Monthly Archives: April 2014
Superconducting Materials for Electric Machines
Classes of Materials. Superconductors divide themselves into two major groups: the lowTc and the highTc materials. The lowTc superconductors (LTSs), the earlier ones, are industrial materials with high performance in terms of current capacity under high fields (1 T—5 T, Fig. 2). However, they operate in general only at temperatures near 4.2 K, which require a complex but wellcontrolled cryogenic system. The most common lowTc superconducting multifilamentary composites use niobium titanium (NbTi) with typical cost of about 1 $/kAm to 2 $/kAm. This figure of merit, the cost of 1 m of wire carrying 1 kA, enables comparisons between materials. The compound niobium tin (Nb3Sn) is only used for very highfield applications (> 8 T) and is less used for electric machines. Its use is more complicated than that of NbTi, due to the long thermal treatments required after winding it, and its cost is higher (5 $/kAm to 10 $/kA m).
HighTc superconductors (HTSs), assuming similar costs and performance to those of NbTi, will lead to a reduction in cryogenic costs (capital and especially operating costs), but the main advantage is the improvement in the stability of the superconducting state, which leads to higher reliability. At 20 K the specific heats are 200 times higher than at 4 K. The specific heat, being the amount of heat input necessary to raise the temperature, represents the materials inherent brake on temperature rise. HTSs are thus less sensitive to thermal disturbances.
Even with the large research effort focused on HTSs, these materials have not yet achieved the state of development stage of NbTi. HTSs are very complex anisotropic ceramic materials, difficult to fabricate in a conventional wire or cable. Intrinsically brittle, they are sensitive to mechanical stresses, and their transport properties under fields are still much poorer than those of NbTi (Fig. 2), except for highly oriented, essentially epitaxial films. They are also very expensive materials, and the current price is the main barrier to their
economic development. Their cost must be lowered to 10 $/kA m to be competitive (4). At present it is nearly 50 times higher.
There are two main routes to fabricate HTS wires. The more advanced one is the (powderintube) (PIT) technique (5,6) based on bismuthcompound filaments embedded in a silver or silver alloy matrix (Fig. 2). Lengths of BiPIT tapes as long as 1 km are produced routinely by several companies throughout the world, and their typical critical current densities are shown in Fig. 2. Still higher critical current densities are obtained on small samples (Jc = 760 MA/m2 at 77 K, 0 T; Je & 250 MA/m2). Some specialists think nevertheless that the limits have almost been reached. The pure silver matrix unfortunately is not suitable for ac applications, due to the high ac coupling losses, and new PIT wires are under development for ac applications (5) (silver alloys, resistive barriers, etc.).
The second route consists of socalled coated conductors (7) and has much potential. Yttrium compounds are deposited in thick films (a few micrometers) on industrial flexible textured metallic substrates through a buffer layer. Very good performance has been obtained with these coated conductors, but only for short lengths. The engineering current density (overall current density including substrate) is large in liquid nitrogen (on the order of 200 megamperes per square meter at 77 K at present), and its decrease under field is small. A lot of difficulties must be overcome to fabricate long, highperformance coated conductors, and there is now no lowcost industrial deposition technique. High quality Y superconductor bulk pellets, up to 100 mm in diameter (8), have been processed, and they can be used in some special machines (hysteresis, reluctance, trappedfield, etc.).
(Fig. 3). As shown in Fig. 3, this minimum work increases rapidly at low temperatures. In order to take into account the real cycle and the imperfections of the thermodynamic transformations, this ratio should be divided by the efficiency factor of the refrigeration system: 
Ac losses. One of the most spectacular properties of a superconductor is its absence of resistive losses. This is true, however, only for nontimevarying electromagnetic quantities (dc conditions). As soon as the magnetic induction varies with respect to time, ac losses appear in superconducting wires. The magnetic induction can be external or due to the current in the wire (selffield). The ac losses have two main consequences. On the one hand, they induce a temperature rise in the superconductor. Since the temperature margin is very small (< 1 K) for lowTc materials (NbTi for example) such a rise can easily quench the superconducting coil, that is, destroy its superconductivity. On the other hand, ac losses are very expensive energetically, since they are dissipated at low temperatures. They therefore greatly reduce the advantage of using superconductors. From the second law of thermodynamics, the removal of energy at a cold temperature (Tc), requires work at a high temperature (T0), usually room temperature. For an ideal closed cycle the ratio of the minimum required work (Wmin) at T0 to the energy (Q) to be removed at Tc is given by Carnot’s expression
This depends mainly on the cold power and little on the cold temperature (Fig. 3, Ref. 9).
