3.1. Method
The problem to be solved is to find the coil parameters which maximize SNR with the contraint of
-homogeneity
. The intensive use of FEM in the optimisation process would be very time consuming. Indeed, the accuracy of each model must be controlled to a high level at each step. Thus, an analytical model is preferrably used in an optimisation routine [
40,
41]. In order to make a parsimonious use of FEM, we combine FEM with the design of experiment (DOE) approach. This method permits to determine an interpolation function of the three coil parameters:
w,
h and
s, for any response of interest (SNR,
R, …). The interpolation function must have a simple form to be readily incorporated in an optimisation routine. We used a polynomial function derived from a Taylor expansion up to the third order around the point
of coordinates (
,
,
). Such an interpolation function is valid only in the vicinity of
, so that several interpolation functions are necessary to cover a large domain. The third order Taylor expansion comprises 20 terms, but the six interactions of quadratic terms with linear terms (
,
, …) proved to be not significant, so that they were discarded.
Linear regression is used to determine the 14 coefficients of the interpolation function. To have the 14 coefficients with the minimum error, we calculate within a cubic domain the responses of interest at 20 particular points. Thus, 6 degrees of freedom are associated to residuals. An algorithm based on the theory of optimal experimental design of Fedorov [
42] is used to find the 20 optimal points, represented in
Figure 6. The algorithm is implemented in the function
optFederov() included in the R package “AlgDesign” [
43,
44]. The algorithm selects 20 points among a list of
candidate points in the cubic box, corresponding to the 5 levels: −2, −1, 0, 1, 2 tested on each axis. The design of experiment (DOE) comprises 20 points which can be readily executed through a parametric study in COMSOL Multiphysics
®. The responses of interest are exported to calculate the 14 coefficients of the interpolation function by linear regression. Residuals are inspected to verify the prediction accuracy of the interpolation function. It is then incorporated in the optimisation routine to find the point (
,
,
) which maximise SNR. If the optimum point is found at a boundary of the cubic box, it means that the optimum has not yet been found. A new interpolation function is calculated which is valid in the new domain next to the preceding cube. If the optimal solution is found within the current domain, then the program is stopped. With this step-by-step method, the evolution of the optimisation process is traced from the beginning to the end. More details about the method presented here can be found in the
Appendix A.
3.2. Result of Optimisation at 200 MHz for a Nanoliter Sample
The number of turns being fixed initially, the method of optimisation described in the preceding section was applied. The optimal coil parameters were calculated for a sample of 1 nL and a coil working at 200 MHz. The results are presented in
Table 2 for the number of turns varying from 2 to 15. The optimum trace widths
are in the range of 2–7 μm, which are of the order of the skin-depth
at
MHz. The optimum trace height
are in the range of 80–115 μm, which matches the sample volume. Surprisingly, we found no optimum for the parameter
s, so the results of
Table 2 are presented for the arbitrary value
μm. In fact, SNR linearly increases as
s tends to zero in the vicinity of the point (
,
). The decrease of
s from
μm to
μm increases SNR marginally of about
in the worst cases. No significant variation of SNR with
s exists for coils of less than 5 turns. It turns out that scroll microcoils do not suffer proximity effect. This feature permits to choose the smallest possible value of
s to have the maximum SNR. However, a very low value of
s would marginally enhance SNR, but would not ensure a good electrical insulation between the conductors. Also, the coil self-capacitance, which increases as
as
s tends to zero, would be considerably increased for a very small value of
s. The consequence of a very low value of
s is the increase of the capacitance, the lowering of the self-resonance frequency and an increase of dielectric losses through the lowering of the parallel resistance
in Equation (
8).
Table 2 shows that SNR and
-homogeneity increase monotonically with the number of turns from 2 to 10, but begins to decrease after 10 turns. The 12-turns scroll coil has a resistance of about
, which is sufficient to preserve the intrinsic SNR of the coil when it is connected to the readout circuit [
16]. The self-capacitance and the self-inductance of the 12-turns coil with
μm are
(
) and
, giving the self-resonance frequency
GHz. The self-resonance frequency is high enough to use the coil for broadband multinuclei detection [
16].
Figure 7 shows contour plots of SNR and
H as a function of
w and
h, for
μm around the optimum solutions, for 4 and 5 turns.
Figure 7b,d show the effect of an additional resistance of
, due to connexion wires, possible sample losses or dielectric losses.
Figure 7a,c concern the intrinsic coil resistance. SNR is less affected by this additional resistance when the resistance has the greatest value.
The optimum trace width
does not depend on the additional resistance, contrary to the optimum trace height
which tends to decrease. To minimize the effect of additional resistance, the intrinsic coil resistance
R should be high. In the perspective of using scroll microcoils for broadband multinuclei NMR, where the low noise amplifier can be directly connected to the coil without impedance matching and without tuning, it is mandatory to have the highest coil resistance in order to preserve the intrinsic SNR of the coil [
16,
20].
Beyond 10 turns, the SNR begins to decrease. As , the decrease in SNR means that the denominator increases faster than the numerator . Indeed, the contribution to the -field of outer windings is becoming smaller and smaller as the windings radii increase, whereas the resistance of these windings increases. From 10 to 15 turns, the increase of is about , which is almost compensated by the increase of of . This results in a decrease in SNR of only from 10 to 15 turns. To decide which coil is optimum, other criteria than SNR must be applied. In order to preserve the intrinsic SNR, it may be judicious to favour a large coil resistance and thus to favour a coil with the largest number of turns. In addition, a large number of turns is beneficial to -homogeneity. However, the addition of turns increases the inductance of the coil, which lowers the self-resonance frequency and risks compromising the use of the coil for broadband NMR spectroscopy. Anyway, depending on the intended application, the optimum number of turns would be between 10 and 20. A large number of turns gives the coil a good mechanical resistance which avoids the use of a support, so that the coil could serve as a container for the sample.