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Rev Sci Instrum. 2016 年 8 月；87(8)：084703。

# Meta-metallic coils and resonators: Methods for high *Q*-value resonant geometries

超金属线圈和谐振器：高 Q 值谐振几何形状的方法

^{1,2}J. W. Sidabras,

^{1}and J. S. Hyde

^{1}

R. R. Mett,

^{1,2}J. W. Sidabras,

^{1}和 J. S. Hyde

^{1}

### R. R. Mett

^{1}Department of Biophysics, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, USA

^{2}Department of Physics and Chemistry, Milwaukee School of Engineering, Milwaukee, Wisconsin 53202, USA

### J. W. Sidabras

^{1}Department of Biophysics, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, USA

### J. S. Hyde

^{1}Department of Biophysics, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, USA

作者信息 文章注释 版权和许可信息 PMC 免责声明

^{1}Department of Biophysics, Medical College of Wisconsin, Milwaukee, Wisconsin 53226, USA

^{2}Department of Physics and Chemistry, Milwaukee School of Engineering, Milwaukee, Wisconsin 53202, USA

## Abstract 摘要

A novel method of decreasing ohmic losses and increasing *Q*-value in metallic resonators at high frequencies is presented. The method overcomes the skin-depth limitation of rf current flow cross section. The method uses layers of conductive foil of thickness less than a skin depth and capacitive gaps between layers. The capacitive gaps can substantially equalize the rf current flowing in each layer, resulting in a total cross-sectional dimension for rf current flow many times larger than a skin depth. Analytic theory and finite-element simulations indicate that, for a variety of structures, the *Q*-value enhancement over a single thick conductor approaches the ratio of total conductor thickness to skin depth if the total number of layers is greater than one-third the square of the ratio of total conductor thickness to skin depth. The layer number requirement is due to counter-currents in each foil layer caused by the surrounding rf magnetic fields. We call structures that exhibit this type of *Q*-enhancement “meta-metallic.” In addition, end effects due to rf magnetic fields wrapping around the ends of the foils can substantially reduce the *Q*-value for some classes of structures. Foil structures with *Q*-values that are substantially influenced by such end effects are discussed as are five classes of structures that are not. We focus particularly on 400 MHz, which is the resonant frequency of protons at 9.4 T. Simulations at 400 MHz are shown with comparison to measurements on fabricated structures. The methods and geometries described here are general for magnetic resonance and can be used at frequencies much higher than 400 MHz.

提出了一种在高频下降低金属谐振器欧姆损耗和提高 Q 值的新方法。该方法克服了射频电流流过截面的集肤深度限制。该方法使用厚度小于集肤深度的导电箔层和层之间的电容间隙。电容间隙可以基本均衡流过每一层的射频电流，从而导致射频电流流过的总横截面尺寸比集肤深度大很多倍。分析理论和有限元模拟表明，对于各种结构，如果层总数大于总导体厚度与集肤深度之比的平方三分之一，则与单一厚导体相比，Q 值增强接近总导体厚度与集肤深度之比。层数要求是由于周围射频磁场导致的每一层箔中的反向电流。我们将表现出这种类型的 Q 增强结构称为“超金属”。此外，由于射频磁场环绕箔的末端，末端效应会大幅降低某些类别的结构的 Q 值。 讨论了 Q 值受此类末端效应显著影响的箔结构，以及五类不受影响的结构。我们特别关注 400 MHz，这是质子在 9.4 T 时的共振频率。显示了 400 MHz 的模拟，并与对制造结构的测量进行了比较。此处描述的方法和几何形状适用于磁共振，并且可以在远高于 400 MHz 的频率下使用。

## I. INTRODUCTION I. 引言

In magnetic resonance, the coil or resonator quality factor *Q*, which is defined as 2*π* multiplied by the ratio of electromagnetic energy stored to dissipated per cycle,^{1} is of great importance. Higher signal can be achieved in coils and resonators with higher *Q*. It is therefore advantageous to maximize *Q* for a given coil or resonator design. In addition, for maximum signal-to-noise ratio, rf dissipation in the sample should typically be comparable to the dissipation in the coil or resonator. This can result in an optimum *Q*-value outside of the range that is possible for typical structures. This paper describes technical advances that lead to higher *Q*-values for a variety of resonant structures.

在磁共振中，线圈或谐振器品质因数 Q（定义为电磁能量与每个周期耗散能量之比乘以 2π， ^{1} ）非常重要。在 Q 较高的线圈和谐振器中可以获得更高的信号。因此，对于给定的线圈或谐振器设计，最大化 Q 是有利的。此外，为了获得最大的信噪比，样品中的射频耗散通常应与线圈或谐振器中的耗散相当。这可能导致超出典型结构可能的范围之外的最佳 Q 值。本文介绍了导致各种谐振结构 Q 值更高的技术进步。

In the design of surface coils for human MRI, the free space *Q*-value is of little importance because the condition of dominant loading, i.e., *Q*_{body} ≪ *Q*_{coil}, is readily satisfied.^{2,3} However, for small animal imaging, even at high magnetic fields such as 9.4 T, satisfaction of the dominant loading condition is problematic. This is because body parts are small, and surface coils that correspond to dimensions of body regions are correspondingly small. The purpose of this paper is to present a novel approach to the design of small surface coils. The experimental parameter of interest in the work is the free-space *Q*-value for a coil resonating at 400 MHz with diameter of the order of 1 cm. Novelty lies in formation of the coil by multiple layers of conducting foil of a thickness that is of the order of the skin depth or less separated by carefully designed layers of dielectric, which we label with the adjective “meta-metallic.”^{4}

在用于人体 MRI 的表面线圈设计中，自由空间 Q 值并不重要，因为主导负载条件，即 Q _{body} ≪ Q _{coil} ，很容易得到满足。 ^{2,3} 但是，对于小动物成像，即使在 9.4 T 等高磁场下，满足主导负载条件也是有问题的。这是因为身体部位很小，并且对应于身体区域尺寸的表面线圈也相应很小。本文的目的是提出一种设计小型表面线圈的新方法。这项工作中感兴趣的实验参数是直径约为 1 cm、在 400 MHz 下谐振的线圈的自由空间 Q 值。新颖之处在于通过多层导电箔形成线圈，其厚度与集肤深度相当或更小，并由精心设计的介电层隔开，我们用“超金属”这个形容词来标记它。 ^{4}

As the size of coils and resonators is reduced, the *Q*-value tends to decrease. This is because the electromagnetic energy is proportional to volume, the dissipated power for metallic structures is proportional to surface area, and the ratio decreases with structure size. The *Q*-value can also be expressed in terms of the inductance *L* and resistance *R*,

随着线圈和谐振器尺寸的减小，Q 值趋于减小。这是因为电磁能与体积成正比，金属结构的耗散功率与表面积成正比，并且该比率随着结构尺寸的减小而减小。Q 值也可以用电感 L 和电阻 R 来表示，

where *ω* is the radian frequency. For metallic structures the resistance can be expressed as

其中 ω 是弧度频率。对于金属结构，电阻可以表示为

where *r _{i}* is the inner radius,

*l*is the axial length,

*σ*is the conductivity, and

*T*is the current flow thickness dimension. It is well-known that rf fields and currents tend to reside on the surfaces of metallic conductors with a characteristic exponential decay length with depth called the skin depth,

^{5}

其中 r

_{i}是内半径，l 是轴向长度，σ 是电导率，T 是电流流动厚度尺寸。众所周知，射频场和电流倾向于驻留在金属导体的表面上，其深度具有称为趋肤深度的特征指数衰减长度，

^{5}

where *μ*_{0} is the magnetic permeability of free space and *f* is the rf frequency. For frequencies above several MHz, conductor thicknesses are typically large compared to a skin depth, and *T* = *δ*. However, if multiple layers of conducting foil of thickness less than a skin depth support substantially equal rf currents,

其中 μ _{0} 是自由空间的磁导率，f 是射频。对于高于几 MHz 的频率，导体厚度通常与集肤深度相比很大，并且 T = δ。但是，如果厚度小于集肤深度的多层导电箔支持基本相等的射频电流，

*T*=

*N*

*t*>

*δ*,

T = Nt > δ，

where *N* is the number of foil layers and foil thickness *t* < *δ*. If the foil layers together have a similar inductance as the thick resonator or coil, the *Q*-value of the foil structure *Q _{f}* is enhanced compared to the thick (solid) structure

*Q*by the factor

_{s}*T*/

*δ*,

其中 N 是箔层数，箔层厚度 t < δ。如果箔层与厚谐振器或线圈具有相似的电感，则箔结构的 Q 值 Q

_{f}与厚（实心）结构 Q

_{s}相比，通过因子 T/δ 增强，

At 400 MHz, *δ* = 3.3 *μ*m for copper. Consequently, at high frequencies, many thin foil layers can substantially enhance the *Q* of a typical coil or resonator.

在 400 MHz 时，铜的 δ = 3.3 μm。因此，在高频下，许多薄箔层可以大幅度增强典型线圈或谐振器的 Q 值。

Litz wire has a similar rationale, except that it uses thin strands of non-resonant wire separated by an insulator.^{6} Litz wire is not used at rf frequencies above about 2 MHz because the rf currents are imbalanced due to skin depth effects and capacitance between wires.

利兹线具有类似的原理，只不过它使用由绝缘体分隔的非谐振细线。 ^{6} 利兹线不用于约 2 MHz 以上的射频，因为射频电流由于集肤效应和线之间的电容而失衡。

Cryo-coil technology is a possible alternative approach to obtain high *Q*-values. However, this approach is seldom used in fMRI experiments that require a high bandwidth, 400 MHz, for example. In that case, the *Q*-value of the loaded resonator should not exceed 1000. Other applications that can benefit from the type of *Q* enhancement discussed in this paper include resonator design for NMR, EPR, rf filters, rf receivers, and low-loss transmission lines.

低温线圈技术是获得高 Q 值的另一种可能方法。然而，这种方法很少用于需要高带宽（例如 400 MHz）的 fMRI 实验。在这种情况下，负载谐振器的 Q 值不应超过 1000。其他可以从本文讨论的 Q 值增强类型中受益的应用包括 NMR、EPR、射频滤波器、射频接收器和低损耗传输线的谐振器设计。

## II. BASIC META-METALLIC STRUCTURAL ELEMENTS

II. 基本金属结构构件

It is found that capacitive gaps between foil layers can result in substantially equal currents in each layer if the overlapping area between layers is substantially equal. Such a structure, which we call a folded-gap loop (FGL), is shown in Fig. Fig.1.1. The structure consists of 10 sets of 10 foil layers that form a loop. Each foil set wraps 51^{∘} and overlaps with the next set on each end for 15^{∘}. The capacitance of the overlapping regions was designed to resonate^{7} with the inductance of the loop at a frequency of 400 MHz. The structure was simulated using the finite-element computer program Ansys High Frequency Structure Simulator (HFSS) (Canonsburg, PA) version 15. The rf current magnitude inside the foil layer detail of Fig. Fig.11 is shown in Fig. Fig.2.2. The current in each layer is directed primarily around the loop. The current is maximum in the non-overlapping regions, decreases in the overlapping regions, and goes to zero on the ends of the foils. The current magnitude in a foil layer is substantially proportional to the area of overlap with adjacent foils. This is why the current in the top left and bottom right foil layers in Fig. Fig.22 is about half the values of the others. For illustration purposes, the foil material was chosen to be stainless steel with a conductivity of 1.1 MS/m, which has a skin depth of 24 *μ*m at 400 MHz. The foil thickness is 11 *μ*m. The magnetic field magnitude is shown in Fig. Fig.3.3. The magnetic field is largest on the inside of the loop and is weaker (and oppositely directed) on the outside. Notice that the magnetic field steps across the foil layers. The magnetic field zero is near the third outermost foil layer. Figure Figure44 shows the electric field magnitude in the structure. The electric field is nonzero only in the overlapping foil regions. The inner loop diameter is 10 mm, the outer loop diameter is 11.4 mm, and the distance between overlapping foils is 25 *μ*m. A conducting shield was placed at a diameter of 20 mm. The *Q*-value of the structure is 587. This can be compared to a simulated *Q*-value of 242 for a one-loop–one-gap loop gap resonator (LGR)^{8,9} of the same inner and outer diameters in the same shield and with a lossless capacitive gap. The ratio of the *Q*-values is 2.4 and this can be compared to a theoretical enhancement factor, Eq. (5), of

