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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 33, NO. 5, AUGUST 2023 5900605
Modeling Eddy Current Losses in HTS Tapes
Using Multiharmonic Method
J. Ruuskanen ,M.Lyly , A. Halbach, T. Tarhasaari, V. Lahtinen ,T.Salmi , and P. Rasilo , Member, IEEE
Abstract—Due to the highly nonlinear electrical resistivity of
high temperature superconducting (HTS) materials, computing the
steady-state eddy current losses in HTS tapes, under time-periodic
alternating current excitation, can be time consuming when using
a time-transient method (TTM). The computation can require
several periods to be solved with a small time-step. One alternative
to the TTM is the multiharmonic method (MHM) where the Fourier
basis is used to approximate the Maxwell fields in time. The method
allows obtaining the steady-state solution to the problem with one
resolution of the nonlinear problem. In this work, using the finite
element method with the H−ϕformulation, the capabilities of
the MHM in the computational eddy current loss modeling of HTS
tapes are scrutinized and compared against the TTM.
Index Terms—AC losses, finite element methods, super-
conducting tapes.
I. INTRODUCTION
REBCO based high temperature superconductors (HTS)
are gaining interest in increasing amount due to their
high operation temperature compared to their lower temperature
superconductor counterparts [1]. The ability to operate at higher
temperature can result in significant cost and energy savings
when developing, testing and operating superconducting appli-
cations especially in the larger scale.
For the large scale applications such as particle accelerator
magnets [2] or energy applications [3],[4], the individual RE-
BCO tapes are typically assembled into a cable consisting of
multiple tapes. There are several different REBCO tape based
cable types for different applications such as Roebel cable [5],
CORC [6] or STAR wire [7]. The tape configuration in the cables
reduces the induced losses due to a changing magnetic field.
When designing such cables from the electromagnetic point
of view, numerical computational methods are essential. One of
Manuscript received 7 November 2022; revised 17 January 2023; accepted 1
February 2023. Date of publication 6 February 2023; date of current version 15
March 2023. This work was supported in part by the Academy of Finland under
Grant 324887 and in part by the Ulla Tuominen Foundation. (Corresponding
author: J. Ruuskanen.)
J. Ruuskanen, T. Tarhasaari, T. Salmi, and P. Rasilo are with the Electrical
Engineering Unit, Tampere University, 33720 Tampere, Pirkanmaa, Finland
(e-mail: janne.ruuskanen@tuni.fi; timo.tarhasaari@tuni.fi; tiina.salmi@tuni.fi;
paavo.rasilo@tuni.fi).
M. Lyly is with the Electrical Engineering Unit, Tampere University, 33720
Tampere, Pirkanmaa, Finland, and also with Quanscient Oy, 33100 Tampere,
Finland (e-mail: mika.lyly@tuni.fi).
A. Halbach and V. Lahtinen are with Quanscient Oy, 33100 Tam-
pere, Finland (e-mail: alexandre.halbach@quanscient.com; valtteri.lahtinen@
quanscient.com).
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/TASC.2023.3242619.
Digital Object Identifier 10.1109/TASC.2023.3242619
the most popular methods is the finite element method (FEM)
used to solve Maxwell’s magnetoquasistatic equations coupled
with the material equations. Typically, the main interest from the
electromagnetic point of view are the Joule losses which act as a
direct heat source for the superconducting system. The heating
decreases the efficiency of the application and in the worst case
can lead to quenching of the superconducting state.
Due to the highly nonlinear electrical resistivity of the HTS
materials, computing eddy current losses in HTS tape based
applications can be time consuming and convergence problems
can occur. In the typical approach, the time-dependent models
are solved with a time-transient method (TTM), where the time-
evolution of the fields to be solved are computed one time-step
at a time. Obtaining the steady-state solution to the problem with
highly nonlinear materials with TTM can require several periods
of excitation to be solved with a small time-step.
One alternative to the TTM is the multiharmonic method [8],
[9] (MHM), or the harmonic balance method, where the Fourier
basis is used to approximate the solution in time. The method
allows obtaining the steady-state solution with one resolution
of the nonlinear problem. The potential drawback in using the
MHM is that when approximating behavior not natural for the
chosen global basis in time, the number of required harmonics
can increase beyond feasible limit in terms of computational
burden.
In order to gain understanding on the capabilities of the MHM
method applied to HTS modeling, the H−ϕformulation [10]
was implemented to model AC losses in a HTS tape using
FEM. The modeling domain and the details about the modeling
task are presented in Section II. In Section III, the results
comparing the predictions given by the MHM and the TTM are
shown.
