Current imbalance in superconducting strand-to-strand joint and its relaxation in multistage...

Available online at www.sciencedirect.com

www.elsevier.com/locate/physc

Physica C 468 (2008) 417–425

Current imbalance in superconducting strand-to-strand jointand its relaxation in multistage cable-in-conduit conductor

Sangil Lee a,c,*, Sangkwon Jeong b, Sung-Min Choi a

a Department of Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology (KAIST),

373-1 Guseong-dong, Yusong-gu, Daejeon 305-701, Republic of Koreab Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Guseong-dong, Yusong-gu,

Daejeon 305-701, Republic of Koreac National Fusion Research Institute (NFRI), 52 Eoeun-dong, Yusong-gu, Daejeon 305-333, Republic of Korea

Received 19 September 2007; received in revised form 4 December 2007; accepted 3 January 2008Available online 15 January 2008

Abstract

This paper presents non-uniformity or imbalance in the transport current of the superconducting toroidal-field coil (TFC) made ofmultistage cable-in-conductor (CICC) due to inductive current in a strand-to-strand (STS) soldered joint of cylindrical shape and also therelaxation of the inductive current in the conductor. Using an NbTi CICC consisting of 486 strands, a joint sample is fabricated andexperimented. An external magnetic field of symmetric trapezoidal waveform is vertically applied to the sample joint. The field maximumand the ramp rate (dB/dt) maximum are 1 T and 0.25 T/s, respectively. Voltage taps and Hall probes are utilized to measure the inductivevoltage and current. Experimental results are analyzed numerically using an infinite electrical transmission line model, which is found toexplain our sample system well. The flattop induced current is proportional to the dB/dt. Being different from our expectation that theinductive current will distribute widely over the cross-sectional surface of the joint, it localizes to a narrow region; a triplet located in theouter-most side of the joint, for instance. The relaxation phenomenon in the induced current is observed experimentally. From numericalanalysis, the relaxation length is found to increase logarithmically with decreasing dB/dt and saturate at a certain value of dB/dt.� 2008 Elsevier B.V. All rights reserved.

PACS: 84.71.Ba; 84.71.Fk; 85.25.Am

Keywords: Current imbalance; Cable-in-conduit conductor; Relaxation length; Joint

1. Introduction

Large-scale superconducting magnets are often con-structed with multistrand cables to avoid too high drivingvoltage that is associated with large inductance of the mag-net. The strands are normally twisted to decrease inter-strand coupling current. The operating current, however,may not be uniformly distributed among the cable espe-

0921-4534/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.physc.2008.01.002

* Corresponding author. Address: Department of Nuclear and QuantumEngineering, Korea Advanced Institute of Science and Technology(KAIST), 373-1 Guseong-dong, Yusong-gu, Daejeon 305-701, Republicof Korea. Tel.: +82 42 710 1823; fax: +82 42 710 1809.

E-mail address: [email protected] (S. Lee).

cially when the operation mode is dynamic. The currentnon-uniformity or current imbalance can be a serious prob-lem for stable operation of the magnets. Not only does itreduce the stability margin [1–3] but also prevents the mag-net from fast ramping [4–8]. The current imbalance iscaused by non-uniform resistance distribution among theindividual strands contacting the joint in the DC operationof the magnet. In the AC operation mode, however, it isadditionally caused by inductive voltage created due tonon-uniform field variation along the cable. This voltageinduces a long lasting current loop of so-called ‘supercur-rent’ other than the coupling current loop [8,9].

All the concerns up to now have mostly involved theinductive voltage source existing in superconducting cables.

mailto:[email protected]

418 S. Lee et al. / Physica C 468 (2008) 417–425

However, such a voltage source can exist also in a joint.The unique soldered joint, which is called a strand-to-strand (STS) joint in the KSTAR (Korea SuperconductingTokamak Advanced Research) device [10], is a good exam-ple of this.

