- Email: [email protected]
Effect of the sheath resistivity and tape geometry on eddy current loss in Bi(2223) tapes
Physica C 310 Ž1998. 208–212
Effect of the sheath resistivity and tape geometry on eddy current loss in Bi ž2223 / tapes Nadia Nibbio ) , Svetlomir Stavrev, Bertrand Dutoit Circuits and Systems, Swiss Federal Institute of Technology— Lausanne, DE-CIRC, EPFL, CH-1015 Lausanne, Switzerland
Abstract We have made a software electromagnetic simulation of eddy current effects in BiŽ2223.Ag sheathed tapes with applied AC transport current in a broad frequency range. We have used simple models of monocore and multicore tapes. Calculated is the power distribution ŽWrm3 . within the metal sheath using the geometry of real tapes. The numerical simulation is carried out with Finite-element-method software. The paper also describes the influence of the tape cross-sectional architecture and the sheath resistivity on the eddy current loss. q 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: BiŽ2223. tape; Eddy current loss; Finite-element-method
1. Introduction One of the future large scale industrial applications of HTS materials, such as BiŽ2223., will be their usage in electric power transmission systems. In the self-field of an applied transport current only, the AC losses in the HTS conductors are mainly hysteretic, in the BiŽ2223. core, and eddy current, in the metal sheath w1x. Hysteresis loss is dominating in the power frequency range Ž40–180 Hz., while eddy current loss has an increasing contribution at frequencies above 200 Hz w2x. The theoretical derivations of eddy current loss in self-field have different dependence on the tape geometry and transport current amplitude Žw1,3x., so it is necessary to evaluate the eddy current loss contributions by other means, e.g., by software electromagnetic simulations.
2. Finite element method simulation For the specific geometry of a HTS tape and its components, there is no analytical solution for the current distribution yet, so it is necessary to solve this problem by using a 2-D numerical analysis. To perform the simulation of the electromagnetic behavior of the tape, a Finite-element-method software package has been used—FLUX2D w4x. The numerical computation is performed using a formulation of the Maxwell’s equations, where A is potential vector, J is current density and V is voltage gradient. The introduction of the potential vector in Faraday’s and Ampere’s laws leads to the equation below:
== )
Corresponding author. E-mail: [email protected]
ž
1
m
/
==A qs
EA Et
sJ
where s is the conductivity of the material.
0921-4534r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 6 3 - 8
Ž 1.
N. Nibbio et al.r Physica C 310 (1998) 208–212
The 3-D numerical solution of Eq. Ž1. in terms of finite element expression is: 1
HV
m
Ž = = W . Ž = = A. q
1
m
Ž = P W . Ž = P A.
qj vs WA q s W= V d V s 0.
Ž 2.
And
HV Ž jvs= w P A q s= w= V . d V s 0
Ž 3.
where the integration is over the domain V ; w and W are weighting parameters, and m the permeability w5x. The total power loss is then calculated as: Qe s
Hcycled tHV E P JdÕ.
Ž 4.
3. Electrical model of Bi(2223) r Ag tape The primary goal of our study is the evaluation of eddy current loss in the metal sheath of BiŽ2223. tapes. For this reason the nonlinear hysteretic behavior of the BSCCO core was not taken into account. We have used the following simple electrical model for the simulation of applying a transport current through a BiŽ2223. tape and the resulting self-field effects: The simulated system is either a monocore or a multicore BiŽ2223. tape. The model consists of a superconductive core in parallel with the Ag ŽAg-alloy. sheath. The tape is characterised by means of the resistivity for each material ŽAg and BSCCO.. The simulation set-up consists of a current source which is connected to two parallel massive conductors, respectively the BSCCO material and the Ag sheath. Both materials are assumed isotropic. The main parameters of the electromagnetic simulation are the resistivity r, as well as the amplitude and the frequency of the transport current. There is neither a current nor a magnetic field inside the tape prior to applying the AC transport current Žzero initial conditions.. The main difficulty of the numerical simulation is the evaluation of the electrical and magnetic properties of the superconducting core. The relation-
209
ship between B and H is considered linear — B s m 0 H, i.e., the relative permeability of the superconductor has been given a value of 1. The ’effective’ resistivity of the BSCCO is calculated from the total measured loss in real BiŽ2223. tapes: P TOTAL s
1 2
2 R eff Irms .
Ž 5.
