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In a steady state, Ampere’s law, Eq. (9.14), holds for a closed line, C, with different surfaces on it, S1 and S2, as shown in Fig. 11.1a, b. If the magnetic field \(\boldsymbol{H}\) is integrated on C in opposite directions as drawn in Fig. 11.1a, b, the sum of the two integrations is naturally zero. At the same time, the sum of the surface integrals of the current density \(\boldsymbol{i}\) on S1 and S2 is also zero. This sum becomes the surface integral on closed surface S12 composed of S1 and S2 (see Fig. 11.1c). Thus, we have

We know that there exists a force between currents. This phenomenon is similar to the Coulomb force between electric charges. Hence, we can presume that currents also make some field in space similar to the electric field made by electric charges. This field is called the magnetic field.

Based on a statistic model describing the critical current distribution, we proposed a method to describe the current-transport properties in HTS. The fitting error between the analytical and measured current-voltage characteristics was analyzed. It has been shown that the analytical form can be fitted with the measurement in wide conditions of magnetic field and temperature. These results indicate that the statistic model is suitable to analyze the transport properties in HTS. Using this method, we can estimate the homogeniety of the material as well as the pinning strength.

One of the features in high-T c superconductors is that the transition from the superconducting state into the resistive state is not as clear as in low-T c superconductors. On the other hand, it has been shown that the electric field (E) vs. current density (J) characteristics in high-Tc superconductors follow the Glass-Liquid (G-L) transition. Since this transition line is close to the irreversibility line, the flux pinning phenomenon near this line is important for practical application of high T c superconductors. We analyzed the vortex dynamics in a medium with randomly distributed pins using Monte-Carlo simulation method. In this report, the correlation between the flux lattice and distributed pins is discussed in the vicinity of the transition line.

To investigate the anisotropic characteristics in YBCO superconductors, we measured the electric field (E) vs. current density (J) characteristics at various conditions such as temperature, T, magnetic field, B, and field angle, θ. We extracted the parameters related to the distribution of the critical current density, J c and estimated the anisotropic properties with respect to the scaling characteristics of the pinning force densities and the transition fields.

Ag-sheathed Bi2223 multifilamentary tapes with and without a layer of Sr6V2O11(SVO) barrier outside and inside each filament were prepared. The SVO barrier layers have no bad effects on the critical current density Jc of the tapes. Especially, Jc increases obviously in the tapes coated with SVO layers both inside and outside each filament, due to the enhancement of Bi2223 phase formation by the reaction of the inside-coated SVO with Bi-based oxide core. This results suggest that even if the SVO barriers react with Bi oxide core due to the break of Ag sheath, the reaction have no bad effect, rather good effect on the Jc values. SVO will be one of most important candidates for barriers.

To submit residual resistance ratio (RRR) measurement method of Cu/Nb3Sn composite conductor as one of new work items to IEC-TC90 in the near future, inter-comparison tests have been carried out. Four laboratories joined the tests using the bronze-processed Cu/Nb3Sn with the external stabilizer as a sample. Laboratory W reacted and measured samples A, B and C, then they were distributed to the other three laboratories (X, Y and Z). Laboratories X, Y and Z measured samples A, B and C, respectively and those data were compared with those of W. As a result, coefficient of variation, COV, of RRR values of 6.7% was obtained. Thermoelectric voltages were also measured in the tests and it was found that it did not affect scattering of RRR measurement if we changed the polarity. If allowable target precision is extended up to around 10%, we will reach the goal of the standardization in RRR measurement method of Cu/Nb3Sn in a year.

We have essentially improved the “LC resonator” method for high-resolution measurements of penetration depth λ of radio frequency magnetic field into thin films and small-size plate-like high-Tc superconductors by replacing the solenoid testing coil by the flat one driven by a highly stable-frequency and low-power tunnel diode oscillator. The advantages of the new technique such as high resolution-Δλ~ 1–3A (Δλ/λ ~ 2–10-6), ability of reliable operation in wide ranges of temperature and magnetic field, etc. enable to use it, in particular, for many-sided studies of the peculiarities of vortex dynamics in thin-film high-Tc superconductors with high resolution. We used it for determination of some material parameters in high-Tc cuprates. Namely, the measurements of the field-dependent radio frequency (~23MHz) penetration depth λ(H,T) allowed to determine the temperature dependencies of the upper critical field, Hc2(T), the Labusch parameter, α(T), and the depinning field value in YBaCuO thin-film ring specimen.

