CN101404193A - Design method for coaxial bidirectional transmission direct-current high-temperature superconducting cable body - Google Patents

Abstract

The invention relates to a design method used for transmitting DC high-temperature superconducting cable body coaxially and bidirectionally, belonging to the field of power system transmission and transformation; the design method comprises the steps as follows: according to the mechanical characteristic and the radius of enwinding framework of the superconducting strips, parameters of the superconducting cable body such as the enwinding helix angle and the enwinding pitch are determined; the magnetic field distribution of all layers of superconducting cable body are determined according to the operation current, thus determining the critical current on all layers of the cable; according to the critical current and the operation redundancy, iterative calculation is carried out for operation current and the magnetic field, thus finally gaining the critical current, best operation current, enwinding layer number and number of the cables. The designed superconducting cable body increases no enwinding process difficulty and achieves bidirectional transmission of DC current; the utilization ratio of the superconducting strip critical current reaches more than 90%; the quantity and low-temperature heat loss of the superconducting cable low-temperature container are respectively reduced by half. The design method has the advantages of large transmission capability, no loss, compact structure, certain current limiting capability, stable mechanical structure, self-shielding, no electromagnetic interference and the like.

Description

A kind of method for designing of coaxial bidirectional transmission direct-current high-temperature superconducting cable body

Technical field

The invention belongs to electric power system power transmission and transformation field, relate to a kind of method for designing of coaxial bidirectional transmission direct-current high-temperature superconducting cable body.

Background technology

Since the high-temperature superconductor of liquid nitrogen temperature was electrically found, because it has the zero resistance characteristic under the direct current transmission condition, so high temperature superconducting materia had potential application prospect in application on power system.Bi system high-temperature superconducting material the without hindrance transmission current of liquid nitrogen temperature can force rate ordinary copper conductor transmittability high two more than the order of magnitude.Compare with traditional cable, the high-temperature superconductor transmission cable has little, in light weight, the advantages such as loss is low, no disaster hidden-trouble of volume.Therefore, high-temperature superconductive cable has broad application prospects as aspects such as electrolysis, electroplating industry, the transmission of mesolow direct current aspect big electric current mesolow electric power transfer.

All cables of high-temperature superconductive cable research and development at present all adopt forward and reverse helical angle winding method, eliminate the influence of axial magnetic field to superconductor substantially, existing a plurality of prototype test operations.Current equalizing method is all adopted in existing high-temperature superconductive cable design, and promptly every layer of method that upward the superconducting tape transmission current is equal designs.Because high temperature superconducting materia is in the increase of liquid nitrogen temperature along with magnetic field, the critical current decay is very severe, and hyperconductive cable outermost layer superconducting tape is in the strongest magnetic field like this, and the critical current decay is maximum.Though this design is for the number of plies less (less than 4 layers), the hyperconductive cable of electric current little (less than 10kA) is effective, and is more for the number of plies, runs on liquid nitrogen temperature, and electric current does not appear in the newspapers greater than the hyperconductive cable design of 10kA.Because make every superconducting tape engineering current density equate on this design principle, and each layer gone up the superconduction belt material critical current difference, outermost layer critical current minimum, innermost layer critical current maximum, the ratio of transmission current and critical current increases from the internal layer to the skin successively, causes the superconducting tape application efficiency very low.For big electric current hyperconductive cable design, this forward and reverse helical angle winding method is not the best approach.

Summary of the invention

Technical problem to be solved by this invention is the shortcoming that exists in the existing hyperconductive cable design described in the background technology, electromagnetic property according to superconductor, at the requirement of big electric current, a kind of method for designing of coaxial bidirectional transmission direct-current high-temperature superconducting cable body is proposed, its technical scheme is:

According to hyperconductive cable rated current capacity, coiling skeleton radius, replace coiling design by the coaxial two-direction transmission current, reaching the suffered magnetic field of each layer minimum, the critical current maximum improves the utilization ratio of high temperature superconducting materia, and concrete steps are as follows:

1) according to the mechanical property and the coiling skeleton radius of superconducting tape,, determine the scope and the coiling pitch scope of the coiling helixangle of hyperconductive cable body;

2) utilize superconduction belt material critical current I _cWith the changes of magnetic field characteristic, determine upward magnetic field size and Orientation of each layer of hyperconductive cable body according to the running current size and Orientation, determine the critical current on described each layer of hyperconductive cable body;

3) according to critical electric I _cWith operation nargin (being coefficient of safety), running current and magnetic field are carried out iterative computation, obtain the critical current and the optimum operation electric current of each layer of cable, finally the radical of the number of plies of the superconducting tape of definite cable and each layer superconducting tape.

The scope of coiling helixangle is in the described step 1):

When coiling skeleton radius during less than band critical bends radius, promptly during r＜R,

${\sin }^{-1}\left(\sqrt{\frac{{\mathrm{\&epsiv;}}_t-{\mathrm{\&epsiv;}}_s-{\mathrm{\&epsiv;}}_p}{{\mathrm{\&epsiv;}}_r-{\mathrm{\&epsiv;}}_p}}\right)\&le;\mathrm{\&theta;}<{\sin }^{-1}\left(\sqrt{\frac{r}{R}}\right)\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;1$

ε wherein _tBe the free percent thermal shrinkage of band, ε _sBe the strain of band in cooling procedure, ε _pBe the rate of change of pitch, ε _γBe the radial shrinkage ratio of conductor layer, r is a band coiling radius, and R is the critical bends radius of band.

When coiling skeleton radius during more than or equal to band critical bends radius, promptly during r 〉=R,

Described step 2) the critical current I in _cFor:

I _c(B)＝f(B)............3

Belt material of high temperature superconduct reduces in the increase of liquid nitrogen temperature critical current with magnetic field, i.e. critical current I _cBe the monotonic decreasing function in magnetic field, function f (B) is provided by the producer of belt material of high temperature superconduct, can be curve, and also available calculating formula provides, and the superconducting tape of producing as American Superconductor Corp. is critical current being changed to magnetic field of 77K temperature:

$I_c\left(B\right)=\frac{I_{c0}\left(0\right)}{48-6.8e^{-\frac{B_{//}}{B_0}}-40.2e^{-\frac{B_{\&perp;}}{B_0}}}$

Wherein, I _C0(0) be to be under zero situation in magnetic field, the critical current of superconduction band, B ₀Be fitting constant, B _//And B _⊥Be respectively the parallel and vertical component of external magnetic field on the superconduction band.