KSTAR is a kind of superconducting tokamak fusiondevice, whose 16 toroidal-field coils (TFC) and 14 poloi-dal-field coils (PFC) are all superconducting [10]. TheTFC carry high transport current of tens of thousands ofamperes and their 32 joints generate significant amountsof heat steadily during operation. To minimize heat gener-ation, a joint with low resistance is required and the STSjoint has been developed so that its overall length is shorterand its DC resistance is much less than the conventionallap joint for joining the PFC. Due to good stability prop-erties of cable-in-conduit conductor (CICC), the TFCand buslines are made of this kind of conductors and theirspecifications are shown in Table 1. The TFC CICC [11] isa square-shaped conductor with round corners, whereasthe busline is a circular-shaped one. Both CICCs havethe same number of superconducting strands but are madeof Nb3Sn and NbTi strands, respectively. The TFC are

Table 1Specifications of the KSTAR TFC and busline conductors

TFC

Superconductor Nb3SnStrand diameter 0.78 mm after Cr platingCr plating thickness 1 lmCritical current density 750 A/mm2 (12 T, 4.2 K)AC loss 250 mJ/cm3 (±3 T cycle)Cu/non-Cu ratio 1.5RRR 100Strand twist pitch 10 mmCabling pattern (2SC + 1Cu) � 3 � 3 � 3 � 6Cable twist pitch 40, 73, 157, 227, 355 mmConduit material Incoloy 908Conduit dimension w25.65, h25.65, t2.86 mm

Fig. 1. Schematic of the STS joint: Nb3Sn CICC from the magnet side

connected to each other via superconducting buslines.The STS joint is used for joining the coil and busline. Asshown in Fig. 1, the joint is uniquely designed so that thecables coming from the magnet and the busline areuntwisted to the triplet level, being straightened, juxta-posed, and soldered over the straight part of the supportcylinder. To make joint resistance as low as possible,chrome plating of all the strands in the straight section isremoved after the untwisting work. Joint pieces such as cyl-inder and conical tubes are used to support superconduc-ting strands in the straight section and in the transitionsections of the joint, respectively. All of the pieces are madeof cupronickel to have the normal AC loss reduced.

The design issue of this kind of joint is, however, thecurrent imbalance under changing magnetic field condi-tion. When time-varying magnetic fields from the PFCare exerted vertically on the joint, a loop current may beinduced. Due to the cable untwisting in the joint region,the induced current will not be negligible and cause largeimbalance in transport current among strands and localheating, which may result in a quench depending on thesignificance of the imbalance. In addition, the current will

Busline

NbTi 2900 A/mm2 (5 T, 4.2 K) 2.8 (2SC + 1Cu) � 3 � 3 � 3 � 6 + (3Cu) � 3 � 3 � 340, 73, 157, 227, 307 mmSS 316Lod32.6, id23.6, t4.5 mm

(on the right) and NbTi CICC from the busline side (on the left).

S. Lee et al. / Physica C 468 (2008) 417–425 419

diffuse into the CICC up to the high field region of themagnet where the stability margin is comparatively small,which may also result in a quench.

A joint sample simulating the real behavior of thedesigned STS joint in the KSTAR environment has beenfabricated and experiments have been performed to inves-tigate the current imbalance at the joint and its relaxationcharacteristic. Numerical analysis using a transmission-linemodel has also been carried out in this paper to understandthe underlying physics. The model is validated by compar-ing numerical and experimental results. Finally, it isassessed how the current imbalance is significant.

2. Experiment

2.1. Design of experiment

2.1.1. Sample designGenerally, when a time-varying external magnetic field

is vertically exerted on a cylindrical conductor, currentsare induced on both side surfaces to screen the field in itsinterior region. They follow a cosine distribution, whichmeans that the majority of the current exists in the 60� sec-tor on each side. In case of the STS joint, this sector corre-sponds approximately to one sub-cable volume. Based onthis deductive reasoning, a rather simple joint sample thatcan be inserted in a magnet, which generates controlledtime-varying field and provides the field to the joint sample,has been designed for the study and is shown in Fig. 2a.Due to the lack of space inside the magnet bore, onlyone CICC is utilized to make the sample. SS-316LN NbTiCICC is used, which has the same dimension and geometryas the TFC CICC since it is difficult to use the Nb3Sn

Fig. 2. (a) Schematic diagram of the experimental system, (b

CICC due to the necessity of heat treatment and the brittle-ness of strands after the heat treatment. The NbTi strand isthe same one as shown in Table 1. Two types of sensors areinstalled to investigate the induction and the propagationphenomena. Hall sensors are to measure the induced cur-rent near the joint and its propagation through the CICC.Voltage taps are installed to measure the emf (electromo-tive force) voltage that is generated at the joint and itspropagation through the CICC.