发现，如果层之间的重叠区域基本相等，则箔层之间的电容间隙会导致每层中的电流基本相等。我们称之为折叠间隙环路 (FGL) 的这种结构如图 1.1 所示。该结构由 10 组 10 个箔层组成，形成一个环路。每组箔层包裹 51 ^{∘} ，并在每端与下一组重叠 15 ^{∘} 。重叠区域的电容设计为在 400 MHz 的频率下与环路的电感产生谐振。使用有限元计算机程序 Ansys 高频结构模拟器 (HFSS)（宾夕法尼亚州坎农斯堡）版本 15 对该结构进行了仿真。图 2.2 中显示了图 11 中箔层细节内的射频电流幅度。每层中的电流主要围绕环路流动。电流在非重叠区域中最大，在重叠区域中减小，并在箔片的末端变为零。箔层中的电流幅度与与相邻箔片的重叠面积基本成正比。这就是图 1 中左上角和右下角箔层中的电流22 大约是其他值的一半。为了说明，选择箔材为电导率为 1.1 MS/m 的不锈钢，其在 400 MHz 时具有 24 μm 的集肤深度。箔材厚度为 11 μm。磁场大小如图所示。图 3.3。磁场在环路的内侧最大，在外侧较弱（且方向相反）。请注意，磁场跨越箔层。磁场零点靠近第三个最外箔层。图 44 显示了结构中的电场大小。电场仅在重叠的箔区域中非零。内环直径为 10 mm，外环直径为 11.4 mm，重叠箔之间的距离为 25 μm。导电屏蔽放置在 20 mm 的直径处。该结构的 Q 值为 587。这可以与一个环路-一个间隙环形谐振器 (LGR) ^{8,9} 的模拟 Q 值 242 进行比较，该谐振器在相同的屏蔽中具有相同的内径和外径，并且具有无损耗电容间隙。Q 值的比率为 2.4，这可以与理论增强因子方程式进行比较。 (5), of

If the foil thickness is on the order of a skin depth or less, the typical metallic boundary condition relating the current per unit width **J** to the surface normal vector **H** just outside the conducting surface

如果箔片厚度与趋肤深度相同或更小，则典型的金属边界条件将单位宽度电流 J 与表面法向量

no longer applies.^{5} However, this boundary condition is the default for high frequency finite-element computer programs including Ansys HFSS. If this boundary condition is used for the foil structure of Figs. 1-4, a *Q*-value of 3.4 results. This is because rf current flows in opposite directions on the inner and outer sides of each foil and is much larger in magnitude than the current in each foil in Fig. Fig.2.2. In order to obtain proper numerical solutions, it is necessary to solve for the fields inside the metal foil and use a mesh with elements of size smaller than a skin depth. Small mesh can make the simulations computationally intense. In addition, for the foil and LGR structures considered here, the ends of the structures were not simulated; a perfect magnetic boundary was used. As such, the resonance frequency and *Q*-value are independent of axial length, see Appendix and Ref. ^{8}. Further discussion of the simulations appears in Sec. IV. Effects of the ends are simulated and discussed in Sec. IV B.

不再适用。 ^{5} 但是，此边界条件是包括 Ansys HFSS 在内的高频有限元计算机程序的默认设置。如果此边界条件用于图 1-4 的箔结构，则 Q 值为 3.4。这是因为射频电流在每个箔的内侧和外侧以相反的方向流动，并且比图 2.2 中每个箔中的电流大得多。为了获得适当的数值解，有必要求解金属箔内的场，并使用元素尺寸小于趋肤深度的网格。小网格会使仿真在计算上变得很繁重。此外，对于此处考虑的箔和 LGR 结构，没有仿真结构的末端；使用了完美的磁边界。因此，谐振频率和 Q 值与轴向长度无关，请参见附录和参考文献 ^{8} 。仿真在第 IV 部分中有进一步讨论。末端的影响在第 IV B 部分中进行仿真和讨论。

## III. RF CURRENT IN CONDUCTING LAYERS

III. 导电层中的射频电流

In order to quantify the *Q*-factor enhancement in meta-metallic foil structures, an analytic theory of the electromagnetic fields in the foils was developed. In the limit of a single thick foil, the analysis reduces to the treatment of the penetration of electromagnetic fields into a good conductor given in Sec. 4.12 of Ref. ^{5}. Consider a single metallic foil of thickness *t* with planar surfaces in Cartesian coordinates (*x*, *y*, *z*) occupying the space 0 < *x* < *t*. The fields vary only as a function of *x*. For *z*-directed currents, the magnetic field is *y*-directed and Ampere’s law ∇ × **H** = **J** can be written as

为了量化金属箔结构中的 Q 因子增强，开发了箔中电磁场的解析理论。在单层厚箔的极限情况下，分析简化为第 4.12 节中给出的电磁场穿透良好导体的处理 ^{5} 。考虑厚度为 t 的单层金属箔，其平面表面在笛卡尔坐标系 (x, y, z) 中占据空间 0 < x < t。场仅随 x 变化。对于 z 向电流，磁场为 y 向，安培定律 ∇ × H = J 可写为

The conduction current density is related to the electric field in the foil by Ohm’s law

导电电流密度与箔中的电场通过欧姆定律相关

*J*

_{z}=

*σ*

*E*

_{z}.

With time harmonic fields varying as *e*^{jωt}, Faraday’s law,

对于随 e ^{jωt} 变化的时间谐波场，法拉第定律

This equation has solutions of the form

此方程组有以下形式的解

*H*

_{y}=

*C*

_{1}

*e*

^{−τx}+

*C*

_{2}

*e*

^{τx},

where *τ*, defined by *τ*^{2} ≡ *jωμ*_{0}*σ*, is a complex parameter that can be written in terms of the skin depth,

其中 τ 由 τ ^{2} ≡ jωμ _{0} σ 定义，是一个复数参数，可以用趋肤深度表示，

and *C*_{1} and *C*_{2} are integrating constants. Boundary conditions that mimic the stepped magnetic field seen in Sec. II can be imposed, *H _{y}*(

*x*= 0) =

*H*,

_{a}*H*(

_{y}*x*=

*t*) =

*H*. These boundaries on Eq. (10) result in the following expression for the magnetic field inside the foil:

_{b}和 C

_{1}和 C

_{2}是积分常数。可以施加模拟在第 II 节中看到的阶梯磁场的边界条件，H

_{y}(x = 0) = H

_{a}，H

_{y}(x = t) = H

_{b}。方程式 (10) 上的这些边界导致箔内磁场的以下表达式：

Substituting this into Eq. (7) gives an equation for the current density in the foil,

将此代入方程式 (7) 给出了箔中的电流密度方程式，

It is possible to show, either from Eq. (7) or from Eq. (13), that the net current per unit *y*-length in the foil is exactly equal to the step (or difference) in magnetic field across the foil,

从方程式 (7) 或方程式 (13) 可以看出，箔中每单位 y 长度的净电流恰好等于箔两侧磁场的阶跃（或差值），

The ohmic power dissipation per unit volume is given by

每单位体积的欧姆功率耗散由

and the power dissipated per unit foil surface area is determined by^{10}

给出，每单位箔表面积耗散的功率由 ^{10}

### A. Thick foil limit A. 厚箔极限

In the thick limit *t* ≫ *δ*, *H _{a}* = 0, and

*H*=

_{b}*H*

_{max}, using Eq. (13),

在厚极限 t ≫ δ 中，H

_{a}= 0，且 H

_{b}= H

_{max}，使用方程式 (13)，

Using trigonometric identities, the power dissipation per unit volume can be reduced to

使用三角恒等式，每单位体积的功率耗散可以简化为

and the power dissipation per unit area can be found,

，每单位面积的功率耗散可以找到，

Equations (17)-(19) match the standard skin depth results, Ref. ^{5}.

方程式 (17)-(19) 匹配标准集肤深度结果，参考文献 ^{5} 。

### B. Thin foil series expansion

B. 薄箔级数展开

When the foil thickness is less than a skin depth, *t* < *δ*, the hyperbolic sine and cosine functions can be expanded in powers of *τt*, Eq. (11),

当箔厚度小于集肤深度时，t < δ，双曲正弦和余弦函数可以展开为 τt 的幂，方程式 (11)，</End>

*τ*

*t*=

*τ*

*t*+ ⅙(

*τ*

*t*)

^{3}+ ⋯ ,

*τ*

*t*= 1 + ½(

*τ*

*t*)

^{2}+ ⋯

and the current density, Eq. (13), can be written as

电流密度，方程 (13)，可以写成

The first term in Eq. (22) represents a constant current throughout the foil. It is a low-frequency term that persists in the steady-state or direct current (DC) limit. The second term in Eq. (22) is caused by counter-currents (eddy currents). It can be seen that for the second order term, the current density reverses direction on each side of the foil: at *x* = 0 the second term is −⅙*τ*^{2}*t*(2*H*_{a} + *H*_{b}) and at *x* = *t* the second term reads ⅙*τ*^{2}*t*(*H*_{a} + 2*H*_{b}). The strength of these counter-currents is proportional to the magnetic field magnitude on the surface of the foil. The counter-current strength is also proportional to foil thickness. Consequently, the counter-currents can be reduced compared to the low-frequency term by reducing the foil thickness.

方程 (22) 中的第一项表示整个箔中的恒定电流。这是一个低频项，在稳态或直流 (DC) 极限中持续存在。方程 (22) 中的第二项由反向电流（涡流）引起。可以看出，对于二阶项，电流密度在箔的每一侧都会反转方向：在 x = 0 时，第二项为 −⅙τ ^{2} t(2H _{a} + H _{b} )，在 x = t 时，第二项为 ⅙τ ^{2} t(H _{a} + 2H _{b} )。这些反向电流的强度与箔表面的磁场强度成正比。反向电流强度也与箔厚度成正比。因此，可以通过减小箔厚度来减小反向电流与低频项相比。

The ohmic power dissipation in a single foil layer caused by each of the current terms can be found from Eqs. (15), (16), and (22),

由每个电流项引起的单箔层中的欧姆功率耗散可以从方程 (15)、(16) 和 (22) 中找到，

and

The dissipation in multiple foil layers can now be found by summing Eq. (24) appropriately. Assuming equal current per unit length *i _{t}* in each layer with

*N*layers and a total current per unit length in all

*N*layers together,

现在可以通过适当求和方程式 (24) 来找到多层箔中的耗散。假设每层中单位长度的电流相等，N 层中单位长度的总电流为，

*I*

_{T}≡

*N*

*i*

_{t}=

*H*

_{max},

Eq. (14) can be used to evaluate

方程式 (14) 可用于评估

The last term in Eq. (24) can be evaluated by assuming the magnetic field *H _{a}* starts at zero on the first layer and recognizing the incremental increase in

*H*(and

_{a}*H*) over the sum,

_{b}方程式 (24) 中的最后一项可以通过假设磁场 H

_{a}在第一层从零开始并识别 H

_{a}（和 H

_{b}）在求和中的增量来评估，

where the finite series was evaluated using Ref. ^{11}. Taking the results of Eqs. (26) and (27) into Eq. (24), we find

其中有限级数使用参考文献 ^{11} 进行评估。将方程式 (26) 和 (27) 的结果代入方程式 (24)，我们发现

where *T* = *Nt*, Eq. (4), has also been used. For many layers, we set ^{12}, in the context of low-frequency (60 Hz) coil design. However, the leading term of Eq. (26) of Ref. ^{12} scales *proportional* to *N* rather than inversely with *N* due to different circuit topology.