Under circular cross-section construction of cable condition, flow through magnetic field that the electric current of superconducting tape produces along the cable circumferencial direction, be that magnetic field is parallel to the superconducting tape surface, be parallel to sense of current magnetic field to not influence of critical current, the axial magnetic field component is owing to adopt the forward and reverse technique for coiling of cable floor, the superconduction tape thickness is compared very little with the coiling radius, the axial magnetic field component is almost cancelled out each other, and can ignore.

In the described step 3) according to critical electric I _cWith operation nargin (being coefficient of safety), running current and magnetic field are carried out iterative computation, obtain the critical current and the optimum operation electric current of each layer of cable, finally determine the number of plies of superconducting tape of cable and the radical of each layer superconducting tape, its calculate and implementation procedure as follows:

Each layer magnetic field of employed 2n layer coaxial bidirectional transmission direct-current high-temperature superconducting cable body: N ₁, N ₂... N _2nBe respectively the 1st layer of conductor, the 2nd layer ... superconducting tape radical on the 2n layer, get N ₁=N ₂, N ₃=N ₄... N _2n-1=N _2nI ₁, I ₂... I _2nBe respectively the 1st layer of the conductor of flowing through, the 2nd layer ... the electric current of 2n layer, get I ₁=I ₂, I ₃=I ₄... I _2n-1=I _2n

Coiling helixangle and used superconduction strip length L have following relation,

$L=\frac{L_0}{\cos \mathrm{\&theta;}}\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;4$

L wherein ₀Be respectively the clear length of cable body and the single physical length of used superconduction band with L, the coiling helixangle is big more, and the physical length of used superconduction band is long more.Therefore, under the DC operation condition, the more little superconduction band of saving more of helixangle.

According to formula 1 or formula 2 described coiling helixangle scopes, determine coiling helixangle size.According to the hyperconductive cable coiling skeleton radius r of intending choosing ₀And the coiling helixangle of determining, determine coiling pitch L _p:

$L_p=\frac{{2\mathrm{\&pi;r}}_0}{\tan \mathrm{\&theta;}}\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;5$

After coiling helical angle and coiling pitch are determined,, carry out the design of the number of plies, radical, coiling direction, critical current and optimum operation electric current according to specified transmission current.

Magnetic field B along each layer of cable circumferencial direction _{I θ}Magnetic field B with axial direction _IzFor:

$B_{\mathrm{i\&theta;}}={\mathrm{\&mu;}}_0(\frac{1}{{2\mathrm{\&pi;r}}_{\mathrm{ip}}}\underset{k=1}{\overset{i-1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_k{I}_k+\left(\frac{r_{\mathrm{ip}}^2-r_{\mathrm{ii}}^2}{r_{\mathrm{io}}^2-r_{\mathrm{ii}}^2}\right)\frac{I_i}{{2\mathrm{\&pi;r}}_{\mathrm{ip}}}{\mathrm{\&alpha;}}_i)\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;6$

$B_{\mathrm{iz}}={\mathrm{\&mu;}}_0(\underset{k=i+1}{\overset{2n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_k{\mathrm{\&beta;}}_k\frac{I_k}{L_{\mathrm{pk}}}+\left(\frac{r_{\mathrm{io}}-r_{\mathrm{ip}}}{r_{\mathrm{io}}-r_{\mathrm{ii}}}\right)\frac{I_i}{L_{\mathrm{pi}}}{\mathrm{\&alpha;}}_i{\mathrm{\&beta;}}_i)\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;7$

Wherein, β _iAnd β _jRepresent i layer and j layer coiling direction respectively, get 1 or-1 respectively, β _iβ _j=-1, represent that the i layer is opposite with j layer coiling direction, β _iβ _j=1, represent that the i layer is identical with j layer coiling direction; With β _iAnd β _jValue is similar, α _iAnd α _jRepresent the i layer and the j layer sense of current respectively, get 1 or-1 respectively, α _iα _j=-1, represent i layer and j layer current opposite in direction; α _iα _j=1, represent that the i layer is identical with the j layer sense of current, r _IoAnd r _IiOuter radius and the inside radius of representing the hyperconductive cable i layer superconducting tape of coiling respectively, r _IpRepresent the position of required calculating magnetic field on the i layer superconduction band and the distance between the cable axis, L _PiAnd θ _iBe the coiling pitch and the coiling helical angle of i layer.

Be parallel to superconducting tape surface and with the magnetic field B of current vertical direction _I//For,

B _i//＝B _izsinθ _i-B _iθcosθ _i .........8

The magnetic field B that is parallel to superconduction belt current direction _I00For,

B _i00＝B _izcosθ _i+B _iθsinθ _i ..........9

Because magnetic field B _I00Parallel with electric current, to not influence of critical current.Because superconduction tape thickness t (being generally less than 0.5mm) is far smaller than coiling radius r (generally greater than 15mm), so the coiling pitch L of each layer _PiBe more or less the same, consider the forward and reverse technique for coiling of cable floor, component opposite in sign in the formula 7, cancel out each other in magnetic field, so axial magnetic field component B _IzWith cable circumferencial direction magnetic field β _{I θ}Compare, can ignore.

After above-mentioned coiling helical angle and coiling pitch were determined, the total superconducting tape radical of cable and the concrete steps of optimum operation current design were as follows:

The first step is determined on every layer of cable the radical N of superconducting tape that can coiling ₁, N ₂, N ₃... ..N _2n:

According to coiling skeleton radius r ₀, superconducting tape thickness t and layer insulation thickness d, the inside radius r of every layer of superconduction band _iFor,

r _i＝r ₀+d+(t+d)×(i-1) ..........10

The radical N of coiling superconduction band on the i layer _iSatisfy following identity,

$r_i\tan \left(\frac{\mathrm{\&pi;}}{N_i}\right)=\frac{g_i}{2\cos \left(\frac{\mathrm{\&pi;}}{N_i}\right)}+\frac{w\cos {\mathrm{\&theta;}}_i}{2}\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;11$

W is the width of superconduction band, and t is the thickness of superconduction band, g _iBe the coiling gap between the i layer superconduction band, r _iBe i layer superconduction band coiling radius, N _iBe the radical of coiling on the i layer, θ _i, be the helical angle of i layer superconduction band coiling.According to superconducting tape mechanical property at low temperatures, gap g _iAlso must satisfy with lower inequality,

$g_i\&GreaterEqual;\frac{{\mathrm{\&epsiv;}}_t}{N_i}2{\mathrm{\&pi;r}}_i(1+c{\tan }^2{\mathrm{\&theta;}}_i)\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;12$

ε wherein _tIt is the free percent thermal shrinkage of superconducting tape.