Following the design, the joint sample is fabricated asshown in Fig. 2b. The dimension of the supporting cylinderis as follows; outer diameter 48 mm, thickness 6.8 mm, andlength 60 mm. The length of the supporting conical tube is25 mm. To locate the sample joint at the center of the bore,rounded-square-through holes are made on the cylindricaltube and the CICC is pushed through them.

Since we use only one CICC, fabricating a joint samplefrom the single CICC may be rather complicated. As inFig. 3, two sub-cables (each 27 triplets) that are facing eachother diagonally are extracted down from the CICC. Eachone is divided into the two cable bundles containing 13 and14 triplets, which are insulated in order for the current notto interfere, after passing by the Hall sensor HS#1. Fig. 3ashows the schematic current flow diagram where two 13-triplet cables are sent together to the left while the others,to the right. Fig. 3b shows the cross section of the samplejoint, where we see how the triplets are arranged on the sur-face of the cylindrical tube. The 14 triplets are placed onthe inner surfaces of the grooves – the 40 grooves on thesurface are made for the purpose of easy fabrication –while the 13 triplets are between them. It should be notedthat the sample is composed of the two independent cur-rent circuits. The two arrows in the CICC of this figure

) completely fabricated joint sample with GFRP fixtures.

Fig. 3. (a) Schematic current flow diagram: The small arrows are thedirections of the induced current when the increasing external fields aregoing into the paper. The italicized numbers 1 and 2 denote the potentialsE1 and E2, respectively, so that |E2 � E1| is the half of the loop voltage. (b)The arrangement of the triplets on the surface of the cylindrical tube. Thegray-colored and the white-colored circles denote the strands for the 14and the 13 triplet bundles, respectively. Solder fills vacancies amongstrands, which is not shown here.


relate to the circuits. Each circuit is made of the two tripletbundles located on the same side of the joint. The two cir-cuit currents merge before HS#1 and thus this sensor mea-sures twice the induced loop current.

Table 2Parameters and values of the applied pulses

B0 0.5 T 1 Ns 2, 4, 8, 16, 32, 60 s 8, 20 s

Fig. 4. Symmetric trapezoidal waveform.

2.1.2. Current measurement using Hall sensors

Transverse type Hall sensors are used to measure thecurrent indirectly. Suppose current flows up and downthrough two infinitely long and parallel wires and a Hallprobe is laid between the wires at equal distances fromthe center. Then, a magnetic field generated by the currentaccording to the following relationship is produced.

B ¼ l0Ipd

; ð1Þ

where B is the magnetic field generated by the up and downcurrent I, and d is the distance between the centers of thewire and the Hall probe. Since small errors in d may resultin significant errors and the value simply calculated fromthe known geometry is possibly uncertain, we thoroughlycalibrated the Hall probes after finishing all the plannedmeasurements. The same sample was cut and the fieldwas measured when the known value of the current was

supplied. As a result, the distance of the Hall probe fromthe sub-cable, d was determined to be 7.3 mm with±1.1% deviation, which verified our current-measuringscheme.

2.1.3. Applied waveform

Under the KSTAR reference operation scenario, theSTS joint goes through the five times of linear field changeof PFCs, whose ramping times are within 0.06–30 s. As aresult, the perpendicular component of the maximum fieldand its change rate are 0.44 T and 0.22 T/s, respectively.The waveform used in this experiment is trapezoidal asshown in Fig. 4, whose flattop value and ramp rates areto simulate the KSTAR operation conditions as close aspossible. In the experiment of this paper, eight differentkinds of pulses are applied to the sample, whose valuesare summarized in Table 2. The ramp rate parametersrange from 0.0083 to 0.25 T/s.

2.2. Measurement system

The fabricated joint-sample assembly is inserted in themagnet and fixed using GFRP fixtures. Then, they areinstalled in the cryostat and cooled with liquid helium(LHe). The signal wires are extracted from the cryostatand connected to the data-acquisition (DAQ) system withsampling rates up to 100 Hz. A bipolar magnet power sup-ply (MPS), capable of supplying up to 500 A at 20 V, isremotely controlled by the DAQ system. The signals fromthe sample are monitored on the screen and archived in theDAQ system at the same time. The stored data can beretrieved afterwards and processed for further analysis.The system setup is depicted in Fig. 5.

2.3. Experimental results

Fig. 6 shows some results of typical measurements. Thecurrent signals in Fig. 6a are those that are converted from

Fig. 5. Instrumentation, control, and DAQ system setup.