The ratio Wmin/Q in real conditions [Eq. (3)] is called the specific work, and its reciprocal the coefficient of performance. To calculate the cost of refrigeration, the losses at low temperature must be multiplied by the specific work. For an efficiency factor of 10%, it amounts to 740 W/W and 29 W/W for cold temperatures of 4
0.1 
^ 100 І .9 =g 10 (D Q_ tf) W 1 1 ra О 0.1 
– 
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77 K:2.9 

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10 100 1000 Cold temperature Гс (K) 
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–  Nlllll 1 ГІ1 1H[ 1 ТТГГІЧ 1 
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0 _______________________ M. 1 1 

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Z 

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Old 

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■ ■30 – 90 К 
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….. 
!»»■’■< ……… и 
10 
0.1 1 10 100 1000 104 10s 106 Cold power (W) 
(a) 
Fig. 3. (a) Carnot’s specific work and (b) efficiency factor as functions of the cold power (9).
K and 77 K, respectively. These two figures illustrate the advantage of operating at high temperatures from a cryogenic point of view and again underlines the interest in using HTSs. The ac loss cost is especially high for LTSs, and it must be reduced to an ultralow level for the system efficiency to be acceptable.
A simple way to understand the ac losses is to consider the MaxwellFaraday law (curl E = — dB/dt). This shows that an electric field appears as soon as the magnetic induction varies with time. The induced electric field associated with a current density (transport current or persistent currents) results in losses. The losses per unit volume are the scalar product of these two vectors.
If it is not possible to suppress the ac losses, it is possible to reduce them by a suitable multifilament structure. This will depend on the field configuration (selffield, transverse or axial field), but ultralowacloss superconducting strands are generally achieved with very fine twisted filaments embedded in a highresistance matrix or with resistive barriers between filaments. The strand diameter should be low as well. Ac NbTi wires have small (< 0.2 mm) elementary strands with hundreds of thousands of filaments (< 0.2 ^m) in a CuNi resistive (0.4 /xQm) matrix (Fig. 2). The first NbTi lowacloss composites were developed only in the eighties when the technology for fine filament fabrication was sufficiently developed (10). Those strands have greatly extended the potential range for superconductivity (11). For highTc materials the requirements are less severe, since the cost of removing the acloss heat is reduced (29 W/W at 77 K compared to 740 W/W at 4 K). Nevertheless, no oxide superconducting tape actually fulfils them with present HTS wire technology.
The ac losses explain why superconducting devices are confined to applications with dc current and without or with timevarying fields, but in the latter case, protected from them.
OPENCIRCUIT VOLTAGE
It should be noted that the vector effective height h is defined for the situation where the antenna is used for transmission. Let us consider how h is related to receiving antenna operation.
Figure 3 shows two antenna systems in which antenna 1 with vector effective height h is used as a transmitted antenna in (a) and as a receiving antenna in (b). 101 is the terminal current and V01 is the open terminal voltage (opencircuit
voltage).
Antenna 2 is an infinitesimal dipole antenna with vector effective height hd = lz, which is used as receiving and transmitting antennas in (a) and (b), respectively.
The radiation field E1 from antenna 1 induces an open – circuit voltage at antenna 2
V02 — E1 ‘ ^Z
jkR
= (J 30 k—— IQ1h) • hd
Using Eq. (1), the radiation field from antenna 2 with terminal current 102 is written as
E2 — — j 30 k — /02hd
The opencircuit voltage V01 at antenna 1 induced by the radiation field from antenna 2 satisfies the relationship according to the [reciprocity theorem (1)]
^01I01 — V02I02
Substituting Eq. (4) into Eq. (6) yields
Пі = – j 30 k
Using Eq. (5) and replacing V01 and E2 with V0 and Einc, respectively, Eq. (7) is written as
V0 — Einc • h
(a) 
L Г 
02 
Ant. 2, h 
d 
Figure 3. Determination of opencircuit voltage at antenna 1. (a) Antenna 1 for transmission. Antenna 2 is an infinitesimal dipole for reception. (b) Antenna 1 for reception. Antenna 2 is an infinitesimal dipole antenna for transmission. 
