Temporal Enhancement of Trapped Field in a Compact NMR Magnet Comprising YBCO Annuli

Abstract

Temporal “enhancement” of trapped fields was observed in the central region of a compact NMR magnet comprising a stack of 2800 YBCO “square” annuli (YP2800), field-cooled at 4.2 K. This paper presents an analytical model to simulate the trapped field enhancement in YP2800. First, based on an inverse calculation technique, the current distributions in the 560 5-plate modules of YP2800 were computed from the measured trapped field distribution. Then, YP2800 was modeled as a set of “three magnetically-coupled subcoils”: the “bottom” coil (C_B, 140 modules); the “middle” coil (C_M, 280 modules); and the “top” coil (C_T, 140 modules). With the index resistance of each coil included, the circuit model shows that the average current in C_M “slowly” increases, induced by “fast” current decays in C_B and C_T. As a result, the center field in YP2800, dominated by the C_M currents, increases in time. The simulation agrees reasonably well with the measurement, which validates the analytical model.

I. INTRODUCTION

SINCE 2009, the MIT Francis Bitter Magnet Laboratory has conducted a 2-phase program to develop a compact desktop NMR (nuclear magnetic resonance) magnet comprising YBCO annuli [1]–[6]. In Phase 1, the main target is to design and construct a 100–200 MHz NMR magnet having a 9-mm room temperature (RT) bore, and operate it in solid nitrogen (SN2) at 10–15 K. The magnet, named YP2800, has been constructed with 2800 YBCO “square” annuli [4], [5]; each square annulus is 40- or 46-mm wide, 0.08-mm thick, and has a machined inner hole of ϕ26 mm. Recently, trapped field tests of YP2800 in a bath of liquid helium at 4.2 K have been completed. Currently, a customized NMR probe is being constructed, which will fit into the 9-mm RT bore of YP2800.

While testing YP2800 at 4.2 K, we observed temporal “enhancement” of trapped fields at and near the center of YP2800. We reported a similar field enhancement in YP1070 [6], a predecessor of YP2800. Patel, et al. also report a similar trapped field enhancement in a stack of high temperature superconductor (HTS) plates operated at “low (< 20 K)” temperatures [7]. To date, the field enhancement, either in YP1070 or YP2800, has not been observed in any tests performed in a bath of liquid nitrogen (LN2) at 77 K [2]–[5]. The temporal behavior of magnetic field is particularly important for high-quality NMR measurements [8]–[14].

This paper presents 4.2-K trapped field test results, focusing on trapped field enhancement in the YP2800 central region. An analytical model is proposed to simulate the temporal behavior of the trapped fields in YP2800, field-cooled at 4.2 K. Simulation results agree reasonably well with measured ones, which validates the proposed analytical approach.

II. TRAPPED FIELD TEST OF YP2800 AT 4.2 K

A. YP2800

YP2800 is a stack of 2800 YBCO “square annulus” plates, each a width of 40 or 46 mm and a thickness of 0.08 mm. Each square plate, originally cut from 40- or 46-mm wide YBCO tape manufactured by AMSC, was machined to a “square” annulus with a ϕ26-mm hole [2]. Five plates were grouped as a single “module” and the trapped field capacities of the 560 modules were individually tested in a bath of LN2 at 77 K [4]. Then, the 560 modules were stacked, to form YP2800, to place the best trapped-field-capacity module at the center and moduli of the descending capacities toward both ends [4]. Table I summarizes key parameters of YP2800.

Table I.

KEY PARAMETERS OF YP2800

Parameters		Values
Square width	[mm]	40 or 46
Inner hole diameter	[mm]	26
Thickness of each plate	[mm]	0.08 ± 0.005
Total number of plates		2800
Number of 5-plate modules		560
Overall height	[mm]	234
Peak trapped field at center at 77	K[T]	0.65

B. Test Setup and Field-Cooling Procedure

Fig. 1 shows a schematic drawing of the setup for the 4.2-K “field-cooling [15], [16]” experiments. A 5-T/300-mm (RT bore) magnet [2] provides a field of 3.5 T for field cooling at 4.2 K, with a discharge rate of 1.0 mT/sec.

Fig. 1.

Test setup of YP2800 trapped field experiments.

C. Field Measurement Accuracy

To measure an axial distribution of trapped fields along the 9-mm RT bore, a search coil, controlled by a step motor in Fig. 1, is used. The i.d., o.d., height, numbers of turns, and moving speed of the search coil are, respectively, 6.0 mm, 8.38 mm, 2.54 mm, 120 and 33.4 mm/s. To investigate the search coil uncertainty, a magnetic center field from the 5-T background magnet was repeatedly measured at a given operating current of the background magnet. Table II summarizes the search coil accuracy test results; the measurement error of the search coil approach is estimated to be ≤ 0.14%.

Table II.

