An Empirical Expression to Predict the Resonant Frequencies of Archimedean Spirals

Abstract

This work presents an empirical formula to accurately determine the frequencies of the fundamental and higher order resonances of an Archimedean spiral in a uniform dielectric medium in the absence of a ground plane. The formula is based on method-of-moments simulations which have been experimentally validated. This empirical formula is widely applicable to a broad range of spirals from thin-ring to disk-shaped (ratio of inner to outer radii 0 to 1), with 10 or more turns.

I. Introduction

Planar spiral coils are used in many applications as inductors or resonators. Examples of these applications include but are not limited to planar inductors [1], wireless power transfer [2], meta-materials [3], filters [4], and nuclear magnetic resonance (NMR) probe coils [5]. Specifically, when spirals are used as high temperature superconducting (HTS) NMR probe coils, it is important to accurately predict not only the self-resonant frequency (SRF) but also the frequencies of the higher order modes. Many currently available formulae that describe the behavior of spirals are either only valid up to the SRF or are only applicable to a limited class of spiral designs. Presented in this work is an empirical expression which predicts the resonant frequencies of the modes with excellent accuracy for spirals with widely varying size, pitch, filling factor, and ratio of inner and outer radii. The expression is based on the simulated mode frequencies of spirals in a uniform dielectric medium.

The resonant frequencies of spirals have been investigated by others in several recent papers. In [6], an empirical formula was presented by Yun et al. for calculating the fundamental resonant frequency of a planar spiral. The formula they present is reasonably accurate in determining the SRF, but does not predict the higher order modes. In [7], Breitkeurtz and Henke solved a discretized transmission line model of a spiral to obtain the mode frequencies as well as the current distribution at the SRF and higher order modes. Other circuit models were introduced to model the behavior of spirals in [8] and [9], though the resonant behavior was not the focus. Even more recently, it was shown by Maleeva et al. [10] that when the inner radius approaches the outer radius, i.e., in the thin- spiral limit, the resonant current distribution and mode frequencies can be calculated analytically. In distinction to previous work, our project is to derive a simple formula for spiral resonances which covers the broadest possible range of geometric parameters. A detailed analysis of the effect of a dielectric substrate is outside of the scope of the present study, but would be of great practical interest.

This work was organized into three stages. The first step was to validate a simulation approach by comparing simulated resonances to those measured experimentally or reported in the literature. This is described in Section III. The second step was to simulate the spectra for spirals with a wide range of parameters. Observations from this collection of spirals are given in Section IV. Details on a subset of the collection are given in Table I. Finally, in Section V, a formula was developed to fit the simulated resonances.

TABLE I.

DETAILS OF REPRESENTATIVE SIMULATED SPIRALS

	Mode number	Freq. Simulated (MHz)	Freq. Prediction (MHz)
Coil #1 N = 20 ri = 0.09 mm ro = 2.49 mm P = 0.12 mm	1 2 3 4 5	706 1658 2591 3524 4455	717 1651 2584 3517 4450
Coil #2 N = 25 ri = 0.12 mm ro = 2.12 mm P = 0.08 mm	1 2 3 4 5	649 1518 2370 3222 4076	652 1507 2362 3217 4072
Coil #3 N = 5 ri = 5.25 mm ro = 10.25 mm P = 1.0 mm	1 2 3 4 5	415 1071 1722 2388 3053	406 1061 1715 2370 3024
Coil #4 N = 5 ri = 5.5 mm ro = 15.5 mm P = 2.0 mm	1 2 3 4 5	330 811 1285 1775 2260	323 806 1289 1772 2255
Coil #5 N = 10 ri = 5.125 mm ro = 10.125 mm P = 0.5 mm	1 2 3 4 5	202 522 835 1159 1480	201 524 847 1170 1492
Coil #6 N = 25 ri = 5.01 mm ro = 6.01 mm P = 0.04 mm	1 2 3 4 5	86 271 438 617 787	90 264 438 612 786
Coil #7 N = 10 ri = 5.025 mm ro = 6.025 mm P = 0.1 mm	1 2 3 4 5	224 698 1128 1581 2020	229 676 1122 1569 2016
Coil #8 N = 50 ri = 5.025 mm ro = 10.025 mm P = 0.1 mm	1 2 3 4 5	39.5 102 163 226 289	39.1 102 165 227 290
Coil #9 N = 10 ri = 50.25 mm ro = 60.25 mm P = 1.0 mm	1 2 3 4 5	22.1 69.7 113 159 203	23.2 67.7 112 157 201