Knowing the filling factor of the tape and the measured resistivity of the Ag sheath, the effective resistivity of the superconductor is then deduced. Thus the superconductive material is modeled as a conductor with very low resistivity calculated from the measured total AC loss for a given value of the current amplitude at a given frequency. For example, from the above equation, for a monocore tape of dimensions 2.75 = 0.32 mm with filling factor 0.45, the effective resistance of the superconductor is calculated to be 1.12 mV, while the resistance of the Ag-sheath is 5.3 m V Žat 59 Hz and Ip s 4 A.. There is a difference of 3 orders of magnitude between the resistance of the superconductive core and the metal sheath. In this way the portion of the transport current flowing through the sheath would be around 1r1000 of the applied transport current. Although in reality there indeed exists a partitioning of the current between the core and the sheath, it is probably of much smaller magnitude. The calculated power dissipation in the sheath would inevitably include a resistive portion resulting from the transport current flow. To avoid that, an additional resistance Ž R add in Fig. 1. with a value much higher than R Ag was included in the simulation model to make sure that practically no transport current flows in the Ag sheath. It has been found that for a value of R add G 0.1 V, the power loss in the sheath remains constant whatever the increase of the additional resistance,
Fig. 1. Electrical model of the BSCCO tape simulation
210
N. Nibbio et al.r Physica C 310 (1998) 208–212
which means that all the loss there is a result of eddy current flow only. The eddy current loss distribution in the sheath depends on the current density and magnetic field profiles in the BiŽ2223. tapes. In order to evaluate the relevance of the obtained power loss distribution, we have made a comparison between the simulated current density and magnetic field distributions with the analytical expressions for a HTS strip of infinitesimal thickness with applied transport current, given by Brandt and Indenbom w6x. In Brandt’s analysis the quantity Jc is given in Arm. In order to have the same dimension for the current density, the calculated value of J w Arm2 x has been integrated along the thickness of the tape. The results for i s 0.25 are shown in Fig. 2 Ž i s IprIc , Ic being the critical current.. As can be seen, the results from the simulation are very close to the theoretical current density and magnetic field distributions. This has confirmed the aptitude of using the simplified electrical tape model from Fig. 1. However, for values of i ) 0.5, there is an increased current density at the superconductor– silver interface, and this model is not suitable anymore w7x. Therefore, the value of the transport current amplitude, which has been used in all simulations, is 4 A. This means i is always - 0.5 since all simulated
Fig. 2. Comparison between the simulated current density and magnetic field distributions with the analytical ones derived by Brandt. Simulated is a monocore sample tape.
Fig. 3. Current density distribution in a 37-filament tape.
tapes have critical currents above 10 A, and for this choice of i the simulated field and current density are close to the analytical ones. Fig. 3 shows the distribution of the current in a BiŽ2223. multi-filamentary tape. The current density, hence the magnetic field amplitude is higher in the outer filaments, close to the Ag-BSCCO interface.
4. Effect of the edge geometry and the sheath resistivity The effect of the tape geometry has been studied with respect to the variation of two parameters — the sheath thickness, and the width of the sheath edge. The ’thin’ and ’thick’ tapes from Fig. 4 are
Fig. 4. Density of power dissipation ŽWrm3 . in the sheath for a 37-filamentary tape. Q —total power dissipation ŽmWrm.. Ža. ordinary tape Žsheath., 3.0=0.36 mm; Qs 0.022 mWrm; Žb. thin tape Žsheath., 3.0=0.32 mm; Qs 0.019 mWrm; Žc. thick tape Žsheath., 3.0=0.44 mm; Qs 0.026 mWrm; Žd. narrow tape Žsheath., 2.85=0.36 mm; Qs 0.012 mWrm; Že. wide tape Žsheath., 3.2=0.36 mm; Qs 0.055 mWrm.