^ Ic: 
Ant. 2, h 
d 
(b) 
(9) 
Equation (8) means that the opencircuit voltage is given by the inner product of the incident wave field and the antenna vector effective height. Example. Let us obtain the maximum opencircuit voltage for a halfwavelength dipole antenna (L = A/2) located on the z axis. The maximum open circuit voltage is obtained when the polarization of an incident wave Einc is parallel to the dipole; that is, the incident wave illuminates the dipole from the в = 90° direction. When the halfwavelength dipole has a current distribution of I(z’) = I0 cos kz’ over the antenna conductor from z’ = —A/4 to z’ = +A/4, Eq. (3) yields S = I0 A/wz using r • R = 0. Hence, h = A/wz. From Eq. (8), the maximum opencircuit voltage is V0 = EincA/w. EQUIVALENT CIRCUIT OF A RECEIVING ANTENNA The original receiving antenna problem shown in Fig. 4(a) can be handled by superimposing two cases, shown in Fig. 4(b) and (c). In Fig. 4(b) an EM plane wave illuminating the antenna induces an opencircuit voltage of V0 = Einc • h. In Fig. 4(c) the antenna acts as a transmitting antenna with terminal current 1. Superimposing these two cases leads to a relationship of 
Vrec — V0 Vtrans 
where Vrec = ZiI and Vtrans = ZAI, with ZA defined as the antenna input impedance. From Eq. (9), the terminal current 1 
(15) (16) 
I 
I 
Vre ^inc 
+ 
V Einc 
Z 
+ 
L 
+ 
Vn 
Vtrans 
E. • h2 w = L = w L 4 r L max E. • h2 E. • h2 ‘ inc ^ * inc ^ 
E 4(^а+га) 4Rl 
(a) 
I = 
(11) (12) 
Figure 4. A receiving antenna. (a) Original receiving antenna problem. (b) Antenna with open terminals. (c) Transmitting antenna.
is given as
V0
(10)
Therefore, the equivalent circuit for Eq. (10) is as shown in Fig. 5, where the opencircuit voltage V0 is used as a voltage generator with an internal impedance ZA.
Let us express the load impedance ZL and the antenna input impedance ZA as
ZL – RL + jXL
ZA – (RA + rA) + jXA
It follows that half of the power provided by the generator is delivered to the antenna load. Note that WLmax in Eq. (15) is the maximum power that can be delivered to the load ZL, because the antenna is perfectly matched to the load. Also, note that WR is recognized as the scattered (reradiated) power from the receiving antenna (1,4,5).
Example. The relationship WR = WL obtained when the impedance is matched can be checked by a numerical technique called the method of moments (3), where the incident power Winc and the received power WL are calculated. The scattered power is obtained from Winc — WL. For a centerload dipole antenna (4), the relationship WR = WL is obtained when the antenna length L is less than 0.8A. Limits on the validity of the equivalent circuit shown in Fig. 5 are discussed elsewhere (1,4,5).
RECEIVING CROSSSECTION AND APERTURE EFFICIENCY
The receiving crosssection Ar of an antenna is defined as the ratio of the power WL [Eq. (13)] received by the antenna to the Poynting power (power density of the incident wave, Pn = EiJ2/Zo, where Zo = 120Ш):
_Emc. h2RL Z0 r lzA +ZLI2 Emc2 
where Ra and rA are the radiation resistance and the loss resistance (not contributing to the radiation), respectively. Then, the power delivered to the antenna load is given as 
Ar = 
(17) 
Using an impedance mismatch factor Mimp defined as WL WLmaxMimp, Eq. (17) is rewritten as 
V02 Rl 
(13) 
ZA + ZL2 and the power consumed in the generator is given as V02(RA+rA) A Iza+zlI2 
l^ncbl2 M 4 (RA+rA) 
Zn 
Ar – 
Eind2 zo Eincl2 
(18) 
.Einc • h2 
Mm 
4R« 
(14) 
where Rl = Ra + rA = Ra/^ (Appendix Eq. (a1)) is used. Using the absolute gain Ga [see Eq. (a5)], Eq. (18) is written as 
When an impedance matching condition is satisfied [i. e., when Zl is complex conjugate of Za(Rl = RA + rA and XL = 
= AlG lEnchl2 r 4я a Emc2h2 
Mim 
(19) 
Because the received power is proportional to the square of the opencircuit voltage, that is, Einc • h2, the third factor Einc • h2/Einc2h2 in Eq. (19), is the reduction factor of the received power from polarization mismatch. Let the third factor be denoted as Mpoi, which has a maximum value of 1 for the case in which h is a real constant multiplied by the complex conjugate of Einc. Eq. (19) now can be written as 
Ar = t— GMM 
(20) 
a pol imp 
4n 
The maximum receiving crosssection, which is called the effective area Aeff, is obtained when Mpol = Mimp = 1. 