In this table the predicted data were obtained by the empirical formula described in Section V. The simulated values above were obtained from EM simulations described in Section III. In the simulations the ground plane was kept at a distance of at least 9 radii. All the spirals in the table above have a filling factor =50%. Each spiral was simulated for filling factors 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90%, with the same pitch and radii as stated in the table.

II. Background

A. Application

The motivation for this study is the authors’ use of spiral resonators as a building block for NMR probes. Nuclear magnetic resonance is an experimental technique which is widely used to identify compounds and their molecular structure. NMR provides a wealth of important information about molecular bonds that cannot be obtained by other methods, but it suffers from inherently low sensitivity. In NMR spectroscopy, samples are placed in a high magnetic field, which causes nuclei with non-zero spin to be resonant at frequencies proportional to this static magnetic field. At the heart of an NMR spectrometer is a sample probe which contains coils that generate uniform RF magnetic fields at the specific frequencies to excite and detect the NMR signals. Resonant coils with Q-values of several hundred are typically used to improve excitation efficiency and detection sensitivity. A further boost in sensitivity can be obtained from the use of high temperature superconducting coils instead of normal metal coils [11]. These HTS coils are patterned out of thin film yttrium–barium–copper–oxide (YBa₂Cu₃O_7-δ or YBCO) deposited as self-resonant structures on planar dielectric substrates such as sapphire [12]. Using self-resonant structures circumvents the need for lossy capacitors and other elements that would reduce the Q-values of the coils. Superconducting coils such as these typically possess matched Q-values ranging between 5,000 and 20,000. Planar spiral resonators are a useful class of designs for these HTS probe coils. Spirals possess a strong magnetic dipole moment and they resonate at lower frequencies than designs which use interdigital capacitors, because the conductor is long relative to the overall coil size. In these probes, the spiral coils are coupled and tuned via movable wire loops. A full description of this technology is given in [13].

NMR experiments require a uniform RF magnetic excitation field. To generate this field, a pair of coils is placed on opposite sides of the sample in a configuration similar to a Helmholtz pair. These coils must be very precisely tuned to resonate at the specific frequencies to which the nuclei of the isotope of interest will respond. The most informative NMR experiments excite multiple isotopes. Some of the most common are ¹H, ¹³C, and ¹⁵N. Also, the NMR response of solvents with enriched levels of deuterium (²H) is typically used to regulate the static magnetic field to the 10^-10 stability required over the duration of a signal-averaged experiment. In order to generate the necessary RF magnetic fields in an HTS probe a separate pair of coils is included for each of these isotopes. These coil pairs are nested orthogonally around the sample to minimize interactions between the various channels. Fig. 1 shows a cross sectional view of a typical coil arrangement within a probe.

Fig. 1.

Cross-section layout of an HTS NMR probe showing orthogonal nesting of Helmholtz pairs of resonators around a 1.5 mm sample [5].

B. Motivation

Effective design of HTS NMR probes requires a high level of control over where the fundamental and higher resonances of each coil occur. The fundamental spiral resonance should occur at the NMR resonance of the relevant isotope. But also, the higher order modes of the spiral must not interfere with the NMR resonances of higher frequency isotopes. The resonant frequency of ¹H nuclei is roughly four times that of ¹³C nuclei and 10 times that of ¹⁵N nuclei, so ¹⁵N and ¹³C coils with resonances in a harmonic series have modes at frequencies that will interfere with the probe’s ¹H channel. Method-of-moments simulation has been previously shown to be an effective way to predict the mode frequencies of spiral coils for NMR probes [11], but such simulations are relatively expensive and time consuming and by themselves provide little insight into the underlying relationships between the spiral parameters and the resonances. A simple and accurate predictive equation would accelerate the design process and provide coil designers with a tool to suitably adjust the mode spectrum.

