3.1 Current density distribution

Thus, we will consider a more general current density distribution in coils: the current magnitude in a coil is independent of the ones in other coils (see Fig. 2). Then, by extending the method, the current density distribution (illustrated by Fig. 3) will be expressed as

where δ(θ) is the one-dimensional Dirac distribution as a function of the angle θ. Then, by applying a reasoning analog to the one developed in Coillot et al. (2016), the Fourier series coefficient (a_2k−1) of j(θ) can be calculated. The current density function (j(θ)) can then be re-written as a Fourier series:

3.2 Magnetic field generation

The magnetic field produced for the in planta portable NMR experiments requires the plant to be placed in the center of the coil. That leads us to focus only on the even SHS coils case (i.e., without a central coil; thus, the term I₀ vanishes) composed of N pairs of coils. Each coil has N_n conductors flown by a current I_n. The magnetic field generated at the center of the SHS coil pair (B_n), based on the magnetic field at the center of a symmetric pair of coils, can be expressed either as a function of the coil radius (R_n), see Eq. (4), or as a function of the radius supporting circle (R, such as in R_n=Rcos(θ_n)), defined in Fig. 2; see Eq. (5).

The total magnetic field (B_SHS) produced by the N pairs of coils results from the sum of the magnetic fields produced by each pair of coils:

Figure 2Electrical conductors of unknown angular positions (θ_n), flown by currents I_n, distributed along the circumference.

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Figure 3Illustration of the space current density distribution for non-identical currents flowing through the coils.

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3.3 Formulation of the homogenous magnetic field coil design problem

In the context of N turn coil pairs without the central coil, we will have to determine the N angles and currents allowing us to produce the target magnetic field (B_SHS=8 mT in our design case) and satisfying the harmonic suppression. That gives us 2N equations allowing us to suppress (2N−1) harmonics. The first equation of the equation set (Eq. 7) allows the magnetic field goal to be reached (namely Eq. 6), while the other equations are deduced from harmonics suppression (Eq. 3):

This set of equations offers the possibility of designing the coil generating the required magnetic field and allowing us to enhance the homogeneity by means of the harmonics suppression technique.

3.4 Numerical resolution aspects

The angle solutions of Eq. (7) are obtained using a numerical resolution tool. Nevertheless, for a large number of variables, the algorithm may converge to an aberrant solution with some angles higher than π∕2, which satisfies the harmonic cancellation but does not maximize the magnetic field. Thus, it is recommended in such a case to solve the designed problem using optimization formalism. The objective function will be a compromise between the maximization of the magnetic field production g(u) (left member of the first equation of Eq. 7) and the minimization of function f(u), ensuring the cancellation of the harmonics (remaining equations of Eq. 7):

where u is the set of variables (θ_n and I_n).

Next, the optimization formalism will permit us to limit the feasible space solution research using inequality constraints. Thus, an example of optimization formalism for the SHS coil design will be

α and β are weighting parameters permitting us to guide the optimization resolution according to the designer priorities. They also have the role of normalizing different goals. In our case, the harmonics suppression is the primary objective, so we choose a high value for the β parameter (β=1000), while the α is used to normalize the function ($\mathit{\alpha }={\mathit{\mu }}_{\mathrm{0}}/R$).

Lastly, the angle solutions of the optimization problem are obtained using a numerical optimization tool.

To conclude this section, we report the angle solutions (expressed in radians) and homogeneity efficiency (see Table 2) obtained for SHS 8 and SHS 10 by means of the optimization formalism of Eq. (10):

• SHS 8 (θ₁=0, θ₂=0.439, θ₃=0.609 and θ₄=1.061);

• SHS 10 (θ₁=0.163, θ₂=0.264, θ₃=0.573, θ₄=0.7901 and θ₅=1.159).

The optimization formalism opens the way to a flexible solution where other kinds of constraints can be implemented (given range of value for instance).