其中 T = Nt，方程式 (4) 也已使用。对于许多层，我们设置 ^{12} 的方程式 (26)，在低频 (60 Hz) 线圈设计的上下文中。然而，由于不同的电路拓扑，参考文献 ^{12} 的方程式 (26) 的首项按比例缩放为 N，而不是与 N 成反比。

### C. The meta-metallic effect: Minimum layer number and maximum thickness

C. 超金属效应：最小层数和最大厚度

In comparing the first term of Eq. (28) to the thick foil limit Eq. (19), it is seen that, to first order, multiple layers have reduced dissipation by the factor *δ*/*T*, compared to a single thick conductor. This is consistent with Eq. (5). However, the second term in Eq. (28) provides the condition for this to be true. In order for the series to converge, the second order term must be smaller than the first. This puts a minimum constraint on the number of layers,

将方程式 (28) 的第一项与厚箔极限方程式 (19) 进行比较，可以看出，与单个厚导体相比，一级近似，多层已将耗散降低了 δ/T 倍。这与方程式 (5) 一致。然而，方程式 (28) 中的第二项提供了此条件为真的条件。为了使级数收敛，二阶项必须小于一阶项。这给层数施加了最小约束，

or

This constraint can be interpreted as the minimum number of layers for a given total conductor thickness *T*. From Eq. (4), this constraint can also be expressed in terms of the maximum layer thickness for a given number of layers,

此约束可以解释为给定总导体厚度 T 的最小层数。根据方程式 (4)，此约束也可以根据给定层数的最大层厚度表示，<Keep This Symbol>请注意方程式 (24) 中的二阶项是平方。当 N 超过最小值（或 t 低于最大值）时，这会产生反向电流耗散的快速衰减。

Notice from Eq. (24) that the second order term is squared. This produces rapid decay of the counter-current dissipation as *N* is increased beyond the minimum (or *t* is decreased below the maximum).

According to this analysis, there is no theoretical limit to the *Q* enhancement due to multiple thin conducting layers. Table TableII contains some practical cases for copper at different frequencies for a given *Q* enhancement, *T*/*δ*. A major and perhaps counter-intuitive implication of this result is that the *Q*-value does *not* monotonically increase with the number of foil layers at a fixed foil thickness. Rather, the *Q*-value will exhibit a *maximum* as the number of layers approaches 3(*δ*/*t*)^{2}, Eq. (31). This is because the first- and second-order dissipation terms scale differently with *t*. Consequently, there is an optimum layer number that produces a maximum *Q*-value for a given foil thickness. These results have been confirmed by Ansys HFSS simulations. Similarly, at fixed foil layer number, increasing *t* also produces a maximum in *Q*-value near *t*_{max}, Eq. (31). In both cases the maximum occurs when the second-order counter-current dissipation is balanced with the first-order DC dissipation.

根据此分析，由于多个薄导电层，Q 增强没有理论限制。表 II 表包含了给定 Q 增强 T/δ 的不同频率下铜的一些实际情况。此结果的一个主要且可能违反直觉的含义是，Q 值不会随着固定箔厚度的箔层数单调增加。相反，Q 值将表现为最大值，因为层数接近 3(δ/t) ^{2} ，方程式 (31)。这是因为一阶和二阶耗散项随 t 的变化不同。因此，有一个最佳层数，可为给定的箔厚度产生最大 Q 值。这些结果已通过 Ansys HFSS 模拟得到证实。类似地，在固定箔层数下，增加 t 也会在 t _{max} 附近产生 Q 值的最大值，方程式 (31)。在这两种情况下，最大值发生在二阶逆流耗散与一阶直流耗散平衡时。

### TABLE I. 表 I.

400 MHz | 9.5 GHz | ||||
---|---|---|---|---|---|

N_{min} | δ (μm) | t_{max} (μm) | δ (μm) | t_{max} (μm) | |

5.5 | 10 | 3.3 | 1.8 | 0.68 | 0.37 |

7.8 | 20 | 1.3 | 0.26 | ||

10 | 33 | 0.99 | 0.20 | ||

20 | 130 | 0.50 | 0.10 | ||

100 | 3300 | 0.099 | 0.020 |

From these results, the *Q* enhancement factor observed in the simulation of Sec. II can be understood. If the foil thickness *t* is significantly less than the maximum given by Eq. (31), the *Q* enhancement factor is accurately given by *T*/*δ*, Eq. (5) since the counter-current dissipation is small compared to the first-order dissipation. However, if the foil thickness is near the maximum, which is the case in Sec. II, the *Q* enhancement factor is about one-half of *T*/*δ* because the first- and second-order dissipation are nearly equal, doubling the total ohmic dissipation.

从这些结果中，可以理解到在第 II 节的仿真中观察到的 Q 增强因子。如果箔厚度 t 显著小于方程式 (31) 给出的最大值，则 Q 增强因子由 T/δ 精确给出，方程式 (5) 因为与一阶耗散相比，逆流耗散很小。然而，如果箔厚度接近最大值（第 II 节中的情况），则 Q 增强因子约为 T/δ 的一半，因为一阶和二阶耗散几乎相等，使总欧姆耗散加倍。

By differentiating Eq. (28) with respect to *t* and setting the result equal to zero, a theoretical foil thickness for minimum ohmic dissipation can be obtained. The result is

通过对方程式 (28) 关于 t 求导并将结果设为零，可以获得用于最小欧姆耗散的理论箔厚度。结果是

as reported in Ref. ^{12}. The foil thickness *t*_{opt} is about 32% lower than the maximum thickness *t*_{max}, Eq. (31). Numerical simulations indicate, for structures with minimal foil edge currents, that the foil thickness for maximum *Q*-value lies between the two thicknesses. This is further discussed in Sec. IV.

如参考文献 ^{12} 中所述。箔厚度 t _{opt} 比最大厚度 t _{max} 低约 32%，等式 (31)。数值模拟表明，对于箔边缘电流最小的结构，最大 Q 值的箔厚度介于两个厚度之间。这将在第 IV 节中进一步讨论。

## IV. SIMULATIONS IV. 模拟

In addition to the meta-metallic effect described in Sec. III C, it was found that the geometry of the foils strongly influences the *Q*-value enhancement. If the rf magnetic fields wrap around the ends of the foil layers and have a significant component perpendicular to the foil edges, intensified rf currents flow along the layer edges, and cause significant additional ohmic dissipation. These end effects are discussed in Sec. IV B. Structures having these end effects do not follow the predicted meta-metallic *Q*-value enhancement over thick conducting structures. However, five types of thin foil structures have been envisioned where end effects are minimal and these are described in Sec. IV C. These structures exhibit *Q*-value enhancements consistent with the theory of Sec. III C.

除了第 III C 节中描述的类金属效应外，还发现箔的几何形状对 Q 值增强有很大影响。如果射频磁场环绕箔层的末端并垂直于箔边缘具有显着分量，则沿层边缘会流过强化的射频电流，并导致显着的附加欧姆耗散。这些末端效应在第 IV B 节中讨论。具有这些末端效应的结构不会遵循预测的类金属 Q 值增强，超过厚导电结构。然而，已经设想出五种薄箔结构，其中末端效应最小，这些结构在第 IV C 节中进行了描述。这些结构表现出与第 III C 节理论一致的 Q 值增强。

All simulations were made using Ansys HFSS, Sec. II. The structures have a resonance frequency near 400 MHz, metallic components are copper, and the material between foil layers is polytetrafluoroethylene (PTFE) with a relative dielectric constant *ϵ _{r}* of 2.1. In order to isolate the meta-metallic effect, the resistivity of the conducting shield and the loss-tangent of the dielectric were set equal to zero. The influence of dielectric loss is described in Appendix and Sec. IV C 1. A Dell Precision Tower 7910 with 24 Intel Xeon dual-core processors with Hyper-Threading and 512 GB of RAM was used to make the numerical simulations. The foils were drawn using many 0.1 mm or 0.25 mm wide adjacent duplicate structures on axis. The boundaries between adjacent foils facilitated meshing. Most of the simulations were done using the Eigenmode solution type. For some of the larger structures, including the toroidal loop, either the memory limit was reached or solutions were simply not found. In these cases, the driven modal network analysis solution type was used. The structure was coupled using a single mode lumped port defined by a planar face placed symmetrically between two adjacent foils in an overlapping region. The electric field vector integration line was perpendicular to and between the foil surfaces. In Figs. 2-4, ,6,6, ,10,10, ,12,12, ,13,13, ,15,15, and and16,16, the plotted quantities correspond to a power of 1 W input to the whole structure. If perfect magnetic boundary conditions were used, Figs. 2-4 and and6,6, the axial length was 1 cm.

所有仿真均使用 Ansys HFSS，第 II 节。这些结构的谐振频率接近 400 MHz，金属组件为铜，箔层之间的材料为聚四氟乙烯 (PTFE)，相对介电常数 ϵ

_{r}为 2.1。为了隔离超金属效应，导电屏蔽的电阻率和介电质的损耗角正切被设定为零。介电损耗的影响在附录和第 IV C 1 节中进行了描述。使用带有 24 个 Intel Xeon 双核处理器（带有超线程）和 512 GB RAM 的 Dell Precision Tower 7910 来进行数值仿真。箔片使用许多 0.1 mm 或 0.25 mm 宽的相邻重复结构在轴线上绘制。相邻箔片之间的边界促进了网格划分。大多数仿真使用本征模态求解类型完成。对于某些较大的结构，包括环形回路，要么达到内存限制，要么根本找不到解决方案。在这些情况下，使用了激励模态网络分析求解类型。 该结构使用单模集总端口耦合，该端口由放置在重叠区域中两个相邻箔之间对称的平面面定义。电场矢量积分线垂直于箔表面且位于箔表面之间。在图 2-4、6、10、12、13、15 和 16 中，绘制的数量对应于输入整个结构的 1 W 功率。如果使用完美的磁边界条件，图 2-4 和 6，轴向长度为 1 cm。

### A. Axial coils with no end effects

A. 无端部效应的轴向线圈

FGL and self-resonant spiral (SRS) structures were simulated using perfect magnetic boundaries at the top and bottom. Both types of structures are found to follow closely the theory of Sec. III C.

使用顶部和底部的完美磁边界模拟了 FGL 和自谐振螺旋 (SRS) 结构。发现这两种类型的结构都紧密遵循了 III C 部分的理论。

#### 1. Folded-gap loop 1. 折叠间隙环路

The foil configuration for an FGL with two sets of 10 foil layers is shown in Fig. Fig.5.5. Dimensions are shown in Table TableIIII with symbol definitions consistent with the Appendix. By constructing the overlapping foil region with a constant distance *d _{ov}* instead of angle as in Sec. II, the capacitance is constant, producing more uniform currents across the layers. The magnetic field magnitude is shown in Fig. 6(a) and the electric field magnitude in Fig. 6(b). They are similar to those presented in Sec. II. The structure resonates at 393 MHz and, for a foil thickness of 1.6

*μ*m, has a

*Q*-value of 5514. This can be compared to a

*Q*-value of 1407 for an LGR of the same inner and outer radius and the same metal. The resulting

*Q*-enhancement ratio, Eq. (5), is 3.9 and can be compared to

*T*/

*δ*= 4.8, Table TableII.II. The eddy current dissipation lowers the

*Q*-enhancement as expected. The inductance of the FGL is approximately (7.9/5)

^{2}= 2.5 times higher than the inductance of the LGR (Appendix) and this factor has the opposite effect, raising the

*Q*-ratio.