N. Nibbio et al.r Physica C 310 (1998) 208–212
obtained by decreasing and increasing the sheath thickness respectively by 1r2 and by 2. And the ’narrow’ and ’wide’ tapes — by decreasing and increasing the edge width by the same factors Ž1r2 and 2.. The simulated results show clearly that most of the power dissipation in the sheath occurs at the edges, where the self-field magnetic field has highest amplitude. The importance of the sheath thickness is much less pronounced. The tape with the wider edges has the highest loss, while the tape with the thinner edges has the lowest loss. The effect of doubling or cutting the edge by a factor of two leads to a double Žor half. loss dissipation. On the other hand, if one doubles Žor cuts. the sheath thickness, the increase Ždecrease. in eddy current loss is only 20–30%. Next, Fig. 6 shows the eddy current loss as a function of the frequency for the four simulated tapes with different number of filaments, based on real tapes geometry and dimensions Žsee Fig. 5.. From the current density and magnetic field distributions, it is expected that the highest loss would be obtained for the tape with largest edges. In addition, the multicore tapes are supposed to have higher eddy current loss than the monocore one. Fig. 6 shows that all curves follow a slope of 2 on a log–log frequency-loss plot. This is exactly the f-square dependence derived in the theoretical calculations for eddy current loss w1,3x, and proves that the power dissipation in the sheath is a result from eddy current flow and not from transport current flow Žits resistive loss would have been frequency independent.. The 1 and 7-filamentary tapes have the largest edges. The 7-filamentary tape indeed exhibits the
Fig. 5. Typical cross-sections for 1, 7, 19 and 37-filament tapes used in the simulations.
211
Fig. 6. Eddy currents loss as a function of the frequency for tapes with different filament number.
highest losses. The monocore loss is between the loss lines for the 19 and 37-filamentary tapes. The effect of eddy currents inside the Ag-sheath has been evaluated for several frequencies Ž59,500 and 1000 Hz, and for 3 different values of the metal sheath resistivity. Fig. 7 shows the expected dependence Ž1rr . of eddy current loss on the sheath composite resistivity w1,3x. The measured values of
Fig. 7. Eddy current loss as a function of the metal sheath resistivity. Plotted are the eddy current-loss factors, which multiplied by f 2 , give the power dissipation in Wrm.
212
N. Nibbio et al.r Physica C 310 (1998) 208–212
Žfor pure Ag, AgMgNi and AgAu-alloy sheaths are respectively 2.6 = 10y9 V m, 12.8 = 10y9 V m, and 15.1 = 10y9 V m.
main geometry parameter determining the eddy current loss in BiŽ2223. tapes.
References 5. Conclusions The electromagnetic simulations have shown that the eddy current loss in the AgrAg-alloy sheath is proportional to the square of the transport current frequency and inversely proportional to the resistivity of the metal. The geometry of the sheath is very important for the magnitude of the power dissipation in the silver. Most power is dissipated in the sheath edges, therefore it is the edge width, which is the
w1x H. Ishii, S. Hirano, T. Hara, J. Fujikami, K. Sato, Cryogenic 36 Ž1996. 697. w2x T. Fukunaga, A. Oota, Physica C 251 Ž1995. 325. w3x K.-H. Muller, Physica C 281 Ž1997. 1. w4x Flux2d, CEDRAT, 38246 Meylan, France. w5x G. Guerin, Determination des courants de Foucault dans les cuves des transformateurs, Ph.D. Thesis, ENSIEG, 1995, p. 43–44. w6x E.H. Brandt, M. Indenbom, Phys. Rev. B 48 Ž1993. 12893. w7x N. Nibbio, S. Stavrev, B. Dutoit, presented at X International Symposium on Superconductivity, 1997, Gifu, Japan.