According to $N_i=\frac{2\mathrm{\&pi;}{r}_i}{\frac{w}{\cos {\mathrm{\&theta;}}_i}}$ Calculate N _i, and formula 11 calculating g are brought in round numbers into _iIf, the g that formula 11 calculates _iDo not satisfy formula 12, so with the N in the formula 11 _iWith N _i-1 replaces, and till the condition that satisfies formula 12, determines the radical N of coiling superconduction band on the i layer _iDetermine on every layer of cable the radical N of superconducting tape that can coiling with this ₁, N ₂, N ₃... N _2n

Second step: the critical current and the running current i that determine the 1st layer of cable superconduction single strips ₁:

Make the running current I of the 1st layer of cable conductor superconducting tape ₁=N ₁I _c(0), I _c(0) is critical electric current value under the 1st layer of single strips null field of conductor, the magnetic field B of calculating on the cable ground floors with formula 6 and formula 8 ₁, calculate the 1st layer of critical current I that goes up superconducting tape by formula 3 _C1(B ₁), reduce I then successively ₁, up to the 1st layer of running current i that goes up single superconducting tape ₁Satisfy,

i ₁＝k ₀k ₁I _c(0) ..........13

K wherein ₀Be operation nargin, k ₁Be cable ground floor superconduction belt material critical current degeneration factor, its size is the 1st layer of critical current I that goes up superconducting tape _C1(B ₁) and this superconducting tape null field under critical current I _c(0) ratio.

Running current on the single superconduction band of ground floor cable $i_1=\frac{I_1}{N_1},$ Running current on the single superconduction band of second layer cable is i ₂, get i ₁=i ₂, the estimated value of the required total superconducting tape radical N of odd-level and even level is respectively,

$N=\frac{I}{i_1}=\frac{I}{i_2}$

Wherein I is the specified running current of cable, and the used superconductor band of hyperconductive cable estimates that total radical is 2N.

The 3rd step: critical current and the running current of determining the 2nd layer of cable superconduction single strips:

Because the second layer and ground floor current opposite in direction (there is insulation the centre), the magnetic direction that is produced is reverse, and both can offset half.Because second layer superconducting tape is in the magnetic field that the ground floor band produces on the cable, with magnetic field, second layer thickness of strip center as its suffered magnetic field of whole band.Second layer superconduction band magnetic field with formula 6 and formula 8 calculating, compare with the ground floor superconduction band magnetic field that second step calculated, magnetic field reduce near half, therefore the critical current of the 2nd layer of superconduction single strips is bigger than ground floor certainly, in order to be cancelled out each other in magnetic field, it is identical with ground floor to get second layer running current, and it is corresponding to ground floor to move nargin simultaneously.

The 4th step: determine the optimum operation electric current of the single superconducting tape of each layer of cable, actual total radical and the number of plies:

Because first and second layer superconduction band radical is identical with electric current, current opposite in direction, promptly at place, the 3rd layer of present position, the magnetic field that first and second layer electric current produces is very little to the 3rd layer of influence.Because the influence of axial magnetic field component is very little, can ignore.The 4th layer is similar with the second layer, and layer 5 and the 3rd layer are similar .... and the like, can go on always.The superconduction band magnetic field of each layer that calculates according to formula 6 and formula 8, the optimum operation current i of the single superconduction band of each layer that calculates by formula 13 again ₁, i ₂... i _2n, select wherein reckling i _iSpecified running current (promptly supposing the single running current minimum of i layer) as the single superconducting tape of hyperconductive cable.According to N=I/i _i, determine last actual total radical N, again according to every layer can coiling on the determined cable of the first step radical N ₁, N ₂, N ₃... .N _2n, and get N ₁=N ₂, N ₃=N ₄... ..N _2n-1=N _2n, determine the actual radical of every layer superconducting tape according to following formula,

N＝N ₁+N ₃+N ₅+....N _2n-1＝N ₂+N ₄+N ₆+N _2n ..........14

Since along with the increase of the number of plies, the corresponding increase of cable radius, and magnetic field is corresponding to be reduced gradually, so i ₁It is the minimum running current of single strips on the whole hyperconductive cable.

Transmission current

I＝(N ₁+N ₃+N ₅+…+N _2n-1)i ₁?..........15

N ₁, N ₃, N ₅... ..N _2n-1Be the 1st, 3 respectively, superconducting tape radical on the 5....2n-1 layer.

Return current

I＝(N ₂+N ₄+N ₆+…+N _2n)i ₁ ..........16

N ₂, N ₄, N ₆... ..N _2nBe the 2nd, 4 respectively, superconducting tape radical on the 6....2n layer.Determine total number of plies 2n according to formula 15 or formula 16.

The operation nargin that more than relates to is enough big, and integral body satisfies the engineering design requirement.

Beneficial effect of the present invention is, the present invention is the method for designing of coaxial bidirectional transmission direct-current high-temperature superconducting cable, make full use of the mechanical performance and the superconducting tape electromagnetic performance of superconducting tape, the hyperconductive cable body of design does not increase the technique for coiling difficulty, can improve the utilization ratio of hyperconductive cable band critical current, its efficient reaches more than 90% of null field lower critical electric current; In addition, than hyperconductive cable with two one-way transmissions, few one times of the quantity of its low-temperature (low temperature) vessel and low temperature heat loss by exhaust.The coaxial bidirectional transmission direct-current high-temperature superconducting cable body of this method design has also that transmission capacity is big, lossless, compact conformation, itself have that current limiting capacity, mechanical structure are stable, self-shileding and do not have advantage such as electromagnetic interference.Therefore the present invention has very big using value for the research and development of the big capacity hyperconductive cable of direct current transmission.