具有两组 10 层箔层的 FGL 的箔配置如图所示。图 5.5。尺寸显示在表中，表 III，符号定义与附录一致。通过以恒定距离 d

_{ov}而不是像在第 II 节中那样以角度构建重叠箔区域，电容保持恒定，从而在各层上产生更均匀的电流。磁场大小如图 6(a) 所示，电场大小如图 6(b) 所示。它们类似于第 II 节中介绍的那些。该结构在 393 MHz 时产生共振，对于 1.6 μm 的箔厚度，Q 值为 5514。这可以与具有相同内半径和外半径以及相同金属的 LGR 的 Q 值 1407 进行比较。所得的 Q 增强率，方程式。(5)，为 3.9，可以与 T/δ = 4.8，表 II.II 进行比较。涡流耗散会降低 Q 增强，正如预期的那样。FGL 的电感大约为 (7.9/5)

^{2}= 比 LGR 的电感高 2.5 倍（附录），这个因素具有相反的效果，提高了 Q 值。

### TABLE II. 表 II.

Structure 结构 | r_{i} | r_{o} | r_{s} | d | d_{ov} | l | f (MHz) | Q_{f} | Q_{s} | N | t (μm) | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

FGL w/no end effects 无端部效应的 FGL | 5 | 10.8 | 28 | 0.3 | 1.2 | n/a | 393 | 5 514 | 1407 | 10 | 1.6 | 4.8 | 3.9 |

5 | 11 | 28.5 | 0.15 | 0.18 | n/a | 412 | 8 105 | 1403 | 20 | 1.1 | 6.7 | 5.8 | |

SRS w/no end effects 无末端效应的 SRS | 3 | 8 | 20 | 1.25 | 104 | n/a | 407 | 2 169 | 809 | 4 | 2.2 | 2.7 | 2.7 |

3 | 8 | 20 | 0.36 | 86 | n/a | 410 | 4 139 | 809 | 14 | 1.4 | 5.9 | 5.1 | |

1 cm length FGL 1 cm 长度 FGL | 5 | 10.9 | 25 | 0.3 | 3 | 10 | 374 | 731 | 1156 | 10 | 3.9 | n/a | 0.63 |

1 cm length FGL w/dielectric ends 1 cm 长度 FGL 带电介质末端 | 5 | 10.8 | 25 | 0.3 | 2.4 | 10 | 411 | 12 780 | 1586 | 10 | 1.6 | 4.8 | 8.1 |

Toroidal loop 环形线圈 | 5 | 8.5 | 14 | 0.076 | 1.77 | 3.5 | 391 | 1 953 | 1131 | 10 | 1.4 | 4.2 | 1.7 |

5 cm length coax 5 cm 长度同轴电缆 | 5 | 7.4 | 28 | 0.13 | 0.9 | 5 | 396 | 19 718 | 5302 | 10 | 1.6 | 4.8 | 3.7 |

A numerical study of the *Q*-value dependence on the foil thickness was made, Fig. Fig.7.7. Shown are results for the 10-foil FGL, along with several other structures discussed subsequently. Also shown is the maximum conductor thickness given by Eq. (31) for 10 and 20 foil layers. It can be seen that the *Q*-value peaks about 20% below *t*_{max}, consistent with theory, and is the thickness used above. It is interesting to note that the optimum foil thickness *t*_{opt}, Eq. (32) falls about 10% further below the numerically obtained thickness for maximum *Q*-value. Nonuniform currents across the foil layers are the likely cause.

对 Q 值对箔片厚度的依赖性进行了数值研究，图 图 7.7。显示了 10 层箔片 FGL 的结果，以及随后讨论的几个其他结构。还显示了方程式 (31) 给出的 10 层和 20 层箔片的最大导体厚度。可以看出，Q 值峰值约为 t _{max} 以下的 20%，与理论一致，并且是上面使用的厚度。有趣的是，最佳箔片厚度 t _{opt} ，方程式 (32) 大约比数值获得的最大 Q 值厚度低 10%。箔片层上的非均匀电流可能是可能的原因。

Ansys HFSS has a minimum physical dimension of 1 *μ*m. In order to overcome this limitation, some simulations were done at 10 times the size and one-tenth the conductivity. It can be verified analytically that this produces an identical *Q*-value for any structure at one-tenth the frequency. This was also verified for several numerical simulations of different structures. The same *Q*-value to six digits was obtained in each case.

Ansys HFSS 的最小物理尺寸为 1 μm。为了克服此限制，一些模拟以 10 倍尺寸和十分之一电导率进行。可以分析验证，这会为任何结构在十分之一频率下产生相同的 Q 值。这也针对不同结构的多个数值模拟进行了验证。在每种情况下都获得了六位数的相同 Q 值。

A 20-foil FGL structure was also simulated with dimensions shown in Table TableII.II. The structure resonates at 412 MHz and, for a foil thickness of 1.1 *μ*m, has a *Q*-value of 8105. This can be compared to a *Q*-value of 1403 for an LGR of the same inner and outer radius. The resulting *Q*-enhancement ratio, Eq. (5), is 5.8 and can be compared to *T*/*δ* = 6.7, Table TableII.II. A scan of *Q*-value with thickness for this structure is shown in Fig. Fig.7.7. The maximum *Q*-value occurs at a foil thickness about 20% below *t*_{max} as for the 10-foil FGL.

还模拟了一个 20 箔 FGL 结构，尺寸如表 TableII.II 所示。该结构在 412 MHz 时产生共振，对于 1.1 μm 的箔厚度，Q 值为 8105。这可以与具有相同内半径和外半径的 LGR 的 1403 Q 值进行比较。产生的 Q 增强率，方程式 (5)，为 5.8，可以与 T/δ = 6.7，表 TableII.II 进行比较。图 Fig.7.7 中显示了此结构的厚度 Q 值扫描。最大 Q 值出现在箔厚度约为 t _{max} 以下 20% 的位置，与 10 箔 FGL 相同。

#### 2. Self-resonant spiral

Another multilayer foil structure that can show *Q*-enhancement consistent with the meta-metallic effect, Eq. (5), is an SRS. Unlike the FGL, the capacitance associated with the overlapping foil area charges to the voltage difference between one foil layer and an adjacent foil layer and, as it discharges, drives a current in parallel with all the layers. The resulting rf current profile in the foil layers produces rf magnetic fields very similar to those of the FGL, Fig. 6(a). Filamentary analogs of the foil SRS are of current interest for meta-material meta-atoms.^{13,14} These references calculate resonance frequency but not *Q*-value.

2. 自谐振螺旋体 另一种多层箔结构可以显示出与超金属效应一致的 Q 值增强，即方程式 (5)，是 SRS。与 FGL 不同，与重叠箔区域相关的电容会对一个箔层和相邻箔层之间的电压差进行充电，并且在放电时，会驱动与所有层平行的电流。箔层中产生的射频电流分布会产生与 FGL 非常相似的射频磁场，图 6(a)。箔 SRS 的细丝类似物是超材料超原子当前关注的焦点。

^{13,14}这些参考计算了谐振频率，但没有计算 Q 值。

Conditions for resonance of single and multiple foil SRS are given in the Appendix. Shown in Fig. Fig.88 is a four-turn spiral foil configuration with dimensions shown in Table TableII.II. The SRS was simulated with results also shown in the table. The SRS *Q*-value can be compared to a *Q*-value of 809 for an LGR of the same inner and outer radius. This corresponds to an actual *Q*-enhancement ratio, Eq. (5), of 2.7 compared to *T*/*δ* = 2.7, Table TableII.II. One might expect that the eddy current dissipation would lower the *Q*-enhancement for the foil thickness near one skin depth. However, the SRS has an approximately (5.5/3)^{2} = 1.8 times the inductance of the LGR (Appendix) and this factor raises the *Q*-ratio. The nearly sinusoidal (nonuniform) current distribution with foil length in the spiral does not have much effect in lowering the *Q*-enhancement. For the different structure types that have been simulated, it was found that close attention to balancing the currents by equal capacitance does not significantly impact the *Q*-value.

单层和多层箔 SRS 的谐振条件在附录中给出。图 88 中显示的是一个四匝螺旋箔配置，其尺寸在表 II.II 中给出。SRS 已模拟，结果也显示在表中。SRS Q 值可以与具有相同内半径和外半径的 LGR 的 Q 值 809 进行比较。这对应于实际 Q 增强比，方程式 (5)，为 2.7，与 T/δ = 2.7，表 II.II 相比。人们可能会认为，涡流耗散会降低接近一个趋肤深度的箔厚度的 Q 增强。然而，SRS 的电感大约是 LGR 的 (5.5/3) ^{2} = 1.8 倍（附录），这个因子提高了 Q 比。螺旋中箔长度的近似正弦（非均匀）电流分布对降低 Q 增强没有太大影响。对于已经模拟的不同结构类型，发现通过相等的电容来平衡电流并不会显著影响 Q 值。

It is possible to increase the number of foil layers by duplicating the single foil spiral in azimuth as shown in Fig. Fig.9.9. The addition of the duplicate foils has a small effect on the resonance frequency relative to the single foil resonance frequency, see the Appendix. The structure shown has the same inner and outer radii as the single foil spiral; however, each foil has 3.5 turns and thickness 1.39 *μ*m. The resonance frequency is 410 MHz and *Q*-value of 4139. The LGR comparison is the same, giving an actual *Q*-enhancement ratio, Eq. (5), of 5.1 compared to *T*/*δ* = 5.9, Table TableIIII.

通过在方位角上复制单箔螺旋，可以增加箔层的数量，如图 9.9 所示。添加重复箔对谐振频率的影响相对于单箔谐振频率很小，请参阅附录。所示结构具有与单箔螺旋相同的内半径和外半径；但是，每个箔有 3.5 匝，厚度为 1.39 μm。谐振频率为 410 MHz，Q 值为 4139。LGR 比较相同，与 T/δ = 5.9 相比，给出了实际 Q 增强比，方程式 (5) 为 5.1，表 III。

### B. Axial coils: End effects

B. 轴向线圈：端部效应

A 1 cm axial length of an FGL similar to that shown in Sec. IV A 1 was simulated. The perfect magnetic boundary conditions (Sec. IV A) were removed and the structure was centered in a conducting cylindrical boundary of axial length 40 mm and a radius 25 mm. A side-view of the magnetic field magnitude for the upper half of the structure is shown in Fig. Fig.10.10. Intense rf magnetic fields at the axial ends of the foil layers are seen. The intensification is caused by strong rf currents that flow along the foil edges, which, in turn, cause increased dissipation and a significant decrease in *Q*-value compared to the structure with no end effects. The intensification of rf current on the edges of conductors has been extensively studied in relation to microstrips^{15} and is caused by a singularity in the rf fields at sharp edges. The effect has recently been called edge singularity^{16} but has also been called strong skin effect at the edge.^{17} When the rf current distribution in a conductor is influenced by one or more nearby conductors, there is also typically an intensification of rf currents and an increase in ohmic losses. Proximity effect^{6} has been used in this case. We prefer “end effects.” The structure dimensions are shown in Table TableII.II. The larger overlap distance (3 mm vs. 1.2 mm) is needed to compensate for the reduced inductance vs. Eq. (A9) resulting from the finite length. By scanning the foil thickness, a maximum *Q*-value of 731 was obtained at a thickness of 3.9 *μ*m. This thickness is about 2.4 times the thickness that gives maximum *Q*-value for the 10-foil structure with no end effects, Sec. IV A 1. Results are shown in Fig. Fig.7.7. The theory of Sec. III C cannot be applied due to the end effects. This compares to a *Q*-value of 1156 for an LGR of the same inner and outer radius and length and a *Q*-enhancement ratio, Eq. (5), of 0.63, Table TableII.II. Similar results were obtained for 1 cm axial length SRSs.