Because of the large heat capacity, the influence of local disturbances on the stability is quite small in high Tc superconductors (HTSC). In addition, smooth voltage-current characteristics make resistive state in HTSC very stable. However, a design of HTSC coils using the critical current density conventionally defined by usual criterions is difficult, because these characteristics change nonlinearly with temperature and/or magnetic field. Therefore, it is necessary to estimate the thermal stability of HTSC coil based on the transport property and the quench property. In this study, we analyze the thermal stability in a Bi-2223 superconducting solenoid coil by using the finite element method with taking account of the nonlinear resistance.

An international round robin test of the residual resistance ratio (RRR) measurement was carried out on a Cu/Nb-Ti composite superconductor following a Working Draft for the test method prepared by Working Group 4 in IEC/TC90. Thirteen institutes from five countries participated in the round robbin test. The effect of the bending strain on the resistance at cryogenic temperatures was corrected using an empirical formula. The average and the standard deviation of RRR were 178.5 and 4.4, respectively. As a result the coefficient of variation (COV) was 2.44 %. Since the target of COV is 5 %, it is concluded that RRR can be correctly measured following the method in the Working Draft.

A new method for the study of the magnetization of thin films of magnetic materials is described. It is based on detecting of the inductance change of an Open-Flat Coil (OFC) which leads to the frequency changes of tunnel diode oscillator. The method is sensitive to small changes of distances with the 1A resolution. It can detect the changes ∼10−12H of HTSC film’s magnetic inductance originated from the variation of its internal flux distribution. We present the basic principle and describe the initial version of OFC magnetometer. The measurement results on the YBCO thin film magnetization and the advantages of the new test method are discussed also.

Electric-field versus current-density (E-J) characteristics and scaling properties in YBa2Cu3O7-δ (YBCO) thin films have been studied as a function of magnetic-field under applying DC and AC transport currents. It was shown that the behavior of magnetic-field dependent In E-lnJ curves for the AC measurement was almost the same as the DC measurement, and the E-J characteristics were dominated by the flux-flow compared to the flux-creep in the electric-field range of 10-5 V/m to 101 V/m.

The residual resistance ratio (RRR) measurement method for Cu/Nb3Sn was selected to be one of the next new work proposal items for submission to IEC-TC90 in the near future. Inter-comparison tests have been carried out for some years in Japan and show that the RRR of Cu/Nb3Sn for the external Cu type is very sensitive to heat-treatment conditions and has large scattering. On the other hand, a relatively low scattering factor of under 9% was obtained in the test in which the scattering problems at heat-treatment were removed. If pre-reacted Cu/Nb3Sn is treated, the goal of the standardization in RRR measurement method will be attained before long.

The irreversibility field and the apparent pinning potential associated with the magnetic relaxation rate are observed for a Bi-2212 single crystal in the three-dimensional state of flux lines. It is tried to explain these irreversible properties by the flux creep theory using the same pinning potential. Fairly good agreements are obtained between theory and experiment. It is concluded that the flux creep theory correctly describes the irreversible properties of high-temperature superconductors.

For different Hg-based superconducting ceramics (Hg-1223 and (Hg, Pb)-1223), the irreversibility lines were determined by means of ac and dc magnetization measurements. In the case where the peak values of the imaginary parts of the ac susceptibility are used under certain conditions a double peak structure occurs in these parts, affecting the irreversibility line. It seems that this result is evident for different kinds of pinning, occurring in two phases with different superconducting properties.

A small superconducting transformer of 500 A class was designed and fabricated using a Bi-2223 tape for a compact AC current source for loss measurement. For the primary winding, a copper wire of 0.2 mmϕ was wound on a bobbin of 54 mm in diameter by 300 turns. A silver-sheathed Bi-2223 multifilamentary tape with the critical current of 45 A at 77 K was used for the secondary winding. Ten tapes connected in parallel were wound by 2 turns to attain a sufficient secondary current. An iron core was used to get a good coupling between the primary and secondary windings. The peak secondary current reached 841 A at 35 Hz when the primary current was 9.28 A in liquid nitrogen. A distortion was not detected in the secondary current in the frequency range of 35–2000 Hz. The AC current energy loss density was measured for a Y-123 bulk superconductor prepared by QMG method using the transformer.

