Design and analysis of an isotropic two-dimensional planar Composite Right/Left-Handed waveguide structure

(1)

www.adv-radio-sci.net/9/73/2011/ doi:10.5194/ars-9-73-2011

Advances in

Radio Science

Design and analysis of an isotropic two-dimensional planar

Composite Right/Left-Handed waveguide structure

M. A. Ebersp¨acher, M. Bauer, and T. F. Eibert

Lehrstuhl für Hochfrequenztechnik, Technische Universität München, Germany

Abstract. A two-dimensional isotropic Composite Right/Left-Handed (CRLH) waveguide structure is proposed which is designed for operation inX-band. The balanced structure possesses left-handed behaviour over a large band-width from 7.5 GHz up to its transition frequency at 10 GHz. Above this region, the unit cell behaves in a right-handed manner up to 13.5 GHz. Operating the structure within these bands yields a frequency dependent index of refraction rang-ing from−2.5≤n≤0.8. Isotropic characteristics are ob-tained between 8.5GHz≤f ≤12GHz resulting in −1.5≤ n≤0.8. The planar CRLH structure is designed based on transmission line theory, implemented in microstrip technol-ogy and optimized using full-wave simulation software. An equivalent circuit model is determined describing the elec-tromagnetic behaviour of the structure whose element values are obtained by even and odd mode analysis. The design of the unit cell requires an appropriate de-embedding process in order to enable an analysis in terms of dispersion character-istics and Bloch impedance, which are performed both.

1 Introduction

A structure possessing at least one frequency band in which the vectorsE, H andkg form a left-handed triad while a right-handed one is found in the remaining frequency range is known as a Composite Right/Left-Handed structure. Op-eration of the guiding structure in the left-handed region re-sults in wave propagation with negative phase velocity. In contrast, the group velocity associated with the flow of en-ergy is positive in all regions. Since materials possessing such a behaviour have not been found in nature yet, they have to be produced artificially. One of the first structures

Correspondence to: M. Ebersp¨acher

([email protected])

exhibiting CRLH behaviour was realised by split-ring res-onators located in the proximity of a resonant wire arrange-ment (Shelby et al., 2001). Due to the loose coupling of the resonators, this configuration suffers from high losses and only a small operative bandwidth of approximately 10% is achieved. It was shown that these problems can be overcome by synthesizing CRLH structures utilizing the transmission line approach (Lay et al., 2004) which is based on periodi-cally arranged unit cells emulating a homogeneous structure. If mutual coupling between the unit cells is not considered, the waveguiding characteristics of the entire cascade is solely governed by the characteristics of the constitutive unit cells. As a consequence, the most challenging part in the artifi-cial material synthesis is the design of appropriate unit cells which fulfil predefined requirements. Besides meeting the electrical characteristics, the structures must be feasible with an acceptable complexity. It has been shown that cascading planar CRLH unit cells in a specific manner enables alter-native concepts for a multitude of applications like power dividers, beam forming concepts and antennas (Eccleston, 2010), (Caloz and Itoh, 2006).

2 Theory 2.1 Synthesis

In order to realise a two-dimensional waveguiding structure which behaves within a certain bandwidth isotropically, the unit cell is supposed to be a fully symmetric and reciprocal four-port with a pair of ports for each direction of space. Us-ing scatterUs-ing matrix representation, this can be written as

S=1

2e −jφ



  

−1 1 1 1 1−1 1 1 1 1−1 1 1 1 1−1



  

(1)

(2)

74 M. Ebersp¨acher et al.: Design and analysis of an isotropic 2-D planar CRLH waveguide structure

M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

3 an interdigital capacitor whose arms supporting the

individ-ual fingers are extended to open-ended stubs in order to

be-have inductively. Thus, this configuration realises a shunt

in-ductor and a series capacitor simultaneously. Both elements

are completed by the occurring parasitics to a series resonant

circuit and a parallel resonant circuit, respectively.