模拟了与第 IV A 1 节中所示类似的 FGL 的 1 cm 轴向长度。去除了完美的磁边界条件（第 IV A 节），并将结构置于轴向长度为 40 mm、半径为 25 mm 的导电圆柱边界中。图中显示了结构上半部分的磁场大小的侧视图。图 10.10。可以看到箔层轴向末端的强射频磁场。这种增强是由沿箔边缘流动的强射频电流引起的，这反过来又会导致耗散增加，并且与没有末端效应的结构相比，Q 值会显著降低。导体边缘射频电流的增强已在微带线 ^{15} 方面得到了广泛的研究，并且是由尖锐边缘处的射频场奇异性引起的。这种效应最近被称为边缘奇异性 ^{16} ，但也被称为边缘处的强趋肤效应。 ^{17} 当导体中的射频电流分布受到一个或多个附近导体的干扰时，通常也会出现射频电流增强和欧姆损耗增加的情况。 本例中使用了 ^{6} 邻近效应。我们更喜欢“端效应”。结构尺寸如表所示表II.II。较大的重叠距离（3 mm 对比 1.2 mm）是用来补偿由于有限长度导致的电感与等式 (A9) 相比的减少。通过扫描箔厚度，在 3.9 μm 的厚度下获得了 731 的最大 Q 值。该厚度大约是无端效应的 10 箔结构的最大 Q 值厚度的 2.4 倍，见第 IV A 1 节。结果如图所示图 7.7。由于端效应，无法应用第 III C 节的理论。这与具有相同内半径、外半径和长度的 LGR 的 Q 值 1156 以及 Q 增强比（等式 (5)）0.63 相比，见表 II.II。对于 1 cm 轴向长度 SRS，获得了类似的结果。

### C. Structures that minimize end effects

C. 最小化端部效应的结构

In this section, we describe five different structures that have minimal end effects and consequently can have significant meta-metallic *Q*-enhancement. The first three structures described in Secs. IV C 1-IV C 3 use an additional structure to provide a boundary condition to the meta-metallic structure that mimics a perfect magnetic boundary condition. The fourth and fifth structures described in Secs. IV C 4 and IV C 5 have geometries in which the foil edge currents approach zero.

在本节中，我们描述了五种不同的结构，它们具有最小的端部效应，因此可以显著提高超金属 Q 值。在第 IV C 1- IV C 3 节中描述的前三个结构使用附加结构为超金属结构提供边界条件，该边界条件模拟了完美的磁边界条件。在第 IV C 4 和 IV C 5 节中描述的第四和第五个结构具有箔边缘电流接近零的几何形状。

#### 1. FGL (or SRS) with dielectric ends

By treating an FGL as the central section of a uniform field (UF) resonator,^{18–21} it is found that a dielectric region placed on each end of an FGL can significantly reduce end effects. The physical principle is that a quarter-wavelength thickness of dielectric converts an electric short at the top of the dielectric to an open impedance, which is presented to the foil edges. The rf open is equivalent to a perfect magnetic boundary condition, the same spatial boundary condition required to keep the rf currents uniform along the axial length of the foil. The configuration is shown in Fig. Fig.11.11. The FGL consists of four sets of 10 foils. The two additional gap regions compared to the FGL of Sec. IV B are found to be needed to couple the foil to the dielectric TE_{01δ} mode.^{22} The foil and dielectric are spaced apart by 0.5 mm and placed inside a conducting shield. The rf magnetic field profile is shown in Fig. Fig.12.12. It is seen that the effect of the dielectric is to nearly eliminate end effects. The combined structure resonates at 411 MHz and, for a foil thickness of 1.6 *μ*m, has a *Q*-value of 12 780. The *Q*-value is maximum at the same foil thickness that produces maximum *Q*-value for the FGL with no end effects of Sec. IV A 1. The *Q*-value is significantly larger than the FGL of Sec. IV A 1 because the dielectric loss tangent was set equal to zero. In reality, the structure would have a maximum *Q*-value somewhat larger than the inverse of the dielectric loss tangent, see the Appendix. Dimensions of the FGL are shown in Table TableII.II. The dielectric radius is 21.2 mm and length 10 mm. The relative dielectric constant of the dielectric end regions is 760. Dielectrics of larger sizes with relative dielectric constants of 100, 200, and 400 have also been shown to couple to the FGL of the same size. Similar *Q*-values were obtained. The relative dielectric constant values of 100-200 are similar to some ceramics.^{23} Larger diameter foils and higher resonance frequencies can accommodate dielectrics with even lower relative dielectric constant values. The FGL of Figs. Figs.1111 and and1212 is centered in a conducting cylinder of radius 25 mm and length 31 mm. A one-loop–four-gap LGR of the same inner and outer radius and length as the FGL can be coupled to identical dielectric ends. The resulting *Q*-value is 1586. The *Q*-enhancement ratio, Eq. (5), is 8.1 and can be compared to *T*/*δ* = 4.8, Table TableII.II. The larger enhancement is due to the additional inductance of the FGL compared to the LGR, (7.9/5)^{2} = 2.5, see the Appendix and Sec. IV A 1, and with this factor is favorably consistent with the theory of Sec. III C. Using low-loss dielectrics, this structure can be used to make resonators with *Q*-values exceeding 10 000. The foil permits a concentration of the rf magnetic field into much smaller volumes and into shapes that are not possible using dielectrics alone.

1. 带电介质端的 FGL（或 SRS）通过将 FGL 视为均匀场（UF）谐振器的中心部分，

^{18–21}发现放置在 FGL 每一端的电介质区域可以显著减少端部效应。物理原理是，四分之一波长的电介质厚度将电介质顶部的电气短路转换为开路阻抗，该阻抗呈现给箔边缘。射频开路等效于完美的磁边界条件，这是保持射频电流沿箔轴向长度均匀所需的相同空间边界条件。配置如图所示。图 11.11。FGL 由四组 10 个箔片组成。与第 IV B 节的 FGL 相比，发现需要两个额外的间隙区域才能将箔片耦合到电介质 TE

_{01δ}模式。

^{22}箔片和电介质相隔 0.5 毫米，并放置在导电屏蔽内。射频磁场分布如图所示。图 12.12。可以看出，电介质的作用几乎消除了端部效应。组合结构在 411 MHz 时产生共振，对于 1.6 μm 的箔片厚度，Q 值为 12 780。 Q 值在产生无 IV A 1 节末端效应的 FGL 的最大 Q 值的相同箔厚度处最大。Q 值明显大于 IV A 1 节的 FGL，因为介电损耗角正切被设为零。实际上，该结构的最大 Q 值将略大于介电损耗角正切的倒数，请参阅附录。FGL 的尺寸显示在表中表 II.II。介电半径为 21.2 毫米，长度为 10 毫米。介电末端区域的相对介电常数为 760。相对介电常数为 100、200 和 400 的较大尺寸的介电材料也已显示与相同尺寸的 FGL 耦合。获得了类似的 Q 值。100-200 的相对介电常数值类似于一些陶瓷。

^{23}较大的直径箔和较高的谐振频率可以容纳具有更低相对介电常数值的介电材料。图中的 FGL。图 1111 和图 1212 居中于半径为 25 毫米、长度为 31 毫米的导电圆柱体中。 一个单环四间隙 LGR 的内半径、外半径和长度与 FGL 相同，可以耦合到相同的介电末端。产生的 Q 值为 1586。Q 增强比，等式。(5)，为 8.1，可以与 T/δ = 4.8，表 TableII.II 进行比较。更大的增强是由于 FGL 与 LGR 相比的附加电感，(7.9/5)

^{2}= 2.5，请参阅附录和第 IV A 1 节，并且这个因素与第 III C 节的理论非常一致。使用低损耗介电材料，此结构可用于制造 Q 值超过 10 000 的谐振器。该箔允许将射频磁场集中到更小的体积中，并集中到仅使用介电材料无法实现的形状中。

SRSs in place of the FGL have been simulated with dielectric ends with similar results. It is found that a larger gap between the dielectric and the foil (1 mm) is required for the SRS than for the FGL to prevent capacitive loading of the foil ends by the dielectric. This is due to the larger voltage between the foil ends of the SRS. The loading causes enhanced rf currents on the foil edges. Such a structure could be used as an NMR resonator, but has limited applicability as an MRI surface coil. We present MRI-suitable structures in Secs. IV C 2 and IV C 3.

用具有类似结果的电介质端模拟了 FGL 中的 SRS。发现 SRS 与 FGL 相比，电介质和箔片之间需要更大的间隙（1 mm）以防止电介质对箔片端的电容性负载。这是由于 SRS 的箔片端之间的电压较大。负载会导致箔片边缘的射频电流增强。这种结构可以用作 NMR 谐振器，但作为 MRI 表面线圈的适用性有限。我们在第 IV C 2 和 IV C 3 节中介绍了适合 MRI 的结构。

#### 2. SRS (or FGL) with dielectric on one side

Surprisingly, it was found that a dielectric region of about twice the size of a uniform field end section described in Sec. IV C 1 placed on *one* side of the SRS or FGL can also suppress the end effects on *both* sides of the foil. This is because the axial length of the foil is much smaller than a wavelength. It is found that when the resonance frequency of the combined structure (dielectric and SRS) is near the resonance frequency of the SRS (or FGL) alone with perfect magnetic boundaries, see Sec. IV A, the end effects are substantially eliminated and the *Q*-value is maximized. A three-turn spiral of inner radius 4.25 mm, outer radius 7.25 mm, and axial length 2.5 mm embedded in PTFE was found to have a resonant frequency of 402 MHz with perfect magnetic axial boundary conditions. With a 3 *μ*m foil thickness the *Q*-value is 2506. With a rutile dielectric cylinder, *ϵ _{r}* = 100, radius and axial length 53.6 mm placed coaxially to the spiral at a distance 1 mm away from the edge of the foil, the combined structure has a resonance frequency of 409 MHz. The spiral was centered in a conducting boundary of radius 53.6 mm and axial length 111.7 mm. The

*Q*-value of the combined structure was 84 750. The

*Q*-value is higher than the SRS alone because there is a large portion of stored energy in the dielectric. The rf magnetic field strength was about the same in the spiral center as the dielectric center. In reality, the

*Q*-value is limited by the loss tangent of the dielectrics as indicated in the Appendix and Sec. IV C 1.