A study of freestanding, circular Archimedean spirals was chosen because they are the simplest of all spirals. However, HTS NMR coils, as well as many other applications for spiral inductors, use dielectric substrates to support the coil conductors. The effect of a dielectric substrate can be approximated to some extent by the use of an effective dielectric constant as in [14]. Also, HTS NMR coils are typically rectangular rather than circular. The objective of this work is to develop a basic understanding of spiral resonance behavior using these simplified designs. It would be useful to extend our approach to include the effect of substrates and noncircular shapes.

III. Methods

A filamentary circular spiral is described in polar coordinates (r, θ) by the equation r = r_i+αθ, where r_i is the inner radius and α sets the distance between turns. A few other simple and interrelated parameters will be used to describe the spirals in this paper. These include the wire width w, the spacing between wires s, the number of turns N, and the outer radius r_o. To remove ambiguity, the inner radius is measured from the center of the conductor at the start of the first turn and the outer radius is measured from the center of the conductor at the end of the last turn. These parameters are depicted in Fig. 2 for the reader’s convenience. It is also convenient to use the derived parameters of pitch P = w + s = 2πα and filling factor F = w / P.

Fig. 2.

Illustration of a spiral with relevant parameters annotated. These parameters include the wire width w, the turn spacing s, the pitch P, the inner radius r_i and the outer radius r_o.

The basic approach of this work was to use an efficient simulation process to evaluate the resonance frequencies of a collection of spirals and use the resulting data to determine relationships between the spiral parameters and the resonances. The spiral collection consisted of more than 80 different designs exhibiting filling factors ranging from 10% to 90%, numbers of turns varying between 5 and 50, ratios of inner and outer radii from 0.03 to 0.88, and a variety of pitch values. The SRF and the frequencies of the next five higher order modes were all determined for each spiral through simulation. The spiral spectra were then analyzed to characterize the deviation of the modes from a purely harmonic series. The spectra were also used to derive an empirical formula to predict frequencies of modes of spirals of arbitrary parameters.

A. Coupling to the resonators

In this investigation, resonant frequencies were determined by measuring or simulating the reflection coefficient into a single loop loosely coupled to the spiral and excited by a 50-Ω source. When measuring the resonances by experiment, a single coupling loop connected to a vector network analyzer was used. The setup is shown in Fig. 3. The mutual inductance between the loop and the spiral induces some shift in the apparent resonant frequency. Froncisz et al. in [15] derived an expression for this shift which is stated in (1).

$$f_{measured}=f_0{\left(1-\frac{2\pi f_0{L}_c}{Z{Q}_0}\right)}^{-\frac{1}{2}}$$ (1)

Here, f₀ represents the resonant frequency of the spiral in the absence of the coupling loop, f_measured is the observed resonant frequency under a “matched” condition, L_c is the inductance of the coupling loop, Q₀ is the quality factor of the coil, and Z is the impedance to which the coil is matched by the coupling loop. For high-Q devices matched to high impedances (i.e., loosely coupled to 50 Ω) through a coupling loop with small self-impedance L_c, the frequency shift is negligible. The measurement of the resonant frequencies could also have been made using two loops and observing the transmission coefficient, but the mutual inductance between the two coupling loops would also have created a shift in the measured frequency as described in [16]. The spirals were placed at least nine radii from the ground plane so that resonance shifts due to eddy currents in the ground plane would be negligible. It can be noted that upper modes are largely unaffected by even a relatively nearby ground plane due to their small magnetic dipole moments.

Fig. 3.

On the left is a picture of the measurement setup. It consists of a network analyzer connected to a coupling loop. On the right is an illustration of a spiral along with a coupling loop. The coupling loop is attached to the end of a coaxial cable, the other end of which is connected to a network analyzer. The details of this coil are listed in Table I as coil #7.

B. Simulation

Simulations used to determine the resonant frequencies were performed using the Hyperlynx (Mentor Graphics) software package. Through routine modeling, it was observed that this package is very accurate in simulations of inductively coupled planar NMR probe coils.

