)]}^{1 + 1 / n (y)}, \end{aligned}

(25)

J_{c} (y) \equiv J_{c} (B_{0} (y))

⁠,

n (y) \equiv n (B_{0} (y))

⁠, and the integral in can be found using . We refer to the model described by Eqs. and as a “model-1.” It is also possible to construct an alternative model that is based on the hysteretic AC energy loss, derived in Ref. 33 for a thin SC layer. In this case, instead of

p (y)

in , we use a different

p_{alt} (y) = 8 f a^{2} B_{0} (y) J_{c} (y) [\frac{2}{g} \ln (\cos h (g)) - \tan h (g)],

(26)

where

g = B_{0} (y) / B_{c} (y)

and

B_{c} (y) = μ_{0} J_{c} (y) / d

with the same

B_{0} (y)

and

J_{c} (y)

as in model-1. We refer to the model described by Eqs. and as a “model-2.”

Finally, the AC power loss due to eddy currents

P_{N}^{ed}

in the stack that takes into account magnetic field screening, is a straightforward generalization of the corresponding expression for this kind of loss in a single tape.²⁷ For an even N,

P_{N}^{ed} = F^{ed} (1 + 2 \sum_{l = 1}^{N / 2 - 1} {\hat{B}}_{l}^{2} + {\hat{B}}_{N / 2}^{2})

(27)

and for an odd N,

P_{N}^{ed} = F^{ed} (1 + 2 \sum_{l = 1}^{(N - 1) / 2} {\hat{B}}_{l}^{2}),

(28)

where

B_{l}

is the magnetic field (screened by the SC layers) between the lth and (l+1)th SC layers,

{\hat{B}}_{l} = B_{l} / B_{0},

and

B_{N - l} = B_{l}

due to symmetry. The factor

F^{ed} = (π^{2} / 6) w^{3} f^{2} B_{0}^{2} (t_{Ag} σ_{Ag} + t_{Cu} σ_{Cu} + t_{H} σ_{H}

+ t_{K} σ_{K}),

where

t_{S}

⁠,

t_{Cu}

⁠,

t_{H}

⁠, and

t_{K}

are the thicknesses of the silver, copper, Hastelloy, and Kapton layers, whereas

σ_{Ag}

⁠,

σ_{Cu}

⁠,

σ_{H}

⁠, and

σ_{K}

are the corresponding electric conductivities. Due to the fact that

σ_{H}

and

σ_{K}

are by several order of magnitude smaller than

σ_{Ag}

and

σ_{Cu}

⁠, one can neglect eddy loss in the Hastelloy and Kapton layers. Thus, in the experimental series 244,

F^{ed} \approx (π^{2} / 6) w^{3} f^{2} B_{0}^{2} t_{Ag} σ_{Ag}

due to

t_{Cu} = 0

and in the series 305,

F^{ed} \approx (π^{2} / 6) w^{3} f^{2} B_{0}^{2} (t_{Ag} σ_{Ag} + t_{Cu} σ_{Cu})

⁠. The total AC loss in the stack is

P_{N}^{tot} = P_{N}^{hys} + P_{N}^{ed}

⁠.

IV. RESULTS AND DISCUSSION

In Figs. 2–5,

T = 77.2 K

and

f = 110 Hz

⁠. Figure 2 shows results of calculation of the AC power loss per unit length for the 244 stack series at four different magnetic field penetration depths

λ

following model-1. As one finds, the AC losses become independent of N for

N > 2 - 3

if

λ ≲ 1 μ m

⁠. This happens because for large enough N,

B_{0} (y)

in the interior of

N d

SC layers becomes negligible together with the integrand , and the

B_{0} (y)

values at y close to 0 or

N d

are essentially independent on N, as is illustrated also in Fig. 3. Model-2 shows similar results. It is important to notice that for both hysteretic and total AC losses,

\frac{P_{N}}{P_{1}} = N \frac{P_{N}^{vol}}{P_{1}^{vol}} = N \frac{P_{N}^{1}}{P_{1}^{1}},

(29)

where

P_{N}^{vol}

is the power dissipated in the stack of tapes per unit volume of the stack and

P_{N}^{1}

is the power per one tape

(P_{N}^{1} = P_{N} / N)

⁠. A modified version of the presented in Sec. II models can be constructed by comparison of the experimental value

P_{1}^{1 \exp}

for just one tape (when it is available) and

P_{1}^{tot}

⁠, and the resulting N dependence is determined as

P_{N}^{tot, m} = (P_{1}^{1 \exp} - P_{1}^{ed}) P_{N}^{hys} / P_{1}^{hys} + P_{N}^{ed},

(30)

where

P_{N}^{hys}

is the result from model-1 or model-2. The corresponding N dependences for

P_{N}^{tot, m}

are also shown in Figs. 4 and 5 as “model-1m” and “model-2m.” These modifications of the models are usually more accurate.

FIG. 2.