2. 一侧带电介质的 SRS（或 FGL）令人惊讶的是，发现放置在 SRS 或 FGL 一侧的电介质区域（其尺寸约为第四部分 C 1 中描述的均匀场末端的两倍）也可以抑制箔片两侧的末端效应。这是因为箔片的轴向长度远小于波长。发现当组合结构（电介质和 SRS）的谐振频率接近 SRS（或 FGL）的谐振频率时，并且具有完美的磁边界（参见第四部分 A），末端效应会基本消除，Q 值最大化。发现内半径为 4.25 mm、外半径为 7.25 mm、轴向长度为 2.5 mm 的三匝螺旋嵌入 PTFE 中，在完美的磁轴向边界条件下具有 402 MHz 的谐振频率。箔片厚度为 3 μm 时，Q 值为 2506。当电介质圆柱体金红石，ϵ

_{r}= 100，半径和轴向长度为 53.6 mm，同轴放置在螺旋中，距离箔片边缘 1 mm 时，组合结构的谐振频率为 409 MHz。螺旋居中放置在半径为 53.6 mm、轴向长度为 111 的导电边界中。7 毫米。组合结构的 Q 值为 84 750。Q 值高于单独的 SRS，因为介电质中储存了大量的能量。螺旋中心和介电质中心的射频磁场强度大致相同。实际上，Q 值受介电质的损耗角正切限制，如附录和第 IV C 1 节所示。

In the coupled dielectric-meta-metallic structure, the lowest-frequency mode is where the rf magnetic field in the spiral and dielectric is in-phase. This is consistent with the lowest frequency parallel mode described in a dielectric-cavity coupled system.^{22} We call the dielectric cylinder an “equalization” element for the meta-metallic SRS or FGL component. Because the rf magnetic fields of the dielectric and meta-metallic are in-phase, the rf magnetic fields from each element add constructively, which increases the inductance of both components and lowers the resonance frequency of the combined structure compared to each individually.

在耦合介电质-超材料金属结构中，最低频率模式是螺旋和介电质中的射频磁场同相的情况。这与介电质-腔耦合系统中描述的最低频率并行模式一致。 ^{22} 我们将介电质圆柱体称为超材料金属 SRS 或 FGL 组件的“均衡”元件。由于介电质和超材料金属的射频磁场同相，因此来自每个元件的射频磁场会相长叠加，从而增加两个组件的电感，并降低组合结构的谐振频率，使其低于单独的每个组件。

#### 3. SRS (or FGL) with equalization coil

It was found that a resonant coil placed near the SRS can produce substantially the same effect as the dielectric equalization element described in Sec. IV C 2. The condition for maximum *Q*-value is the same. We call such a coil an equalization coil for the meta-metallic component. The rf magnetic fields for the coupled system are shown in Figs. 13(a) and 13(b).

3. 带均衡线圈的 SRS（或 FGL）发现放置在 SRS 附近的谐振线圈可以产生与第四部分 C 2 中描述的介电均衡元件基本相同的效果。最大 Q 值的条件是相同的。我们称此类线圈为超金属元件的均衡线圈。耦合系统的射频磁场如图 13(a) 和 13(b) 所示。

Here, the three-turn copper spiral has an inner radius 4.25 mm, outer radius 7.25 mm, axial length 5 mm, foil thickness 3 *μ*m and is embedded in PTFE. The spiral is coaxial with a toroidal equalization loop made of (thick) silver with major radius 17 mm and minor radius 5 mm. The bottom edges of both coils are coplanar. The toroidal loop has a capacitive gap of thickness 0.6 mm filled with a dielectric of *ϵ _{r}* = 10. The capacitive gap thickness was adjusted so that the coupled system resonated near 400 MHz. The

*Q*-value of the coupled system is 2220 at 381 MHz. Further adjustments would increase the

*Q*-value slightly. This can be compared to a

*Q*-value of 2441 for the three-turn spiral alone with no end effects. Even for only three turns, the theoretical

*Q*-enhancement due to the meta-metallic effect is 2.7, Eqs. (5) and (31), over a thick conducting structure of similar size. With no equalization coil and with end effects, the

*Q*-value becomes 500 at 539 MHz. The enhanced rf magnetic field due to the end effects is shown in Fig. 13(c) and is significantly larger than those with the equalization coil, Fig. 13(b). The

*Q*-value of the equalization coil alone is 3070 with the capacitive gap decreased to resonate at 399 MHz. Therefore, most of the losses are due to the equalization coil and not the SRS. If the equalization coil is made lossless, the

*Q*-value of the coupled system is 4640. This

*Q*-value is higher than the SRS alone with perfect magnetic boundaries due to the additional stored energy near the equalization coil.

在此，三匝铜螺旋的内半径为 4.25 毫米，外半径为 7.25 毫米，轴向长度为 5 毫米，箔厚度为 3 μm，并嵌入 PTFE 中。螺旋与由（厚）银制成的环形均衡回路同轴，主半径为 17 毫米，副半径为 5 毫米。两个线圈的底部边缘共面。环形回路具有厚度为 0.6 毫米的电容间隙，其中填充了介电常数为 ϵ

_{r}= 10 的电介质。电容间隙厚度经过调整，使耦合系统在 400 MHz 附近产生共振。耦合系统的 Q 值在 381 MHz 时为 2220。进一步调整会略微增加 Q 值。这可以与仅三匝螺旋的 Q 值 2441 进行比较，该螺旋没有端部效应。即使只有三匝，由于超金属效应，理论 Q 值增强 2.7 倍，方程式 (5) 和 (31)，超过了类似尺寸的厚导电结构。如果没有均衡线圈且有端部效应，Q 值在 539 MHz 时变为 500。由于端部效应而增强的射频磁场如图 13(c) 所示，并且明显大于带有均衡线圈的磁场，图 13(b)。 当电容间隙减小到在 399 MHz 处产生共振时，均衡线圈单独的 Q 值为 3070。因此，大部分损耗是由均衡线圈造成的，而不是由 SRS 造成的。如果均衡线圈无损耗，则耦合系统的 Q 值为 4640。由于均衡线圈附近的附加储存能量，此 Q 值高于具有完美磁边界的 SRS。

This coupled system can be used as a practical surface coil for MRI. Because the rf magnetic fields of the SRS and the equalization coil are in phase, the depth sensitivity below the SRS is enhanced by the equalization coil. The *Q*-value can be tailored to whatever it needs to be to produce dominant loading. Dominant loading is where the subject to be imaged absorbs at least as much power from the coil as the power losses in the coil itself. A wide variety of different types of equalization elements could be used. The equalization coil could also be used as a coupling loop.

此耦合系统可用作 MRI 的实用表面线圈。由于 SRS 和均衡线圈的射频磁场同相，因此均衡线圈增强了 SRS 下方的深度灵敏度。Q 值可以根据需要进行调整，以产生主导负载。主导负载是指被成像的物体从线圈吸收的功率至少与线圈本身的功率损耗一样多。可以使用各种不同类型的均衡元件。均衡线圈也可以用作耦合环。

#### 4. Toroidal loop

Another structure that minimizes end effects is a folded-gap toroidal loop. The structure has an overall shape of a torus (e.g., a thick single loop of wire) but the symmetry of the folded gaps is in the poloidal direction instead of the axial direction of the FGLs. A picture of the foils is shown in Fig. Fig.14.14. The foils have the shape of concentric rings. The structure has 10 sets of 10 foils with overlapping and non-overlapping regions distributed azimuthally exactly the same as the FGL shown in Fig. Fig.1.1. The rf currents are directed primarily around the loop.

4. 环形线圈 另一种最小化端部效应的结构是折叠间隙环形线圈。该结构的整体形状为环形（例如，一根粗的单线圈），但折叠间隙的对称性在极向方向上，而不是 FGL 的轴向方向上。箔片的照片如图所示。图 14.14。箔片呈同心环状。该结构有 10 组 10 个箔片，重叠和非重叠区域沿方位角分布，与图 1.1 中所示的 FGL 完全相同。射频电流主要围绕线圈流动。

The rf magnetic field distribution is shown in Fig. Fig.15.15. Outside of the outermost foil, the rf magnetic field distribution is similar to a thick loop of wire carrying an rf current around the loop. Inside the foils, the rf magnetic field magnitude steps down across the foils similar to that seen for the FGL, e.g., Fig. Fig.6.6. The magnetic field on the inside of the innermost foil is zero. The major radius of the toroidal loop is 6.78 mm, the minor radius is 1.74 mm, the spacing between foil layers is 76 *μ*m, Table TableII.II. The structure was centered in a cylindrical conducting boundary of radius and length 14 mm. Results of a numerical study of the dependence of the *Q*-value of the structure with foil thickness is shown in Fig. Fig.7.7. It can be seen that the maximum *Q*-value is obtained at a foil thickness of about 1.4 *μ*m, nearly the same as for the FGL with no end effects. The maximum *Q*-value is 1953. This *Q*-value can be compared to that of a thick conducting loop of copper of the same major and minor radii with a gap and centered in a conducting boundary of the same size. With the gap capacitance adjusted to produce a resonance frequency of 400 MHz, simulations show a resulting *Q*-value of 1131. This corresponds to a *Q*-enhancement factor, Eq. (5), of 1.7 and can be compared to *T*/*δ* = 4.2, Table TableII.II. The *Q* enhancement factor is about half of what one would expect based on the theory of Sec. III C.

射频磁场分布如图所示。图 15.15。在最外层箔的外部，射频磁场分布类似于一根粗电线环绕回路携带射频电流。在箔内，射频磁场幅度跨箔阶梯状下降，类似于 FGL 所见，例如图 6.6。最内层箔内侧的磁场为零。环形回路的主半径为 6.78 mm，次半径为 1.74 mm，箔层之间的间距为 76 μm，表 II.II。该结构居中于半径和长度为 14 mm 的圆柱形导电边界内。图 7.7 显示了对结构的 Q 值与箔厚度的依赖性的数值研究结果。可以看出，在约 1.4 μm 的箔厚度下获得最大 Q 值，几乎与没有端部效应的 FGL 相同。最大 Q 值为 1953。该 Q 值可以与具有相同主半径和次半径且具有间隙并居中于相同尺寸的导电边界内的粗铜导电回路的 Q 值进行比较。 通过调整间隙电容以产生 400 MHz 的谐振频率，仿真显示产生的 Q 值为 1131。这对应于 Q 增强因子，方程式 (5)，为 1.7，并且可以与 T/δ = 4.2，表 II.II 进行比较。Q 增强因子大约是根据 III C 部分的理论预期值的一半。

The reason for this has to do with the rf current distribution in the foils. Examination of the current distribution in the foils of the structure using Ansys HFSS reveals significant poloidally directed currents caused by the relatively large overlapping regions of the outer gaps compared to the inner gaps. The rf currents flow from these capacitive regions poloidally to the innermost regions of the foils and then back poloidally to the next capacitor. The rf current paths are inefficient compared to those in the thick conducting loop and the *Q*-value enhancement ratio is decreased. This effect can be reduced by reducing the ratio of the minor radius to the major radius.

造成这种情况的原因与箔中的射频电流分布有关。使用 Ansys HFSS 检查结构箔中的电流分布，发现与内间隙相比，外间隙的重叠区域相对较大，从而导致了明显的极向电流。射频电流从这些电容区域极向流向箔的最内层区域，然后极向返回到下一个电容器。与厚导电环路中的射频电流路径相比，射频电流路径效率较低，并且 Q 值增强比降低。可以通过减小次半径与主半径的比率来降低这种影响。

A simpler method of constructing the toroidal foil configuration than that shown in Fig. Fig.1414 is to replace each set of 10 groups of foils with a single poloidally spiraled 10-turn foil. This single foil must then be cut azimuthally in order to break the poloidal currents that tend to flow from inner foil layers to outer foil layers. The cut should be placed where the rf magnetic field is weakest in order to minimize rf currents along the foil edges created by the cut. It can be seen in Fig. Fig.1515 that the foils should be cut where they meet the bottom edge of the figure at the largest distance from the axis. This structure has been simulated and produces similar *Q*-values to the structure of Fig. Fig.14.14. The data point is shown in Fig. Fig.7.7. The slightly lower *Q*-value is due to additional rf currents that flow along the cut edges.