The mode frequencies of two spiral resonators are then compared with experimental measurements to validate the simulation technique. In the first test, the spiral described by Kurter et al. was simulated and the obtained results were compared to the measurements reported in [3]. The frequencies reported in [3] are 74, 219, 355, 498, and 636 MHz. The results obtained through simulation are 80, 219, 352, 494, and 631 MHz. A comparison is shown in Fig. 5. In a second test, an Archimedean spiral was patterned on an FR4 substrate, shown in Fig. 4, and measured its resonance frequencies for comparison to our simulations. These results are also given in Fig. 5. An exact value for the dielectric constant of the FR4 board was not available, but reported values for FR4 substrates range between 4.2 and 4.8. A value of 4.3 was used, which provided a good match between the SRF and the simulation. A single, small pick-up loop was coupled to the spiral and the reflection coefficient was measured to determine the resonances. Modes were observed at 68, 149, 220, 290, and 359 MHz. The corresponding simulated modes were in close correspondence at 68, 151, 221, 289, and 353 MHz. Aside from the uncertainty in the dielectric constant of the FR4, the agreement between the simulations and measurements is at the level of 1%, which is sufficient for the present purposes.

Fig. 5.

Comparison of measured data against simulated results. The data include results reported by Kurter et al. in [3] and measured results from the spiral in Fig. 4, and the results from simulations of both spirals.

Fig. 4.

Archimedean spiral used to verify the accuracy of simulations. The spiral was fabricated from a copper clad FR4 circuit board. Simulations of this device match well with measured experimental results.

Using the verified simulation approach, a collection of more than 80 spiral resonators having a wide range of parameters were simulated up to the sixth resonant mode. The spiral from Fig. 4 is not included in this collection since its substrate would affect the results. Selected results are listed in Table I. This collection of spiral modes was used to develop and test the empirical formula in section V. Subsets of this collection were used for the plots in section IV.

IV. Observations

The modes observed in the studied spirals were analogous to the TEM modes seen in transmission line resonators: the number of maxima in the magnitude of current along the conductor is equal to the mode number of the resonance. All spirals in this investigation were inductively coupled, which enforced the boundary condition that the current must be zero at each end. As a result, modes that required nonzero current at the ends of the spiral were not allowed.

The spiral from Fig. 3, (coil #7 in Table I) is chosen as an example to illustrate some of the general properties of planar spirals. Fig. 6 shows the resonance frequencies of the first eight modes of coil #7. This resonance behavior shows features commonly exhibited by the wide range of spirals that were simulated. Fig. 6 also includes for comparison, the resonances of a uniform transmission line of the same physical length. As is well known, the spectrum of the uniform transmission line is harmonic, meaning that the frequency of the n^th mode is exactly n times the SRF, where n is a natural number. In a mode plot such as Fig. 6, a trendline through the modes of a uniform transmission line will have a y-intercept of zero. The mode plot of the spiral is similar in having resonances that lie along a straight trendline. However, both the slope and y-intercept of the trendline are somewhat different from the uniform transmission line. The y-intercept of the spiral trendline is negative, which results in a lower SRF for the spiral by about 50% compared to the corresponding mode of the uniform transmission line. Also, the slope of the spiral trendline is greater than that for the uniform transmission line, so the frequency gap between resonances for this spiral is about 4% larger than for a uniform transmission line. The variation in the slope of the spiral trendline appears to be related to the number of turns. In Fig. 7, the normalized slope of the spectrum has been plotted against the number of turns for the nine spirals of Table I. The quantity (c/2L) was used to normalize each value, where c is the speed of light and L is the conductor length.

Fig. 6.

A comparison between the modes of a harmonic transmission line and an Archimedean spiral resonator is shown. The conductor length of the spiral is the same as the length of the transmission line (347 mm).

Fig. 7.

The normalized slope of the mode spectrum depends on the number of turns. Spirals with roughly 30 and 40 turns have the same slope as a transmission line of the same length, while spirals with 5 - 30 turns were observed to have a greater slope and spirals with more turns, a smaller slope.

The quantity D is now introduced to describe the deviation from a harmonic spectrum. D is defined by (2), where b is the y-intercept of the trendline, and m is its slope.

$$D=\frac{b}{m}$$ (2)

Since m will always be positive, the sign of D depends only on the sign of b. D was observed to be negative for each the Archimedean spirals in uniform media that were analyzed, and the SRF was observed to be consistently less than the SRF for a straight wire of the same length. Previously reported measurements indicate that, for the cylindrical helix in a uniform medium, D can be positive for some structures and mode frequencies are sometimes higher than for a straight transmission line of equivalent length [17].