Description of drawings

Fig. 1 is that high temperatures typical superconducting tape 77K warm area critical current is with parallel external magnetic field attenuation curve;

Fig. 2 is a kind of coaxial two-direction transmission current hyperconductive cable body schematic diagram of eight layers of superconducting tape coiling;

Fig. 3 is a kind of coaxial two-direction transmission current hyperconductive cable body side-looking sense of current schematic diagram of eight layers of superconducting tape coiling;

Fig. 4 is a kind of coaxial two-direction transmission current hyperconductive cable body end connection diagram of eight layers of superconducting tape coiling;

Fig. 5 is the coaxial two-direction transmission current hyperconductive cable body schematic diagram of another kind of eight layers of superconducting tape coiling;

Fig. 6 is the coaxial two-direction transmission current hyperconductive cable body side-looking sense of current schematic diagram of another kind of eight layers of superconducting tape coiling;

Fig. 7 is the coaxial two-direction transmission current hyperconductive cable body end connection diagram of another kind of eight layers of superconducting tape coiling;

Embodiment

Be described further below in conjunction with accompanying drawing with to method for designing of the present invention:

Fig. 1 is a characteristic curve under the parallel magnetic field outside of high temperatures typical superconducting tape 77K warm area critical current.In Fig. 1, abscissa represents that (unit:, ordinate is represented the normalization critical current to field intensity tesla).As shown in Figure 1, superconducting tape is under parallel field, and critical current is decayed with the increase in magnetic field.Use this belt material of high temperature superconduct in the design, can from then on scheme to calculate the critical current of superconducting tape.

Embodiment 1

Fig. 2 is a kind of schematic diagram of coaxial two-direction transmission current hyperconductive cable body embodiments of eight layers of superconducting tape coiling.In Fig. 2,201 expression ground floor superconducting tapes, 202 is second layer superconducting tape, 203 is the 3rd layer of superconducting tape, and 204 is the 4th layer of superconducting tape, 205 expression layer 5 superconducting tapes, 206 is the layer 6 superconducting tape, and 207 is the layer 7 superconducting tape, and 208 is the 8th layer of superconducting tape.Wherein, 201,203,205 and 207 layers of electric current in the same way, for removing electric current; 202,204,206 and 208 layers of electric current are return current in the same way, and interlayer insulating film satisfies insulating requirements.Coiling helixangle among the figure is meant the angle of superconducting tape coiling direction and cable axis, and under circular cross-section construction of cable condition, magnetic field is along circumferencial direction, and promptly magnetic field is parallel to the superconducting tape surface.The coiling helixangle of each layer is expressed as respectively: θ ₁, θ ₂, θ ₃...., θ ₈

As shown in Figure 2, the ground floor of superconducting tape is identical with second layer coiling direction, current opposite in direction; The 3rd layer identical with the 4th layer of coiling direction, current opposite in direction; Layer 5 is identical with layer 6 coiling direction, current opposite in direction; Layer 7 is identical with the 8th layer of coiling direction, current opposite in direction; The coiling angle guarantees θ ₁=θ ₂, θ ₃=θ ₄, θ ₅=θ ₆, θ ₇=θ ₈But coiling angle θ ₁, θ ₃, θ ₅, θ ₇(θ ₂, θ ₄, θ ₆, θ ₈) size can be identical, also can be different.

Fig. 3 is cross section each layer superconducting tape sense of current schematic diagram of present embodiment.

Represent that from outside to inside corresponding diagram 2 expressions from right to left;

Represent that from inside to outside corresponding diagram 2 expressions from left to right.

Fig. 4 is a present embodiment coaxial two-direction transmission current hyperconductive cable end connection diagram.601,602,603,604 represent the metal copper bar that ground floor, the 3rd layer, layer 5, layer 7 superconducting tape are connected with the end respectively; The busbar that confluxes of above 4 copper bars of 605 expressions; 606 is the insulating barrier of ground floor superconducting tape 201 and coiling skeleton, and 607 are the coiling skeleton.613 expression loads.609,610,611,612 represent that respectively the second layer, the 4th layer, layer 6, the 8th layer of superconducting tape are connected the metal copper bar with the end; The busbar that confluxes of above 4 copper bars of 608 expressions.

The present embodiment coiling is simple, and each layer insulation according to the working voltage standard, adopts corresponding insulating material.Odd even interlayer current opposite in direction, high electronegative potential is staggered to carry out, but every layer of needs are drawn totally 8 of metal copper bars separately, and four copper bars 601,602,603,604 are connected with the 201st, 203,205,207 4 layer of superconduction band respectively with 605 busbars that conflux and are connected the delivery transmission current; In addition four copper bars 609,610,611,612 are connected with the 202nd, 204,206,208 4 layer of superconduction band respectively with 608 busbars that conflux and are connected, and deliver return current.

Embodiment 2

Fig. 5 is the schematic diagram of the coaxial two-direction transmission current hyperconductive cable body embodiments of another kind of eight layers of superconducting tape coiling.In Fig. 5,301 expression ground floor superconducting tapes, 302 is second layer superconducting tape, 303 is the 3rd layer of superconducting tape, and 304 is the 4th layer of superconducting tape, 305 expression layer 5 superconducting tapes, 306 is the layer 6 superconducting tape, and 307 is the layer 7 superconducting tape, and 308 is the 8th layer of superconducting tape.Coiling helixangle among the figure is meant the angle of superconducting tape coiling direction and cable axis, and under circular cross-section construction of cable condition, magnetic field is along circumferencial direction, and promptly magnetic field is parallel to the superconducting tape surface.Wherein, 301,304,305 and 308 layers of electric current in the same way, for removing electric current; 302,303,306 and 307 layers of electric current are return current in the same way, and interlayer has insulation between 301 and 302, can naked between 302 and 303; Between 303 and 304 insulation is arranged; Can naked between 304 and 305; Interlayer has insulation between 305 and 306; Can naked between 306 and 307; Between 307 and 308 insulation is arranged.