A comparative study of dc and ac magnetization measurements on the (Hg,Pb)-1223 cuprate superconductor with T c = 133 K is presented. Hg0.7Pb0.3Ba2Ca2Cu3O8+δ bulk material was synthesized directly from the metal oxides without any precursors and characterized by X-ray diffraction and SEM. The irreversibility line is determined by different experimental techniques and the results are discussed in the framework of flux creep theory. Here B irr(t) α (I-t 2 ) n with n = 2m/(3-2γ) and t = T/T c is predicted. From the dc magnetization hysteresis the pinning parameter m and γ were obtained by assuming a scaling law Δm α (l-t 2 )m B γ-1 at low temperatures or low fields. We found m = 8.8 and γ = 0.6 leading to n = 9.8. This is in good agreement with the experimental finding at low temperatures. Furthermore the critical current density jc(T,B) was estimated by applying Beans critical state model where jc(T,B) α Δm. The in-plane penetration depth λ ab (t) was deduced by analyzing the linear behaviour of the reversible magnetization as a function of InB. A temperature dependence λ ab (t) = λ ab (0) (I-t) -1/2 with λ ab (0) = 1960 A is found.

The longitudinal and transverse flux bundle sizes are estimated for a three-dimensional single crystal Y-123 superconductor under a magnetic field along the c-axis and the result is compared with previous results of Bi-2212 and Bi-2223. It is considered that the longitudinal flux bundle size is determined only by the flux pinning strength independently of the dimensionality of superconductor. The number of flux lines in the flux bundle,g 2 , which is related to the transverse flux bundle size, is estimated to be larger than 8. This number is much larger than those in Bi-2212 and Bi-2223.

We tested a Bi-2212 based, multifilament high temperature superconducting tape in the silver matrix. The tape formed as a coil was instrumented by few potential taps and thermometers. The test was performed at the temperatures from 13K to 40K. Many potential taps attached, permit us to measure the critical current in different sections of the coil. We found that critical currents are sufficiently different along the tape length. The maximum critical currents are in the central part of the coil. Critical currents reduce almost ten times toward the joints with current leads. The thermometers attached to the sample confirmed the uniform temperature along the sample. It completely excludes the hypothesis about heating of the sample from the joints. We suggest that current transfer from the matrix to the superconductor may lead to the critical currents’ non-uniformity along the sample.

Electric field vs. Current density (E-J) characteristics in Y1Ba2Cu3O7-δ thin films were measured over more than 13 decades of electric field by use of the four probe method and the magnetization method. It was shown that even in the so-called glass state determined by the four probe method, concave InE-InJ curves were observed in the low electric field region obtained by the magnetization method. This means that the scaling is valid only in a limited electric field region. These results can be qualitatively interpreted by the dependence of the observable hopping rate of fluxoids on the electric field sensitivity from the view point of the percolation limit.

Force-displacement characteristic of flux lines in a superconducting Bi-2212 single crystal was measured using the Campbell method at low temperatures and low magnetic fields where the peak effect of the critical current density was observed. It was found that the interaction distance, d i, had a peak around the peak field, while the Labusch parameter, α L, increased monotonously with increasing magnetic field. The peak of d i means an increment of the reversible limit of motion of pinned flux lines. This seems to be caused by an introduction of some disorder in the flux line lattice at the so-called 3-dimensional to 2-dimensional transition field. However, the characteristic length of the deformation along the c-axis is considered to be much longer than the distance between the CuO2 planes.

The crystal axis of Hg-1223 powders was aligned in epoxy resin under a high magnetic field and the irreversibility field, B i, was estimated from the onset of the temperature dependence of the fundamental and third harmonic AC susceptibilities under a magnetic field parallel to the c-axis. When the temperature dependence of B i is expressed as [1 — (T/T c )2]n, n was large at low temperatures and reduced to 1.5 at high temperatures. This behaviour is the same as other oxide superconductors and indicates that two kinds of pinning centers work collectively. The flux creep theory with assuming two kinds of pinning centers successfully explains the temperature dependence of B i. Comparing the present result with the pinning force density dependence of B i in various oxide superconductors, the dimensionality of Hg-1223 was found to be close to that of Bi-2223 and Hg-1212.

It is known that the irreversibility line and the transition line between the so-called glass and liquid states of flux lines are different, although those are closely correlated to each other. This difference is speculated to be associated with the distribution width of the flux pinning strength. The two characteristic lines are analyzed as a function of the distribution width of the flux pinning strength using the flux creep-flow model for a Bi-2212 superconducting tape. Discussion is given on the difference between the two characteristic lines.