Adjust-ing the resonant frequencies as well as the Bloch impedance

can be achieved easily by modifying either the stublength or

the fingers of the arrangement allowing to meet given

spec-ifications and balancing the structure. Ensuring the artificial

structure to behave homogeneously, the length of the unit cell

must not exceed the quarter wave length limit (Lay et al.,

2004). Dealing with 2d structures, this requirement must be

fulfilled in both directions. Along with the desired isotropic

characteristic this results in a quadratic footprint of the unit

cell possessing a sidelength which is smaller or equal to a

quarter of the wavelength. Although the absence of interlayer

connections makes the initial design easy to fabricate and

hence preferable, it cannot be extended to a two-dimensional

unit cell in a straightforward manner, since the open-ended

stubs cannot be reduced to fit into the footprint without

los-ing its inductive behaviour. Therefore, the open-ended stubs

are substituted by short-circuited ones which require to be

reduced in length to maintain their electromagnetic

proper-ties. This modification influences strongly the field

distribu-tion within the interdigital capacitor. Whereas in the original

case the maximum of capacitance was realised close to the

open ends, in the modified version this appears close to the

hostline. In order to reduce the amount of interlayer

connec-tions, these are placed at the corners of the unit cell so that

they may be used by adjacent cells simultaneously, as seen

in figure 1.

3.1 Analysis

As seen in figure 1 and 2, the boundary of the unit cell is

located right between the interdigital capacitor.

In terms

of an equivalent circuit model this can be considered as two

capacitors connected in series representing together the

ca-pacitance of the interdigital capacitor separated by the cell

boundary. However, excitation of the structure may only be

performed with a quasi TEM mode bound to the microstrip

lines. Consequently, the reference plane for determining the

scattering parameters cannot be set correctly. As a result of

this, the determinable scattering parameters always contain

contributions of adjacent cells, in case of a periodically

ar-ranged cascade, or contributions of the feeding network, in

case of analysing a single section. Thus, to obtain the

prop-erties of the unit cell itself the reference plane for the

scatter-ing matrix analysis must be placed between two inseparable

capacitors which is not possible in practice. This difficulty

can be overcome by modelling the unwanted contributions

with an equivalent circuit model which is used to de-embed

the unit cell parameters. For this, the values of the elements

L

R

/

2 ,

2 L

L

,

C

R

/

2 ,

2 R

P

,

R

S

/

2 and

2 C

L

must be determined

Port 1 Port 2

Port 3

Port 4

Fig. 1. Picture of the fabricated structure which is produced on

Rogers RO4350B substrate (thicknessh = 0.762 mmandr =

3.66). The 2d CRLH unit cell is within the square marked by the dashed line (side lengthl = 4.54 mm ≈λ/4). Parts located out-side the square belong to either adjacent cells or the feeding net-work.

Fig. 2.Equivalent circuit model of the structure seen in figure 1 with

the series elements CL, LR/2 andRS/2 and the shunt elements

CR/2,2LLand2RP. The equivalent circuit model of the 2d CRLH

unit cell is within the region marked by the dashed line. Elements located outside the marked region belong to either adjacent cells or the feeding network.

based on the simulated or measured network response which

can be done by performing even and odd analysis.

Starting with the impedance matrix representation of the

four-port network referred to as extended unit cell

Z

¯

euc

,

which comprises the unit cell and the feeding network,

al-Fig. 1. Picture of the fabricated structure which is produced on Rogers RO4350B substrate (thicknessh=0.762mm andr=3.66).

The 2d CRLH unit cell is within the square marked by the dashed line (side length l=4.54mm≈λ/4). Parts located outside the square belong to either adjacent cells or the feeding network.