Fig. 6. Experimental results under the condition B0 = 1 T and s = 20 s: (a)current signals, (b) voltage signals.


the Hall sensor signals after calibration. There is a delay inthe rise and fall of the HS#2 signal compared with the

HS#1 signal, from which a diffusion current phenomenoncan be confirmed. The average current at the flattop is227 A for HS#1 and 109 A for HS#2 and the ratio betweenthe two is 0.48, which is also maintained during the ramp-ing up and down.

Using the experimental condition and the joint geome-try, the loop voltage during the ramping was calculatedto be 0.23 mV but Fig. 6b shows the inductive voltage ofthe VTS#1 signal is only 0.11 mV, being about the half.Though this looks like a contradiction, this can beexplained from Fig. 3a, which shows that the VTS#1installed at the bottom of the CICC is measuring only halfof the loop voltage. After analyses of all experimental con-ditions, we found that the average flattop voltages as wellas the peak values are almost proportional to the ramprate, from which it is confirmed that the field attenuationdoes not occur.

To compare the signals for different experimental condi-tions, we normalize them in two ways (Fig. 7). In the casesthat the signals are not fully saturated, a common pointamong them was sought and they are normalized by theirown current values at that point. At the very beginningof the ramp up, all the signals had several ripple-like peaks.Since the first peak was the most prominent, it was selectedas the reference point (Fig. 7a). In the other cases, they areeach normalized by their average flattop current (Fig. 7b).

The rising and falling shapes of the normalized signalsare almost identical no matter what the ramp rates are.This observation suggests that the longitudinal resistanceis invariant with respect to the change in ramp rate thoughthere is an additional joint at the top of the sample. That is,the dominant longitudinal resistance comes from the mainjoint only. The decay time constant will be determined pre-cisely with the aid of numerical analysis.

Fig. 7. Normalized signals (HS#1): Data sets are normalized (a) to thefirst individual peak and (b) to the average flattop current.

Fig. 8. Transmission line model: The sample joint is assumed to be locatedat x = 0 as a point.


The current signal starts to decay before reaching flattopwhen the ramping time is shorter than 5 s. In this case, wecan estimate the ‘virtual’ flattop and obtain the flattopvalue as follows. First, we obtain the ratio between the firstpeak and the flattop values of the saturated signal. Multi-plying the first peak value of the non-saturated signal bythe ratio finally gives the ‘virtual’ flattop value. It was1102 A in the case of the 2 s ramp-up whose first peak valuewas 1005 A. Note that the induced current is half of thisvalue or about 500 A since there exist two current loopsin the sample joint.

;

3. Numerical analysis

3.1. Analysis model and method

An electrical transmission-line model is used to analyzethe experimental joint sample system. The mode circuit isshown in Fig. 8 where R, L, G, and C stand for longitudinalresistance, inductance, transverse conductance, and capac-itance per unit length, respectively. The sample joint islocated at the very beginning of the line (x = 0) as the volt-age source is ramped up and down. For this experiment,the capacitance per unit length can be neglected since the

transmission lines contact each other. The governing equa-tions of the model are as follows.

� ovðx; tÞox

¼ Riðx; tÞ þ Loiðx; tÞ

ot;

� oiðx; tÞox

¼ Gvðx; tÞð2Þ

The sample joint as emf voltage generator is assumed to bea point located at x = 0. With the assumption that thewaveform as shown in Fig. 4 does not have the ramp-downprocess, the boundary condition for the voltage is given asfollows:

vð0; tÞ ¼ e0; 0 < t 6 s

¼ 0; t > sð3Þ

where e0 = U/s and U is the magnetic-flux change or theproduct of the magnetic-field change and the flux linkagearea (FLA). The total FLA of the sample joint is4.5 � 10�3 m2. As previously mentioned, the sample iscomposed of two circuits. These can be assumed to be iden-tical if we neglect the difference of the transmission linesmade of the 13-triplet and the 14-triplet bundles whichare nearly half of the sub-cable as shown in Fig. 3. The rel-evant FLA to each circuit is thus half of the total FLA.

