A predictive model of the temperature dependence of AC transport losses in (Bi, Pb) 2Sr2Ca2Cu3Ox tapes

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A predictive model of the temperature dependence of AC transport losses in (Bi, Pb)2Sr2Ca2Cu3Ox tapes

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 Supercond. Sci. Technol. 24 085008 (http://iopscience.iop.org/0953-2048/24/8/085008) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

SUPERCONDUCTOR SCIENCE AND TECHNOLOGY

Supercond. Sci. Technol. 24 (2011) 085008 (4pp)

doi:10.1088/0953-2048/24/8/085008

A predictive model of the temperature dependence of AC transport losses in (Bi, Pb)2Sr2Ca2Cu3Ox tapes Guo Min Zhang1,2 , Liang Zhen Lin1,2 , Li Ye Xiao1,2 , Yun Jia Yu1,2 , Justin Schwartz3 and Sastry V Pamidi4 1

Key Laboratory of Applied Superconductivity, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 2 Institute of Electrical Engineering, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China 3 Department of Materials Science and Engineering, North Carolina State University, Raleigh, NC 27695, USA 4 Center for Advanced Power Systems, Florida State University, Tallahassee, FL 32310, USA

Received 14 February 2011, in final form 24 May 2011 Published 30 June 2011 Online at stacks.iop.org/SUST/24/085008 Abstract Critical currents and AC losses of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox superconducting tapes were measured in self-field as a function of temperature. The experimental data of the temperature dependence of critical current were compared with calculated results. An approach to calculating AC losses as a function of temperature was developed and the calculated AC losses were compared with the measured data. The study shows that AC losses at any temperature can be estimated using the model from the critical parameters or from the measured AC loss factor at a certain temperature, such as 77 K. (Some figures in this article are in colour only in the electronic version)

self-field as a function of temperature ranging from 45 K to about the critical temperature are reported. Furthermore, a theoretical model describing the temperature dependence of the AC losses is proposed based on Norris’s formula and a model of the temperature dependence of the critical current. A semi-empirical formula that calculates accurately the temperature dependence of AC losses is presented. A comparison between the measured data and the model is presented.

1. Introduction In practical power applications, high temperature superconducting (HTS) power devices may operate over a wide range of temperatures. This includes the likely presence of significant temperature gradients in devices such as motors and generators, particularly if the magnet is conduction cooled. Critical performance metrics of HTS conductors, including the critical current density and AC losses, are strongly temperature dependent. Hence, a theoretical or empirical model that describes the temperature dependent critical current density and AC losses is necessary for optimizing the design of both the HTS device and the associated cryogenic systems. Many researchers have studied the temperature dependence of critical currents of HTS tapes both experimentally and theoretically [1–6]. Some experimental studies have been carried out on AC losses of HTS tapes at different temperatures [7–10], but theoretical or empirical models that predict the temperature dependence of AC losses are rare. In this paper, results of critical current and AC transport loss measurements of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox tapes (Bi2223/Ag) in 0953-2048/11/085008+04$33.00

2. Experimental approach The tape samples used in the experiments were from a batch of 55-filament Ag-sheathed, non-twisted (Bi, Pb)2 Sr2 Ca2 Cu3 Ox tape with a 4.1 mm × 0.3 mm cross section manufactured by America Superconductor Corporation. Its critical temperature was 110 K. The length of the sample was 15 cm, and the distance between voltage taps for critical current and AC loss measurements was 5 cm. To maintain a uniform temperature, the sample was placed on a sapphire plate which was thermally anchored to the cold 1

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Supercond. Sci. Technol. 24 (2011) 085008

G M Zhang et al

Figure 1. Calculated and measured self-field critical currents of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox /Ag tape as a function of temperature.