An expression for the quantity D is important because it allows a coil designer to quantify the type of mode spectrum needed for a particular design. Based on simulations of the spirals in Table I among others, it was observed that D is primarily determined by the ratio of the inner radius r_i to the outer radius r_o, and to a lesser extent by the filling factor F and pitch P. A graph depicting the relationship between D and r_i/r_o for different values of F is shown in Fig. 8. Each spiral investigated was simulated with 9 different filling factors ranging from 10% to 90%. As the filling factor was increased from 10% to 90%, D was observed to decrease as is shown in Fig. 8. Based on values of D obtained from the simulated spirals just described, an empirical expression was developed to model the dependence of D on the various parameters of the spiral. Equation (3) relates D directly to the filling factor, the scaled pitch, and the ratio of the radii.

$$D=-\left((0.29+0.043F)\left(\frac{r_i}{r_o}\right)+0.22\right)\left((F-0.5)\left(2.65\left(\frac{P}{r_o}\right)+0.03\right)+1\right)$$ (3)

All coil parameters were defined in section III. Note that for a filling factor of 50%, D is independent of the pitch.

Fig. 8.

Deviation from harmonic behavior seen in a variety of circular Archimedean planar spirals. These spirals have a wide range of pitch and number of turns. It is clear that D depends primarily on the ratio of inner and outer radii.

V. Calculation of resonant frequencies

A. Empirical expression

An expression to calculate the resonant frequencies of an Archimedean spiral was generated by curve fitting the simulated resonances of a collection of spirals, some of which are detailed in the Table I. The expression is given in (4).

$$f_n\left(Hz\right)=\left(\frac{v}{2L}\right)(0.24N^{-0.46}+0.95)(n+D)$$ (4)

In (4), n is the mode number, N is the number of turns in the spiral, and v is the velocity of electromagnetic waves in the surrounding medium. The expression for D is given by (3). Because all the parameters in (4) are either unitless or participate in simple ratios, any consistent choice of length units can be used. It is straightforward to derive (5) which gives the length L of an Archimedean spiral.

$$\begin{array}{cc}\hfill & L=\left(\frac{r_o}{2}\right)\sqrt{\frac{r_o^2}{{\alpha }^2}+1}+\left(\frac{\alpha }{2}\right)\text{ln}\left(\frac{r_o}{\alpha }+\sqrt{\frac{r_2^o}{{\alpha }^2}+1}\right)-\hfill \\ \hfill & \left(\frac{r_i}{2}\right)\sqrt{\frac{r_i^2}{{\alpha }^2}+1}-\left(\frac{\alpha }{2}\right)\text{ln}\left(\frac{r_i}{\alpha }+\sqrt{\frac{r_2^i}{{\alpha }^2}+1}\right)\hfill \end{array}$$ (5)

If the spiral is supported by a dielectric substrate, the velocity v can be calculated by dividing the speed of light by the square root of an effective dielectric constant [18]. For thick substrates a good approximation for an effective dielectric constant can be obtained from (6).

$${\epsilon }_{eff}=\frac{({\epsilon }_r+1)}{2}$$ (6)

When the substrate thickness is small, the modes may not all be affected equally and a more rigorous determination of effective dielectric constant will be required. The threshold of thickness is governed by the parameters of the spiral and the amount electric field fringing that occurs.

The empirical formula in (4) produced an accurate prediction of the resonance frequency for each of the spirals simulated. The average error seen across more than 480 resonant frequencies gathered from 81 different spirals was about 0.87%, the median error was 0.56%. Roughly 46% of the resonance frequencies were predicted to within 0.5% error and about 93% of frequencies were predicted to within less than 2% error.

In the case where F = 50%, (3) takes on a simpler form. When (3) and (4) are combined under this condition, the following equation is produced.

$$f_n\left(Hz\right)=\left(\frac{v}{2L}\right)(0.24N^{-0.46}+0.95)\left(n-\left(\left(0.31\right)\left(\frac{r_i}{r_o}\right)+22\right)\right)$$ (7)

The level of accuracy of (7) is shown in Fig. 9, where the first five resonances simulated for six different spirals with 50% filling factor are plotted along with the prediction curves generated from (7). As shown, there is excellent agreement between the two.