In case R = 0, Eq. (2) is reduced to a diffusion equationand under the ramped field with the boundary conditionabove, its analytical solution for the current is given inRef. [9] for both a finite and infinite lines. In the latter case,the solution for the current is as follows.

iðx; tÞ¼ Imax

ffiffiffits

rexp �LGx2

4t

� ��

ffiffiffiffiffiffiffiffiffiffiffiLGx2

4t

serfc


4t

s0@

1A

8<:

9=;;

0< t6 s

iðx; tÞ¼ Imax

ffiffiffits

rexp �LGx2

4t

� ��


4t

serfc


4t

s0@

1A

8<:

9=;

� Imax

ffiffiffiffiffiffiffiffiffiffits�1

rexp � LGx2

4ðt� sÞ

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLGx2

4ðt� sÞ

serfc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLGx2

4ðt�sÞ

s0@

1A

8<:

9=;

t> s;

ð4Þ


where

Imax ¼ 2U

ffiffiffiffiffiffiffiffiG

psL

r; ð5Þ

erf(n) is an error function of n;

erfðnÞ ¼ 2ffiffiffipp

Z n

0

e�g2

dg;

erfc(n) is a complementary error function of n, defined aserfc(n) = 1 � erf(n). The real STS joint is connected tothe 600-m long CICC. It can be, therefore, considered infi-nite since the relaxation length is normally at most a fewmeters [12]. In this paper, we will analyze the joint samplewith an infinite-line solution.

In case of R 6¼ 0, Eq. (2) is reduced to the following par-tial-differential equation.

oiðx; tÞot

¼ 1

LGo2iðx; tÞ

ox2� R

Liðx; tÞ: ð6Þ

The voltage equation has the same form as the above. Bot-tura et al. [13] derived an analytical solution of the sameequation as the above to get the current distribution inmultistrand superconducting cable. The longitudinal resis-tance of our system is non-zero and we need to solve Eq.(6) numerically. A finite difference method was used andall the partial-differential equations were approximated toexplicit difference equations since non-zero x boundary isopen. The procedures to get the numerical solution are asfollows. First, we solve the voltage equation which is sim-ilar to Eq. (6) with v instead of i under the boundary con-dition of Eq. (3). Second, we obtain the boundarycondition for the current by substituting the solution forthe first equation of Eq. (2). Finally, we get a desired solu-tion of the current using this condition. Once the currentboundary condition is accurate enough, so is its solution.In the first and the second steps above, the mesh sizes ofspace and time are adjusted carefully. To check the exact-ness of our method, a numerically obtained boundary con-dition with R = 0 is compared to the analytically calculatedone using Eq. (4). Its average deviation is �1.3% duringramp-up and 0.013% during the flattop period.

Fig. 9. Numerical solutions to fit the measured induced current:measurement data comes from Fig. 6. L = 5.6 � 10�7 H/m is used forthe calculations.

3.2. Induced current, time constant and relaxation length

Using a short sample CICC which is the same as that ofthe joint sample and from which a pair of sub-cables areextracted to compose the two transmission lines, the induc-tance per unit length L is measured as 5.6 � 10�7 H/m.Though in reality the transmission line is either 13 or 14triplet bundle other than sub-cables, the inductance is littledifferent from that of the sub-cable since the constituenttriplets are twisted through three additional cabling stagesand they are positioned randomly in the sub-cable.

Being different from L, the transverse conductance perunit length G varies depending on the diameter of thetransmission line, which will be discussed later. However,

the analysis shows that even if G varies, the induced currentprofile which is normalized to the flattop value does notchange. This means, in other words, that the time constantcan be determined no matter what the G value is. Using theshort sample CICC, G is measured as 2.74�106 S/m, whichis used as the reference value in determining the decay timeand the longitudinal resistance as described below.

Numerical solutions are used to fit the measurementdata and to find an adequate value of R. For this purpose,the measured data of Fig. 6 is used. The numerical solu-tions are obtained for three different values of R. Whenthey are compared together, it is found that the maximuminduced current or flattop current is inversely proportionalto the square root of R. As shown in Fig. 9, all of the solu-tions are normalized to the average flattop current, whichenables us to see the differences in the numerical solutionsclearly. The solution with R = 400 nX/m is the best fit andthe decay time constant is approximately 1.2 s. Note thatthe fitting curve during the decay period is not perfectly lin-ear in the log plot. This fact suggests that the transverseconductance, as a source of an additional resistance tothe induced current, may play some minor role in the decayperiod.