The coiling helixangle of each layer is expressed as respectively among Fig. 5: θ ₁, θ ₂, θ ₃...., θ ₈

As shown in Figure 5, the ground floor of superconducting tape is identical with second layer coiling direction, current opposite in direction, the second layer is identical with the 3rd layer of sense of current, but the coiling direction is opposite, and the 4th layer identical with the layer 5 sense of current, the coiling direction is opposite, layer 5 is identical with layer 6 coiling direction, but current opposite in direction, layer 6 is opposite with layer 7 coiling direction, the sense of current is identical, layer 7 is identical with the 8th layer of coiling direction, current opposite in direction, and the coiling angle guarantees θ ₁=θ ₂, θ ₃=θ ₄, θ ₅=θ ₆, θ ₇=θ ₈, but coiling angle θ ₁, θ ₃, θ ₅, θ ₇(θ ₂, θ ₄, θ ₆, θ ₈) size can be identical, also can be different.

Fig. 6 is cross section each layer superconducting tape sense of current schematic diagram of present embodiment.

Represent that from outside to inside corresponding diagram 5 expressions from right to left;

Represent that from inside to outside corresponding diagram 5 expressions from left to right.

Fig. 7 is the coaxial two-direction transmission current hyperconductive cable end connection diagram of present embodiment.701,702,703 and 704 represent the metal copper bar that ground floor, the 4th layer, layer 5 and the 8th layer of superconducting tape are connected with the end respectively, and wherein 702 and 703 metal copper bars lead-in wire unites two into one; The busbar that confluxes of above three the metal copper bars of 705 expressions; 706 is the insulating barrier of ground floor superconducting tape 301 and coiling skeleton, and 707 are the coiling skeleton, 713 expression loads.709,710,711 and 712 represent that respectively the second layer, the 3rd layer, layer 6 are connected the metal copper bar with the layer 7 superconducting tape with the end, and wherein, 709 and 710,711 and 712 unite two into one respectively.The busbar that confluxes of above 2 the metal copper bars of 708 expressions.

Present embodiment needn't insulate between all layers, and in Fig. 5 and 6,302 and 303,304 and 305,306 and 307 are connected to together, and 301 and 308 print off separately.Therefore also needn't every layer draw copper bar separately, compare embodiment 1, metal copper bar radical is reduced to 5 by original 8, and wherein 3 and 705 busbars that conflux are connected, and 2 and 708 busbars that conflux are connected.

Coaxial two-direction transmission current hyperconductive cable current capacity is strong, and the magnetic field major part that produces during operation is cancelled out each other, and the critical current of each layer band is subjected to the running current influence degree of hyperconductive cable very little, has improved the utilization ratio of belt material of high temperature superconduct.In addition, owing to realize unit cable two-way communication stream, hyperconductive cable only needs a cover low-temperature (low temperature) vessel can realize common two cover functions that low-temperature (low temperature) vessel has, and low temperature leaks heat and reduces one times.

Embodiment 3

The design example of coaxial bidirectional transmission direct-current high-temperature superconducting cable,

Designing requirement: rated current is 20kA, and operation nargin is 80%,

Design procedure is:

The problem that step 1. designing institute need be considered:

1. the strain that suffered because of thermal contraction and superconduction band bending of limiting hts band;

2. act on singly with on magnetic field should be minimum so that reduce I _cDegree of degeneration;

3. the suffered magnetic field of each layer of coaxial bidirectional transmission direct-current hyperconductive cable minimum, critical current is degenerated minimum, improves the belt material of high temperature superconduct utilance.

The selection of the step 2. coaxial bidirectional transmission direct-current high-temperature superconducting cable number of plies:

1. belt material of high temperature superconduct electromagnetic performance: the Bi2223/Ag5705 stainless steel of selecting American Superconductor Corp. to produce is strengthened band, its sectional dimension: 4.3 * 0.28mm ²77K null field lower critical electric current: I _c(0)=and 125A, minimum bending radius 35mm (being the critical bends radius), electromagnetic property is as shown in Figure 1 under the parallel field.Because belt material of high temperature superconduct has strong anisotropy, increase the critical current decay with magnetic field, in order to make full use of superconducting tape, require magnetic field, superconducting tape present position minimum.

2. from given running current 20kA, calculate cable conductor innermost layer magnetic field: relevant because of magnetic field with radius, be circular for the cross section, vertical magnetic field too I is ignored, and only needs to consider the magnetic field of circumferencial direction.

3. the coiling skeleton determines.According to actual requirement of engineering, select suitable coiling skeleton.Present embodiment selection diameter is that the commercial corrugated stainless steel tubing of 75mm is the coiling skeleton.

4. determining of coiling helical angle:

Selected superconducting tape mechanical property: minimum bending radius: R=35mm; Free percent thermal shrinkage is ε _t=0.26%, critical stretching strain is ε _Cs=0.21%.For safety, get ε _s＜ε _Cs, i.e. ε _s≤ 0.2%.

Corrugated stainless steel tubing is as the coiling skeleton, and its performance is, diameter is 75mm, and stainless steel is 0.28% at the free shrinkage of 77K temperature, because the pitch of corrugated stainless steel tubing is very little with respect to diameter, so its radial shrinkage ratio is 0.28% also, for safety, gets ε _r=0.3%, the pitch rate of change of corrugated stainless steel tubing: ε _p=0.05%.

According to above parameter, because selected coiling skeleton radius r ₀=75/2=37.5mm, any layer of superconduction band radius r _iAll greater than R, i.e. r _i/ R＞1, by the coiling angle θ of formula 2 each layers of calculating,

By the formula 2 left side calculating formulas layer coiling angle θ that can win ₁〉=11.5 °, get at ground floor coiling angle, formula 2 right side: θ ₁≤ 90 °.Here get the identical coiling angle θ of each layer=12.5 °.

5. according to formula 5, radius is r _iThe coiling helixangle of conductor layer band coiling _iWith coiling pitch L _PiRelation, calculate every layer of pitch L that goes up band of coaxial bidirectional transmission direct-current high-temperature superconducting cable conductor _Pi

Step 3. is determined every layer radical N _i, total radical N and number of plies 2n.

