A Coupled Microstrip Line Transverse Electromagnetic Resonator Model For High-Field Magnetic Resonance Imaging

by

G. Bogdanov and R. Ludwig

Department of Electrical and Computer Engineering

Worcester Polytechnic Institute Worcester, MA 01609

*Correspondence Author: R. Ludwig, Dept. of Electrical and Computer Engineering,

Worcester Polytechnic Institute, Worcester, MA 01609. Phone: 508-831-5315, FAX: 508-831-5491,

E-mail: ludwig@wpi.edu

Submitted to

Dr. Felix W. Wehrli Editor in Chief

Journal of Magnetic Resonance in Medicine University of Pennsylvania Medical Center

Department of Radiology 3400 Spruce Street

Philadelphia, PA 19104-4283

2nd Revision

August 6, 2001

2

Abstract The performance modeling of RF resonators at high magnetic fields of 4.7 T and more

requires a physical approach that goes beyond conventional lumped circuit concepts.

The treatment of voltages and currents as variables in time and space leads to a

coupled transmission line model, whereby the electric and magnetic fields are assumed

static in planes orthogonal to the length of the resonator, but wave like along its

longitudinal axis. In this paper a multi-conductor transmission line model was

developed and successfully applied to analyze a 12-element unloaded microstrip line

transverse electromagnetic (TEM) resonator coil for animal studies at 4.7 T. This

model estimates the resonance spectrum, field distributions and certain types of losses

in the coil, while requiring only modest computing resources. Both the theoretical

basis and its engineering implementation are developed and the resulting impedance

predictions are compared to practical measurements. The boundary element method is

adopted to compute all relevant transmission line parameters.

Although this multi-conductor transmission line model is applied to a small bore

animal system of 7.5 cm inner diameter and operated at 200 MHz, it is demonstrated

that lumped circuit models can no longer adequately predict the coil’s resonance

behavior. At 9.4 T, or 400 MHz resonance frequency for proton imaging, the circuit

model breaks down completely.

Keywords: Multi-conductor transmission line model, microstrip lines, TEM

resonator, boundary element method.

3

1. Introduction

The TEM resonator design (1, 5-7) as an RF coil has received heightened

attention as a superior replacement for the standard birdcage coil (4) in high-field 4.7 T to

9.4 T MRI applications. It has been demonstrated (1-3) that at the corresponding

operating frequencies of 200 and 400 MHz, the TEM resonator can achieve better field

homogeneity and a higher quality factor than an equivalent birdcage coil, resulting in

improved image quality.

In our opinion, the primary difference between a TEM resonator and a birdcage

coil is the cylindrical shield, which functions as an active element of the system,

providing a return path for the currents in the inner conductors. In a birdcage coil, the

shield is a separate entity, disconnected from the inner elements, only reflecting the fields

inside the coil to prevent excessive radiation losses. Because of this shield design, the

TEM resonator behaves like a longitudinal multi-conductor transmission line, capable of

supporting standing waves that occur at high frequencies. Unlike a birdcage coil, the

TEM resonator’s inner conductors do not possess connections to their closest neighbors,

but instead connect directly to the shield through capacitive elements. Resonance mode

separation is accomplished though mutual coupling between the inductive inner

conductors. Since all the conductors connect to the shield with tunable capacitive

elements, the field distribution can be adjusted to achieve the best homogeneity.

The first descriptions of TEM resonator structures appeared in patents (5, 6).

Subsequently, several papers (1, 8-11) have been published on the subject of modeling

the TEM resonator. The models are typically based on transmission line concepts.

Vaughan et al (1) derived an estimate for the resonance frequency by treating the

resonator as a section of coaxial transmission line terminated by capacitors. Tropp (8)

developed a lumped circuit model for a TEM resonator, where the inner conductors are

treated as inductors with mutual coupling and the terminating elements are capacitors.

The model predicts all resonant modes of the TEM resonator and shows good agreement

with experimental data. However, the measurements are conducted at a relatively low

frequency of 143 MHz. Röschmann (9) and Chingas el al (10) developed TEM resonator

4

models based on simplified coupled transmission line equations. Theses models are

accurate at predicting particular resonance frequencies, but require complete symmetry of

the structure and its terminating elements.

Baertlein et al (11) developed a resonator model based on multi-conductor

transmission line theory, which included accurate calculation of per-unit-length

parameter matrices, resonance frequencies and field distributions inside the coil. The

TEM coil can be modeled under linear or quadrature drive conditions. The resonance

frequency predictions compared well with measurements and a full-wave Finite-

Difference Time Domain (FDTD) model, developed separately in (12). The transmission

line model is suitable for studying the resonance behavior of an unloaded conventional

TEM resonator. Its inability of estimating the quality factor of the coil is not an issue

because the unloaded resonator losses are primarily dominated by radiation, while the

load dominates the losses in the loaded coil. Additionally, the model does not provide for

the inclusion of different dielectrics in the TEM resonator cavity.

In this paper, the multi-conductor transmission line model for the TEM resonator

is developed to its full potential. Its capabilities are expanded because of the requirement

to integrate the structure with an animal system. This new coil is small, utilizes a plastic

former and uses microstrips for the inner conductors. Although the design is inexpensive

compared to a conventional TEM resonator, it is not as efficient. Unlike the tubular

resonator case (1), inserting a load into the microstrip TEM coil does not cause a large

drop in quality factor. This indicates that coil losses are significant in the presence of a

load. To improve its performance, the coil needs to be carefully optimized using an

accurate simulation model. The complete multi-conductor transmission line (MTL)

model for the TEM resonator, discussed in the subsequent sections, includes support for

multiple dielectrics (plastic former), and computes the coil’s quality factor based on

losses due to the dielectrics, conductors and terminating capacitors. Moreover, the MTL

model computes the S11 frequency response, the current, voltage and energy distributions

along the length of the coil, and the field patterns at any cross-section at any frequency.

The coil drive circuit is complete with a matching capacitor and the option of quadrature

drive. The model places no symmetry restrictions on the cross-sectional geometry or the

terminating elements.

5

The MTL model can be used to efficiently design the unloaded coil behavior.

Small changes introduced by the biological load are primarily compensated by an

adjustment in the matching network and minor changes in a single tuning capacitor. We

attribute the fact that the load does not significantly perturb this coil to the highly

distributed capacitance and inductance as well as a low filling factor. Furthermore,

radiation effects that the model neglects are generally not a dominant source of losses for

shielded, loaded coils employed in in-vivo studies.