II. METHODOLOGY
A. Finite Element Model
In this paper, the MHM is compared against the TTM in a
2-D modeling domain consisting of an HTS tape cross-section
surrounded by an air domain having a radius of 8 cm. The 4 mm
wide and 95 μm thick tape consists of several layers of different
materials as depicted in Fig. 1. At the top and bottom there are
20 μmthick copper (Cu) layers. The steel alloy (Hast) has a
thickness of 50 μmand each of the two silver (Ag) layer have
the thicknesses of 2 μm. The superconducting layer (HTS) is
1μm thick [11]. The mesh of the air domain surrounding the
tape is also depicted in the figure. Note that all the material
layers in the tape are meshed and modeled individually, hence
e.g. homogenisation techniques were not used.
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
5900605 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 33, NO. 5, AUGUST 2023
Fig. 1. Depiction of the modeling domain with the HTS tape cross-section
consisting of the different material layers. Also the cohomology source field
H
H
HIis shown.
The H−ϕformulation is used where we solve for Faraday’s
law
curlE
E
E+∂tB
B
B=0,(1)
where the field quantities E
E
Eand B
B
Bare the electric field strength
and the magnetic flux density. To note, in the 2-D problem
scrutinized in this work, the magnetic field is in the plane and
consequently, the electric field strengthE
E
Eand the electric current
density J
J
Jhave only the out-of-plane component. Ampère’s
law relates the electric current density J
J
Jto the magnetic field
strength H
H
Has J
J
J=curlH
H
H. The constitutive relations used in the
formulation are
E
E
E=ρJ
J
Jand B
B
B=μH
H
H, (2)
where ρis the electrical resistivity and μthe permeability. The
models for these materials are detailed in Section II-B.
In this formulation, the unknown field quantity H
H
H
is decomposed as follows. In the conducting domain,
H
H
H=H
H
Hc+gradϕ+H
H
HIand in the nonconducting domain
H
H
H=gradϕ+H
H
HI, where ϕis a scalar field. H
H
Hcis the magnetic
field strength in the conducting domain. The cohomology source
field H
H
HIis used to impose the total current to the HTS tape [12].
The basis of H
H
HI, consisting of the red edges, is shown in
Fig. 1. Moreover, H
H
HIand H
H
Hcare interpolated with Whitney-1
forms [13], i.e. with edge-elements.
B. Material Models
As shown in, Fig. 1, the modeling domain consists of different
materials. The models for them at 77 K temperature are detailed
next. Copper, silver and Hastelloy are assumed to have con-
stant resistivities: ρCu =2.9·10−9Ωm, ρAg =2.8·10−9Ωm
and ρHast =10
−6Ωm. Moreover, μis assumed to be equal to
the permeability of free space μ0everywhere.
The nonlinear resistivity of the superconducting material is
modeled using the power law
ρ=Ec
Jc|J
J
J|
Jcn−1
,(3)
where |·|is the Euclidean norm, n=30.5, critical electric field
strength Ec=10
−4V/m and Jcis the critical electric current
density. The critical current density is modeled here with the two
different models: the Bean model and the Kim model [11].In
the Bean model, Jcequals to a constant Jc0 (= 2.85 ·1010 A/m)
while in the Kim model, Jcis an anisotropic function of H
H
H:
Jc=Jc0 1+ μ0
B0
|H
H
Hk|β
,(4)
where H
H
Hk=[kHx,H
y,0]T,k=0.29515,B0=42.65 mT
and β=−0.7as reported in [11].
C. Models in Time
The two different models in time under comparison in this
work are the TTM and the MHM. In the TTM, the simulation
problem is solved at discretized instants of time ti+1 =ti+Δt,
where Δtis the length of the time-step. In this work, constant
Δtwas used. Hence, the time derivative of a field F
F
Fcan be
approximated using for example the Backward Euler Method as
∂tF
F
F(x
x
x, ti+1)=F
F
F(x
x
x, ti+1)−F
F
F(x
x
x, ti)
Δt.(5)
In the multiharmonic method, a truncated Fourier series is
used as the global basis in time to approximate a time-varying
field F
F
Fas
F
F
F(x
x
x, t)=F
F
F0(x
x
x)+
N
k=1
F
F
Fsk(x
x
x)sin(kωt)+F
F
Fck(x
x
x)cos(kωt).