一种比图中所示的构建环形箔配置更简单的方法。图 1414 是用一个单一的极向螺旋 10 圈箔代替每组 10 组箔。然后必须将这个单箔沿方位角切割，以打破从内箔层流向外箔层的极向电流。切割应放置在射频磁场最弱的位置，以最大程度地减少切割产生的箔边缘沿线的射频电流。在图中可以看到。图 1515 中的箔应在它们与图中底边的最大距离处相遇时进行切割。该结构已经过模拟，并产生与图中的结构类似的 Q 值。图 14.14。数据点如图所示。图 7.7。稍低的 Q 值是由于沿切割边缘流动的附加射频电流。

#### 5. Coaxial length

A 5 mm length of coaxial cable was simulated with the inner conductor replaced by two sets of 10 axially overlapping foils. Dimensions are shown in Table TableII.II. With each end of the coaxial cable shorted, the capacitance between foils *C* was designed for a resonance frequency near 400 MHz using the transmission line impedance equation given by Eq. (A22) of the Appendix. A cut view of the cable showing the foils and the rf magnetic field magnitude profile are shown in Fig. Fig.16.16. At a foil thickness of 1.6 *μ*m, the *Q*-value of the structure has a maximum value of 19 718. This can be compared to a *Q*-value of 5302 for the same coaxial length with the foils replaced by a thick inner conductor with an outer radius the same as the *average* radius of the foils, 6.3 mm. In order to resonate the cable, a 0.5 mm gap was created in the center conductor centered at 2.5 mm axial distance and a capacitive element of the same value calculated above (105 pF) was added across the gap. The corresponding *Q* enhancement factor is 3.7 and can be compared to *T*/*δ* = 4.8, Table TableII,II, consistent with the theory of Sec. III C. The *Q*-value could be further improved by adding capacitive gaps to the outer shield. However, the additional *Q* enhancement would be smaller by the ratio of the inner conductor radius to the outer conductor radius. Such a structure could be used as a low-loss transmission line that would have similar loss characteristics in a much more compact geometry than traditional rectangular waveguide. A small section shorted at both ends could also be used as a resonator for NMR.

A much longer structure with no shorted ends and many capacitive regions could be used as a coaxial cable transmission line. As such, it would exhibit some dispersion due to the capacitive regions. This property is unlike a coaxial cable with a thick inner conductor, which carries a pure TEM mode, but is similar to standard waveguide. The amount of dispersion can be adjusted through the capacitance.

一个没有短路端且具有许多电容区域的更长的结构可以用作同轴电缆传输线。因此，它会由于电容区域而表现出一些色散。此特性不同于具有厚内导体的同轴电缆，该同轴电缆承载纯 TEM 模式，但类似于标准波导。色散量可以通过电容进行调整。

## V. RESULTS V. 结果

### A. Fabrication A. 制造

Spiral foil structures were fabricated by winding strips of foil-dielectric laminate and inserting them into PTFE holders. The laminate can been made by three methods: (1) electrodeposition of copper onto PTFE (CuFlon, Polyflon Company, Norwalk, CT), (2) application of heavy gilding foil onto PTFE, and (3) physical vapor deposition (PVD) onto dielectrics.

螺旋箔结构是通过缠绕箔-电介质层压条并将其插入 PTFE 支架中制造的。层压可以通过三种方法制成：(1) 将铜电沉积到 PTFE 上（CuFlon，Polyflon Company，诺沃克，康涅狄格州），(2) 将重镀金箔应用到 PTFE 上，以及 (3) 物理气相沉积 (PVD) 到电介质上。

In general, dielectric materials used in making the meta-metallic structures must be low-loss, see the Appendix and Sec. IV C. Practical dielectric materials for use between foil layers include PTFE, polyethylene, polypropylene, polystyrene, paraffin wax, silicon dioxide, glass, sapphire, and Rogers RT/duroid® 5880. Materials with large dielectric constants can also be used. Any adhesive used to apply the foil to the substrate is also a dielectric and must therefore also be low loss. Alternatively, if the adhesive is highly conducting it then adds to the foil thickness.

一般来说，用于制造金属结构的电介质材料必须是低损耗的，请参见附录和第四部分 C。用于箔层之间的实用电介质材料包括 PTFE、聚乙烯、聚丙烯、聚苯乙烯、石蜡、二氧化硅、玻璃、蓝宝石和 Rogers RT/duroid® 5880。也可以使用具有大介电常数的材料。用于将箔应用到基材上的任何粘合剂也是电介质，因此也必须是低损耗的。或者，如果粘合剂是高导电性的，那么它会增加箔的厚度。

To date, coils have been made by the first two methods. An issue surrounding method 1 is that control of the foil thickness is poor. The foil thickness tends to run larger than specification by up to a factor of two and vary by up to 50% between different parts of the laminate panels. Foil thickness has been calculated by two techniques: (1) weight, density, and dimensions; and (2) four-point probe voltage and current measurements. Of the two techniques, the four-point determination is more accurate because it is insensitive to any non-conducting adhesive layers between the foil and substrate.

迄今为止，线圈已经通过前两种方法制成。围绕方法 1 的一个问题是箔厚度的控制很差。箔厚度往往比规格大两倍，并且在层压板的不同部分之间变化高达 50%。箔厚度已通过两种技术计算：(1) 重量、密度和尺寸；以及 (2) 四点探针电压和电流测量。在这两种技术中，四点测定更准确，因为它对箔和基材之间的任何非导电粘合剂层不敏感。

In method 2 (gilding foil), the foil thickness is within a few percent of specification as determined by both weight/dimension and four-point probe measurement techniques. Foil can be ordered in 10 cm^{2} in copper, silver, and gold with any thickness between 2 and 10 *μ*m. Gold can be ordered down to 0.1 *μ*m. However, it is difficult to apply the foil to the dielectric substrate. In traditional gilding, the size (glue) sets up a tack that lasts for at least an hour. This permits time to apply the foil. However, commercial gilding size (both acrylic and oil-based) is too lossy to be used to make meta-material laminates. An adhesive that is known to work is Q-dope (GC Electronics), which is a polystyrene glue. However, Q-dope dries very rapidly. Two methods to deal with this problem have been identified. One is to apply the glue thickly and then press the foil between two blocks. The flow of the glue to the edges straightens the foil and the amount of pressure can be used to control the glue thickness. The other method is to mix the polystyrene glue with a lower vapor pressure solvent such as ethylbenzene or propylbenzene. The thinned glue can be applied more thinly and the solvent evaporates slowly enough for the foil to be applied.

在方法 2（镀金箔）中，箔的厚度在几个百分点内符合规格，这是通过重量/尺寸和四点探针测量技术确定的。可以订购 10 cm ^{2} 的铜、银和金箔，厚度在 2 到 10 μm 之间。可以订购厚度低至 0.1 μm 的金箔。然而，很难将箔应用于电介质基板。在传统的镀金中，大小（胶水）会产生一种粘性，这种粘性至少持续一个小时。这有时间来应用箔。然而，商业镀金大小（丙烯酸和油基）的损耗太大，无法用于制作超材料层压板。已知有效的粘合剂是 Q-dope（GC Electronics），它是一种聚苯乙烯胶水。然而，Q-dope 干燥非常快。已经确定了两种解决此问题的方法。一种方法是将胶水涂抹得很厚，然后将箔压在两个块之间。胶水流向边缘会使箔变直，并且可以使用压力量来控制胶水的厚度。另一种方法是将聚苯乙烯胶水与乙苯或丙苯等较低蒸汽压溶剂混合。 稀释后的胶水可以涂得更薄，溶剂蒸发得足够慢，以便贴上箔片。

In method 3, discussions with scientists in the PVD field indicate that the proper foil thickness can be achieved on up to 15 cm^{2} substrates.

在方法 3 中，与 PVD 领域的科学家的讨论表明，可以在高达 15 cm ^{2} 的基板上实现适当的箔片厚度。

It is found that if the foil is affixed to thin (50 *μ*m) dielectric sheets, buckling or cracking of the foil does not occur when the laminate is bent. Thin foil-PTFE laminate can be cut cleanly into strips using a rotary cutter and cutting mat.

发现，如果将箔片贴在薄的 (50 μm) 介电薄片上，则在弯曲层压板时不会发生箔片的翘曲或开裂。薄箔-PTFE 层压板可以使用旋转刀具和切割垫整齐地切成条状。

### B. Characterization B. 表征

*Q*-value measurements were made on SRS foil structures using an Agilent Technologies E8363C PNA Network Analyzer. It was calibrated using Electronic Calibration Module N4691-60001. The network analyzer was connected to a 16 mm diameter coupling loop at the end of a 22 cm length of 3 mm outer diameter 50 Ω semi-rigid coaxial cable with an SMA connector. Critical coupling to the coil was achieved by adjusting the axial distance between the coupling loop and the coil. *Q*-value measurements were made by observing the frequency of the *S*_{11} −6 dB points on either side of resonance.

使用安捷伦科技 E8363C PNA 网络分析仪对 SRS 箔片结构进行 Q 值测量。使用电子校准模块 N4691-60001 对其进行校准。网络分析仪连接到 22 cm 长、3 mm 外径、50 Ω 半刚性同轴电缆末端的 16 mm 直径耦合环，该电缆带有 SMA 连接器。通过调整耦合环和线圈之间的轴向距离来实现与线圈的临界耦合。通过观察共振两侧 S _{11} -6 dB 点的频率来进行 Q 值测量。

A *Q*-value of 623 was measured for a stand-alone SRS of copper foil thickness 5.5 *μ*m as determined by a four-point probe. For the measurement, the SRS was centered in a copper shield of diameter 11.4 cm and height 10.2 cm. This *Q*-value can be compared to the 10-foil FGL simulations as shown in Fig. Fig.7.7. The spiral is 21/3 turns with an outer diameter of 16.8 mm and an inner diameter of 12.6 mm. The structure was made from a CuFlon panel with 51 *μ*m thickness PTFE and a specified copper cladding weight of 1/16 ounce/square foot, which has a corresponding nominal thickness of 2.2 *μ*m. However, the actual thickness is believed to be 5.5 *μ*m, based on four-point probe measurements. The CuFlon was cut into a 11.5 cm long 1 cm wide strip using a rotary cutter and plastic mat. This strip was sandwiched between two 0.51 mm thickness PTFE strips (with no cladding) of the same length and width, wound and placed inside a cylindrical PTFE holder. The spacing between turns is 1.07 mm. Simulations of the built structure yielded a *Q*-value of 715 with a PTFE loss tangent of 1.5E-4 and conducting walls. The *Q*-value of this structure is lower than it can be due to end effects described in Sec. IV B.

通过四点探针测得厚度为 5.5 μm 的独立 SRS 的 Q 值为 623。在测量过程中，SRS 居中放置在直径为 11.4 cm、高度为 10.2 cm 的铜屏蔽罩中。此 Q 值可与图 7.7 中所示的 10 层 FGL 模拟进行比较。螺旋为 21/3 圈，外径为 16.8 mm，内径为 12.6 mm。该结构由厚度为 51 μm 的 CuFlon 面板和指定铜包覆重量为 1/16 盎司/平方英尺的 PTFE 制成，其对应的标称厚度为 2.2 μm。然而，根据四点探针测量结果，实际厚度被认为是 5.5 μm。使用旋转刀和塑料垫将 CuFlon 切割成长度为 11.5 cm、宽度为 1 cm 的条带。该条带夹在两条长度和宽度相同的 0.51 mm 厚 PTFE 条带（无包覆）之间，缠绕并放置在圆柱形 PTFE 支架内。圈与圈之间的间距为 1.07 mm。对已建结构的模拟得出 Q 值为 715，PTFE 损耗角正切为 1.5E-4，且导电壁。由于第四部分 B 中描述的端效应，此结构的 Q 值低于其可能达到的值。

The method of using a spiral of a thin sheet of CuFlon sandwiched between two layers of PTFE as described above could also be used to fabricate an FGL. Because CuFlon can be ordered like a printed circuit board, the sizes of clad bars and gaps on the CuFlon panel can be specified such that when cut and spiraled they produce the appropriate overlap of adjacent layers. The non-clad regions would make up almost one full turn of each layer.