The MTL model does not simulate the non-TEM magnetic field distribution or the

eddy-current losses in a biological load at high frequency. Full wave models, such as

FDTD (28, 29), are necessary for this task. Although the capabilities of the MTL model

are somewhat limited compared to an FDTD model, its computational requirements are

light. A full simulation completes in approximately 5 minutes on an average PC (20

seconds for a frequency sweep), compared to several hours with FDTD.

6

2. Theory

2.1. Microstrip TEM resonator

The TEM resonator in its functional form is shown in Figure 1a. Similar to

traditional designs (1, 7), this resonator consists of multiple longitudinal conductors

arranged in a cylindrical pattern and enclosed by a cylindrical shield. However, unlike

previous designs it uses microstrip conductors. Moreover, the inner conductors connect

to the shield with commercially available fixed and trimmer capacitors instead of custom-

made open-circuited coaxial line sections. A hollow cylindrical coil former made of

dielectric material supports the TEM resonator structure.

To describe the electromagnetic TEM behavior a wave propagation model is

developed that treats the microstrip conductors as coupled transmission lines with wave

propagation in the longitudinal direction (13-15). To achieve a closed-form solution, the

model is based on the quasi-TEM approximation (13). Under this assumption, the

electric and magnetic fields in the structure have no longitudinal or z-directed

components. The TEM assumption is truly valid only for systems homogeneous in the

longitudinal direction with perfect conductors, a single dielectric, and an electrically

small cross-section that is much smaller than the line’s length. However, previous works

have successfully applied multi-conductor transmission line (MTL) techniques based on

the quasi-TEM approximation to modeling the conventional TEM resonator (9-11). TE

and TM modes do not occur in the unloaded resonator because the operating frequency is

below the cutoff frequency of these modes even in human coils (11). The following

sections develop the essential steps in constructing a full-featured MTL model of an

arbitrary TEM resonator.

2.2. Frequency domain solution of coupled microstrip lines

In the frequency domain, the generalized multi-conductor transmission line

(MTL) equations can be written in the following matrix form involving spatially varying

voltages and currents (13)

( )( )

( )( )

−−

=

zz

zz

dzd

IV

0YZ0

IV

[1]

7

Here LRZ ωj+= and CGY ωj+= are the per-unit-length impedance and admittance

matrices, respectively, which characterize the multi-conductor transmission line

configuration, and fπω 2= is the angular frequency. The above system has the solution

( )( ) ( ) ( )

( )( ) ( )( ) ( )

( )( )

=

=

00

00

2221

1211

IV

ΦΦΦΦ

IV

ΦIV

zzzz

zzz

[2]

where ( )zΦ is the so-called chain-parameter matrix defined in terms of the matrix

exponential

( ) AeΦ zz = ;

−−

=0YZ0

A ; l++++=!3!2

3322 AAAEe A zzzz [3]

with E being the identity matrix. Expanding [3], we obtain (13)

( )( ) ( )

( ) ( )

+−−

−−+=

=

−−−−

−−−

11

1

2221

1211

21

21

21

21

YeeYeeZYZ

YZYeeee

ΦΦΦΦ

ΦZYZYZYZY

ZYZYZYZY

zzzz

zzzz

z [4]

where the matrix square root is defined according to AAA = . Although this

definition does not yield a unique matrix square root, it will not influence the final result

as the expansion of the matrix exponential in [3] produces only integer powers of ZY.

One possible approach of finding the frequency-domain solution involves

determining the unknown vectors I(0) and V(0) in [2] from the terminating (boundary)

conditions at the source and load sides. The resonator’s terminating conditions at z=0

and z=L are found from the generalized Thévenin equivalent circuit expression

( ) ( )00 IZVV SS −= ( ) ( )LL LL IZVV +=

[5]

where ZS and ZL are the impedance matrices characterizing the passive networks at the

source and load sides, and VS and VL are the Thévenin voltage vectors.

Substituting [5] into [2], the source side current vector I(0) is found to be

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) [ ]LSLLSLS LLLLLL VVΦZΦΦZZΦZΦZΦI −−+−−= −2111

1222112110 [6]

In addition, the source side voltage vector V(0) is obtained from [5], thus making the

solution in [2] complete.

Another useful solution can be derived from [2], when the input impedance

matrix for the MTL system is expressed in terms of the known load impedance matrix. If

8

it is assumed that the load side contains no sources, we can make the terminating

conditions

( ) ( )00 IZV in= ( ) ( )LL LIZV =

[7]

where the unknown Zin is the input impedance matrix of the MTL of length L terminated

by a load network ZL. Substituting [7] into [2] and asserting that the result must be valid

for arbitrary I(0), the input impedance becomes

( ) ( )[ ] ( ) ( )[ ]LLLL LLin 12221

2111 ΦΦZΦZΦZ −−= − [8] The advantage of expression [8] is that no knowledge of the source side is

required. This removes the need for a Thévenin or Norton equivalent circuit

representation at the source side, thereby allowing more flexibility in simulating complex

networks. Once the input current I(0) is calculated, the expression for the current

anywhere along the line follows from [2] and [7]:

( ) ( ) ( )[ ] ( )01211 IZΦΦI inzzz += [9]Further reduction of the computational requirements is possible if the

transmission line system is symmetric such that the matrices Z and Y are circulant, which

implies that the subsequent column may be obtained by rotating the previous column

downward by one. The TEM resonator is an example of a system with symmetric,

circulant Z and Y matrices. The improvement in computation efficiency is due to the fact

that the eigenvectors of circulant matrices are fixed. The diagonalizing transformation

matrix with elements mnT of a circulant matrix can be written in a form equivalent to the

Discrete Fourier Transform (DFT) scaled by N1 for normalization purposes (16)

( )( )1121 −−−

=nm

Nj

mn eN

Tπ

[10]

In addition, the eigenvalues of a circulant matrix may be more directly obtained through

the DFT of the first column (or row) of that matrix

( )( )

∑=

−−−=

N

m

nmN

j

mn eZZ1

112

1,ˆ

π

[11]

where nZ is the nth eigenvalue of Z with n=1, 2,…N. The Fast Fourier Transform (FFT)

algorithm makes the evaluation of the eigenvalues of a circulant matrix computation very

9

efficient. Here, the matrix square root and exponential operations reduce to element-by-

element computations of the eigenvalues

1

11

1 −

−−

−

=

=

TTee

TMTTTMMTTM aa

[12]

where M is a circulant matrix, a is an arbitrary scale factor, and T is defined in [10]. The

actual transformations from the matrix to its eigenvalues, and vice versa, are performed

through the FFT and inverse FFT.

2.3. Transmission Line Parameters

A typical TEM transmission line system, such as the TEM resonator, consists of

parallel conductors aligned along the z-axis and separated by dielectrics. For this system,

the per-unit-length parameters Z and Y may be separated into (mutual) inductance L,

resistance R, capacitance C, and conductance G matrices. The coefficients for these

matrices are obtained by solving a two-dimensional static field problem based on

Laplace’s equation (13).