(6)
The coefficients F
F
F0,F
F
Fsk and F
F
Fck are spatially unknown vector
fields to be solved with FEM. The total number of the fields in
the full approximation equals to 2N+1. The angular frequency
is ω=2πf0, where f0is the fundamental frequency. Different
harmonics are defined by the integer k=1,...,N. Moreover,
a harmonic is odd when kis an odd number and even when k
is an even number. In the simulations carried out in this work,
the source current will be driven at the fundamental frequency,
i.e., at odd frequency. Consequently, using the material models
defined in Section II-B, the multiharmonic solution consists
only of the odd harmonics [9]. Consequently, in this work, the
multiharmonic field approximations are of the form
F
F
F(x
x
x, t)=
N
k=1,3,5,...
F
F
Fsk(x
x
x)sin(kωt)+F
F
Fck(x
x
x)cos(kωt)(7)
Therefore, the total number of unknown fields is reduced to
N+1, where Nis odd.
Other global bases for the time domain could be used. For
simulation tasks, where the total current is ramped up linearly,
the Chebyshev polynomials having a linear component in the
basis functions can be a better choice than the Fourier basis
functions.
D. Newton Method for Solving the Nonlinear Problem
In this work, the electric field E
E
E, depending nonlinearly on
J
J
Jand H
H
His approximated using the Taylor series at the known
RUUSKANEN et al.: MODELING EDDY CURRENT LOSSES IN HTS TAPES USING MULTIHARMONIC METHOD 5900605
fields (J
J
J0,H
H
H0)as
E
E
E≈E
E
E(J
J
J0,H
H
H0)+∂E
E
E
∂J
J
J(J
J
J−J
J
J0)+ ∂E
E
E
∂H
H
H(H
H
H−H
H
H0),(8)
where the current field value of E
E
Eand the derivatives of it with
respect to J
J
Jand H
H
Hare evaluated at the known field values
(J
J
J0,H
H
H0)solved at the previous iteration.
This linearisation (8) is substituted into the strong formu-
lation (1). From there on, standard FEM is applied to solve
the equation iteratively until the desired tolerance is met. In
the multiharmonic method, the nonlinear expression E
E
Eand its
derivatives are transformed into frequency domain using the
discrete Fourier transform. Consequently, the assembled matrix
equation of the weak formulation consists of the spatial degrees
of freedom (DoF) for all the fields in the series. In the MHM,
the spatial solution as a function of time is obtained by solving
the nonlinear problem iteratively only once. In the TTM, the
nonlinear problem is solved at each time-step.
III. RESULTS AND DISCUSSION
The modeling domain geometry and the corresponding mesh
were created using Gmsh [14], which was used for solving the
cohomology basis for setting the tape’s total current. The MHM
and the TTM solvers were implemented using the open-source
FEM C++ library Sparselizard [15].
In the simulations, the main focus was in comparing the
steady-state AC losses obtained with the MHM and the TTM.
The two different models for Jc, detailed in Section II-B,are
compared: the Bean and Kim models. In all the simulations, total
current of
I(t)=0.8Ic0 sin(ωt)(9)
was enforced to flow through the tape-cross section, where Ic0 =
114 A and ω=2π50 rad/s. The resulting AC losses Pover the
cross-section of the tape, were computed as a function of time
as
P(t)=Ωtape
E
E
E(t)·J
J
J(t)dΩ,(10)
where Ωtape is the area of the tape cross-section.
The reference solutions for the comparisons between the TTM
and the MHM were obtained using the TTM, with the two Jc
models. Only one period of the excitation was computed and for
the TTM, the steady-state AC loss over the cycle converged with
∼900 time-steps. Note that the TTM solution was assumed to be
in the steady-state after the first half of the cycle. Furthermore,
the reference solutions obtained with the TTM were compared
against the predictions given by the alternative simulation ap-
proach using the MHM. The first comparison was based on the
relative difference computed as
|LTTM −LMHM|
LTTM
,(11)
where LTTM is the steady-state loss over the cycle computed
with the TTM and LMHM is the loss over cycle computed with
the MHM.
Fig. 2shows the relative difference between the steady-state
losses over the cycle predicted with MHM in comparison to the
losses given by the TTM. Based on the results, the accuracy
Fig. 2. Relative difference between the MHM and the TTM in the steady-state
AC losses as a function of number of fields in the Fourier series approximation.
Left: Bean model. Right: Kim model.
Fig. 3. Comparing simulation time as a function of the number fields in the
Fourier series.
of the multiharmonic method increases with increasing number
of fields in the Fourier series used to approximate the electro-
magnetic fields. Using 10 fields (five harmonics), the relative
difference to the TTM is ∼1.5% with the both Bean and Kim
models. Using 14 fields (7 harmonics), the relative differences
of 0.4% (Bean) and 0.2% (Kim) were obtained.
