ac Losses in Superconductors
Authors: Wipf S. L. 
Published: February 2004

Since the discovery of technologically interesting high-field superconductors in 1961 more than 50 experimental and theoretical publications have appeared which are concerned with the relevant ac loss mechanisms. These papers are reviewed, and certain of the experimental findings are unified in the light of the present day understanding. In many cases seemingly disconnected and even apparently contradictory results are brought into consonance. In this review, theories and models as well as experiments are discussed, and brief mention is made of actual and proposed ac applications. For frequencies with which we are mainly concerned (&lt;104 sec−1) and for fields less than Hc and Hc1, respectively, for type I and type II superconductors, ideal homogeneous materials are loss free. Likewise, in dc background fields up to Hc3, losses remain negligible if the amplitude of the ac field stays below a value determined by intrinsic critical surface current densities. For larger amplitudes cyclical flux movement takes place inside the superconductor (in the intermediate or mixed state for type I or II, respectively) and flux-flow and eddy-current losses appear. In real nonideal superconductors much larger losses appear. These larger losses arise from flux pinning, which also accounts for the large dc transport currents which inhomogeneous materials can support in the mixed state. Although flux-pinning theories are mostly semiempirical and are incompletely developed they nevertheless provide a good understanding of many loss phenomena. Elastic properties of the fluxoid structure can also lead to resonant losses, as has been predicted and subsequently observed. In addition to direct measurement of losses, experimental work has also been concerned with related factors such as critical currents and fields. Various methods to measure power losses have been employed; in order of popularity they are: (1) Helium boil-off, (2) Phase shift between current and voltage, (3) Magnetization measurements (hysteresis of B-H curves), (4) Calorimetry, (5) Q of L-C circuits, (6) Q of cavities, and (7) Mechanical force or torque measurements. With the exception of (6), these methods are suitable for low frequencies and the first four yield most of the quantitative data. The materials most frequently investigated include pure Nb, Nb alloys with Zr or Ti, and pure Pb. One can distinguish different loss regimes depending on the amplitude Ha of the ac fields: (1) Ha&lt;Hc1, (2) Hc1&lt;Ha&lt;Hp, (3) Hp&lt;Ha&lt;Hc2, and (4) Hc2&lt;Ha&lt;Hc3. Here Hp is the field of complete penetration, i.e., the field above which the total volume of the superconductor sees an ac field. In general, only the first two regions have been studied extensively because the losses become very large for Ha&gt;Hp. Two important generalizations emerge from the present review of existing data: (1) The losses depend on the peak field to which the superconductor is subjected; it does not matter whether these fields are due to an ac current in the superconductor or are externally applied. (2) The loss per cycle is practically independent of frequency for frequencies less than or approximately equal to 104 sec−1. In agreement with these two points are loss measurements carried out by quasistatic cycling of the magnetic field and results which show that losses are independent of the ac waveform. A plot of all loss measurements as loss per cycle per unit surface area vs peak ac field at the surface exhibits a consistent trend all the way from 10−13 to 10−3 J/cm2/cycle. It shows reasonable agreement with theoretical models for Hc1&lt;Ha&lt;Hp, where the loss per cycle, in J/cm2, is 4.22×10−9 (Ha−ΔH)3/jc, with the peak field Ha in oersteds and the critical current density jc in A/cm2. In this formula ΔH is a field step at the surface, due to a superconducting surface current density. Like jc, ΔH may be different from specimen to specimen, and is field dependent, having roughly the size of Hc1 at Hc1 and getting smaller as the field increases. The measured losses in Nb alloys are almost all in this regime; only a few results are available for Ha&gt;Hp and then the losses are proportional to the volume. The much smaller losses in the Meissner effect regime (Ha&lt;Hc1) depend strongly on the amount of flux frozen into the specimen. A qualitative discussion connects these losses with flux pinning in the surface. Measurements for pure Nb are usually in this region. These results can be applied in more complicated experimental situations. Superimposed dc fields modify the losses by changing jc and ΔH; a varying temperature has a similar influence. In fact the ac critical current is often limited by thermal runaway. The losses for a solenoid have to be computed by considering the local fields.