MEASUREMENT ACCURACY OF SEARCH COIL (UNIT: T)

Trial 1	Trial 2	Trial 3	Trial 4	Mean	SD [%]
3.0497	3.0482	3.0497	3.0477	3.0488	0.089
3.2483	3.2485	3.2463	3.2452	3.2471	0.14
3.4933	3.4949	3.4968	3.4937	3.4947	0.14

D. Test Results

Fig. 2 shows measured axial field distributions in the central region of YP2800 (|z| ≤ 30 mm); z is defined as axial displacement from the YP2800 center. t = 0 is set to 5 minutes after the background magnet is completely discharged, i.e., the magnet current reached down to zero. In Fig. 2, the trapped field at the center (z = 0) “increases” from 3.5115 T at t = 0 to 3.5302 T at t = 4 hrs. In contrast, Fig. 3 shows measured axial field “decay” at 80 mm < z < 100 mm. The trapped field enhancement in YP2800 is similar to the reported for YP1070 [6]. Note that field enhancement was not observed in any tests at 77 K, either for YP2800 or YP1070 [2]–[4].

Fig. 2.

Temporal “increase” of trapped fields at and near the YP2800 center.

Fig. 3.

Temporal decay of trapped fields at 80 mm < z < 100 mm along the YP2800 axis.

III. SIMULATION OF TEMPORAL BEHAVIOR OF TRAPPED FIELDS IN YP2800 AT 4.2 K

A. Estimation of J_ce at 77 K

Fig. 4 shows measured axial trapped field distribution of YP2800 (black squares) in a separate LN2 test at 77 K. For the field cooling, the background field was 2 T, > 3 times larger than a measured peak trapped field of 0.65 T. This implies that the 0.65 T is practically the maximum trapped field of YP2800 at 77 K and thus all the plates in YP2800 are “saturated” in terms of current carrying capacity. We may define an average critical engineering current density, J_ce for each 5-plate module as the maximum current of a module divided by its overall cross section area (A_M).

Fig. 4.

Axial trapped fields along the YP2800 center: Black squares: measured at 77 K; blue circles: calculated from the estimated J_ce at 77 K in Fig. 5; black diamonds: measured at 4.2 K; blue triangles: calculaed from the estimated J_e at 4.2 K in Fig. 5.

Using an “inverse calculation technique [17]”, J_ce of the 560 modules at 77 K (red triangles in Fig. 5) can be estimated to minimize the object function of (1), where $J_{ce}^i$, B_z|m, B_z|cal, and 2z₁ are, respectively, average critical engineering current density of the ith module, measured axial fields, calculated axial fields with a given $J_{ce}^i$ distribution, and the target axial space; due to the “small (9-mm)” RT bore size, we focused only on the “axial” field distribution in |z| < 100 mm. Fig. 4 presents good agreement between the measured (black squares) and calculated (red circles) axial field distributions at 77 K; Fig. 5 shows the estimated J_ce of the 560 modules at 77 K by the inverse technique for the calculated axial fields (red circles) in Fig. 4

$$f\left(J_{ce}^i\right)=\underset{-z_1}{\overset{z_1}{\int }}{\left\{B_{z|m}-B_{z|cal}\left(J_{ce}^i\right)\right\}}^{2}dz,i=1,\dots ,560.$$ (1)

Fig. 5.

Estimated current densities in the 560 5-plate modules by the inverse technique; red triangles: J_ce at 77 K; blue circles: J_ce at 4.2 K; black squares: J_e at 4.2 K.

B. Estimation of J_e and J_ce at 4.2 K

Black diamonds in Fig. 4 corresponds to the measured axial trapped fields at |z| < 100 mm from the LHe experiment. The “plateau” of the measured distribution at 4.2 K in Fig. 4, not observed in those from the LN2 test (black squares), indicates that a number of YBCO plates around the YP2800 center are “not saturated”, i.e., the average engineering current density, J_e in each plate is smaller than its J_ce. Thus, using the measured fields at 4.2 K (black diamonds) in Fig. 4, the same inverse calculation technique was applied to estimate J_e of each module at 4.2 K, black squares in Fig. 5, not the J_ce.

In our previous report [6], the maximum trapped field of YP1070, a predecessor to YP2800, at 4.2 K is 4.3 T, ~10 times larger than that of 0.43 T at 77 K. Based on the result, we assume that, for simplicity, J_ce values at 4.2 K (blue circles in Fig. 5) is ~10 times those at 77 K.

C. Equivalent Circuit Analysis With “Three-Coil Model”

Based on the results in Fig. 5, the 560 modules in YP2800 may be grouped into “three magnetically-coupled subcoils” [Fig. 6(a)]: the first coil (C_B) consists of the “bottom” 140 saturated (J_e ≃ J_ce) modules; the second coil (C_M) the “middle” 280 unsaturated (J_e < J_ce) modules; and the last coil (C_T) the “top” 140 saturated modules. The total turns of a “coil” is assumed to be the number of modules; the coil current is defined as J_e multiplied by A_M, the cross section area of a single 5-plate module.

Fig. 6.

(a) Picture of YP2800 with the “three subcoils” indicated; (b) equivalent circuit diagram of the “three-coil” model.