We also compared the simulation time of MHM (tMHM)asa
function of the number of fields to the simulation time required
to solve the problem with the TTM (tTTM). The comparison is
showninFig.3as the fraction tMHM /tTTM. With 10 unknown
fields, MHM was 5-6 times faster with the relative difference of
∼1.5% in both cases. This comparison is only indicative since
the TTM and the MHM are very different methods by nature.
However, it can be concluded that the less fields can be used, the
faster the MHM is in comparison to the TTM.
To get a more detailed comparison, Fig. 4compares the AC
loss as a function of time for the Bean and Kim models in case
of 14 fields. In the results obtained with the TTM, a transient
phase can be seen in the first half of the excitation period. For
both the Bean and Kim models, the MHM agrees well with the
TTM given results. Hence, we can conclude that the MHM based
approach is able to predict succesfully the AC losses in the given
simulation task.
A point-wise comparison of the norm of the magnetic flux
density |B
B
B|obtained with the MHM and the TTM is shown in
Fig. 5, where |B
B
B|is interpolated in the center of the HTS layer
along the width of the tape. The |B
B
B|-profiles are shown at 3/4T,
where Tis the duration of the period. Hence, the total current is
at its minimum (−0.8Ic0). The comparison is shown for both the
5900605 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 33, NO. 5, AUGUST 2023
Fig. 4. Comparison of the AC losses as a function of time. Left: Bean model.
Right: Kim model.
Fig. 5. Comparing the norm of the magnetic flux density in the center of the
HTS layer along the width of the tape. Left: Bean model. Right: Kim model.
Fig. 6. Comparing the out-of-plane component of J
J
J, normalized with Jc0,in
the center of the HTS layer along the width of the tape. Left: Bean model. Right:
Kim model.
Bean and Kim models. The results show that the MHM is able to
predict the magnetic flux density profile in the given simulation
task using the Bean and Kim models.
Fig. 6compares the current density profile in the HTS tape
predicted with the MHM and the TTM using the Bean and the
Kim models. For the comparison, the out-of-plane component
of the current density, normed with Jc0, was interpolated in the
center of the HTS layer along the width of the tape. The results
show an agreement between the MHM and the TTM for the
both Jcmodels. However, J
J
Jdoes not seem to be as smooth in
the saturated region as in the case of the TTM. This can be related
to the truncated Fourier series and/or different requirements on
the mesh density or interpolation order in comparison to the
TTM.
As a final comparison, Fig. 7shows average loss in the super-
conducting tape as a function of the excitation frequency. Results
were computed using both the MHM and TTM with the Bean and
Fig. 7. Average loss as a function of source current frequency simulated using
the MHM and the TTM with Bean and Kim models.
Kim models. Good agreement was observed between the MHM
and TTM. Furthermore, the average losses increased linearly
with increasing frequency, and higher losses were obtained with
the Kim model than the Bean model.
IV. CONCLUSION
In this paper, the multiharmonic method using global basis
functions in the time domain was demonstrated by predicting
computationally the eddy current losses and field quantities in
REBCO HTS tape cross-section using the finite element method.
The approach was succesfully validated in a test simulation case
against the simulation results given by the conventional time-
transient method. In the comparison, the H−ϕformulation
was used for both approaches with two different Jcmodels: Bean
and Kim, where in the Kim model the anisotropic dependency
of the magnetic field on the resistivity of the superconducting
material was taken into account.
In the test case, total electric current at 50 Hz frequency was
applied through the HTS tape cross-section. The main interest
in the comparisons was how accurately the MHM approach
was able to predict the steady-state AC losses over one period
of excitation. As expected, the accuracy of the MHM based
approach increased when the number of fields in the Fourier
series was increased. With 14 fields, the relative difference to
AC losses per cycle obtained with the TTM was 0.2% with Kim
model and 0.4% with Bean model.
Even though the degrees of freedom in the FEM problem to be
solved for 14 fields was 14 times the DoFs needed for the TTM
(5001), the multiharmonic method was still faster in the simula-
tion test case investigated in this work. With 10 fields, the MHM
was 5-6 times faster than the TTM with only ∼1.5% relative
difference in the steady-state AC losses per cycle. An additional
succesful comparison was also carried out where MHM and
TTM were compared by simulating average losses as a function
of applied current density. The results showed that the average
losses increased linearly with the increasing frequency.
As the future work, the efficiency of the MHM approach, in
comparison to the TTM, should be tested by simulating a stack
of tapes in 2-D and 3-D modelling domains. In addition, other
global bases for the time domain could be investigated such as
the Chebyshev polynomials.
RUUSKANEN et al.: MODELING EDDY CURRENT LOSSES IN HTS TAPES USING MULTIHARMONIC METHOD 5900605
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