Matrix C accounts for the capacitative effects between all metallic conductors in

the transmission line system, characterizing the electric field energy storage in the MTL.

If jiC ,~ is defined as the per-unit-length capacitance between any two conductors

numbered i and j in the system, including a reference conductor labeled i = 0, then the

capacitance matrix C is (13)

≠−

==∑

≠=

jiC

jiCC

ji

N

ikk

ki

ji

,~

,~

,

0,

, [13]

The ith row of this matrix for an arbitrary MTL system can be straightforwardly

calculated from the solution of the following two-dimensional Laplacian in the x-y plane

( ) 0=Φ∇⋅∇ tt ε subject to: 0V=Φ , on the ith conductor’s surface 0=Φ , on all other conductors

[14]

Here, Φ is the electric potential, yy

xxt ˆˆ

∂∂+

∂∂=∇ is the gradient operator in the

transverse plane, 0εεε r= is the dielectric constant, and V0 is a reference voltage

10

(normally set to 1V). If multiple dielectrics are encountered, then the appropriate

interface conditions must be enforced on the boundaries between different dielectrics.

Solving [14], we can next calculate the ith row of the C matrix from the electric charge on

each conductor:

∫==jl s

jji dlq

VVQ

C00

,1 [15]

where lj represents the contour around the jth conductor. This follows from the fact that

nEDq nns ∂

Φ∂−=== εε is the surface charge density, and Dn and En are the normal

components of the electric flux density and the electric field, respectively. Although

many methods for the numerical evaluation of [14] are available (17), the Boundary

Element Method (BEM) (17, 18) appears to be the most appropriate if no restrictions on

the geometry are imposed. This method is efficient since only the boundaries and

dielectric interfaces have to be discretized. The result of the solution is an approximation

of the potential and its normal derivative on all boundaries and interfaces. The normal

derivative can directly be used in a numerical integration for the evaluation of [15].

The conductance matrix G describes the power losses in the dielectric media, i.e.

the losses associated with the electric field. It can be implicitly obtained together with

the capacitance matrix C if losses are incorporated in terms of a complex dielectric

constant

( )δεεε tan10 jr −= [16]where εr is the relative dielectric constant, and tanδ is the loss tangent of the material.

Unlike the lossless case, the resulting capacitance matrix C′ is now complex, leading to

an admittance matrix Y according to (13)

[ ] [ ]CGCCCY ′−=′=′= Im,Re, ωωj [17] The inductance matrix L contains the self-inductances of the conductors on the

diagonal, and the mutual inductances between conductors in the off-diagonal terms.

More generally, it defines the magnetic field energy storage. In the high-frequency limit,

i.e. the skin depth is sufficiently small such that current flow occurs only on the surface

of the conductors, the inductance matrix L can be obtained from a special capacitance

matrix C ′′ by using [14] and [15] with all dielectrics set to free space (ε0). The

inductance matrix in terms of C ′′ is (13)

11

100

−′′= CL εµ [18]where the magnetic permeability µ0 is constant throughout the system.

The resistance matrix R characterizes the resistive losses occurring in the

conductors, but more generally any losses associated with the magnetic field in the MTL.

Unfortunately, under the high-frequency limit, the resistive losses cannot be cast into the

simple framework of the MTL equations. Matrix R needs to satisfy

( ) ( ) ( )zzjzdzd RILIV −−= ω [19]

but also the power dissipation relation

( ) ( ) ( )zzzP RII H

21= [20]

where P(z) is the average per-unit-length power dissipated in the conductors at spatial

location z along the MTL and I(z)H is the Hermitian transpose of phasor I(z).

It is readily seen that under the high-frequency limit R cannot satisfy both [19]

and [20]. For example, if R satisfies [19], it can be written as (13)

≠=+

=jirjirr

R iji ,

,

0

0, [21]

where ri is the per-unit-length resistance of the ith conductor, where i = 0, 1,…N, and r0

being the resistance of the reference conductor. Unfortunately, R in [21] does not satisfy

the power loss relation [20] because of additional power losses due to eddy currents. To

demonstrate this, let current I1 flow in conductor 1 with the return current flowing in the

reference conductor and zero currents in all other conductors. Then the power loss per-

unit-length according to [20] and [21] should be ( )012

121 rrI + . However, the current

density on all the other conductors is not zero because of eddy currents. These eddy

currents do not contribute to any voltage change in z-direction as per [19]. Nonetheless,

they dissipate additional power that is not included in [20] and [21]. The eddy currents

become particularly important when conductors are located close to each other relative to

the dimensions of the conductors. Furthermore, any lossy dielectric (i.e. with non-zero

conductivity) supports eddy currents, which add to the total power dissipation.

When deciding between satisfying the MTL equation [19] or the power

dissipation relation [20], the latter is the preferred choice. At high frequencies, the

12

contribution of the R matrix to the MTL equation is nearly negligible, except for its

significance in resistive losses. If only the losses in conductors are considered and the

high-frequency assumption is in effect, the resistance matrix satisfying [20] can be

specified as (19)

[ ] [ ]∫=l jsissji dlJJR

IR ˆˆ1

20

, [22]

where l represents the collection of all the conductor outlines in the x-y plane cross-

section of the MTL configuration, [ ] isJ is the surface current density distribution

resulting from the current I0 flowing in the ith conductor, the return current flowing in the

reference conductor and zero current in all other conductors. In [22] Rs is the surface

resistance σπµδσ /)( 01 fRs == − , with σ being the conductivity, and )(1 0σµπδ f=

is the skin layer depth. In the special case where the eigenvectors of the R matrix are

known, the power dissipation relation [20] can be used to calculate the eigenvalues.

Under high-frequency assumption, the ith eigenvalue of R is given by the integral

∫=l ssi dlJR

IR 2

2ˆ1ˆ [23]

where ∑=

=N

jjII

1

2ˆ is the modal current, typically 1A if the eigenvector is normalized, Js

is the surface current density resulting from a current distribution dictated by the ith

eigenvector, l represents the collection of all conductor contour outlines in the x-y plane

cross-section.