View large Download slide

Hysteretic (left) $P_{N}^{hys}$ and total (right) $P_{N}^{tot}$ AC power losses per unit length dissipated in the 244 stack series at different values of $λ$ ⁠. Here, $B_{0} = 0.57 T$ ⁠, T = 77.2 K, $d = 1.6 μ m$ ⁠, and $f = 110 Hz$ .

FIG. 3.

Dependence of B ≡ B 0 ( y ) on the relative distance from the top of the stack (in units of d): λ = 0.4 μ m (left) and λ = 1 μ m (right).

View large Download slide

Dependence of $B \equiv B_{0} (y)$ on the relative distance from the top of the stack (in units of $d$ ⁠): $λ = 0.4 μ m$ (left) and $λ = 1 μ m$ (right).

FIG. 4.

View large Download slide

Total AC power loss as a function of the number of tapes in the 244 stack series calculated using the semi-analytical models. Also shown are experimental data from series 244 and the AC loss from the analytical model. The left figure shows absolute losses, while the right figure shows relative values.

FIG. 5.

View large Download slide

Same as in Fig. 4 for series 305 and for the absolute values of the total power loss.

Figures 4 and 5 compare results from the semi-analytical approaches (24)–(28) and (30) with the experimental results obtained by our group for experimental series 244 and 305 and also with an analytical model introduced in Ref. 34. The later model is based on the dilute superconductor model proposed in Ref. 35. Here, we use $λ$ as a fitting parameter in constructing Figs. 4–6. As one finds, all our models fit the experimental loss well for both absolute and relative values.

FIG. 6.

View large Download slide

Fit of our models to the experiment data from Ref. 9 and comparison with numerical results from Ref. 9 for the normalized energy loss per tape and per cycle at $T = 77 K$ ⁠.

Figure 6 compares the results given by our models with the experimental data from Ref. 9 at $T = 77 K$ and the numerical model also from Ref. 9 for the normalized energy losses. In that work, AC losses per cycle and per tape of the stack $Q_{N}$ were considered at frequencies $f \leq 2 Hz$ ⁠, thus the eddy loss can be disregarded. For the stack of tapes with width $w = 3 mm$ (Fig. 6), results of all our models are at least qualitatively close to the experimental values and show decreasing dependence of $Q_{N}$ with N unlike the numerical results shown also in Ref. 9.

Thus, as follows from our models, the power loss per tape decreases when the number N of tapes increases and these results are validated by our in-house experiments and also by experimental data from another research group. It is important, however, that the tapes in the stack must be placed on the top of each other without spaces. This is consistent with our assumption that the magnetic field penetrates the spaces between SC layers only along the direction perpendicular to the tapes. As expected, when distances D between the SC layers increase, the side penetration of the magnetic field into the interlayer spaces becomes more significant. In this case, the stack starts to behave more like a collection of independent tapes and the observed N dependencies have a tendency to reach plateaus deviating up from our theoretical curves (see Figs. 4–6). Eventually, when $D \to \infty$ ⁠, the power loss per tape approaches the power loss of a single tape. In this case, the total power loss is proportional to N, similar to the results from Refs. 10 and 11. This limit can be also achieved when $λ \to \infty$ ⁠, which is illustrated in Fig. 2. Indeed, when $λ \to \infty$ ⁠, there is no shielding of the magnetic field by the SC layers and the effect of the magnetic field on each tape will be the same and, again, the total AC power losses will be proportional to N.

In Figs. 7–12, results of applying model-1 to the stacks of 244 series for lower temperatures (⁠ $T = 20, 50,$ and 65 K) and higher magnetic fields (⁠ $B_{0} = 2 T, 3 T,$ and 5 T) are presented. Using (16), one can find that $r ≪ 1$ for these values of T and $B_{0}$ ⁠. Our numerical estimates show that despite r increases when T or $B_{0}$ decreases, $4 \times 10^{- 6} ≲ r ≲ 10^{- 3}$ for the whole range of T and $B_{0}$ considered here. The strong inequality $r ≪ 1$ breaks if $B_{0} < 0.1 T$ for $T = 65 K$ ⁠, if $B_{0} < 0.2 T$ for $T = 50 K$ ⁠, and if $B_{0} < 0.5 T$ for $T = 20 K$ ⁠. In order to extend the developed models to describe AC losses at lower temperatures, one can use the following dependence of $λ$ on temperature and magnetic field. Indeed, assuming that $λ \sim B / μ_{0} j_{c}$ and following,³⁶ one finds that $λ (T, B) = λ_{0} / (1 - T / T_{c})$ ⁠, where $λ_{0} = λ (0, B)$ and $T_{c} = T_{c} (B)$ is the critical temperature. Field dependence in $T_{c}$ was measured in Ref. 37 and can be approximated as $T_{c} (B) \approx 91.2 - 0.96 B$ ⁠, where B is measured in T. Approximately the same dependence $T_{c} (B)$ was experimentally found in Ref. 38. As an example, $λ_{0}$ can be found using that $λ (T = 77.2 K, B_{0} = 0.57 T) = 2 μ m$ as follows from our model-1 for series 244 (see Fig. 4), and the results for the different temperatures and magnetic fields are shown in Figs. 7–12. As one finds, both the hysteretic part of AC loss (Figs. 7–9) and the total (Figs. 10–12) AC loss per tape also decrease (as in Figs. 4–6) with N due to screening of the external magnetic field by SC layers. In fact, the screening effect is even more pronounced for the eddy current loss, because the latter it is proportional to $B_{0}^{2} (y)$ ⁠.