We extract the average induced currents at flattops fromall the measured data according to the various ramp ratesand compare them with the numerical calculations(Fig. 10). The characteristic values used in the calculationsare L = 5.6 � 10�7 H/m and R = 4.0 � 10�7 X/m. In orderto fit the experimental data at the HS#1 position, the trans-verse conductance is adjusted to 4.24 � 105 S/m using thefact that for fixed U, s, and L, the maximum induced cur-rent is proportional to the square root of G as shown in Eq.(5). The meaning of this value will be discussed in the fol-lowing section.

As shown in Fig. 10, the induced current increases line-arly with increasing ramp rate. The calculation resultsagree well with the measurement data at the HS#1 posi-tion. The calculated current differs slightly from the mea-sured one and the discrepancy almost linearly increases

Fig. 10. Comparison of the measured and calculated average flattopinduced currents and estimations of relaxation lengths: legend #1 and #2stands for the data from HS#1 and HS#2, respectively.


from �1.6% to 3.1% with an increase in the ramp rate,which is consistent with the shielding effect of the cupro-nickel tubes. The HS#2 was installed to know how muchthe induced current would attenuate during propagationthrough the CICC. At this sensor position, however, theerror is as much as 62.2% and remains constant with theramp-rate variation. The ratio between the HS#1 andHS#2 is 0.78 from the numerical result, while it is 0.48from the experimental one, which means that the currentis more attenuated than expected from the numerical anal-ysis. We found later that on one end of the CICC weextracted a different pair of sub-cables than from the oneextracted on the opposite end. The error seems to be attrib-uted mainly to this and partly to the difference between thefinite sample length and the analysis which assumes an infi-nite one.

Using the numerical results, the relaxation length of theinduced current is calculated. We define the relaxationlength as the length within which the maximum inducedcurrent reduces by 1/e or 36.8%. It is 2.4 m for ramp rates

Fig. 11. (a) Loci of randomly selected strands from the two sub-cables, one frindicate starting points. (b) Variation of distance between the strands in the twfor a 500 mm cable.

up to 0.031 T/s, after which the length starts to decreaseslightly and has a steeper slope as the ramp rate increasesfurther. It is also found that almost all the current disap-pears within 10 m under any ramp-rate condition, whichagrees with the conclusion in Ref. [12].

4. Discussion

In the previous section, we found that a smaller trans-verse conductance than that of the sub-cable pair isrequired in fitting the experimental data, which means thatthe induced current flows through a smaller unit of the sub-cable. In a transmission line composed of two identicaltwisted wires, the transverse conductance per unit lengthis expressed as follows [8];

G ¼ 2

RcLp

; ð7Þ

where Rc is the inter-wire contact resistance and Lp is thetwist pitch. And the resistance Rc is expressed as;

Rc ffi qc

dpaLp

; ð8Þ

where qc is the contact resistivity, d is the distance betweenthe two wires, and a is the wire radius. Combining Eqs. (7)and (8), the following equation can be derived;

G ffi paqcd

: ð9Þ

The conductance G is inversely proportional to the distancebetween the two wires and proportional to the diameter ofthe wires. When we consider a long-length conductor, d inEq. (9) can be assumed to be the distance between the wiresin two separate sub-cables since the wires constituting thesub-cable are twisted around the center of the sub-cable.Fig. 11 shows the loci and distance variation of the twostrands randomly selected from the two sub-cables, wherewe find the pre-stated assumption to be applicable. There-fore, the conductance is affected only by the radius or

om each sub-cable, when the cable proceeds out of the paper: Black dotso sub-cables along the length of the cable: The average distance is 13.7 mm


diameter of the cable. The fitted conductance is 6.46 timessmaller than that of the sub-cable pair. Since the diameterof a sub-cable is 9.85 mm, the diameter of the transmissionline should be reduced to 1.52 mm, which is close to that ofa triplet or 1.65 mm. This fact is different from our expec-tation that the inductive current will distribute widely overthe cross-sectional surface of the joint. We think that thetwo-triplet pairs placed in the outer-most side of the cupro-nickel cylinder are actively involved in the induced currenttransmission. Not only experimental data but also numer-ical analysis shows that the induced current does not de-pend on the magnitude of external field but on the ramprate, which means that the degree of external-field shieldingis determined only by the ramp rate. From this, we infer thefollowings: Initially, the inductive current will start to flowthrough the outermost strand where the emf voltage ishigher than the inside strands. If the ramp rate becomesso high that the required shielding current cannot be sus-tained by one strand, one additional neighboring strandshould participate in the current flow, resulting in the in-crease of conductance G and the decrease of resistance R.In that case, the slope of the average flattop induced cur-rent against the ramp rate in Fig. 10 would change dis-cretely. In our case, since a triplet is the smallestuntwisted unit, it will behave as if it were a strand and,in the given ramp-rate regime, only one triplet is able toflow the required shielding current without any problem.