The integrations in [22] or [23] require a mechanism for setting up an arbitrary

current distribution in the conductors. This procedure is similar to setting up a desired

charge distribution with the capacitance matrix in [14] and [15]. The magnetic and

electric fields in a TEM transmission line exhibit a high degree of duality, allowing the

use of the same two-dimensional equation solver for both problems. The static magnetic

field can be expressed in terms of the magnetic vector potential (19)

AH

×∇=0

1µ

[24]

In the TEM case, only the z-component of the magnetic potential is nonzero, satisfying

the following PDE in the x-y plane

13

02 =∇ zt A [25]The inductance matrix can thus be calculated by solving [25] with boundary conditions

similar to those in [14]

Az = ψ0, on the ith conductor surface Az = 0, on all other conductor surfaces [26]

Here ψ0 is a reference magnetic potential, normally set to unity. By iterating i, the current

distribution matrix I is collected row by row, giving

[ ]∫=jl isji dlJI , [27]

where nAHJ z

ts ∂∂−==

µ1 is the surface current density (z-component) on the conductors,

and lj again represents the contour around the jth conductor. The current distribution

matrix I is similar to the special capacitance matrix C ′′ in [18], the only difference is a

multiplying constant. For each i, the current density distribution [ ]isJ on all conductors

is stored for further processing. The current distribution I and the inductance matrix L are

linked through the following relationship (13)

LIΨ = [28]where ΨΨΨΨ the form of a magnetic potential at each conductor of the MTL, assuming that

the magnetic potential at the reference conductor is zero. During the calculation of I the

magnetic potential matrix ΨΨΨΨ was prescribed by [26] as E0ψ , where E is again the

identity matrix. Therefore, the expression for the inductance matrix is:

10

−= IL ψ [29]which is equivalent to [18].

With the availability of the inductance matrix, any desired current distribution can

be obtained by imposing the magnetic potential boundary conditions given by [28] on

[25]. A more efficient method involves the superposition of the surface current density

distributions obtained during acquisition of the I matrix:

[ ] [ ]∑=

=N

iisidesireds JJ

10

1 ψψ

[30]

where ψi is the magnetic potential at the ith conductor computed using a desired current

distribution in [28], and [ ]isJ is the current density on the conductors resulting from

imposing the boundary conditions [26] on [25].

14

Now that any arbitrary current distribution can be represented, the integration in

[22] or [23] can be carried out efficiently. If the eigenvalues of R are acquired, the

resistance matrix is then straightforwardly reconstructed using its eigenvectors.

Alternatively, the resistance matrix can be reconstructed directly from the surface current

density distributions [ ]isJ collected during acquisition of I in [25] – [27]. First, a per-

unit-length power loss matrix is computed

[ ] [ ]∫=l jsissji dlJJRP

21

, [31]

where [ ]isJ is the surface current density on the conductors resulting from substituting

[26] into [25]. The power loss matrix P still obeys the power loss relation [20], or

IRIP T

21= [32]

Equation [32] directly yields R in the form

( ) LPLIPIR TT20

11 22ψ

== −− [33]

Besides resistive losses, the magnetic field inside the conductors carries internal

reactance (inductance). The skin layer approximation is taken into account by the

impedance boundary condition (20)

HnZE st

×= ˆ [34]where ( ) ss RjZ += 1 is the surface impedance, n is the outward pointing surface normal,

and ( )EnnEt

××−= ˆˆ is the tangential component of the electric field. From this, the

complete impedance matrix Z of the MTL is:

( )RLZ jj ++= 1ω [35]where L contains external inductances only (i.e. L is the inductance matrix for the perfect

conductor case), as given by [18] or [29]. It is important to point out that the solution of

[25] for the calculation of R requires modifications if microstrip conductors of finite

thickness are considered. The edge shapes of the microstrip conductors significantly

affect the resulting R matrix (20). In contrast, the C, G and L matrices may be computed

with reasonable accuracy as if the microstrips are of zero-thickness.

Another successful method of computing the R matrix is called Wheeler’s

incremental inductance rule (21-23). It takes advantage of the fact that the magnetic field

15

in the skin layer produces the same amounts of reactance and resistance. The resistance

matrix is obtained from the estimated internal reactance (23)

n∂

∂= LR2δω [36]

This method suffers from the drawback that it requires very accurate calculation of the L

matrix. In our computations the 2D PDE solver was only powerful enough to compute

the diagonal elements of the L matrix to 5-digit precision. By using the incremental rule,

only the diagonal elements of R could be estimated with reasonable accuracy. For this

reason, the method based on the integration of the surface current density is preferred.

2.4. Drive Circuit

The multi-conductor transmission line model for the TEM resonator is shown in

Figure 2. The TEM resonator structure can be thought of as a system with a passive

network at the load side and a passive network containing sources on the source side with

the strips and the shield relying on a distributed model. The remaining circuit

components are assumed lumped because of their small size. With this assumption in

effect, the impedance matrix of the network on the load side is

[ ]

≠

==

ji

jiCjZ iLjiL

,0

,1

_,ω [37]

where CL_i is the capacitance terminating the ith line at the load side.

Given the load impedance matrix and the per-unit-length parameters of the MTL,

the input impedance of the transmission line can be calculated using [8]. After the input

impedance matrix Zin is obtained, the source side of the TEM resonator is simulated as a

lumped circuit. The following KVL equations need to be solved to complete the model

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

=

++−

+

+

−+

SS

N

SMS

S

NSNNinNinNin

NinS

inin

SNinin

Sin

Vii

ii

CjCjR

Cj

CjZZZ

ZCj

ZZ

CjZZ

CjZ

0

00

11001

01

01

11

2

1

1_1_

_,2,1,

,22_

2,21,2

1_,12,1

1_1,1

ωωω

ω

ω

ωω

[38]

16

where CS_i is the capacitance terminating the ith line at the source side, CM is the matching

capacitance, RS is the source impedance, i1…N are the input currents to the transmission

lines, iS is the source current, and VS is the source voltage.

Since losses are an important consideration in MRI coil designs, capacitor losses

must be included in the calculations. The capacitor’s quality factor Q is typically

specified by the manufacturer and can be modeled by the following complex capacitance

−=′

QjCC 1 [39]

Since Q varies with frequency and capacitance, datasheets usually include graphs of Q

against frequency and capacitance. Using either interpolation or curve fitting, these

relationships can be incorporated as part of the simulation strategy.

Once the system [38] is solved, the source current iS allows the calculation of the

input impedance and the factor S11 (reflection coefficient) as seen from the RF generator.

In practice, a network analyzer can directly provide this information. The input

impedance at the RF port is

SS

Sin R

iVZ −= [40]

The reflection coefficient is (24)

Sin

Sin

RZRZS

+−=11 [41]

where RS is the source and cable impedance, typically 50Ω.

2.5. Field Calculation

The transmission line input current distribution I(0) = (i1, i2, … iN)T obtained by

solving [38] can be used in [9] to compute the current distribution anywhere along the

line. The voltage distribution is also obtained straightforwardly, for example by

substituting [7] into [2]. From the current and voltage distributions, the TEM magnetic

and electric field pattern at any position z along the MTL can be obtained. An efficient

method for field calculations relies on the superposition of the field distributions that

occurred during the calculation of the C and L matrices.