The distribution of pinning strength is estimated from measured current-voltage curves using the Weibull function for a Bi-2223 silver-sheathed tape wire under a magnetic field parallel to the c-axis. The obtained result is compared with a simple distribution function assumed in the analysis of the scaling of current-voltage curves using the flux creep-flow model. It is found that the simple distribution function describes well. The wide distribution seems to be caused by percolating current paths due to weak lines.

Ac transport current loss is measured for a QMG (Quench and Melt Growth) processed Y-123 bulk superconductor up to 300 A at 77.3 K. For this purpose a simple current transformer is used. The frequency of ac current is 400 and 550 Hz. The hysteresis loss extrapolated to zero Hz is compared with the prediction of the Irie-Yamafuji model where the critical current density is estimated from the hysteresis of dc magnetization. The obtained hysteresis loss is slightly smaller than the theoretical prediction. This is considered to be caused by the reduction in the critical current density due to the flux creep in the dc magnetization measurement.

Interlaboratory comparison tests of hysteresis loss measurements in Cu/Nb-Ti composite superconductors using vibrating-sample magnetometers (VSM) or SQUID magnetometers have been carried out. Two Cu/Nb-Ti wires with different filament diameters of 61 µm and 4.7 µm were prepared for the tests. The resulting coefficients of variation of hysteresis losses for 61 µm and 4.7µm wires were 5.2% and 2.6%, respectively, under the unified conditions: magnetic field amplitude of ± 1 T, measurement temperature of 4.5 K and sample length of above 30 mm. It was noted that the hysteresis losses for the 61 µm wire were strongly influenced by the sample length, whereas the losses for the 4.7 µm wire showed little influence.

Pinning potential depth U 0 estimated from Anderson-Kim model is generally known to increase with increasing temperature T at low T range. This anormalous behavior is considered to result from the model which assumes the linear dependence of the potential barrier on J and can be explained qualitatively by taking into account the nonlinear current dependent barrier energy U(J). Experimental confirmation of the U(J) relation in a wide range of J, however, will need an astronomically long observation time. By cooling a superconductor magnetized at higher temperature T o + ΔT down to an operating temperature T o (HTM), the flux distribution at T o+ΔT which nearly corresponds to a low J state at temperature T o is forced to transfer to low T o regime. From the observations of the flux creep after HTM on YBaCuO sample, the long term magnetic relaxation corresponding to over 10 days are obtained and the U(J) relation is discussed.

In order to investigate the effect of reversible motion of fluxoids, the 3rd harmonic of complex AC susceptibility was measured for Y-based superconducting powder and QMG-processed bulk specimens. The critical current density was estimated from the experimental results using the theoretical model of Campbell for the reversible motion of fluxoids and compared with the estimation from DC magnetization. Deviation from the estimated value from DC magnetization is remarkable when the applied AC magnetic field is small. It is considered that the reversible fluxoid motion at low AC magnetic fields is much more prominent than expected in the present theory.

The electric field E vs. current density J in two superconducting PbBi samples with quite different J c were measured at various magnetic fields and temperatures by using a usual resistive method. A nonlinear E-J characteristic similar to that in high T c superconductors was observed. Each E-J curve collapsed onto one of the two universal branches and a scaling behavior was recognized. It turned out that the flux pinning characteristic influences E-J curve and the scaling significantly. The stronger flux pinning makes T g or B g higher.

The structures of superconducting order parameter, magnetic flux density and the current density around a single flux line are calculated using the Ginzburg-Landau theory for various values of the Ginzburg-Landau parameter, κ. It is found that the effective radius of the normal core is much larger than the coherence length, ξ, expected usually in the local model. A new approximate formula of order parameter is proposed. The elementary pinning force of a wide superconducting-normal interface is approximately estimated with this result. The obtained elementary pinning force is about 1.7 times as large as that estimated from the local model.

The current-voltage curves are measured for two Bi-2223 tape wires at various temperatures under the magnetic field parallel to the c-axis. It is found that the current-voltage characteristics are scaled on two master curves by normalizing as predicted in the vortex glass-liquid transition theory. The obtained dynamic critical index, z, increases and the static one, v, decreases with increasing magnetic field. The transition line shifts to higher temperature with increasing critical current density of the wire. These experimental results are compared with the flux creep-flow theory in which the distribution of flux pinning strength is taken into account. It is found that the scaling behavior, the critical indices and the transition line are well describable by the flux creep-flow theory.