M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure 3

an interdigital capacitor whose arms supporting the individ-ual fingers are extended to open-ended stubs in order to be-have inductively. Thus, this configuration realises a shunt in-ductor and a series capacitor simultaneously. Both elements are completed by the occurring parasitics to a series resonant circuit and a parallel resonant circuit, respectively. Adjust-ing the resonant frequencies as well as the Bloch impedance can be achieved easily by modifying either the stublength or the fingers of the arrangement allowing to meet given spec-ifications and balancing the structure. Ensuring the artificial structure to behave homogeneously, the length of the unit cell must not exceed the quarter wave length limit (Lay et al., 2004). Dealing with 2d structures, this requirement must be fulfilled in both directions. Along with the desired isotropic characteristic this results in a quadratic footprint of the unit cell possessing a sidelength which is smaller or equal to a quarter of the wavelength. Although the absence of interlayer connections makes the initial design easy to fabricate and hence preferable, it cannot be extended to a two-dimensional unit cell in a straightforward manner, since the open-ended stubs cannot be reduced to fit into the footprint without los-ing its inductive behaviour. Therefore, the open-ended stubs are substituted by short-circuited ones which require to be reduced in length to maintain their electromagnetic proper-ties. This modification influences strongly the field distribu-tion within the interdigital capacitor. Whereas in the original case the maximum of capacitance was realised close to the open ends, in the modified version this appears close to the hostline. In order to reduce the amount of interlayer connec-tions, these are placed at the corners of the unit cell so that they may be used by adjacent cells simultaneously, as seen in figure 1.

3.1 Analysis

As seen in figure 1 and 2, the boundary of the unit cell is located right between the interdigital capacitor. In terms of an equivalent circuit model this can be considered as two capacitors connected in series representing together the ca-pacitance of the interdigital capacitor separated by the cell boundary. However, excitation of the structure may only be performed with a quasi TEM mode bound to the microstrip lines. Consequently, the reference plane for determining the scattering parameters cannot be set correctly. As a result of this, the determinable scattering parameters always contain contributions of adjacent cells, in case of a periodically ar-ranged cascade, or contributions of the feeding network, in case of analysing a single section. Thus, to obtain the prop-erties of the unit cell itself the reference plane for the scatter-ing matrix analysis must be placed between two inseparable capacitors which is not possible in practice. This difficulty can be overcome by modelling the unwanted contributions with an equivalent circuit model which is used to de-embed the unit cell parameters. For this, the values of the elements

LR/2,2LL,CR/2,2RP,RS/2and2CLmust be determined

Port 1 Port 2

Port 3

Port 4

Fig. 1. Picture of the fabricated structure which is produced on Rogers RO4350B substrate (thicknessh = 0.762 mmandr = 3.66). The 2d CRLH unit cell is within the square marked by the dashed line (side lengthl= 4.54 mm≈λ/4). Parts located out-side the square belong to either adjacent cells or the feeding net-work.

Fig. 2.Equivalent circuit model of the structure seen in figure 1 with the series elementsCL,LR/2andRS/2and the shunt elements CR/2,2LLand2RP. The equivalent circuit model of the 2d CRLH

based on the simulated or measured network response which can be done by performing even and odd analysis.

Starting with the impedance matrix representation of the four-port network referred to as extended unit cell Z¯euc, which comprises the unit cell and the feeding network,

al-Fig. 2. Equivalent circuit model of the structure seen in figure 1 with the series elementsCL,LR/2 andRS/2 and the shunt elements CR/2, 2LLand 2RP. The equivalent circuit model of the 2d CRLH

whereas the phase variation experienced by a wave travel-ling through the cell is described by the exponential function e−jφ. For the synthesis process the impedance matrix repre-sentation of S is of further interest, which can be determined by

ZS=Z0(I+S)(I−S)−1 (2)

4 M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

lows to calculate

Z

even

= ¯

Z

euc

(1

,

1) + ¯

Z

euc

(1

,

2) + 2 ¯

Z

euc

(1

,

3)

(13)

and

Y

odd

= 1

/

_¯

Z

euc

(1

,

1)

−

Z

¯

euc

(1

,

2)

.

(14)

It is assumed here that

Z

¯

euc

can be described by a matrix

possessing the same characteristics as the matrix shown in

equation (9).