Figure 2. AC losses of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox /Ag tape, at 50 Hz, as a function of normalized transport current ( Im /Ic (T )) for different temperatures. Here Im is the peak value of the transport current, whereas Ic (T ) is the critical current at temperature T .

head of a cryocooler. The experimental chamber was filled with helium gas to minimize the temperature gradient along the sample. The sample temperature was monitored by a Cernox temperature sensor mounted on the sample surface. The temperature was controlled by a Lakeshore temperature controller and a resistive heater mounted on the sample holder. The temperature fluctuation during the measurements was less than ±0.2 K. Critical current of the tape was measured using the four-probe method with a 1 μV cm−1 electric field criterion. AC transport losses were measured using the electrical method with a lock-in amplifier [11, 12]. Details of the measurement procedures were reported previously [13, 14].

with γ = 1.76, (where γ is calculated from the critical current value at 45 K and equation (3)). The calculated values fit the measured results well. AC losses of Bi2223/Ag tape were measured from 45 to 93 K with an AC transport current frequency of 50 Hz. Figure 2 shows the AC losses as a function of normalized transport current and temperature which were published in [13]. The AC transport loss per cycle decreases with increasing temperature for the same normalized current, suggesting that AC losses are temperature dependent. The AC transport loss curves at different temperatures are nearly linear with slopes n of about 3.3, indicating that the AC losses are dominated by hysteresis losses [20, 21]. Dividing the AC transport loss, P(T ), by the square of the corresponding critical current, one obtains the AC loss factor, F(T ). Figure 3 shows the AC loss factor as a function of normalized transport current and temperature [13]. Here, the data shown by symbols are the measured data and the dashed line is calculated from Norris’s elliptical formula [22]:

3. Results The critical current of an HTS tape as a function of temperature ( Ic (T )) is often described by [15–17]:   Tc − T γ Ic (T ) = Ic (Tref ) . (1) Tc − Tref In equation (1), Ic (0) is the critical current at 0 K, γ is a fitting parameter, Tref is the reference temperature or operating temperature, and Tc is the critical temperature. For (Bi, Pb)2 Sr2 Ca2 Cu3 Ox , Tc is 110 K. Typically γ = 1 for low temperature superconductors [18, 19] but typically γ > 1 for Bi2223 tapes. For a commercial HTS tape, the critical currents at 77 and 4.2 K are often standard parameters provided by the vendor. Thus, with Tref = 77 K, equation (1) becomes   Tc − T γ Ic (T ) = Ic (77 K) . (2) Tc − 77

F(T, i ) =

μ0 P(T, i ) = {(1 −i ) ln(1 −i ) + (2 −i )i /2}, (4) 2 Ic (T ) π

where i = Im /Ic (T ) is the normalized current. The solid line is the fitted loss factor at 77 K, which can be described by the expression: log F(77 K) = 3.32 log i − 6.80.

γ can then be determined by measuring the critical current at a certain temperature, such as 4.2 K, or some other temperature of interest using:     I (T ) Tc − 77 K . γ = log log (3) I (77 K) Tc − T

(5)

4. Discussion Although the AC transport losses are temperature dependent, the AC loss factor curves coincide with one another on the same fitted line at 77 K; thus, the AC loss factors are temperature independent. In addition, the loss factors are well predicted by Norris’s formula [13]. These results indicate that

The measured Ic (T ) from 45 to 106 K is shown in figure 1. The solid line represents the values calculated from equation (2) 2

Supercond. Sci. Technol. 24 (2011) 085008

G M Zhang et al

Figure 4. Measured AC losses of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox tape, at 50 Hz, as a function of normalized transport current and temperature, and the corresponding values calculated using equation (6). The symbols represent the measured data from figure 2 and the solid lines represent the calculated values.

Figure 3. AC loss factor for (Bi, Pb)2 Sr2 Ca2 Cu3 Ox /Ag tape, at 50 Hz, as a function of normalized transport current ( Im /Ic (T )) for different temperatures.