^eff – 4jr Ga 
(21) 
ation power density from a reference antenna that has the same input power as the test antenna:
G(e^) — 
(a4) 
W 
Z 
W 
Ga — nD 
Let the receiving antenna gain be defined as the ratio of the effective area of the antenna, Aeff, to the effective area of an isotropic antenna, Aeff iso = A2/4w. Then, from Eq. (21), the receiving antenna gain is equal to the absolute gain when the same antenna is used as a transmitting antenna. Example. An isotropic antenna has a gain of Ga = 1 (=0 dB) by definition. Using Eq. (21), the effective area is calculated to be Aeff = 0.0796A2. An infinitesimal dipole, whose vector effective height is h = lsin в, has a gain of Ga = 1.5 ( = 1.76 dB) at в = 90° and an effective area of Aeff = 0.119A2. A halfwavelength dipole antenna with h = A/w yields an effective area of 0.130A2, because Ga = 1.64 (=2.15 dB). Note that all Gas are calculated using Appendix (a6) and (a7) with ^ = 1. When a receiving antenna has an aperture Aap which is much larger than the wavelength A, the performance of the receiving antenna is evaluated by how efficiently the aperture is utilized for reception. Since the receiving antenna of Aap has the potential to collect an EM wave power of Wap = PincAap, the ratio of the received power PincAeff to Wap is defined as the aperture efficiency ^ap, where 
where E0 is the farfield radiated from the reference antenna, and Win, 0 is the power input to the reference antenna. Note that the maximum value of the gain is conventionally used for the antenna gain if the coordinates (в, ф) for the direction of interest are not specified. When an isotropic antenna (hypothetical antenna radiating with uniform radiation power density in all directions) is chosen as the reference antenna, the gain is called the absolute gain and denoted as Ga. Equation (a4) becomes 
E(R, 9, 0)2/Win E0I2/Wln>0 
4nR2 E(R, Є, ф)^ 
Ga = 
(a5) 
(a6) 
— n 
Zn 
Using Eq. (1) and Eq. (a2), 
4nR2 E(R, Є, ф)^ 
rad 
Advanced Guidance Algorithms
Classic PNG and APNG were initially based on intuition. Modern or advanced guidance algorithms exploit optimal control theory, i. e. optimizing a performance measure subject to dynamic constraints. Even simple optimal control formulations of a missiletarget engagement (e. g., quadratic acceleration measures) lead to a nonlinear twopoint boundary value problem requiring creative solution techniques, e. g., approximate solutions to the associated HamiltonJacobiBellman equation—a formidable nonlinear partial differential equation (23). Such a formulation remains somewhat intractable given today’s computing power, even for command guidance implementations that can exploit powerful remotely situated computers. As a result, researchers have sought alternative approaches to design advanced (nearoptimal) guidance laws. In Reference 20, the authors present a PNGlike control law that optimizes squareintegral acceleration subject to zero miss distance in the presence of a one pole guidancecontrol – seeker system.