Z

even

is the impedance which appears at the

terminals if the network is excited simultaneously at all ports

with an identical signal. In this case no current is flowing

across the nodal point and the equivalent circuit model

sim-plifies to the one seen in figure 3. Likewise, exciting the

structure at each two ports with signals possessing a phase

shift of

180

◦

leads to the odd admittance whose equivalent

circuit model is seen in figure 4. In order to determine the

values of the containing elements both equivalent networks

are simulated with the circuit simulation software AWR

Mi-crowave Office (AWR). The results obtained from the EM

structure are then compared with the results obtained by

cir-cuit analyses and in an iterative process the elements are

modified until both responses match. This fitting sequence

can either be performed with an automatic optimizer or by

tuning the element values manually by performing the steps

as follows: Starting with the observation of

Z

even

shows

that at the resonant frequency of the shunt resonator the

impedance of

Z

even

is expected to peak since the

suscep-tances of the resonator cancel each other out. Consequently,

Zeven

LR/2 RS/2

2_RP

2_LL

CL

CR/2

Fig. 3.

Equivalent circuit model of the impedance

Z

even

obtained

by even mode excitation of the unit cell with the feeding network.

Yodd

LR/2 RS/2

2_RP

2_LL

CR/2

CL

Fig. 4.

Equivalent circuit model of the admittance

Y

odd

obtained by

odd mode excitation of the unit cell with the feeding network.

the product of

L

and

C

R

is immediately known. Due to

the fact that the unit cell realises a band pass structure with

only weak losses, the parallel resistance

R

P

must be highly

resistive whereas its series counterpart

R

S

must be of low

resistance. Hence, we can conclude that

R

P

>> R

S

and

for this reason

< {

Z

even

} ≈

R

P

at the parallel resonant

fre-quency. Now the quality factor of the shunt resonator can be

determined which allows a first estimation of

L

and

C

R

. As

seen in the equivalent circuit, depicted in figure 3, the series

resonance frequency occurs at

ω

rs

=

√

2 /

√

L

R

C

L

which is,

if a balanced structure is assumed, far above the parallel

res-onant frequency. Thus, the series resres-onant circuit behaves

capacitively, depending on the elements

C

L

and

L

R

/

2 , since

it is operated below its resonance frequency. By modifying

these values, the imaginary part of

Z

even

of the equivalent

circuit model may be adjusted until it fits to the desired one.

After these first steps we can evaluate the odd admittance

Y

odd

at the shunt resonance frequency. Obviously, the

par-allel resonator appears as an open-circuit and the remaining

network is almost the series resonator, as it can be seen in

figure 4. Again, utilizing the quality factor and the

reso-nance frequency allows to estimate the remaining element

values. Since both adjustment steps are not decoupled the

approximation of the element values can require some

fur-ther iterations. Figure 5 shows the comparison of the even

impedance and the odd admittance obtained by EM

simula-tions and circuit simulasimula-tions after the element values have

been determined correctly. The determined elements can be

0 1000 2000

9 10 11 12

Z

/

Ω

f GHz

u

tutututtutuutututtuutututtututuutututtuuttuututututututttuutuutut ut tu ut ut ut ut ut

u

t

u

t

u

t ut ut ut ut ut ut

u

tututtuutututututttuuututtututuutututtuttuutututuutututtututuutut ut ut ut ut ut ut

u

t

u

t

u

t

u

t ut ut ut ut

u

t ut

0 0.15 0.30

9 10 11 12

Y

/

Ω

−

1

f GHz

u

ttutututuututututtutuututututututttuuutututtutuututututttuutuututututututututututut ut ut ut ut ut ut ut

u

t

u

t

u

t

u

t ut ut ut tu ut ut ut ut

u

ttutututuututututtutuututututututttuuutututtutuututututttuutuututututututututututut ut ut ut ut ut ut ut

u

t

u

t

u

t

u

t ut ut ut

u

t ut ut ut ut

Fig. 5.