A sharp variation in the resistivity vs temperature was observed in a weakly pinned metallic Nb-Ti superconductor. This is attributed to other mechanisms than the melting of fluxoids, since the melting occurs only in the vicinity of the critical temperature. The experimental results are compared with the theoretical analysis based on the mechanism of flux creep and flow. The reason for such a sharp variation in the resistivity is discussed.

Ac inductive measurement on critical current densities, Jc’s, and flux motions was carried out for two Pb-Bi alloy specimens with different flux pinning strength. The interaction distance, di, was evaluated by fitting the numerically obtained flux profile with the experimental result. It was found that di depends more strongly on temperature in the specimen with weaker flux pinning strength. A discussion was given on the effect of flux creep on the temperature dependence of di.

Recently discovered Hg-based oxide superconductors are found to be less anisotropic than Bi-based ones and expected to have superior irreversible properties. In order to investigate the irreversible property in this material, the critical current density and the irreversibility line in sintered bulk Hg-1212 specimens are measured. The history effect of the critical current density due to the weak link property between grains is observed. The obtained irreversibility field is compared with the results for other various superconductors and it is found that the irreversibility field is as low as that in Bi-2223 superconductors. This result indicates that the superconductivity in the sintered Hg-1212 is as strongly two-dimensional as Bi-2223.

The pinning force density in Bi-2223 tape wires were measured at various temperatures under the magnetic field normal and parallel to the c-axis. It is found that the normalized pinning force density collapses to a master curve when it is potted versus the magnetic field normalized by the irreversibility field for the both field directions. The relation between the maximum pinning force density, Fp(max)(T), and the irreversibility field, Bi(T),is expressed as FP(max) α Bim′. The results are compared with the theoretical analysis based on the flux creep model. It is found that the experimental results are well explained by the theory, although the decrease in the observed pinning force density is more marked than the theoretical prediction.

Irreversibility line was studied for superconducting Pb-Bi alloys with different pinning strength. It was found that the magnitude and temperature dependence of the irreversibility field depend rather slightly on the flux pinning strength. The difference between the irreversibility field and the upper critical field remains finite even in specimens with very strong pinning force. The experimental result is compared with the flux creep theory and found to be explained well by the theory.

A superconducting bearing system using high TC superconductors is newly constructed. The superconducting bearing system consists of two superconducting bearings and a rotor. The rotor of the bearing system is self-centered in the speed range not exceeding 63,000 rpm. The displacements are smaller than 30 μmp − p except critical speeds. This paper describes the static characteristics such as load capacity of the system and the dynamic characteristics such as impulse responses of the rotor and displacements of the spin-down rotor.

The flux pinning strength dependence of the irreversibility field in Bi-2212 superconductors was investigated for a melt-processed bulk specimen and a single crystal specimen. The larger irreversibility field in the melt-processed specimen is attributed to its stronger flux pinning property. The results of the irreversibility field can be approximately explained by the flux creep theory. It is concluded that the irreversibility field is the depinning line due to the thermally activated fluxoids.

After a long period of creep process, the slope of the magnetic flux distribution becomes less steep and the flux distribution seems to be effectively equal to that magnetized at a higher temperature. Then the gentle creep is expected, if the flux distribution at an operating temperature T o is replaced with that at higher temperature T o+ΔT by cooling a YBaCuO superconductor magnetized at higher temperatures (HTM). By this HTM, drastic reductions in the creep rate by one order of magnitude were observed for a temperature increment ΔT ≥ 3K at T o ranging from 40 to 77K in melt-growth YBaCuO superconductors. Time scale shift t off falls the HTM creep data on the normal creep line. An expression for ΔT s is given, where a linear decay rate of critical current (1/J)(dJ/dt) at ΔT > ΔT s becomes small than an allowable linear rate S(sec-1) which is necessary for a superconducting device performance.

In order to investigate the macroscopic properties of weak links in a QMG-processed Y-Ba-Cu-O specimen, the temperature dependence and the magnetic field history dependence of the critical current density and the critical current density under a longitudinal magnetic field were measured. The history effect was observed simultaneously at low temperatures where the critical current density initially decreased sharply with increasing temperature. On the other hand, the longitudinal field effect was observed up to a medium temperature at which the critical current density shows the second rapid fall. Above this temperature region, the temperature dependence of the critical current density approached the common one determined by the flux pinning mechanism, suggesting that the transport current only through strongly bounded region. It is considered from the observed results that two kinds of weak links exist in the present specimen.