Fig. 9.

This plot shows the comparison between the simulated frequencies and those predicted by (7). Coil details are included in the Table I.

The proposed formula was applied to three different Archimedean spirals reported in literature by other investigators. Table II contains the data showing how accurately the resonance frequencies were predicted. Specifically, the percent error for the resonant frequencies determined by the empirical formula relative to those measured by experiment is given. The example from [3] and SR2 reported in [18] were modeled very accurately. These two coil examples are both fabricated from a superconducting film. The remaining example, SR1 from [18] was not modeled nearly as well. In all three examples, the effective dielectric constant was determined using (6). The SR1 example illustrates the limitation of (6) when the substrate is thin relative to the spiral dimension. In the two well-modeled examples, the substrate thickness is at least 20% of the spiral width (r_o-r_i). In the case of SR1, the substrate thickness is less than 5% of this value.

TABLE II.

COMPARISON TO PUBLISHED DESIGNS

mode	% error for SR1 in [18] εr = 3.48	% error for SR2 from [18] εr = 11.45	% error for example in [3] εr = 4.53*
1 2 3 4 5	16% 15% 11% 8.6% 6.7%	0.99% 1.3% 0.83% 1.3% 1.0%	0.58% 3.2% 1.7% 2.5% 2.1%

This table shows the accuracy with which the proposed formula predicts the resonance frequencies of spirals that have been reported by other investigators. The ratio of the frequency predicted by the empirical formula to the frequency measured by the investigators is reported for each of the first 5 modes for the three examples. Accuracy was excellent in the case of the 2 superconducting spirals but significantly worse for SR1 which was fabricated on a copper clad PCB.

^*This dielectric constant was obtained from [19] as none was reported in [3]. This value represents the average of the permittivity values perpendicular and parallel to the anisotropy axis.

B. Limitations of empirical formula

One of the limits of validity of (4) and (7) is when the spiral has less than 10 turns. Consideration of a collection of spirals with the same pitch, wire width, filling factor and outer radius demonstrates the few-turn limit as shown in Fig. 10. Only the number of turns and inner radius are varied. The 10-turn spiral in this set is described in Table I as coil #5. The first three modes of the spirals were simulated, and then they were compared to the result of the empirical formula to calculate the associated error. The number of turns was decreased by removing turns from the center. As this occurred, the error increased. When fewer than five turns were used, the error increased significantly, to as much as 60% when only one turn was present. The decreased accuracy of (4) for fewer than 10 turns is not surprising. When sufficiently many turns are present, wave propagation is influenced by the effect of neighboring turns. When only one turn is present the resonator approximates a uniform transmission line and has a nearly harmonic spectrum. However, for a single turn spiral with r_i approximately equal to r_o, (3) predicts a large magnitude for D, while in actuality, D for a single turn spiral is close to zero. While accurate results may be possible for some spirals with fewer than 10 turns, this empirical formula cannot be recommended for those structures. Spiral resonators in NMR probes typically have more than 10 turns.

Fig. 10.

Percent error in the first 3 modes of 10 different spirals. They all have the same pitch, wire width, and outer radius, but the number of turns varies from 1-10.

The accuracy of (4) beyond about 10 modes has not been explored. Very high order modes are difficult to measure, both in simulation and by experiment, because their RF magnetic fields fall off rapidly with distance.

C. Comparison to other methods

In Fig. 11, the results of four different methods for predicting the resonances for the spiral of Fig. 3 (coil #7) are shown for comparison. The first method, calculating the resonant frequencies based solely on the conductor length, was only accurate for the highest modes. The next method, one presented by Yun in [6], predicted the SRF with an accuracy of about 12% but was not intended for use for higher order modes. The method presented by Maleeva et al. in [10] was derived from first principles for spirals with r_i ≈ r_o and performed very well for coil #7. As noted in [10], thin ring-shaped spirals exhibit a ratio of resonant frequencies of approximately f₁:f₂:f₃:...:f_n = 1:3:5:...:(2n-1). Applying (2) to this mode spectrum results in value of D = −0.5, just as Fig. 7 and (3) predict will be observed for thin spirals. The empirical formula presented here exhibited consistently low error and is highly effective for a wide class of Archimedean spirals.