17

In case of the electric field, the voltage distribution V(z) directly specifies the

electric potential boundary conditions for [14]. The electric potential anywhere in the x-y

plane can be obtained as:

( ) ( ) ( )∑=

Φ=ΦN

iii yxzV

Vyx

10

,1, [42]

where Φi(x,y) is the electric potential at the point (x,y) due to the boundary conditions in

[14], i.e. V0 on conductor i and zero on all other conductors, V0 being a reference voltage.

Thus, the electric field is

( ) ( ) ( ) ( ) yy

yxxx

yxyxyxE t ˆ,ˆ,,,∂

Φ∂−∂

Φ∂−=Φ−∇=

[43]

For the magnetic field, the magnetic potential boundary conditions for [25] can be

obtained from the current distribution I(z) through [28]. The magnetic potential

anywhere in the x-y plane is:

( ) ( ) ( )[ ]∑=

=N

iiziz yxAzyxA

10

,1, ψψ

[44]

where ( )[ ]iz yxA , is the z-component of the magnetic potential at the point (x,y) due to the

boundary conditions in [26], i.e. 0ψ at the ith conductor and zero on all other conductors,

0ψ being a reference magnetic potential, and ( )ziψ is calculated using I(z) in [28]. Then,

the B-field is

( ) ( ) ( ) ( ) yx

yxAxy

yxAyxAyxB zz ˆ,ˆ,,,∂

∂−∂

∂=×∇=

[45]

The derivatives in [43] and [45] can be evaluated numerically with the finite-

difference method if the potential data is organized over a regular grid. Alternatively, if

the electric and magnetic field data is available, the same superposition technique as in

[42] and [44] can be applied directly to the electric and magnetic field.

3. Results

3.1. General Resonator Structure

The TEM resonator is schematically shown in Figure 1a. The functional elements

of the TEM resonator are 12 inner microstrip conductors, distributed in a cylindrical

pattern and connected at the ends with variable capacitors to the cylindrical outer shield.

The physical layout is seen in Figure 1b and consists of an inner and an outer cylinder,

18

connected by two end-caps. The outer cylinder has an outside diameter of 10.5cm as

indicated in Figure 1a, while the inner cylinder has a diameter of 7.25cm. Both cylinders

have a wall thickness of approximately 2.8mm. A segmented copper shield (1.5mil thick

copper) is glued to the surface of the outer cylinder. The 12 segments of the shield are

connected with capacitors as depicted in Figure 1b to assure the outer conductors behave

as a solid cylindrical shield at high frequency. At lower frequencies, the segmented

shield impedes the propagation of eddy currents induced by the switching gradients.

Twelve microstrip conductors 0.25” (6.4 mm) wide and 1.5mil (38µm) thick are glued to

the inner surface of the inner polycarbonate cylinder. Two printed circuit boards (PCBs)

are attached at the front and rear endplates of the former as shown in Figure 1b. The

PCBs hold the microstrip terminating capacitors (both tunable and fixed), the detuning

circuits, and the connectors.

3.2. Determination of transmission line parameters

The per-unit-length parameters for the MTL approximation of the TEM resonator

are obtained by solving [14] and [25]. The basic geometric arrangement, including

boundary conditions for both electric and magnetic potentials, is shown in Figure 3.

Because of symmetry, only half of the TEM resonator cross-section is needed. The cut

straight through the middle of the coil, including the excited conductor, assures field

symmetry using a zero-flux boundary condition.

The Boundary Element Method (17, 18) is employed to solve both the electric and

magnetic problems as seen in Figure 4. The dots in Figure 4 indicate the BEM nodes,

where the potential and the associated normal flux are estimated. The line segments are

boundary elements, lining various boundaries and interfaces in the solution domain. The

thick lines correspond to elements with prescribed potential boundary conditions. The

thin lines represent dielectric interfaces in the mesh for the electric field problem. Both

BEM meshes include between 1300 and 1400 nodes and are generated adaptively. Once

a solution is obtained, the potential at any point can be calculated from the boundary

potential and normal flux. Figure 5 shows the resulting electric and magnetic potential

distributions. As discussed in the previous sections, the integration of the normal flux

over the conductor contours determines the per-unit-length parameter matrices. Table 2

19

lists the first column of the C, G, L and R matrices. This information is sufficient to

reconstruct the complete matrices because they are circulant. The accuracy of the

calculated parameters can be estimated from their convergence as the BEM mesh is

refined. The diagonal elements of the C, G and L matrices have an estimated error of

±0.001%, with lower accuracy for the off-diagonal entries. The diagonal elements of the

R matrix have an estimated error of ±1%.

3.3. The MTL model predictions

An MTL model based on the previously discussed per-unit-length parameters and

lumped terminating circuits was constructed for the purpose of simulating the TEM

resonator. The simulated coil is driven by a voltage source simulating a 50Ω RF

generator. Since it is a frequency domain model, only the steady-state behavior of the

TEM coil applies.

The TEM model is tuned such that mode 1 (which produces a uniform magnetic

field) occurs at 200MHz. This is accomplished by setting all terminating capacitors to

10.455pF with the exception of the driven capacitor. For ATC (Advanced Technical

Ceramics) Type-C capacitors the quality factor is approximated to be 500. Matching to a

50Ω line is accomplished using a 2.415pF matching capacitor connected to an 8.04pF

terminating capacitor. For relatively high-Q coils (Q>100), the field uniformity is

preserved when

01_ CCC SM ≈+ [46]where CM is the matching capacitor, CS_1 is the driven terminating capacitor (see Figure

2) and C0 is the common value of all terminating capacitors.

Figure 6a shows a simulated frequency sweep of S11 at the RF port of the coil.

For comparison, Figure 6b displays an equivalent sweep obtained from a real TEM coil

using a network analyzer (HP 8714ES). The predicted resonance frequencies are in good

agreement with real coil measurements. Although the terminating capacitors on the real

coil cannot be measured accurately once the coil is assembled, the predicted terminating

capacitance of 10.455pF is close to the actual capacitances. Table 2 lists the simulated

and measured resonance frequencies for all 7 resonant modes. The discrepancies in the

20

frequencies are small, and can be attributed to small inaccuracies of the MTL model’s

assumptions as well as a slight tuning dissymmetry of the real coil.