Comparison between even impedance and odd

admit-tance obtained from EM simulation (solid lines) and circuit

sim-ulation (triangles), respectively. Imaginary parts are displayed in

red colour, real parts are drawn in blue colour.

considered either as the feeding network, in case of analysing

a single section, or as a part of adjacent cells in case of a

pe-riodically arranged cascade. Thus, by finding these element

values the equivalent circuit model of the CRLH unit cell is

fully determined.

4 Results

A well balanced unit cell is achieved after optimizing the

structure supported by EM full-wave simulation using CST

Microwave Studio (CST) combined with the procedure

in-Fig. 3. Equivalent circuit model of the impedanceZevenobtained

by even mode excitation of the unit cell with the feeding network.

4 M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

lows to calculate

Zeven = ¯Zeuc(1,1) + ¯Zeuc(1,2) + 2 ¯Zeuc(1,3) (13)

and

Yodd= 1/

_¯

Zeuc(1,1)−Z¯euc(1,2)

. (14)

It is assumed here thatZ¯euc can be described by a matrix

possessing the same characteristics as the matrix shown in equation (9). Zeven is the impedance which appears at the

terminals if the network is excited simultaneously at all ports with an identical signal. In this case no current is flowing across the nodal point and the equivalent circuit model sim-plifies to the one seen in figure 3. Likewise, exciting the structure at each two ports with signals possessing a phase shift of180◦ leads to the odd admittance whose equivalent circuit model is seen in figure 4. In order to determine the values of the containing elements both equivalent networks are simulated with the circuit simulation software AWR Mi-crowave Office (AWR). The results obtained from the EM structure are then compared with the results obtained by cir-cuit analyses and in an iterative process the elements are modified until both responses match. This fitting sequence can either be performed with an automatic optimizer or by tuning the element values manually by performing the steps as follows: Starting with the observation of Zeven shows

that at the resonant frequency of the shunt resonator the impedance of Zeven is expected to peak since the

suscep-tances of the resonator cancel each other out. Consequently,

Zeven

LR/2 RS/2

2_RP 2LL

CL

CR/2

Fig. 3. Equivalent circuit model of the impedanceZevenobtained

Yodd

LR/2 RS/2

2RP 2LL

CR/2

CL

Fig. 4.Equivalent circuit model of the admittanceYoddobtained by

odd mode excitation of the unit cell with the feeding network.

the product of LL andCR is immediately known. Due to

the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistanceRP must be highly

resistive whereas its series counterpart RS must be of low

resistance. Hence, we can conclude that RP >> RS and

for this reason< {Zeven} ≈RP at the parallel resonant

fre-quency. Now the quality factor of the shunt resonator can be determined which allows a first estimation ofLLandCR. As

seen in the equivalent circuit, depicted in figure 3, the series resonance frequency occurs atωrs =

√

2/√LRCLwhich is,

if a balanced structure is assumed, far above the parallel res-onant frequency. Thus, the series resres-onant circuit behaves capacitively, depending on the elementsCLandLR/2, since

it is operated below its resonance frequency. By modifying these values, the imaginary part ofZeven of the equivalent

circuit model may be adjusted until it fits to the desired one. After these first steps we can evaluate the odd admittance Yoddat the shunt resonance frequency. Obviously, the

par-allel resonator appears as an open-circuit and the remaining network is almost the series resonator, as it can be seen in figure 4. Again, utilizing the quality factor and the reso-nance frequency allows to estimate the remaining element values. Since both adjustment steps are not decoupled the approximation of the element values can require some fur-ther iterations. Figure 5 shows the comparison of the even impedance and the odd admittance obtained by EM simula-tions and circuit simulasimula-tions after the element values have been determined correctly. The determined elements can be