Fig. 11.

This figure depicts the error associated with predicting the mode frequencies by 4 different methods. A length based approximation, the method presented by Yun in [6], the method presented by Maleeva in [10], and a new empirical formula are all used to predict the resonant frequencies of a spiral. The empirical formula presented here maintains a consistently low level of error.

VI. Discussion

The utility of the empirical formula can be best illustrated with an example. The NMR frequency of the ¹⁵N isotope at 14.1 T occurs at 61 MHz. Spiral resonator “Design 1” for ¹⁵N was designed to operate with its fundamental resonance at 60 MHz, however additional resonances were found at 151, 240, 333, 425, 518, and at 610 MHz. The additional resonances at 151 MHz and 610 MHz are very close to the resonant frequencies for ¹³C (151 MHz) and ¹H (600 MHz) at 14.1 T, and would be expected to degrade the performance of those channels. This particular coil consisted of 72 turns and a filling factor of 50%. The value of N was chosen to achieve the desired frequency given the outer radius. The wire width and spacing were both 0.019 mm, the inner radius was 2.16 mm and the outer radius was 4.91 mm. These data are listed in Table III.

TABLE III.

DETAILS OF EXAMPLE SPIRAL DESIGNS

	Design 1	Design 2
N ri ro P F	72 2.16 mm 4.91 mm 0.019 mm 50%	42 4.30 mm 4.98 mm 0.0164 mm 50%

This table lists the parameters which describe the two spiral designs used in section VI. Design 1 suffers from modes that happen to occur at frequencies that are important to avoid. Design 2 was specifically designed using the presented techniques to avoid this problem.

By using the empirical equation and data presented above, it was possible to redesign the coil to move the higher order resonances further away from the ¹³C and ¹H frequencies. The design requirement may now be reframed as a resonator that has its fundamental frequency at approximately 60 MHz with the upper modes roughly 120 MHz apart. This corresponds to a value of −0.5 for D. According to Fig. 8, this occurs when the ratio of the radii is approximately 0.9. If a similar outer radius of 4.98 mm is used, then the appropriate pitch is determined to be 0.0164 mm, from (3), assuming that the length is the same as that for a transmission line that resonates at 120 MHz. The corresponding inner radius is 4.3 mm and the number of turns is 42. The predicted resonances according to (7) for “Design 2” are 61, 183, 304, 426, 547, and 669 MHz. Design 2 was simulated, and confirmed the resonance frequencies at 59, 190, 307, 432, 552, and 673 MHz. It is important to note that the Design 2 has no modes near the ¹H or ¹³C frequencies, making it a viable design. This characteristic is illustrated in Fig. 12, where the frequencies of the modes have been plotted together with horizontal lines representing the resonances of the ¹³C and ¹H isotopes. Minor adjustments can be made easily at this stage to fine tune the fundamental frequency. In this example, (3) and (7) were used to design a spiral which had the desired fundamental mode but would not interfere with other channels, without the need for cut-and-try techniques or time-consuming simulations.

Fig. 12.

This figure depicts the resonances of the two designs described in Table III. Also depicted are horizontal lines representing the frequencies where interference with the ¹H and ¹³C resonances would occur. The modes for Design 1 overlap both of these frequencies, while the modes for Design 2 avoid them both.

VII. Conclusion

A new empirical model is presented in this paper that can be applied to Archimedean spirals very easily with good accuracy and will advise designers on the appropriate changes to make in order to produce the desired spectrum of resonant frequencies. The median error observed was approximately 0.56% and the mean was 0.87%. Over 99% of all predictions showed less than 5% error. The quantity D has also been introduced to describe the deviation from a harmonic spectrum exhibited by a spiral. The effect that adjusting the different parameters has on D is also discussed. Specifically, adjusting the ratio of inner and outer radii of spirals was determined to have the most significant effect on this amount of deviation from a harmonic spectrum. The information presented will be useful in designing spirals with resonant frequencies at very specific locations which will reduce the need for cut and try techniques.