1. determine the radical N that every layer on cable can coiling ₁, N ₂, N ₃... ..N _2n: determine every layer of superconducting tape gap of cable g by formula 11 and formula 12 _iWith radical N that can coiling ₁, N ₂, N ₃... ..N _2n, and get N ₁=N ₂, N ₃=N ₄... .., N _2n-1=N _2n

2. determine total radical N and number of plies 2n:

Make I ₁=N ₁I _c(0), I _c(0) is critical electric current value under the cable conductor i layer single strips null field,, calculates the 1st layer of critical current I that goes up superconducting tape by formula 3 according to the Distribution of Magnetic Field that formula 6 and formula 8 calculate on the cable ground floors _C1(B ₁); Reduce I then successively ₁, up to satisfying:

i ₁＝k ₀k ₁k _c(0) ...........13

K wherein ₁=I _C1(B ₁)/I _c(0), k ₀Be operation nargin, move nargin here and require 80%.I ₁Be ground floor superconducting tape running current.Order $i_1=\frac{I_1}{N_1},$ It is the running current on the single superconduction band of cable ground floor.According to rated current, required total superconducting tape radical is $N=\frac{I}{i_1},$ Again according to rated current I and above-mentioned every layer definite radical N _i, by the actual total radical of the superconducting tape of formula 14,15,16 definite cables and total number of plies 2n.

Step 4. is determined the major parameter of coaxial bidirectional transmission direct-current high-temperature superconducting cable conductor: the total radical of the superconducting tape of cable body, total number of plies, each layer radius, radical, coiling pitch, coiling helical angle, superconduction belt material critical current degeneration factor, single superconducting tape optimum operation electric current, the isoparametric calculating of optimum operation electric current and the design result of cable are as follows:

Each layer radius is respectively: r ₁=37.6mm, r ₂=37.98mm, r ₃=38.36mm, r ₄=38.74mm, r ₅=39.12mm, r ₆=39.50mm, r ₇=39.85mm, r ₈=40.26mm.

N ₁＝53，L _p1＝1066mm，θ ₁＝+12.5°，k ₁＝0.926，i ₁＝92.6A

N ₂＝53，L _p2＝1076mm，θ ₂＝+12.5.°，k ₂＝0.923，i ₂＝92.3A

N ₃＝54，L _p3＝1087mm，θ ₃＝-12.5°，k ₃＝0.926，i ₃＝92.6A

N ₄＝54，L _p4＝1097mm，θ ₄＝-12.5°，k ₄＝0.927，i ₄＝92.7A

N ₅＝55，L _p5＝1109mm，θ ₅＝+12.5°，k ₅＝0.926，i ₅＝92.6A

N ₆＝55，L _p6＝1119mm，θ ₆＝+12.5°，k ₆＝0.922，i ₆＝92.2A

N ₇＝56，L _p7＝1130mm，θ ₇＝-12.5°，k ₇＝0.926，i ₇＝92.6A

N ₈＝56，L _p8＝1141mm，θ ₈＝-12.5°，k ₈＝0.927，i ₈＝92.7A

Total number of plies is 8 layers, and total radical is 2 * 218=436, and minimum single running current is 92.2A, and the optimum operation electric current of cable is 218 * 92.2=20100A, reaches 20kA rated current design objective, realizes design object.

And, realize that same transmitted in both directions function of current needs with the super cable body of general one-way transmission electric current high temperature, and total number of plies is 2 * 5=10 layer, and total radical is 2 * 250=500, and minimum single running current is 80A, could realize that same target promptly transmits the electric current of 20kA.

The present invention relates to a kind of method of coaxial bidirectional transmission direct-current high-temperature superconducting cable body.Therefore, consider operation nargin, reasonably the minimum running current that should choose in each layer of running current meets The Principle of Engineering Design.According to the mechanical property of superconducting tape, determine that the coiling helical angle scope of hyperconductive cable body obtains coiling pitch parameter area; Utilize the characteristic of superconduction belt material critical current, determine upward Distribution of Magnetic Field of each layer of hyperconductive cable body, determine the critical current on each layer of cable according to running current with changes of magnetic field; According to critical current and operation nargin, running current and magnetic field are carried out iterative computation, finally obtain the critical current of cable and optimum operation electric current, its coiling number of plies and radical.The hyperconductive cable body of this method design does not increase the technique for coiling difficulty, has improved the utilization ratio of superconduction belt material critical current, and its efficient reaches more than 90% of self-fields lower critical electric current.The present invention has very big using value for the research and development of direct current transmission hyperconductive cable.

Claims (4)

1. the method for designing of a coaxial bidirectional transmission direct-current high-temperature superconducting cable body, it is characterized in that, according to hyperconductive cable rated current capacity, coiling skeleton radius, replace the coiling design by the coaxial two-direction transmission current, to reach the suffered magnetic field of each layer minimum, the critical current maximum, the utilization ratio of raising high temperature superconducting materia, concrete steps are as follows:

1), determines the scope and the coiling pitch scope of the coiling helixangle of hyperconductive cable body according to the mechanical property and the coiling skeleton radius of superconducting tape;

2) utilize superconduction belt material critical current I _cWith the changes of magnetic field characteristic, determine upward magnetic field size and Orientation of each layer of hyperconductive cable body according to the running current size and Orientation, determine the critical current on described each layer of hyperconductive cable body;

3) according to critical current I _cWith operation nargin, running current and magnetic field are carried out iterative computation, obtain the critical current and the optimum operation electric current of each layer of cable, finally the radical of the number of plies of the superconducting tape of definite cable and each layer superconducting tape.