0 1000 2000

9 10 11 12

Z

/

Ω

f GHz

u

ttututututtututuuttututuuutututututtutuuttuutututtutututuutttuuut ut ut ut ut ut ut ut

u

t

u

t

u

t ut ut ut ut ut ut

u

tttuuututututututtuttuuutututututututttuuuutttutuutututututututut ut tu ut ut ut ut

u

t

u

t

u

t ut

u

t ut ut ut ut ut

0 0.15 0.30

9 10 11 12

Y

/

Ω

−

1

f GHz

u

ttututututtututuuututtutuuutttutuututututuuttttuuututututututtuututututttuuutututut ut tu ut ut ut ut ut

u

t

u

t

u

t

u

ttututututtututuuututtutuuutttutuututututuuttttuuututututututtuututututttuuutututut ut tu ut ut ut ut ut

u

t

u

t

u

t ut

u

t ut ut ut ut ut ut ut

Fig. 5. Comparison between even impedance and odd admit-tance obtained from EM simulation (solid lines) and circuit sim-ulation (triangles), respectively. Imaginary parts are displayed in red colour, real parts are drawn in blue colour.

considered either as the feeding network, in case of analysing a single section, or as a part of adjacent cells in case of a pe-riodically arranged cascade. Thus, by finding these element values the equivalent circuit model of the CRLH unit cell is fully determined.

4 Results

A well balanced unit cell is achieved after optimizing the structure supported by EM full-wave simulation using CST Microwave Studio (CST) combined with the procedure

in-Fig. 4. Equivalent circuit model of the admittanceYoddobtained by

with I being the identity matrix and Z0 the reference impedance. Since the four-port network which is to be re-alised is assumed lossless its impedance matrix is purely imaginary. Thus, taking the imaginary part of ZSresults in

={ZS} = −jZ0



    

2cosφ−1 2sinφ

1 2sinφ

2cosφ−1 2sinφ

1 2sinφ

2cosφ−1 2sinφ

1 2sinφ

2cosφ−1 2sinφ



    

.

(3) Now, the goal is given by finding a network comprising lumped elements which emulates ZS, at least within a cer-tain bandwidth. In the simplest case this could be achieved by extending a T-shaped network with the series impedance

Z and the shunt admittanceY to a four-port structure. The resulting network is given by

ZT= 1 2ZI+

1

Y1 (4)

with 1 being a matrix with all elements equal to 1. Conse-quently, the elements of ZTmay be determined and yield

Z=j2tanφ

2 (5)

and

Y=j 2

Z0

sinφ. (6)

Since we want the synthesized structure to appear as a ho-mogenous one, only small phase angles φ are considered.

(3)

M. Ebersp¨acher et al.: Design and analysis of an isotropic 2-D planar CRLH waveguide structure 75 lows to calculate

Zeven= ¯Zeuc(1,1) + ¯Zeuc(1,2) + 2 ¯Zeuc(1,3) (13)

and

Yodd= 1/ _¯

Zeuc(1,1)−Z¯euc(1,2)

. (14)

It is assumed here thatZ¯euc can be described by a matrix possessing the same characteristics as the matrix shown in equation (9). Zeven is the impedance which appears at the terminals if the network is excited simultaneously at all ports with an identical signal. In this case no current is flowing across the nodal point and the equivalent circuit model sim-plifies to the one seen in figure 3. Likewise, exciting the structure at each two ports with signals possessing a phase shift of180◦leads to the odd admittance whose equivalent circuit model is seen in figure 4. In order to determine the values of the containing elements both equivalent networks are simulated with the circuit simulation software AWR Mi-crowave Office (AWR). The results obtained from the EM structure are then compared with the results obtained by cir-cuit analyses and in an iterative process the elements are modified until both responses match. This fitting sequence can either be performed with an automatic optimizer or by tuning the element values manually by performing the steps as follows: Starting with the observation of Zeven shows that at the resonant frequency of the shunt resonator the impedance ofZeven is expected to peak since the suscep-tances of the resonator cancel each other out. Consequently,