4.1 Extended SHS 4 design

We consider the case of two pairs of turns (i.e., N=2) with identical turn numbers (i.e., $N_{\mathrm{1}}=N_{\mathrm{2}}=N$). Thus, we have four variables: two angles and two currents, which allows us to cancel the first three spatial harmonics:

The solution of this set of equations is obtained by means of a numerical minimization and leads to θ₁=0.3142, θ₂=0.9425, I₁=1.7794 A and I₂=1.0997 A ($I_{\mathrm{2}}/I\mathrm{1}=\mathrm{0.618}$), which is slightly different from the solution obtained by Hoult and Deslauriers's coil (cf. Hoult and Deslauriers, 1990) mentioned here (${\mathit{\psi }}^{\prime }=\mathit{\pi }/\mathrm{2}-\mathit{\psi }=\mathrm{0.288}\approx {\mathit{\theta }}_{\mathrm{1}}$, ${\mathit{ϵ}}^{\prime }=\mathit{\pi }/\mathrm{2}-\mathit{ϵ}=\mathrm{0.871}\approx {\mathit{\theta }}_{\mathrm{2}}$ and $I_{\mathrm{2}}/I\mathrm{1}=\mathrm{0.682}$).

Finally, we report, in Table 3, the comparison of the 1 % magnetic field homogeneity for standard SHS, an extended SHS coil and Hoult–Deslauriers coils.

4.2 Extended SHS 6 design

We now consider the case of three pairs of turns (i.e., N=3). We have six variables: three angles and three currents, which allows us to cancel five harmonics:

The solution of this set of equations is obtained by means of a numerical minimization and leads to θ₁=0.2244, θ₂=0.6732, θ₃=1.122, I₁=1.305 A, I₂=1.0465 A and I₃=0.5808 A, while the homogeneity efficiency is reported in Table 4.

Table 3Four-turn axisymmetric coil homogeneity: comparison between the four-turn SHS coils, the four-turn extended SHS and the four-turn Hoult and Deslauriers coil.

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The results of the simulations show that the extended SHS method makes it possible to improve the homogeneity in the region where the NMR measurement was achieved.

4.3 Thermal behavior of SHS coils

Splitting of SHS coils into multiple coils has an important advantage, with respect to the thermal behavior, since it allows us to split the total power on multiple coils (or, in other words, to increase the exchange area of coils). The thermal behavior will be studied with a special focus for the inner coil, which is the closest to the plant stem where the self-heating must be limited. The thermal exchange between neighbor coils is neglected.

4.3.1 Power to be dissipated by SHS coils

The power dissipated by Joule's effect for the nth coil (with resistance Res_n), see Fig. 4, is expressed as

$$\begin{array}{}\text{(13)}& P_n={\mathrm{Res}}_n{I}_n^{\mathrm{2}},\end{array}$$

where I_n is the current flowing through the nth coil and Res_n is its resistance, given by the following equation:

$$\begin{array}{}\text{(14)}& {\mathrm{Res}}_n=\mathrm{2}\mathit{\rho }{\displaystyle \frac{N_n\times R\times \cos \left({\mathit{\theta }}_n\right)}{r_{\mathrm{c}}^{\mathrm{2}}}},\end{array}$$

where ρ is the material resistivity, r_c is the wire radius and N_n is the number of turns of the considered nth coil.

Table 4Six-turn axisymmetric coil homogeneity: comparison between the six-turn SHS coils and the six-turn extended SHS.

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4.3.2 Thermal model

The thermal resistance of the coil will be neglected since a copper conductor is assumed. Next, the power to be dissipated (P_n) will be evacuated by means of natural convection (an average convection coefficient h will be assumed) and radiation through the exchange area of the coil (S_n):

$$\begin{array}{}\text{(15)}& P_n=h{S}_n\left(T_{\mathrm{s}}-T_{\mathrm{\infty }}\right)+\mathit{\sigma }\mathit{ϵ}{S}_n\left(T_{\mathrm{s}}^{\mathrm{4}}-T_{\mathrm{\infty }}^{\mathrm{4}}\right),\end{array}$$

where T_s is the average surface temperature, σ is the Stefan–Boltzmann constant (5.6703e⁻⁸ Wm⁻² K⁻⁴), ϵ is the emissivity of the surface of a material (a black body is assumed, i.e., ϵ=1) and T_∞ is the ambient temperature.

Figure 4SHS coil constituted of N pairs of coils.

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The natural convection coefficient is preliminarily determined by fitting the thermal prediction given by Eq. (15) with thermal measurement on a SHS 4 prototype (whose geometrical parameters are summarized in Table 5 below), supplied with $I_{\mathrm{1}}=I_{\mathrm{2}}=\mathrm{1.4}$ A (in order to produce 8 mT). Consequently the measured temperature of the coil overheating, ΔT<25 K and T_∞=297 K, leads us to estimate h=6 Wm⁻² K⁻¹.