2. the method for designing of a kind of coaxial bidirectional transmission direct-current high-temperature superconducting cable according to claim 1 is characterized in that, the scope of coiling helixangle is in the described step 1):

When r≤R,

${\sin }^{-1}\left(\sqrt{\frac{{\mathrm{\&epsiv;}}_t-{\mathrm{\&epsiv;}}_s-{\mathrm{\&epsiv;}}_p}{{\mathrm{\&epsiv;}}_r-{\mathrm{\&epsiv;}}_p}}\right)\&le;\mathrm{\&theta;}\&le;{\sin }^{-1}\left(\sqrt{\frac{r}{R}}\right)...1$

ε wherein _tBe the free percent thermal shrinkage of band, ε _sBe the strain of band in cooling procedure, ε _pBe the rate of change of pitch, ε _γBe the radial shrinkage ratio of conductor layer, r is a band coiling radius, and R is the critical bends radius of band;

When r＞R,

3. the method for designing of a kind of coaxial bidirectional transmission direct-current high-temperature superconducting cable body according to claim 1 is characterized in that, described step 2) in critical current I _cFor:

I _c(B)＝f(B)…………3

Belt material of high temperature superconduct reduces in the increase of liquid nitrogen temperature critical current with magnetic field, i.e. critical current I _cBe the monotonic decreasing function in magnetic field, function f (B) is provided by the producer of belt material of high temperature superconduct, can be curve, and also available calculating formula provides, and the superconducting tape of producing as American Superconductor Corp. is critical current being changed to magnetic field of 77K temperature:

$I_c\left(B\right)=\frac{I_{c0}\left(0\right)}{48-6.8e^{-\frac{B_{//}}{B_0}}-40.2e^{-\frac{B_{\&perp;}}{B_0}}}$

Wherein, I _C0(0) be to be under zero situation in magnetic field, the critical current of superconduction band, B ₀Be fitting constant, B _//And B _⊥Be respectively the parallel and vertical component of external magnetic field on the superconduction band;

Under circular cross-section construction of cable condition, flow through magnetic field that the electric current of superconducting tape produces along the cable circumferencial direction, be that magnetic field is parallel to the superconducting tape surface, be parallel to sense of current magnetic field to not influence of critical current, the axial magnetic field component is owing to adopt the forward and reverse technique for coiling of cable floor, the superconduction tape thickness is compared very little with the coiling radius, the axial magnetic field component is almost cancelled out each other, and can ignore.

4. the method for designing of a kind of coaxial bidirectional transmission direct-current high-temperature superconducting cable body according to claim 1 is characterized in that, in the described step 3) according to critical current I _cWith operation nargin (being coefficient of safety), running current and magnetic field are carried out iterative computation, obtain the critical current and the optimum operation electric current of each layer of cable, finally determine the number of plies of superconducting tape of cable and the radical of each layer superconducting tape, its calculate and implementation procedure as follows:

Each layer magnetic field of employed 2n layer coaxial bidirectional transmission direct-current high-temperature superconducting cable body: N ₁, N ₂... N _2nBe respectively the 1st layer of conductor, the 2nd layer ... superconducting tape radical on the 2n layer, N ₁=N ₂, N ₃=N ₄... N _2n-1=N _2n, I ₁, I ₂... I _2nBe respectively the 1st layer of the conductor of flowing through, the 2nd layer .... the electric current of 2n layer, I ₁=I ₂, I ₃=I ₄... I _2n-1=I _2n

Coiling helixangle and used superconduction strip length L have following relation:

$L=\frac{L_0}{\cos \mathrm{\&theta;}}...4$

L wherein ₀Be respectively the clear length of cable body and the single physical length of used superconduction band with L;

According to formula 1 or formula 2 described coiling helixangle scopes, determine coiling helixangle size.According to the hyperconductive cable coiling skeleton radius r of intending choosing ₀And the coiling helixangle of determining, determine coiling pitch L _p:

$L_p=\frac{2\mathrm{\&pi;}{r}_0}{\tan \mathrm{\&theta;}}...5$

After coiling helical angle and coiling pitch are determined, according to specified transmission current, carry out the design of the number of plies, radical, coiling direction, critical current and optimum operation electric current,

Magnetic field B along each layer of cable circumferencial direction _{I θ}Magnetic field B with axial direction _IzFor:

$B_{\mathrm{i\&theta;}}={\mathrm{\&mu;}}_0(\frac{1}{2\mathrm{\&pi;}{r}_{\mathrm{ip}}}\underset{k=1}{\overset{i-1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_k{I}_k+\left(\frac{r_{\mathrm{ip}}^2-r_{\mathrm{ii}}^2}{r_{\mathrm{io}}^2-r_{\mathrm{ii}}^2}\right)\frac{I_i}{2\mathrm{\&pi;}{r}_{\mathrm{ip}}}{\mathrm{\&alpha;}}_i)...6$

$B_{\mathrm{iz}}={\mathrm{\&mu;}}_0(\underset{k=i+1}{\overset{2n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_k{\mathrm{\&beta;}}_k\frac{I_k}{L_{\mathrm{pk}}}+\left(\frac{r_{\mathrm{io}}-r_{\mathrm{ip}}}{r_{\mathrm{io}}-r_{\mathrm{ii}}}\right)\frac{I_i}{L_{\mathrm{pi}}}{\mathrm{\&alpha;}}_i{\mathrm{\&beta;}}_i)...7$

Wherein, β _iAnd β _jRepresent i layer and j layer coiling direction respectively, get 1 or-1 respectively, β _iβ _j=-1, represent that the i layer is opposite with j layer coiling direction, β _iβ _j=1, represent that the i layer is identical with j layer coiling direction; With β _iAnd β _jValue is similar, α _iAnd α _jRepresent the i layer and the j layer sense of current respectively, get 1 or-1 respectively, α _iα _j=-1, represent i layer and j layer current opposite in direction; α _iα _j=1, represent that the i layer is identical with the j layer sense of current, r _IoAnd r _IiOuter radius and the inside radius of representing the hyperconductive cable i layer superconducting tape of coiling respectively, r _IpRepresent the position of required calculating magnetic field on the i layer superconduction band and the distance between the cable axis, L _PiAnd θ _iBe the coiling pitch and the coiling helical angle of i layer;

Be parallel to superconducting tape surface and with the magnetic field B of current vertical direction _I//For,

B _i//＝B _izsinθ _i-B _iθcosθ _i………8

The magnetic field B that is parallel to superconduction belt current direction _I00For,

B _i00＝B _izcosθ _i+B _iθsinθ _i ………9

Because magnetic field B _I00Parallel with electric current, to not influence of critical current.Because superconduction tape thickness t is far smaller than the coiling radius r, so the pitch L of each layer _PiBe more or less the same, consider the forward and reverse technique for coiling of cable floor, component opposite in sign in the formula 7, cancel out each other in magnetic field, so axial magnetic field component B _IzWith cable circumferencial direction magnetic field B _{I θ}Compare, can ignore;