Effect of the sheath resistivity and tape geometry on eddy current loss in Bi ž2223 / tapes Nadia Nibbio ) , Svetlomir Stavrev, Bertrand Dutoit Circuits and Systems, Swiss Federal Institute of Technology— Lausanne, DE-CIRC, EPFL, CH-1015 Lausanne, Switzerland
Abstract We have made a software electromagnetic simulation of eddy current effects in BiŽ2223.Ag sheathed tapes with applied AC transport current in a broad frequency range. We have used simple models of monocore and multicore tapes. Calculated is the power distribution ŽWrm3 . within the metal sheath using the geometry of real tapes. The numerical simulation is carried out with Finite-element-method software. The paper also describes the influence of the tape cross-sectional architecture and the sheath resistivity on the eddy current loss. q 1998 Published by Elsevier Science B.V. All rights reserved. Keywords: BiŽ2223. tape; Eddy current loss; Finite-element-method
1. Introduction One of the future large scale industrial applications of HTS materials, such as BiŽ2223., will be their usage in electric power transmission systems. In the self-field of an applied transport current only, the AC losses in the HTS conductors are mainly hysteretic, in the BiŽ2223. core, and eddy current, in the metal sheath w1x. Hysteresis loss is dominating in the power frequency range Ž40–180 Hz., while eddy current loss has an increasing contribution at frequencies above 200 Hz w2x. The theoretical derivations of eddy current loss in self-field have different dependence on the tape geometry and transport current amplitude Žw1,3x., so it is necessary to evaluate the eddy current loss contributions by other means, e.g., by software electromagnetic simulations.
2. Finite element method simulation For the specific geometry of a HTS tape and its components, there is no analytical solution for the current distribution yet, so it is necessary to solve this problem by using a 2-D numerical analysis. To perform the simulation of the electromagnetic behavior of the tape, a Finite-element-method software package has been used—FLUX2D w4x. The numerical computation is performed using a formulation of the Maxwell’s equations, where A is potential vector, J is current density and V is voltage gradient. The introduction of the potential vector in Faraday’s and Ampere’s laws leads to the equation below:
== )
Corresponding author. E-mail: [email protected]
ž
1
m
/
==A qs
EA Et
sJ
where s is the conductivity of the material.
0921-4534r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 4 6 3 - 8
Ž 1.
N. Nibbio et al.r Physica C 310 (1998) 208–212
The 3-D numerical solution of Eq. Ž1. in terms of finite element expression is: 1
HV
m
Ž = = W . Ž = = A. q
1
m
Ž = P W . Ž = P A.
qj vs WA q s W= V d V s 0.
Ž 2.
And
HV Ž jvs= w P A q s= w= V . d V s 0
Ž 3.
where the integration is over the domain V ; w and W are weighting parameters, and m the permeability w5x. The total power loss is then calculated as: Qe s
Hcycled tHV E P JdÕ.
Ž 4.
3. Electrical model of Bi(2223) r Ag tape The primary goal of our study is the evaluation of eddy current loss in the metal sheath of BiŽ2223. tapes. For this reason the nonlinear hysteretic behavior of the BSCCO core was not taken into account. We have used the following simple electrical model for the simulation of applying a transport current through a BiŽ2223. tape and the resulting self-field effects: The simulated system is either a monocore or a multicore BiŽ2223. tape. The model consists of a superconductive core in parallel with the Ag ŽAg-alloy. sheath. The tape is characterised by means of the resistivity for each material ŽAg and BSCCO.. The simulation set-up consists of a current source which is connected to two parallel massive conductors, respectively the BSCCO material and the Ag sheath. Both materials are assumed isotropic. The main parameters of the electromagnetic simulation are the resistivity r, as well as the amplitude and the frequency of the transport current. There is neither a current nor a magnetic field inside the tape prior to applying the AC transport current Žzero initial conditions.. The main difficulty of the numerical simulation is the evaluation of the electrical and magnetic properties of the superconducting core. The relation-
209
ship between B and H is considered linear — B s m 0 H, i.e., the relative permeability of the superconductor has been given a value of 1. The ’effective’ resistivity of the BSCCO is calculated from the total measured loss in real BiŽ2223. tapes: P TOTAL s
1 2
2 R eff Irms .
Ž 5.