3.4 Quality factor

The quality factor of a matched TEM coil at mode 1 can be estimated from the S11

sweep

lu

rcoil ff

fQ−

≈ [47]

where fr is the mode 1 resonance frequency of the coil, fu and fl are the 3dB frequencies

above and below the resonance frequency, where S11 3dB below the base return loss (due

to the cable) is observed. Using this method, the Q factor of the MTL simulated coil is

183, while the Q factor of the real TEM coil is approximately 160. Although the

agreement is quite good, the estimated terminating capacitor Q of 500 is probably overly

pessimistic, and thus the modeled Q is likely to be higher. Additionally, the MTL model

does not incorporate radiation losses. Thus, the coil quality factor will always be

overestimated.

3.5. Field Distributions

As a post-processing step, the MTL model allows calculating the voltage and

current on the conductors at any point z along the transmission line, as already discussed

in the previous sections. Figure 7 (a and b) shows the z dependence of the current and

voltage on the conductors. The current distribution is already non-uniform along the

length of the conductors at 200MHz, following a sinusoid with a peak at the center of the

coil. The voltage distributions also follow approximately a sinusoid, crossing zero at the

center of the coil. The voltage and current in the MTL correspond to electric and

magnetic field intensity. Thus, the magnetic field is highest in the center of the coil,

while the electric field is lowest. Figure 7c plots the per-unit-length energy stored in the

magnetic and electric field as a function of z. Most of the energy is stored in the

magnetic field, with a maximum magnetic and minimum electric energy content

occurring in the center of the coil. This is a favorable energy distribution that avoids

excessive losses due to the electric field in the tissues of the imaged animal.

21

As the final step, the MTL model allows plotting the magnetic and electric field

patterns at any cross-section of the coil. The total field distribution is obtained by

rotation and superposition of the individual fields due to single-element excitation as in

Figure 5. Figure 8 depicts the magnetic field in the coil midpoint (z=3in) cross-section at

the 200MHz resonance. Specifically, Figure 8a contains magnetic potential contours that

translate into magnetic field lines. Figure 8b shows the magnetic field magnitude

distribution in normalized units. As expected, the predicted magnetic field is uniform in

the imaging region. The magnitude of the magnetic field in the imaging region can be

linked to the coil’s filling factor (the ratio of the magnetic field energy in the imaged

sample to the total magnetic energy in the coil) for comparing the SNR performance of

different coil designs. Also, knowing the coil’s loaded Q and the magnetic field intensity

allows estimating the power requirements for a typical 180° RF pulse used in MRI

imaging sequences.

Figure 9 depicts the electric field near the end of the coil (z=0) in the form of the

electric potential. The electric field behaves similar to the magnetic field, although

differing in direction. Electric field distributions can be used to estimate RF tissue

heating in MRI experiments, although this is normally not a factor in animal studies.

The magnetic field distribution in Figure 8 corresponds to a coil with linear

(single-element) drive. Figure 10 shows the magnitude of the magnetic field when the

same coil is driven in quadrature, in this case at two elements separated geometrically by

a 90° angle with sources differing by 90° in phase. The resulting circularly polarized

magnetic field is even more uniform than that in the liner-driven case. Thus, the MTL

model demonstrates the ability of this TEM coil to operate with quadrature drive, as well

as point out its advantages. Quadrature drive is an option for TEM resonators, but it

presents a difficult tuning and matching challenge.

3.6. Comparison with the lumped circuit model

In order to confirm the need for a distributed transmission line model, a simple

lumped circuit model was developed for comparison. Such equivalent lumped circuits

are common in modeling birdcage coils (8, 25, 26) operating at frequencies as high as

140 MHz in human systems and even higher in animal systems. In these models the

22

rungs and rings of the birdcage are treated as inductors with mutual coupling among

them. These inductors are interconnected by lumped capacitors, forming a resonance

circuit model. In general, lumped circuit methods can be very accurate as long as the self

and mutual inductances are calculated correctly and the frequency of operation is low

such that the coil’s electrical length is small. At higher frequencies, circuit models start

to deviate from reality because of the aforementioned transmission line effects.

The equivalent lumped circuit model for the TEM resonator is shown in Figure

11. In order to keep the lumped model simple and compatible with the MTL model, the

same TEM field assumptions are made. This means the self and mutual inductances in

Figure 11 are obtained by multiplying the per-unit-length inductance matrix L (used by

the MTL model) by the length of the coil. The remaining components of the circuit

model are the same as in the MTL model. The KVL equations for the circuit model

assume the following form

=

++−

++

++

−++

SS

N

SMS

S

NLNN

NSNN

NLS

SN

LS

Vii

ii

CjCjR

Cj

CjZ

CjZZ

ZCj

ZCj

Z

CjZZ

CjZ

Cj

0

00

11001

011

011

111

2

1

1_1_

_,

_2,1,

,22_

2,22_

1,2

1_,12,1

1_1,1

1_

ωωω

ωω

ωω

ωωω

[48]

where ( )[ ]ljj RLZ ++= 1ω accounts for inductance and resistance in the conductors, L

and R being the same per-unit-length matrices as in the MTL model, and l being the

length of the coil. Thus, the circuit model can be solved similarly to the MTL model.

Figure 12a compares the S11 frequency responses of the lumped circuit with the

MTL model at 200MHz. The discrepancy in resonance frequency between the two

models is not large, but noticeable, and increasing as the operational frequency increases.

The predicted mode 1 frequencies differ by about 5MHz, which may be marginally

acceptable. Figure 12b shows the z-dependence of the currents in the conductors

according to the MTL model, indicating a considerable deviation from the uniform

current of the circuit model. Figure 12c compares the S11 frequency responses of the two

models when the MTL model is tuned to 400MHz. At this higher frequency, the two

models differ substantially. The predicted mode 1 frequencies differ by 45MHz, and the

lumped circuit model is no longer well matched, pointing towards significant differences

23

in quality factor. The z-dependent MTL current distributions in Figure 12d are very non-

uniform, showing the standing wave nature of the resonance. Consequently, the lumped

circuit model is no longer valid at 400MHz. The MTL model is desirable even at

200MHz because of it higher resonance frequency accuracy and the ability to predict

lossy capacitive effects, allow approximating biological loads, and provide new

possibilities for tuning mechanisms, a topic that will be discussed in a future paper. The

MTL model even becomes desirable at lower frequencies for larger (longer) TEM coils in

human MRI systems. However, it is not as accurate as a good circuit model at low

frequencies because the normal circuit models are free from the quasi-TEM assumption.

3.7. Coil Scaling

It is interesting to investigate how the unloaded TEM coil scales with respect to

frequency and geometric size. As a basis of this comparison we can identify the coil

length divided by the free space wavelength (l/λ) versus the various quality factors.

Specifically, the factors involving the inductive component of the strips, QL, the

terminating ATC 100 C series capacitors, QC, the former dielectric (polycarbonate,

εr=2.9, tanδ=0.012), Qε, and the total quality factor, Qtotal, are computed and compared.