Even for advanced guidance algorithms (e. g., optimal guidance methods), the effects of guidance and control system parasitics must be carefully evaluated to ensure nominal performance and robustness (20). Advanced (optimal)
In contrast with PNG, this expression shows that the resulting APNG acceleration requirements decrease with time rather than increase. From the expression, it follows that increasing N increases the initial acceleration requirement but also reduces the time required for the acceleration requirements to decrease to negligible levels. For N — 4, the maximum acceleration requirement
for APNG, acAPNGmax — — Nat, is equal to that for PNG,
]at. For large N — 5, APNG requires a
acA png(,) — 2 N 
N2 
N 
it can be shown (under the simplifying assumptions given earlier) that
Variants of PNG
Within Reference 20, the authors compare PNG, APNG, and optimal guidance (OG). The zero miss distance (stability) properties of PPNG are discussed within Reference 24. A nonlinear PPNG formulation for maneuvering targets is provided in Reference 27. Closed form expressions for PPNG are presented in Reference 28. A more complex version of PNG that is “quasioptimal” for large maneuvers (but requires tgo estimates) is discussed in Reference 29. Twodimensional miss distance analysis is conducted in Reference 21 for a guidance law that combines PNG and pursuit guidance. Within Reference 30, the authors extend PNG by using an outer LOS rate loop to control the terminal geometry of the engagement (e. g., approach angle). Generalized PNG, in which acceleration commands are issued normal to the LOS with a bias angle, is addressed in Reference 31. Threedimensional (3D) generalized PNG is addressed within Reference 32 using a spherical coordinate system fixed to the missile to better accommodate the spherical nature of seeker measurements. Analytical solutions are presented without linearization. Generalized guidance schemes are presented in Reference 33, which result in missile acceleration commands rotating the missile perpendicular to a chosen (generalized) direction. When this direction is appropriately selected, standard laws result. Timeenergy performance criteria are also examined. Capturability issues for variants of PNG are addressed in Reference 34 and the references therein. Within Reference 35, the authors present a 2D framework that shows that many developed guidance laws are special cases of a general law. The 3D case, using polar coordinates, is considered in Reference 36.
SUPERCONDUCTING MOTORS, GENERATORS, AND ALTERNATORS
Superconductors are very promising and exciting materials for electric power engineering in general and for electric machines in particular. By allowing very high current densities and by suppressing the Joule losses, superconductors improve the performance of electric machines by reducing weight, improving efficiency and, to a lesser degree, by increasing compactness. The reduction of losses results in longterm savings in capital cost, making superconducting machines attractive from an economic point of view. Since a cryogenic system is required to maintain the superconducting state, these advantages appear only above a critical (breakeven) size or rating, such that where the refrigeration penalty is negligible. Superconductivity is thus attractive only for large electrical machines. Small motors (up to a few kilowatts) will not be superconducting except if roomtemperature superconductors are discovered, which is highly improbable.
The critical size is reduced when the operating temperature increases. Devices with highcritical – temperature superconductors will be attractive for lower ratings than systems cooled with liquid helium.
Electric Machines—General Structure(l)
An electric machine is reversible. It can operate as a motor, converting electrical into mechanical power, or as a generator, converting mechanical input into electricity. The electromagnetic force or torque is produced in two ways. The first one is the interaction between currents, called armature currents, and a variablereluctance structure (variablereluctance machines). The second and more widely used way is the interaction between currents and a magnetic field called the excitation field. The mobile part can be either the armature or the excitation. But as the energy transfer to a moving part creates losses except by electromagnetic way, the armature is preferably stationary.
The torque per unit volume is proportional to the excitation field component perpendicular to the current times the armature ampereturn loading, as shown below. The armature ampereturn loading is the total armature current divided by the mean armature circumference. The expression for the maximum torque (rmax) for a threephase machine is simply:
where
Bo = excitation field component K = armature ampere turn loading ro = mean armature radius L = active length Ns = total series turns per phase kd = winding factor
I = rated armature current
#mach = approximate machine volume
лі*h кг iWh
The excitation field is created either by permanent magnets or by current flow in a winding. Permanent magnets are limited in magnetic induction and are made of very expensive materials, but they are not dissipative. They are well suited particularly for small machines (kilowatt range). The currents in a conventional conductor produce heat through the Joule effect (R i2 where R is the resistance and i the current), dissipating energy. The current capacity is hence limited by the ability to remove heat. Better cooling conditions increase the current capacity but reduce the efficiency. The current density (current per unit crosssectional area) is then limited by thermal and economic factors. The allowable current density in copper is on the order of amperes per square millimeter (5 MA/m2 to 10 MA/m2). With such values the amounts of conductor required to produce magnetic fields without magnetic materials are large, leading to huge Joule losses. Thanks to the peculiar properties of soft magnetic materials (high relative permeability), the total current (ampere turns) required to produce a given magnetic induction is greatly reduced. For this reason practically all electric machines have a magnetic circuit with slots where the windings are embedded.