如上所述，将薄的 CuFlon 薄片夹在两层 PTFE 之间的螺旋方法也可以用于制造 FGL。由于可以像印刷电路板一样订购 CuFlon，因此可以指定 CuFlon 面板上的包覆棒和间隙的尺寸，以便在切割和螺旋时产生相邻层的适当重叠。未包覆的区域将构成每一层的几乎一整圈。

## VI. DISCUSSION VI. 讨论

It is seen that several different types of structures made of many thin layers of metallic foil with capacitive gaps between adjacent layers exhibit enhancement of the *Q*-value compared to similar structures made with a single layer of thick conductor. The *Q* enhancement factor is consistent with the meta-metallic effect, given by Eq. (5) and discussed in Sec. III C. Structures that exhibit such enhancement in *Q*-value have the characteristic that the rf magnetic field lines in the vicinity of the foils are generally parallel to the foil surfaces, particularly the edges. One way this can occur is that the rf currents approach zero near the foil edges as seen in Secs. IV C 4 and IV C 5. Another way this can occur, as described in Secs. IV C 1-IV C 3, is when another nearby structure, such as an equalization coil or dielectric, produces an additional magnetic field parallel to the field generated by the meta-metallic. If the resonance frequency of the coupled system is equal to the resonance frequency of the meta-metallic structure alone with perfect magnetic boundary conditions, the rf current distribution in the foils becomes substantially uniform, eliminating end effects.
重试
错误原因

A practical coil with a *Q*-value of 623 was constructed and characterized at 400 MHz. When combined with an equalization coil and run in parallel mode, simulations suggest that significantly higher *Q*-values can be obtained. We expect the resonance frequency and *Q*-value of any of these structures to be as stable with temperature as the underlying physical properties of the conductivity of the foil layers and the dielectric constant and loss tangent of the dielectric between layers. The foil should be protected from oxidation using a low-loss coating. The SRS with equalization coil is a promising structure for an MRI surface coil. In the limit of dominant loading of the *Q*-value of an MRI surface coil by tissue, increase in *Q*-value of an MRI surface coil tends to be of little benefit. However, for rodent imaging at, for example, 400 MHz using surface coils of 2 cm diameter or smaller, the dominant-loading condition is difficult to achieve. The methods of this paper provide significant advantages for MRI of small animals and also of tissue samples. The problem is acute in murine imaging.^{24} In principle, the resonance condition can be satisfied over a range of small coil diameters by compensating the decrease in single-turn inductance by increase in the number of metal layers as well as increase in capacitance through use of thin dielectric film between layers. Improved methods of forming the coil, including physical vapor deposition, may be required.

在 400 MHz 下构建并表征了一个 Q 值为 623 的实用线圈。当与均衡线圈结合并以并行模式运行时，仿真表明可以获得明显更高的 Q 值。我们预计这些结构的任何一个的谐振频率和 Q 值与箔层电导率的基本物理特性以及层间电介质的介电常数和损耗角正切一样稳定。应使用低损耗涂层保护箔免受氧化。带有均衡线圈的 SRS 是 MRI 表面线圈的一种有前途的结构。在组织对 MRI 表面线圈的 Q 值的主导负载限制内，MRI 表面线圈的 Q 值增加往往几乎没有好处。然而，对于啮齿动物成像，例如，在 400 MHz 下使用直径为 2 cm 或更小的表面线圈，很难实现主导负载条件。本文的方法为小动物的 MRI 以及组织样本的 MRI 提供了显着的优势。这个问题在小鼠成像中很严重。 ^{24} 原则上，通过增加金属层数以及通过使用层间薄介电膜增加电容，可以补偿单匝电感下降，从而在一定范围的小线圈直径上满足谐振条件。可能需要改进线圈的成型方法，包括物理气相沉积。

The technology of meta-metallic coils is promising for use in NMR microscopy. Decrease in coil diameter must be offset by increase in capacitance between foil layers in order to achieve resonance. Spacing between layers can be as small as 10 *μ*m, and the dielectric constant can be substantially higher, providing a significant opportunity to decrease the meta-metallic coil diameter. Using advanced manufacturing techniques, a *Q*-enhancement of 100 seems feasible since the required thickness is 100 nm at 400 MHz and 20 nm at 9.5 GHz, Table TableI,I, both of which are considerably larger than the lattice constant of copper, 3.6 Å. As the foil gets thinner, any residual end effects become stronger, and the structure becomes increasingly difficult to fabricate.

超金属线圈技术有望用于核磁共振显微镜。为了实现谐振，必须通过增加箔层之间的电容来抵消线圈直径的减小。层间间距可以小至 10 μm，介电常数可以大得多，这为减小超金属线圈直径提供了重要机会。使用先进的制造技术，Q 值增强 100 倍似乎是可行的，因为在 400 MHz 时所需的厚度为 100 nm，在 9.5 GHz 时为 20 nm，表 I，表 I，两者都远大于铜的晶格常数 3.6 Å。随着箔变得更薄，任何残余的端效应都会变得更强，并且结构变得越来越难以制造。

## Acknowledgments 致谢

This work was supported by grants P41 EB001980 and R01 EB000215 from the National Institute of Biomedical Imaging and Bioengineering (NIBIB) of the National Institutes of Health (NIH).

这项工作得到了国家生物医学成像和生物工程研究所 (NIBIB) 的国家卫生研究院 (NIH) 的赠款 P41 EB001980 和 R01 EB000215 的支持。

## APPENDIX: ANALYTIC FORMULAS FOR COILS AND RESONATORS

附录：线圈和谐振器的解析公式

One of the most efficient resonators for small samples is the loop-gap.^{8} The inductance of a 1-loop–1-gap LGR is a parallel combination of an inner and outer inductance,^{21,25}

对于小样品来说，最有效的谐振器之一是环形间隙。 ^{8} 1 环 1 间隙 LGR 的电感是内部电感和外部电感并联组合， ^{21,25}

where, neglecting end effects,

其中，忽略端部效应，

where *r _{i}* is the inner radius and

*l*is the axial length,

其中 r

_{i}为内半径，l 为轴向长度，

where *r _{o}* is the outer radius of the LGR and

*r*is the inner radius of the conducting shield. Similarly, the resistance can be written as

_{s}其中 r

_{o}是 LGR 的外半径，r

_{s}是导电屏蔽的内半径。类似地，电阻可以写为

where 其中

and, neglecting the shield conductivity,

并且，忽略屏蔽电导率，

Substituting these equations into Eq. (1) and using Eq. (3), the *Q*-value can be written as

将这些方程代入方程式 (1) 并使用方程式 (3)，Q 值可以写为

This equation reduces to Eq. (4) of Ref. ^{8} in the limit *r _{i}* ≪ (

*r*,

_{o}*r*−

_{s}*r*). The

_{o}*Q*-value is independent of axial length. For the LGR discussed in Sec. II, Eq. (A7) gives 284, which is about 20% higher than simulation due to ohmic dissipation in the gap. The resonance frequency of a coil or resonator is given by

该方程在极限 r

_{i}≪ (r

_{o}, r

_{s}− r

_{o}) 中简化为参考文献

^{8}的方程式 (4)。Q 值与轴向长度无关。对于第二节中讨论的 LGR，方程式 (A7) 给出 284，由于间隙中的欧姆耗散，这比仿真高出约 20%。线圈或谐振器的谐振频率由下式给出

where *L* is the inductance and *C* is the capacitance. For a 1-loop–1-gap LGR, the inductance is given by Eq. (A1) and the capacitance is the gap capacitance.

其中 L 为电感，C 为电容。对于 1 环 1 间隙 LGR，电感由公式 (A1) 给出，电容为间隙电容。

For the FGL with no end effects, Fig. Fig.1,1, the inductance can be estimated from Eq. (A1) where, instead of Eq. (A2), the average radius is used

对于没有端部效应的 FGL，图 1，1，电感可以从公式 (A1) 估算，其中，使用平均半径代替公式 (A2)

The *Q*-value is enhanced due to the multiple current paths of thickness *t*. Equations (4) and (5) apply, although the foil thickness must be thinner than the maximum given by Eq. (31). The *Q*-value can also be estimated from Eq. (1) with

由于厚度 t 的多条电流路径，Q 值得到增强。公式 (4) 和 (5) 适用，尽管箔厚度必须比公式 (31) 给出的最大值薄。Q 值也可以用公式 (1) 估计，其中

where *N* is the number of foil layers in the non-overlapping region. Since this resistance does not account for the eddy current dissipation, it can result in an overestimate of the *Q*-value. The capacitance can be expressed as

其中 N 是非重叠区域中箔层的数量。由于此电阻未考虑涡流耗散，因此可能导致 Q 值高估。电容可表示为

where *N _{ov}* is the number of azimuthal overlapping regions, see Fig. Fig.1,1, and

其中 N

_{ov}是方位重叠区域的数量，见图 1,1，以及

where *ϵ*_{0} is the electric permittivity of free space, *ϵ _{r}* is the relative dielectric constant of the material between the layers,

*A*is the net area of an overlapping region, and

*d*is the distance between adjacent foil layers in the overlapping region. The area can be approximated as

其中 ϵ

_{0}是自由空间的电容率，ϵ

_{r}是层间材料的相对介电常数，A 是重叠区域的净面积，d 是重叠区域中相邻箔层之间的距离。该面积可近似为

*A*= (2

*N*− 1)

*l*

*θ*

_{ov}(

*r*

_{i}+

*r*

_{o})/2,

where *θ _{ov}* is the foil overlapping angle in radians.

其中 θ

_{ov}是以弧度为单位的箔重叠角。

For a single-foil SRS with no end effects, the inductance, resistance, and *Q*-value can be estimated using the same equations as the FGL given above, where *r _{i}* is the minimum foil radius,

*r*is the maximum foil radius, and

_{o}*N*is the number of turns of the spiral. For this structure,

*N*need not be an integer. The resonance frequency is given by Eq. (A8) with

*C*the total capacitance between adjacent layers,

对于没有端部效应的单箔 SRS，电感、电阻和 Q 值可以使用与上面给出的 FGL 相同的方程进行估计，其中 r

_{i}是最小箔半径，r

_{o}是最大箔半径，N 是螺旋的匝数。对于此结构，N 不必是整数。谐振频率由方程式给出。(A8)，其中 C 是相邻层之间的总电容，

where *p* is the pitch of the spiral. For the multiple foil spiral, Fig. Fig.9,9, there is surprisingly little interaction between the individual foils. The result is that nearly the same magnetic field is obtained with individual foil currents reduced by the number of individual foils *N _{f}*. The equations for inductance, resistance,

*Q*-value, and frequency are the same as for the single foil spiral except the resistance given by Eq. (A10) is divided by the number of foils,

其中 p 是螺旋的螺距。对于多箔螺旋，图 9，图 9，各个箔之间的相互作用出人意料地小。结果是，几乎相同的磁场可以通过将各个箔电流减少到各个箔数 N

_{f}来获得。电感、电阻、Q 值和频率的方程与单箔螺旋相同，只是由方程式 (A10) 给出的电阻除以箔数，