The frequency scaling is shown in Table 3 for the coil dimensions reported in Section

3.1. We notice that Qtotal=1/(QC-1+Qε

-1+QL-1) remains remarkably constant at

approximately 180.

Table 4 depicts the geometric scaling for a fixed frequency of 200 MHz. All

dimensions are scaled by the same amount, i.e., coil diameter, length, strip width, and

former wall thickness. For the scaled coil sizes, the boundary element code had to re-

compute the MTL matrix coefficients to account for the scaled size in the orthogonal

plane, resulting in different quality factor predictions, even for the same l/λ ratios. We

observe that an increase in size by a factor of two (2x) improves Qtotal significantly,

before the value begins to drop for the larger size 3x. The reason for this behavior is

attributed to the more prominent influence of the former material. It should also be noted

that for larger l/λ ratios, radiation losses play a more dominant role, a fact that cannot be

modeled with this MTL formulation.

24

4. Conclusions

In this paper an MTL formulation is introduced capable of modeling an unloaded

RF resonator coil at high frequency. Detailed comparisons with a novel linearly driven

12-element microstrip coil underscore the validity of this model and point out the

limitations of the lumped circuit model. Moreover, it is demonstrated that at a frequency

of 400 MHz (i.e. 9.4 T fields for proton imaging) the transmission line model shows

major differences with the equivalent circuit model, underscoring the advantages of

treating voltages and currents as propagating TEM electromagnetic waves. Although

applied to small animal imaging, the MTL approach can easily be scaled to model human

coils as well.

However, even the TEM MTL model has its limitations: only an unloaded coil or

a simple biological load can be modeled, and losses in the load can be incorporated as

part of the electric field only. Losses due to eddy currents in the load, caused by the

magnetic field, are not modeled. Furthermore, one cannot simulate effects such as

standing wave patterns in the load (27) and radiation losses. Despite these shortcomings,

the MTL approach becomes an indispensable tool when dealing with high-field resonator

coil designs due to its accuracy and light computing requirements.

25

References

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coils for clinical NMR imaging and spectroscopy. Magn Reson Med 1994;32:206-218.

2. Adriany G, Vaughan JT, Andersen P, Merkle H, Garwood M, Ugurbil K. Comparison

between head volume coils at high field. In: Proceedings of 3rd SMR and 12th ESMRMB

annual meeting, Nice, France, 1995. p 971.

3. Pan JW, Vaughan JT, Kuzniecky RI, Pohost GM, Hetherington HP. High resolution

neuroimaging at 4.1 T. Magn Reson Imaging 1995;13:915-921.

4. Hayes CE, Edelstein WA, Schenck JF, Mueller OM, Eash M. An efficient highly

homogeneous radio-frequency coil for whole-body NMR imaging at 1.5 T. J Magn Reson

1985;63:622-628.

5. Röschmann P. High-frequency coil system for a magnetic resonance imaging

apparatus. US Patent 4746866, 1988.

6. Bridges JF. Cavity resonator with improved magnetic field uniformity for high

frequency operation and reduced dielectric heating in NMR imaging devices. US Patent

4751464, 1988.

7. Vaughan JT. Radio frequency volume coil for imaging and spectroscopy. US Patent

5557247, 1996.

8. Tropp J. Mutual inductance in the birdcage resonator. J Magn Reson 1997;126:9-17.

9. Röschmann P. Analysis of mode spectra in cylindrical N-conductor transmission line

resonators with expansion to low-, high- and band-pass birdcage structures. In:

Proceedings of the 3rd Annual Meeting of the International Society of Magnetic

Resonance in Medicine, Nice, France, 1995. p 1000.

10. Chingas GC, Zhang N. Design strategy for TEM high field resonators. In:

Proceedings of the 4th Annual Meeting of the International Society of Magnetic

Resonance in Medicine, New York, 1996. p 1426.

11. Baertlein BA, Ozbay O, Ibrahim T, Lee R, Yu Y, Kangarlu A, Robitaille PML.

Theoretical Model for an MRI Radio Frequency Resonator. IEEE Trans Biomed Eng

2000;47:535-545.

26

12. Ibrahim TS, Lee R, Baertlein BA, Kangarlu A, Robitaille PML. Modeling the TEM

resonator in the presence of a human head for high field MRI. In: Proceedings of the 8th

Annual Meeting of the International Society of Magnetic Resonance in Medicine,

Denver, CO, 2000.

13. Paul CR. Analysis of Multiconductor Transmission Lines. New York: Wiley-

Interscience; 1994.

14. Fache N, Olyslager F, Zutter DD. Electromagnetic and Circuit Modeling of

Multiconductor Transmission Lines. Oxford, U.K.: Clarendon; 1993.

15. Nobakht RA, Ardalan SH, Shuey K. An algorithm for computer modeling of coupled

multiconductor transmission line networks. In: IEEE International Conference on

Communications, Conference record, vol. 3, 1989. p 1462-1467.

16. Bracewell RN. Two-Dimensional Imaging. Englewood Cliffs, NJ: Prentice-Hall;

1995.

17. Lapidus L. Numerical solution of partial differential equations in science and

engineering. New York: Wiley-Interscience; 1999.

18. Brebbia CA. The Boundary element method for engineers. London: Pentech; 1984.

19. Tsuk ML, Kong JA. A hybrid method for the calculation of the resistance and

inductance of transmission lines with arbitrary cross sections. IEEE T Microw Theory

1991;39:1338-1347.

20. Barsotti EL, Kuester EF, Dunn JM. A simple method to account for edge shape in the

conductor loss in microstrip. IEEE T Microw Theory 1991;39:98-106.

21. Wheeler HA. Formulas for the skin effect. P IRE 1942;30:412-424.

22. Gentili GG, Melloni A. The incremental inductance rule in quasi-TEM coupled

transmission lines. IEEE T Microw Theory 1995;43:1276-1280.

23. Plaza G, Mesa F, Horno M. Spectral domain analysis of conductor losses in a

multiconductor system via the incremental inductance rule. Electron Lett 1994;30:1425-

1427.

24. Pozar D. Microwave Engineering. New York: John Wiley; 1998.

25. Harpen MD. Equivalent circuit for birdcage resonators. Magn Reson Med

1993;29:263-268.

27

26. Pascone R, Vullo T, Farrelly J, Cahill PT. Explicit treatment of mutual inductance in

eight-column birdcage resonators. Magn Reson Imaging 1992;10:401-410.

27. Carlson JW. Radiofrequency field propagation in conductive NMR samples. J Magn

Reson 1988;78:563-573.