After above-mentioned coiling helical angle and coiling pitch were determined, the total superconducting tape radical of cable and the concrete steps of optimum operation current design were as follows:

The first step is determined on every layer of cable the radical N of superconducting tape that can coiling ₁, N ₂, N ₃... ..N _2n:

According to coiling skeleton radius r ₀, superconducting tape thickness t and layer insulation thickness d, the inside radius r of every layer of superconduction band _iFor,

r _i＝r ₀+d+(t+d)×(i-1)………10

The radical N of coiling superconduction band on the i layer _iSatisfy following identity,

$r_i\tan \left(\frac{\mathrm{\&pi;}}{N_i}\right)=\frac{g_i}{2\cos \left(\frac{\mathrm{\&pi;}}{N_i}\right)}+\frac{w\cos {\mathrm{\&theta;}}_i}{2}...11$

W is the width of superconduction band, and t is the thickness of superconduction band, g _iBe the coiling gap between the i layer superconduction band, r _iBe i layer superconduction band coiling radius, N _iBe the radical of coiling on the i layer, θ _i, be the helical angle of i layer superconduction band coiling.According to superconducting tape mechanical property at low temperatures, gap g _iAlso must satisfy with lower inequality,

$g_i\&GreaterEqual;\frac{{\mathrm{\&epsiv;}}_t}{N_i}2\mathrm{\&pi;}{r}_i(1+c{\tan }^2{\mathrm{\&theta;}}_i)...12$

ε wherein _tIt is the free percent thermal shrinkage of superconducting tape;

According to $N_i=\frac{2\mathrm{\&pi;}{r}_i}{\frac{w}{\cos {\mathrm{\&theta;}}_i}}$ Calculate N _i, and formula 11 calculating g are brought in round numbers into _iIf, the g that formula 11 calculates _iDo not satisfy formula 12, so with the N in the formula 11 _iWith N _i-1 replaces, and till the condition that satisfies formula 12, determines the radical N of coiling superconduction band on the i layer _i, determine on every layer of cable the radical N of superconducting tape that can coiling with this ₁, N ₂, N ₃... ..N _2n

Second step: the critical current and the running current i that determine the 1st layer of cable superconduction single strips ₁,

Make the running current I of the 1st layer of cable conductor superconducting tape ₁=N ₁I _c(0), I _c(0) is critical electric current value under the 1st layer of single strips null field of conductor, the magnetic field B of calculating on the cable ground floors with formula 6 and formula 8 ₁, calculate the 1st layer of critical current I that goes up superconducting tape by formula 3 _C1(B ₁), reduce I then successively ₁, up to the 1st layer of running current i that goes up single superconducting tape ₁Satisfy,

i ₁＝k ₀k ₁I _c(0)………13

K wherein ₀Be operation nargin, k ₁Be cable ground floor superconduction belt material critical current degeneration factor, its size is the 1st layer of critical current I that goes up superconducting tape _C1(B ₁) and this superconducting tape null field under critical current I _c(0) ratio;

Running current on the single superconduction band of ground floor cable $i_1=\frac{I_1}{N_1},$ Running current on the single superconduction band of second layer cable is i ₂, get i ₁=i ₂, the estimated value of the required total superconducting tape radical N of odd-level and even level is respectively,

$N=\frac{I}{i_1}=\frac{I}{i_2}$

Wherein I is the specified running current of cable, and the used superconductor band of hyperconductive cable estimates that total radical is 2N;

The 3rd step: critical current and the running current of determining the 2nd layer of cable superconduction single strips:

Because the second layer and ground floor current opposite in direction (there is insulation the centre), the magnetic direction that is produced is reverse, and both can offset half.Because second layer superconducting tape is in the magnetic field that the ground floor band produces on the cable, with magnetic field, second layer thickness of strip center as its suffered magnetic field of whole band.Second layer superconduction band magnetic field with formula 6 and formula 8 calculating, compare with the ground floor superconduction band magnetic field that second step calculated, magnetic field reduce near half, therefore the critical current of the 2nd layer of superconduction single strips is bigger than ground floor certainly, in order to be cancelled out each other in magnetic field, it is identical with ground floor to get second layer running current, and it is corresponding to ground floor to move nargin simultaneously;

The 4th step: determine the optimum operation electric current of the single superconducting tape of cable, actual total radical and the number of plies,

Because first and second layer superconduction band radical is identical with electric current, current opposite in direction, promptly at place, the 3rd layer of present position, the magnetic field that first and second layer electric current produces is very little to the 3rd layer of influence.Because the influence of axial magnetic field component is very little, can ignore, the 4th layer similar with the second layer, layer 5 and the 3rd layer are similar .... and the like, can go on always, the superconduction band magnetic field of each layer that calculates according to formula 6 and formula 8, the optimum operation current i of the single superconduction band of each layer that calculates by formula 13 again ₁, i ₂... i _2n, select wherein reckling i _iAs the specified running current of the single superconducting tape of hyperconductive cable, according to N=I/i _i, determine last actual total radical N, again according to every layer can coiling on the determined cable of the first step radical N ₁, N ₂, N ₃... .N _2n, and get N ₁=N ₂, N ₃=N ₄... ..N _2n-1=N _2n, determine the actual radical of every layer superconducting tape according to following formula,

N＝N ₁+N ₃+N ₅+....N _2n-1＝N ₂+N ₄+N ₆+N _2n ………14

Since along with the increase of the number of plies, the corresponding increase of cable radius, and magnetic field is corresponding to be reduced gradually, so i ₁Be the running current of single strips minimum on the whole hyperconductive cable,

Transmission current

I＝(N ₁+N ₃+N ₅+…+N _2n-1)i ₁………15

N ₁, N ₃, N ₅... .N _2n-1Be the 1st, 3 respectively, 5....2 _N-1Superconducting tape radical on the layer,

Return current

I＝(N ₂+N ₄+N ₆+…+N _2n)i ₁………16

N ₂, N ₄, N ₆... ..N _2nBe the 2nd, 4 respectively, superconducting tape radical on the 6....2n layer, determine total number of plies 2n according to formula 15 or formula 16.

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