4.4 Constrained extended SHS 6: fixed current and aperture angle θ₁

To solve the thermal issue we will now constrain the formulation of the extended SHS by fixing some parameters like the current intensity flowing into the inner coil (1 A should permit us to limit the coil overheating: ΔT<10 K) and the angle of the central pair of coils (in order to have a sufficient gap to insert the plant stem, namely ≃30 mm). Thus, we will fix I₁=1 A and θ₁=0.2244, which reduces our number of variables to four and thus simplifies the set of Eq. (12). The solutions obtained for the new set of equations are θ₁=0.2244, θ₂=0.4509, θ₃=0.9676, I₁=1 A, I₂=0.8844 A and I₃=1.0017 A.

Table 5Geometric parameters of SHS 4 (supplied with $I_{\mathrm{1}}=I_{\mathrm{2}}=\mathrm{1.4}$ A).

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Table 6Homogeneity comparison between the SHS 4 prototype and the constrained extended SHS 6.

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5.1 SHS 4 coils prototype

As depicted before, the SHS 4 coil could suffer from too much overheating. In order to solve this issue we increased the exchange area (S=0.24 m²) and the nature of material support so as to reduce the temperature. Our goal is to evacuate heat by the natural convection mechanism, so we created a simple heat sink placed around the element to be cooled (coils). To make it, we have chosen a non-magnetic aluminum material with a high thermal conductivity and a black paint to get the emissivity coefficient as close as possible to 1. The improved setup (Fig. 5) exhibits an overheating of about ≃ 12 K, which satisfies the plant integrity.

Our prototype allowed us to measure, with a one-pulse NMR sequence (acquisition time < 3 mn for repetition time about 1 s), the signal at the stem of a sorghum plant directly in the greenhouse (Fig. 6).

The NMR signal of plants measured at 333 kHz (which corresponds to the expected Larmor frequency of ¹H at 8 mT) demonstrates clearly the homogeneity performance: the line width of the NMR signal used to estimate the homogeneity of the prototype is about 900 ppm (300 Hz) in a volume of 4 cm³ corresponding to the internal volume of the clipping RF coil dedicated to the plant stem (diameter = 20 mm and height = 14 mm).

Note that the NMR system comprises a resistive SHS magnet (without gradient coils) flowing by current produced by an enslaved power supply, a homemade coil dedicated to sorghum plants, an amplifier low noise high impedance as suggested in Asfour (2011) and concerning the pulse sequencer, a Chameleon (RS2D Company) was used.

Figure 5Portable NMR prototype based on the SHS 4 model at 333 kHz for agronomic studies (R=100 mm, N_n=320, window of wire winding: 18 mm × 15 layers, r_c=0.4 mm, and the prototype total weight = 7 kg).

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From a practical point of view, it is helpful to improve the homogeneity of the magnetic field starting from identical currents calculated by the SHS method and finally adjusting them by fine changes. This allows, when we are in good experimental conditions, to reach a homogeneity as low as 600 ppm (see Fig. 6).

Figure 6NMR spectrum of a sorghum plant at 333 kHz by the SHS 4 coils prototype in a greenhouse at CIRAD.

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The numerical computation of the homogeneity for the SHS 4 coil obtained by means of the standard deviation extracted from the magnetic field distribution over the centered sphere (performed with a numerical finite-element software) gives a linewidth value of about 456 ppm, which is in good agreement with the experimental one.

5.2 Constrained extended SHS 6 coils prototype

The design validated on the SHS 4 prototype results has encouraged us to go further in the prototype construction. The design of the constrained extended SHS 6 (based on parameters given in Sect. 4.4) seems to offer a promising compromise between homogeneity, weight, overheating and accessibility to the plant stem.

Indeed, increasing the number of coils to produce the same magnetic field is known to be beneficial for the thermal behavior.

The gap is now 30 mm (all the coils of the prototype have the same number of turns equal to 320 turns) and totally compliant with our future experiments.

The constrained SHS 6 prototype (as shown in Fig. 7) nominally supplied has exhibited an overheating limited to about ≃ 6 K around the sorghum stem under test, which is within the acceptable overheating range.

Our prototype allowed us also to measure with a one-pulse NMR sequence (acquisition time < 3 mn for repetition time about 1 s) the signal at the stem of a sorghum plant directly in the greenhouse. The homogeneity of the prototype is improved at 333 kHz (8 mT) by 10 % compared to the prototype SHS 4, but it would be even better if we had not imposed the gap restriction.