FIG. 7.

Frequency dependence of the hysteretic power loss per unit length and per one tape P N , hys 1 = P N hys / N for B0 = 2 T (left), B0 = 3 T (middle), and B0 = 5 T (right) for different numbers of tapes N in the stack at T = 65 K.

View large Download slide

Frequency dependence of the hysteretic power loss per unit length and per one tape $P_{N, hys}^{1} = P_{N}^{hys} / N$ for B₀ = 2 T (left), B₀ = 3 T (middle), and B₀ = 5 T (right) for different numbers of tapes N in the stack at T = 65 K.

FIG. 8.

View large Download slide

Same dependencies as in Fig. 7 for T = 50 K.

FIG. 9.

View large Download slide

Same dependencies as in Fig. 7 for T = 20 K.

FIG. 10.

Frequency dependence of the total power loss per one tape P1N≡PtotN/N per unit length for B 0 = 2 T (left), B0 = 3 T (middle), and B0 = 5 T (right) for different numbers of tapes N in the stack at T = 65 K.

View large Download slide

Frequency dependence of the total power loss per one tape P¹_N≡P^tot_N/N per unit length for $B_{0} = 2 T$ (left), B₀ = 3 T (middle), and B₀ = 5 T (right) for different numbers of tapes N in the stack at T = 65 K.

FIG. 11.

View large Download slide

Same dependencies as in Fig. 10 for T = 50 K.

FIG. 12.

View large Download slide

Same dependencies as in Fig. 10 for T = 20 K.

Results from model-2 at $λ = 2.8 μ m$ are very close to the corresponding results from model-1 at $λ = 2 μ m$ ⁠, as one can find from Fig. 13. This is consistent with the results shown in Fig. 4.

FIG. 13.

View large Download slide

Comparison between results from model-1 (“M1”) and model-2 (“M2”) at $T = 65 K$ (left), $T = 50 K$ (middle), and $T = 20 K$ (right) for N = 5, 10, 20, 40, and $B_{0} = 3 T$ ⁠.

V. CONCLUSIONS

In accordance to the developed semi-analytical models, the AC power loss per tape decreases with increasing the number of tapes in the stack. These results are validated by experiments, with different architecture of (Y, RE)BCO tapes with varying Ag. Cu, and Hastelloy thickness. Using the magnetic penetration depth that was found from comparison between the experiment and model at particular values of the magnetic field and temperature and using the knowing dependence of the magnetic penetration depth on $B_{0}$ and T, described in Sec. IV, one can predict the AC loss at larger values of magnetic field and lower temperatures as illustrated in Figs. 7–12. The screening effect in these cases is more pronounced than for the experimental cases due to a greater contribution of the eddy current loss: in Figs. 4–6, the eddy current loss does not exceed 20% of the total AC loss, whereas in Figs. 7–13 eddy current loss dominates. There is an increasing number of studies on MW-class power electric generators and motors that use Y–Ba–Cu–O tapes for shielding and confining the magnetic field lines within gaps that can double the power density (for example, refer to Refs. 23 and 24). Within the domain of their applicability, the proposed semi-analytical models may provide a basis for fast and reliable prediction of optimal parameters for contemporary generators/motors with expanded range for magnetic fields up 5 T, frequencies up to 2 kHz, and lower temperatures down to 20 K.

ACKNOWLEDGMENTS

This research was supported by the National Research Council (NRC)—Research Associate Program through the fellowship awarded to Dr. G. Y. Panasyuk and the Air Force Office for Scientific Research (AFOSR) grants LRIR's Nos. 18RQCOR100, 23RQCOR008, and 24RQCOR004 awarded to Dr. T. J. Haugan at the AFRL/RQ Aerospace Systems Directorate.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

George Y. Panasyuk: Writing – original draft (equal); Writing – review & editing (equal). Charles R. Ebbing: Writing – original draft (equal); Writing – review & editing (equal). John P. Murphy: Writing – original draft (equal); Writing – review & editing (equal). Nadina Gheorghiu: Writing – original draft (equal); Writing – review & editing (equal). Mike D. Sumption: Writing – original draft (equal); Writing – review & editing (equal). Timothy J. Haugan: Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Semi-analytical modeling AC loss of a flat stack of Y-Ba-Cu-O tapes

I. INTRODUCTION

II. EXPERIMENT

III. MODELING

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Semi-analytical modeling AC loss of a flat stack of Y-Ba-Cu-O tapes

I. INTRODUCTION

II. EXPERIMENT

III. MODELING

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

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Related Content

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