Knowing the filling factor of the tape and the measured resistivity of the Ag sheath, the effective resistivity of the superconductor is then deduced. Thus the superconductive material is modeled as a conductor with very low resistivity calculated from the measured total AC loss for a given value of the current amplitude at a given frequency. For example, from the above equation, for a monocore tape of dimensions 2.75 = 0.32 mm with filling factor 0.45, the effective resistance of the superconductor is calculated to be 1.12 mV, while the resistance of the Ag-sheath is 5.3 m V Žat 59 Hz and Ip s 4 A.. There is a difference of 3 orders of magnitude between the resistance of the superconductive core and the metal sheath. In this way the portion of the transport current flowing through the sheath would be around 1r1000 of the applied transport current. Although in reality there indeed exists a partitioning of the current between the core and the sheath, it is probably of much smaller magnitude. The calculated power dissipation in the sheath would inevitably include a resistive portion resulting from the transport current flow. To avoid that, an additional resistance Ž R add in Fig. 1. with a value much higher than R Ag was included in the simulation model to make sure that practically no transport current flows in the Ag sheath. It has been found that for a value of R add G 0.1 V, the power loss in the sheath remains constant whatever the increase of the additional resistance,
Fig. 1. Electrical model of the BSCCO tape simulation
210
N. Nibbio et al.r Physica C 310 (1998) 208–212
which means that all the loss there is a result of eddy current flow only. The eddy current loss distribution in the sheath depends on the current density and magnetic field profiles in the BiŽ2223. tapes. In order to evaluate the relevance of the obtained power loss distribution, we have made a comparison between the simulated current density and magnetic field distributions with the analytical expressions for a HTS strip of infinitesimal thickness with applied transport current, given by Brandt and Indenbom w6x. In Brandt’s analysis the quantity Jc is given in Arm. In order to have the same dimension for the current density, the calculated value of J w Arm2 x has been integrated along the thickness of the tape. The results for i s 0.25 are shown in Fig. 2 Ž i s IprIc , Ic being the critical current.. As can be seen, the results from the simulation are very close to the theoretical current density and magnetic field distributions. This has confirmed the aptitude of using the simplified electrical tape model from Fig. 1. However, for values of i ) 0.5, there is an increased current density at the superconductor– silver interface, and this model is not suitable anymore w7x. Therefore, the value of the transport current amplitude, which has been used in all simulations, is 4 A. This means i is always - 0.5 since all simulated
Fig. 2. Comparison between the simulated current density and magnetic field distributions with the analytical ones derived by Brandt. Simulated is a monocore sample tape.
Fig. 3. Current density distribution in a 37-filament tape.
tapes have critical currents above 10 A, and for this choice of i the simulated field and current density are close to the analytical ones. Fig. 3 shows the distribution of the current in a BiŽ2223. multi-filamentary tape. The current density, hence the magnetic field amplitude is higher in the outer filaments, close to the Ag-BSCCO interface.
4. Effect of the edge geometry and the sheath resistivity The effect of the tape geometry has been studied with respect to the variation of two parameters — the sheath thickness, and the width of the sheath edge. The ’thin’ and ’thick’ tapes from Fig. 4 are
Fig. 4. Density of power dissipation ŽWrm3 . in the sheath for a 37-filamentary tape. Q —total power dissipation ŽmWrm.. Ža. ordinary tape Žsheath., 3.0=0.36 mm; Qs 0.022 mWrm; Žb. thin tape Žsheath., 3.0=0.32 mm; Qs 0.019 mWrm; Žc. thick tape Žsheath., 3.0=0.44 mm; Qs 0.026 mWrm; Žd. narrow tape Žsheath., 2.85=0.36 mm; Qs 0.012 mWrm; Že. wide tape Žsheath., 3.2=0.36 mm; Qs 0.055 mWrm.