28. Chen J, Feng Z, Jin JM. Numerical Simulation of SAR and B1-Field Inhomogeneity

of Shielded RF Coils Loaded with the Human Head. IEEE Trans Biomed Eng

1998;45:650-659.

29. Collins CM, Smith MB. Signal-to-Noise Ratio and Absorbed Power as Functions of

Main Magnetic Field Strength, and Definition of “90°” RF Pulse for the Head in the

Birdcage Coil. Magn Reson Med 2001;45:684-691.

28

Figure Captions

Figure 1. The microstrip TEM resonator: (a) schematic and (b) physical layout.

Figure 2. Schematic of the MTL model for the TEM resonator.

Figure 3. Geometrical setup and boundary conditions for the PDEs used to solve for the

per-unit-length parameters of the MTL.

Figure 4. Boundary element meshes used to solve for (a) C and G matrixes with former

dielectric present and (b) L and R matrices. Adjacent nodes indicate size of linear

elements.

Figure 5. Field solutions: (a) electric potential for C and G matrices with dielectric of εr =

2.9 in the former, (b) magnetic potential for L and R matrices.

Figure 6. S11 frequency sweeps: (a) MTL model with linear drive, (b) network analyzer

measurement on a TEM coil with linear drive.

Figure 7. Mode 1 current (a), voltage (b) and average per-unit length energy storage (c)

distributions along the length of the coil (z-coordinate) according to the MTL model with

linear drive. Traces are labeled with corresponding conductor numbers.

Figure 8. Magnetic field in the z=3in (center) cross-section of the coil with linear drive

according to the MTL model: (a) magnetic potential contours corresponding to magnetic

field lines, (b) magnetic field magintude normalized to the field intensity in the center.

Figure 9. Electric field represented by the electric potential distribution in the cross-

section of the coil near the end (z=0) according to the MTL model.

Figure 10. Magnetic field magnitude in the center (z=3) cross-section of the coil

according to the MTL model with quadrature drive. The field magnitude is normalized to

its value in the center.

Figure 11. Schematic of the equivalent circuit model for the TEM resonator.

Figure 12. Comparison between the MTL and the equivalent circuit models: (a) S11

spectra and (b) MTL current variation in z-direction when tuned to 200MHz (4.7T), (c)

29

S11 spectra and (d) MTL conductor current against z when tuned to 400MHz (9.4T).

The MTL model is properly tuned and matched. The circuit model uses of the same

capacitors as the MTL model.

30

Table Captions

Table 1. First columns of the C, G, L and R matrices.

Table 2. Measured and predicted TEM coil resonance frequencies.

Table 3. Frequency scaling of modeled loss mechanisms.

Table 4. Geometric size scaling of modeled loss mechanisms at 200 MHz.

31

(a)

Front PCB

Shield

Shield Connecting Capacitors Rear PCB

Solder

Microstrip elements

RF ConnectorFormer

(b)

Figure 1

32

CS_2

CS_1

CS_N

CL_2

CL_1

CL_N

i0

i1

iN

Reference conductor

z = 0 z = L

zCM

RSiS

VS

Line 1

Line 2

Line N

Coupled

Figure 2

33

Figure 3

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

Φ=1

Φ=0

0=∂Φ∂n

34

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(a)

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.050

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(b)

Figure 4

35

0

0.2

0.4

0.6

0.8

-0.05 0 0.050

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(a)

0.2

0.4

0.6

0.8

-0.05 0 0.050

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(b)

Figure 5

36

160 180 200 220 240 260 280-40

-30

-20

-10

0

f [MHz]

S11

[dB

]

(a)

(b)

Figure 6

37

0 1 2 3 4 5 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

z [in]

Cur

rent

am

plitu

de [A

]

1, 7

2, 6, 8, 12

3, 5, 9, 11

4, 10

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

Vol

tage

am

plitu

de [V

]

z [in]

1, 7

2, 6, 8, 12

2, 5, 9, 11

4, 10

(a) (b)

0 1 2 3 4 5 60

1

2

3

4

5

6x 10-9

z [in]

Per-u

nit-l

engt

h en

ergy

sto

rage

[J/m

]

Magnetic energyElectric energy

(c)

Figure 7

38

-0.05 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(a)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.05 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

(b)

Figure 8

39

-3

-2

-1

0

1

2

3

-0.05 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

Figure 9

40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.05 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x [m]

y [m

]

Figure 10

41

CS_2

CS_1

CS_N

CL_2

CL_1

CL_N

i0

i1

iN

CMRS

iS

VS M12

L1

L2

LN

R1

R2

RN

M1N

Figure 11

42

160 180 200 220 240 260 280 300-40

-30

-20

-10

0

f [MHz]

S11

[dB

]

MTL circuit

0 2 4 60

0.01

0.02

0.03

0.04

z [in]

Cur

rent

am

plitu

de [A

]

(a) (b)

350 400 450 500 550 600 650-40

-30

-20

-10

0

f [MHz]

S11

[dB

]

MTL circuit

0 2 4 60

0.01

0.02

0.03

0.04

z [in]

Cur

rent

am

plitu

de [A

]

(c) (d)

Figure 12

43

Table 1

Row C [pF/m] G [1/(Ωm)] @ 200MHz L [µH/m] R [Ω/m] @ 200MHz

1 30.7719 0.00018095 0.56560 0.6348

2 -6.5518 -0.00006483 0.11697 0.0179

3 -0.7571 -0.00000178 0.04701 0.0077

4 -0.3307 -0.00000056 0.02596 0.0064

5 -0.2137 -0.00000034 0.01786 0.0056

6 -0.1698 -0.00000026 0.01451 0.0051

7 -0.1579 -0.00000024 0.01358 0.0050

8 -0.1698 -0.00000026 0.01451 0.0051

9 -0.2137 -0.00000034 0.01786 0.0056

10 -0.3307 -0.00000056 0.02596 0.0064

11 -0.7571 -0.00000178 0.04701 0.0077

12 -6.5518 -0.00006483 0.11697 0.0179

44

Table 2

Mode MTL simulated f [MHz] Measured f [MHz]

0 173.15 180.4

1 200.00 200.0

2 224.40 220.0

3 244.25 238.6

4 258.60 253.6

5 267.20 262.0

6 269.95 268.6

45

Table 3

F, MHz 64 200 400 600

l/λ 0.033 0.102 0.203 0.305

QL 413 730 1031 1263

QC 328 261 285 486

Qε 38000 3915 931 386

Qtotal 182 183 180 184

46

Table 4

scale 0.5x 1x 2x 3x

l/λ 0.051 0.102 0.203 0.305

QL 389 730 1366 1970

QC 160 261 525 1285

Qε 15860 3915 929 385

Qtotal 112 183 269 257