Localized induced current may cause overheating prob-lems. However, localization means lower conductance andhigher resistance compared to the case of wide distribution.Consequently, the induced current is lower than the distrib-uted case since, as mentioned previously, the induced cur-rent is proportional to the square root of G and inverselyproportional to the square root of R. Taking this consider-ation into account, overheating problem is not as serious aswe think. In the case of the maximum dB/dt of 0.25 T/s,heat generation in the joint is calculated to be only12 mW. This amount of heat causes negligible temperaturerise. If the heat is applied to the helium flow of 5 g/s at4.5 K and 5 bar, for example, which is the KSTAR stan-dard condition at the inlets of cooling channels, heliumtemperature will increase only by 0.57 mK.

5. Summary

This paper presents the current imbalance phenomenonthat is observed in a soldered joint under time-varyingmagnetic field and its relaxation in the CIC conductor con-nected to the joint. The KSTAR TFC joint is a good exam-ple for investigating this phenomena and the experimentwith a joint sample is conducted under real operationscenarios.

A train of symmetric trapezoidal pulses with differentramp rates up to 0.25 T/s is applied to the sample. Whenthe ramping time is shorter than 5 s, it is found that theinduced current signal starts to decay before the flattop isreached, but the rising and falling shapes are the same asthe others. Experimental data shows that the flattop cur-rent is proportional to the ramp rate of the magnetic field(dB/dt) which agrees with the result of the numerical anal-ysis. The discrepancy between the measured and the calcu-lated values is only a few percent and increases slightly andalmost linearly with the dB/dt increase, which suggests theexistence of a small attenuation of the field by the jointpieces.

An infinite transmission line model is used in numericalanalysis where the solutions are used to fit the experimentaldata. Characteristic parameters such as longitudinal resis-tance, decay time constant, transverse conductance areobtained. According to the results of the analysis, it isfound that the induced current is not distributed in thesub-cable but localized to a much smaller region such asa triplet. The current imbalance by the induced current inthe joint relaxes by 1/e within 2.4 m even under lowramp-rate conditions and disappears completely within10 m in the case of NbTi CICC.

Acknowledgements

This work was supported by the Korean Ministry of Sci-ence and Technology under the KSTAR project. We aregrateful to Mr. Jongsu Kim, Mr. Yongbok Chang, andDr. Hirofumi Yonekawa for their assistance in the experi-ment. And also Dr. Keeman Kim, Dr. Denis P. Ivanov, Dr.Yeong-Kook Oh, and Dr. Jooshik Bak are highly appreci-ated for their invaluable advice.

References

[1] N. Mitchell, Cryogenics 39 (6) (1999) 539.[2] L. Bottura, Physica C 310 (1998) 316.[3] N. Amemiya, Cryogenics 38 (5) (1998) 545.[4] S. Kim, S. Jeong, Physica C 407 (2004) 62.[5] S. Jeong, M. Takayasu, Cryogenics 42 (2) (2002) 79.[6] S. Jeong, S. Kim, T.K. Ko, IEEE Trans. Appl. Supercond. 11 (1)

(2001) 1689.[7] S. Jeong, J.H. Schultz, M. Takayasu, et al., Cryogenics 36 (8) (1996)

623.[8] L. Krempasky, C. Schmidt, Cryogenics 36 (6) (1996) 471.[9] L. Krempasky, C. Schmidt, Cryogenics 39 (1) (1999) 23.

[10] J.S. Bak, H.L. Yang, Y.K. Oh, et al., Fus. Eng. Des. 81 (2006)2315.

[11] B.S. Lim, J.Y. Choi, S.I. Lee, et al., IEEE Trans. Appl. Supercond. 16(2) (2006) 743.

[12] N. Mitchell, Cryogenics 40 (2) (2000) 99.[13] L. Bottura, M. Breschi, M. Fabbri, J. Appl. Phys. 92 (12) (2002) 7571.

Current imbalance in superconducting strand-to-strand joint and its relaxation in multistage...

Documents

Transcript of Current imbalance in superconducting strand-to-strand joint and its relaxation in multistage...