N. Nibbio et al.r Physica C 310 (1998) 208–212
obtained by decreasing and increasing the sheath thickness respectively by 1r2 and by 2. And the ’narrow’ and ’wide’ tapes — by decreasing and increasing the edge width by the same factors Ž1r2 and 2.. The simulated results show clearly that most of the power dissipation in the sheath occurs at the edges, where the self-field magnetic field has highest amplitude. The importance of the sheath thickness is much less pronounced. The tape with the wider edges has the highest loss, while the tape with the thinner edges has the lowest loss. The effect of doubling or cutting the edge by a factor of two leads to a double Žor half. loss dissipation. On the other hand, if one doubles Žor cuts. the sheath thickness, the increase Ždecrease. in eddy current loss is only 20–30%. Next, Fig. 6 shows the eddy current loss as a function of the frequency for the four simulated tapes with different number of filaments, based on real tapes geometry and dimensions Žsee Fig. 5.. From the current density and magnetic field distributions, it is expected that the highest loss would be obtained for the tape with largest edges. In addition, the multicore tapes are supposed to have higher eddy current loss than the monocore one. Fig. 6 shows that all curves follow a slope of 2 on a log–log frequency-loss plot. This is exactly the f-square dependence derived in the theoretical calculations for eddy current loss w1,3x, and proves that the power dissipation in the sheath is a result from eddy current flow and not from transport current flow Žits resistive loss would have been frequency independent.. The 1 and 7-filamentary tapes have the largest edges. The 7-filamentary tape indeed exhibits the
Fig. 5. Typical cross-sections for 1, 7, 19 and 37-filament tapes used in the simulations.
211
Fig. 6. Eddy currents loss as a function of the frequency for tapes with different filament number.
highest losses. The monocore loss is between the loss lines for the 19 and 37-filamentary tapes. The effect of eddy currents inside the Ag-sheath has been evaluated for several frequencies Ž59,500 and 1000 Hz, and for 3 different values of the metal sheath resistivity. Fig. 7 shows the expected dependence Ž1rr . of eddy current loss on the sheath composite resistivity w1,3x. The measured values of
Fig. 7. Eddy current loss as a function of the metal sheath resistivity. Plotted are the eddy current-loss factors, which multiplied by f 2 , give the power dissipation in Wrm.
212
N. Nibbio et al.r Physica C 310 (1998) 208–212
Žfor pure Ag, AgMgNi and AgAu-alloy sheaths are respectively 2.6 = 10y9 V m, 12.8 = 10y9 V m, and 15.1 = 10y9 V m.
main geometry parameter determining the eddy current loss in BiŽ2223. tapes.
References 5. Conclusions The electromagnetic simulations have shown that the eddy current loss in the AgrAg-alloy sheath is proportional to the square of the transport current frequency and inversely proportional to the resistivity of the metal. The geometry of the sheath is very important for the magnitude of the power dissipation in the silver. Most power is dissipated in the sheath edges, therefore it is the edge width, which is the
w1x H. Ishii, S. Hirano, T. Hara, J. Fujikami, K. Sato, Cryogenic 36 Ž1996. 697. w2x T. Fukunaga, A. Oota, Physica C 251 Ž1995. 325. w3x K.-H. Muller, Physica C 281 Ž1997. 1. w4x Flux2d, CEDRAT, 38246 Meylan, France. w5x G. Guerin, Determination des courants de Foucault dans les cuves des transformateurs, Ph.D. Thesis, ENSIEG, 1995, p. 43–44. w6x E.H. Brandt, M. Indenbom, Phys. Rev. B 48 Ž1993. 12893. w7x N. Nibbio, S. Stavrev, B. Dutoit, presented at X International Symposium on Superconductivity, 1997, Gifu, Japan.