the AC loss factor at any temperature can be replaced by the loss factor at 77 K or that calculated from Norris’s formula. As a result, considering equation (4), the AC transport losses at any temperature can be predicted from Ic (T ) and either the measured or calculated loss factor at 77 K. From equations (1) and (4), and taking Tref as 77 K, we obtain the temperature dependent AC loss formula:   μ0 2 Tc − T 2γ I P(T, i ) = (77 K) π c Tc − 77 × {(1 − i ) ln(1 − i ) + (2 − i )i /2}. (6) Thus, AC losses of Bi2223/Ag at any temperature can be estimated from the critical current at 77 K. Figure 4 compares the measured results of AC losses as a function of temperature with the corresponding calculated values from equation (6). The calculated values fit the measured results well. Some deviation is due to the fact that the measured loss factors are a little higher than that of the theoretical line of Norris’s formula (see figure 3) which is for uniform elliptical shapes while the cross section of the real sample is not rigid elliptical. The match, however, is sufficiently good to be used for device design. Applying the 77 K loss factor formulation, one can obtain a semi-empirical formula to calculate AC losses at other temperatures:

 P(T, i ) = F(77 K)Ic2 (77 K)

Tc − T Tc − 77

Figure 5. The measured AC losses of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox tape, at 50 Hz, as a function of normalized transport current and temperature, and the corresponding calculated values using the fitted loss factor. The symbols are the measured data from figure 2 and the solid lines are from equation (7).

5. Conclusions AC transport losses of (Bi, Pb)2 Sr2 Ca2 Cu3 Ox tape in self-field at different temperatures are dominated by hysteresis losses, which are determined by the critical currents of the tapes. The loss factors at different temperatures fit onto a universal curve that is described well by Norris’s loss factor formula or by the fitted line at 77 K. The temperature dependent critical current can be described by expression (1) and the temperature dependent AC losses can be estimated from the critical temperature, critical current at 77 K, and one straightforward equation. Thus, extensive AC loss characterization is not required in order to design AC HTS devices; instead standard measurements of the critical current and temperature are sufficient.

2γ (7)

where F(77 K) is given by equation (5). The comparison of the measured AC losses for different temperatures with the corresponding calculated values using equation (7) is shown in figure 5. The calculated values match the measured results well because the measured loss factors are well described by the fitted line at 77 K. Hence, the semiempirical formula (7) can be used to predict AC transport losses at other temperatures from the measured data at any reference temperature, and most commonly 77 K.

Acknowledgments This work was supported by the National Natural Science Foundation of China (No. 50977092) and the Chinese 3

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G M Zhang et al

Academy of Sciences, and was partly supported by A Foundation for the Author of National Excellent Doctoral Dissertation of PR China (No. 200750).

[11] Zhang G M, Lin L Z, Xiao L Y and Yu Y J 2003 Cryogenics 43 25 [12] Zhang G M, Lin L Z, Xiao L Y, Qiu M and Yu Y J 2003 IEEE Trans. Appl. Supercond. 13 2972 [13] Zhang G M, Schwartz J, Sastry P, Lin L Z, Xiao L Y and Yu Y J 2004 Supercond. Sci. Technol. 17 1018 [14] Zhang G M, Knoll D J, Nguyen D N, Sastry P V P S S, Wang X R and Schwartz J 2007 IEEE Trans. Appl. Supercond. 17 3874 [15] Wesche R 1995 Physica C 246 186 [16] Curr´as S R, Vi˜na J, Ruibal M, Gonz´alez M T, Osorio M R, Maza J, Veira J A and Vidal F 2002 Physica C 372 1095 [17] Duron J, Grilli F, Antognazza L, Decroux M, Dutoit B and Fischer Ø 2007 Supercond. Sci. Technol. 20 338 [18] Wilson M N 1987 Superconducting Magnets (Oxford: Oxford University Press) [19] Iwasa Y 1994 Case Studies in Superconducting Magnets Design and Operational (New York: Plenum) [20] Ashworth S P 1994 Measurements of ac losses due to transport current in bismuth superconductors Physica C 229 355 [21] Ciszek M, Tsukamoto O, Amemiya N, Ueyama M and Hayashi K 1999 Angular dependence of ac transport losses in multifilamentary Bi-2223/Ag tape on external dc magnetic fields IEEE Trans. Appl. Supercond. 9 817 [22] Norris W T 1970 J. Phys. D: Appl. Phys. 3 489

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