Fig. 6(b) presents an equivalent circuit model for the three coils, of which the circuit equations can be expressed by (2), where L and M are self and mutual inductances, respectively. R is an effective resistance of “index loss” in (3), where V_c and n are, respectively, voltage criterion for critical current (1 μV × coil conductor length) and index value of ReBCO conductor, typically in the range 10–40 [18]–[21], which is set to 30 in this analysis. Table III summarizes key parameters of the “three coils”.

Table III.

KEY PARAMETERS OF “THREE” COILS

Parameters		CB	CM	CT
Number of 5-plate modules		140	280	140
i.d.; o.d.	[mm]		26;49
height	[mm]	58.8	117.6	58.8
Total turns		140	280	140
Self inductance	[mH]	0.2973	0.6921	0.2973
Mutual inductance with CB	[μH]	-	53	1.2
Mutual inductance with CM	[μH]	53	-	53
Mutual inductance with CT	[mH]	1.2	53	-
Index resistance, R at t=0	[μΩ]	1.9	2.8 × 10−7	0.61
Average Je at t=0	[A/mm2]	125.4	196.8	121.4
Average Jce	[A/mm2]	145.4	335.1	141.0
Field constant at z=0 mm	[mT/A]	0.0571	2.84	0.0571
Field constant at z=100 mm	[mT/A]	0.00536	0.134	2.36

$$\left[\begin{array}{ccc}{L}_T& M_{TM}& M_{TB}\\ M_{TM}& L_M& M_{MB}\\ M_{TB}& M_{MB}& L_B\end{array}\right]\left[\begin{array}{c}\frac{d{I}_T}{dt}\\ \frac{d{I}_M}{dt}\\ \frac{d{I}_B}{dt}\end{array}\right]=-\left[\begin{array}{c}{R}_T{I}_T\\ R_M{I}_M\\ R_B{I}_B\end{array}\right]$$ (2)

$$R_i(t)=\frac{V_c}{J_e(t)A_M}{\left(\frac{J_e(t)}{J_{ce}}\right)}^{n}i=1,2,3$$ (3)

D. Simulation Results

The initial current of each coil is calculated by J_e|t₌₀A_M. Fig. 7 shows the temporal variation of currents in the three coils; the inset represents an enlarged view of C_B. Due to inductive coupling among the three coils, C_M current “slowly” increases as C_B and C_T currents “rapidly” decrease. Fig. 8 shows consequent temporal behavior of the trapped fields at the YP2800 center (z = 0), calculated (blue circles) and measured (black squares). From Table III, the field constant of C_M at the center, 2.84 mT/A, is ~50 times larger than those of C_B and C_T. Therefore, the center trapped field is “enhanced” in time as the C_M current increases.

Fig. 7.

Temporal variation of currents in three coils (calculated); black squares: C_B; blue circles: C_M; red triangles: C_T. The inset is an enlarged view of the C_M currents.

Fig. 8.

Temporal enhancement of trapped fields at z = 0; Black squares: measured; blue cicles: calculated.

Fig. 9 shows temporal decay of trapped fields at z = 100 mm, measured (black squares) and calculated (blue circles). The field constant at z = 100 mm is dominated by C_T, 2.36 mT/A, > 17 times larger than those of C_B and C_M (Table III). Thus, the trapped field at z = 100 mm decrease as the C_T current decays in Fig. 7.

Fig. 9.

Temporal decay of trapped fields at z = 100 mm; Black squares: measured; blue circles: calculated.

The difference between calculated and measured results in Figs. 8 and 9 is mainly due to the “simplicity” of the three-coil modeling in Fig. 6, while each “coil” consists of >100 plates. Still, the proposed approach explains reasonably well the two experimental observations, field enhancement at z = 0 mm and field decay at z = 100 mm.

IV. CONCLUSION

Temporal “enhancement” of trapped fields was observed in the central region (|z| < 30 mm) of a compact NMR magnet comprising a stack of 2800 YBCO “square” annuli (YP2800), field-cooled at 4.2 K. Simultaneously, toward the ends of YP2800, |z| > 80 mm, temporal “decay” of trapped fields was observed. An analytical approach was proposed to simulate both enhancement and decay of trapped fields in YP2800. First, based on an inverse calculation technique, the current distributions in the 560 5-plate modules of YP2800 were estimated from measured trapped field distribution. The results showed that the 140 modules at the top and bottom of the YP2800 stack were “saturated” in terms of current-carrying capacity, while the 280 modules in the middle of YP2800 were “unsaturated”. Then, YP2800 was modeled as “three magnetically-coupled subcoils”: he first coil (C_B) consists of the bottom 140 modules; the second coil (C_M) the middle 280 modules; and the third coil (C_T) the top 140 modules. With the index resistance of each “coil” taken into consideration, the circuit model shows that the average current in C_M “slowly” increases, induced by “fast” current decays in C_B and C_T. As a result, the center field, dominated by C_M currents, increases, while the field at |z| > 80 mm decays. The simulation results agreed reasonably well with the measurement ones, which validates the analytical approach.