The armature ampereturn loading is limited by Joule losses and by the current density allowable in the conductors, because the space they can occupy is limited.
The magnetic circuit has other advantages than the reduction ofthe excitation current. It confines the flux within the machine and reduces the stray field to negligible levels. It also prevents magnetic disturbances to other equipment. The magnetic circuit is also very useful from a mechanical point of view. When the conductors are inserted into slots they are subjected only to a reduced electromagnetic force, since the field concentrates itself in the teeth. The electromagnetic force is mainly applied at the interface between the slots and the magnetic teeth. The torque is then essentially supported by the magnetic circuit and not by the conductors. The reduced mechanical stresses on the conductors are an important advantage, because the mechanical strength of copper is low. In a slotted structure, no special care need be taken in order to reduce the eddycurrent losses in the conductors, since they only see low fields. Without magnetic teeth, a strong mechanical support structure must be provided in order to sustain all the electromagnetic torque, and the conductors should follow the finely divided Litz wire configuration to avoid large eddycurrent losses. However, the magnetic circuit is heavy, the increased magnetic induction it provides is limited by saturation, and it creates pulsating torques, because the alternation of magnetic teeth and slots produces local magnetic variations. The magnetic teeth also reduce the space available for conductors and thus the armature ampereturn loading. The slotted structure is also not convenient for insulation, so that the maximum voltage is limited (to about 30 kV).
Superconducting materials show promise for electric machines because they offer the possibility to increase both the excitation field and the armature ampereturn loading (2,3). Superconductors are particularly convenient for producing magnetic fields that are constant in time. Since the current densities in superconductors can be very high (up to a hundred times the allowable value in copper, i. e., hundreds of megamperes per square meter), the required quantity of conductor to produce a given field is greatly reduced from that with conventional conductors, even without the help of magnetic materials. The magnetic circuit is usually nearly removed when using superconductors. Magnetic materials are in general used only to form a magnetic shield in order to avoid large stray fields outside the machine. Current maintenance in a superconducting winding does not cost any energy, due to the absence of losses for constant current and constant external field. The disappearance or large reduction of the magnetic circuit leads to a light and saturationfree structure with
Fig. 1. Schematic cross sections of ac generators. (a) classical; (b) superconducting field winding; (c) fully superconducting. 
more active space for conductors and insulation materials. The ampereturn loading and the voltage can then be increased. The absence of iron teeth will decrease vibration by suppressing torque ripples. Acoustically very quiet electric machines can be designed.
However, the torque is applied directly to the conductors. They must therefore be supported by a suitable structure. An armature without magnetic teeth subjects the conductors to large forces at twice the frequency of rotation, which must be restrained by novel means of support for which high reliability must be maintained. Figure 1 shows the main differences between a conventional machine and superconducting ones (for synchronous machines).
The weak point of a conventional machine is in general its insulation, which degrades badly with time. It is very sensitive to thermal cycling, and overheating strongly affects its lifetime. A cryogenic system is hence very favorable from this point of view: it almost completely avoids thermal cycling in operation. Moreover, at low temperatures all aging process are slowed down. The cryogenic components of superconducting machines should thus last longer, particularly if the machine remains at low temperature. Numerous thermal cycles from room temperature to cryogenic temperature must be avoided. Furthermore, they are costly in time and energy.
The very high current densities in superconductors make them very attractive for the armature by increasing the ampereturn loading. However, the armature currents are in general alternating, so that losses appear in the superconductors. This is an important disadvantage in a cryogenic environment. In order to
Bi2223 PIT 3.5 x 0.35 mm2 (BICC) 36 Bi2233 filaments. Ag matrix Fig. 2. Engineering critical characteristics of superconducting materials and wire cross sections. 
discuss this point and for the sake of completeness, some information about superconducting wires (materials and ac losses) will be given in the following sections.