Zeven

LR/2 RS/2

2_RP 2LL

CL

CR/2

Fig. 3. Equivalent circuit model of the impedanceZevenobtained

Yodd

LR/2 RS/2

2_RP 2LL

CR/2

CL

Fig. 4.Equivalent circuit model of the admittanceYoddobtained by

the product ofLL andCR is immediately known. Due to the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistanceRPmust be highly resistive whereas its series counterpartRS must be of low resistance. Hence, we can conclude thatRP >> RS and

for this reason< {Zeven} ≈RPat the parallel resonant fre-quency. Now the quality factor of the shunt resonator can be determined which allows a first estimation ofLLandCR. As seen in the equivalent circuit, depicted in figure 3, the series resonance frequency occurs atωrs=

√

2/√LRCLwhich is, if a balanced structure is assumed, far above the parallel res-onant frequency. Thus, the series resres-onant circuit behaves capacitively, depending on the elementsCLandLR/2, since it is operated below its resonance frequency. By modifying these values, the imaginary part ofZeven of the equivalent circuit model may be adjusted until it fits to the desired one. After these first steps we can evaluate the odd admittance

Yoddat the shunt resonance frequency. Obviously, the par-allel resonator appears as an open-circuit and the remaining network is almost the series resonator, as it can be seen in figure 4. Again, utilizing the quality factor and the reso-nance frequency allows to estimate the remaining element values. Since both adjustment steps are not decoupled the approximation of the element values can require some fur-ther iterations. Figure 5 shows the comparison of the even impedance and the odd admittance obtained by EM simula-tions and circuit simulasimula-tions after the element values have been determined correctly. The determined elements can be

0 1000 2000

9 10 11 12

Z / Ω f GHz u t u

ttututututtututuuututututtuututututtuttutuuututututuutttutuutut ut tu ut ut ut ut ut

u

t

u

t

u

t ut ut ut ut ut ut

u

t

u

tttuuututututututtutuutttuuututututttuuutututututututtututuutut ut ut ut ut ut ut

u

t

u

t

u

t ut

u

t ut ut ut ut ut

0 0.15 0.30

9 10 11 12

Y / Ω − 1 f GHz u t u

tttuuututuutututtttutuuututuutttutuututututtututuuttutututuututututtuttuuuttuutut ut tu ut ut ut ut ut

u t u t u t u

u

t

u

tttuuututuutututtttutuuututuutttutuututututtututuuttutututuututututtuttuuuttuutut ut tu ut ut ut ut ut

u

t

u

t

u

t ut

u

t ut ut ut ut ut ut ut

Fig. 5. Comparison between even impedance and odd admit-tance obtained from EM simulation (solid lines) and circuit sim-ulation (triangles), respectively. Imaginary parts are displayed in red colour, real parts are drawn in blue colour.

considered either as the feeding network, in case of analysing a single section, or as a part of adjacent cells in case of a pe-riodically arranged cascade. Thus, by finding these element values the equivalent circuit model of the CRLH unit cell is fully determined.

4 Results

A well balanced unit cell is achieved after optimizing the structure supported by EM full-wave simulation using CST Microwave Studio (CST) combined with the procedure

in-Fig. 5. Comparison between even impedance and odd admit-tance obtained from EM simulation (solid lines) and circuit sim-ulation (triangles), respectively. Imaginary parts are displayed in red colour, real parts are drawn in blue colour.

M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure 5

troduced in the previous section. The S-parameters of the resulting unit cell including the feeding network are shown in figure 6. Although this is not the response of the unit cell

-7.5 -5.0 -2.5 0

9 10 11 12

M a g n it u d e / d B f GHz -135 -90 -45 0 45 90 135 180

9 10 11 12

P h a se / D eg re e f GHz

Fig. 6.Simulated scattering parameters of the structure seen in fig-ure 1:S11red line,S21green line,S31orange line,S41blue line.

itself, the isotropic behaviour and the power balance must be conserved anyway. As seen, the result is very close to the the-oretical expectations of an isotropic structure which requires magnitudes of -6 dB, identical phase shifts of all transmis-sion paths as well as a phase difference of 180◦between transmitted and reflected waves. The waveguiding properties of the de-embedded unit cell, possessing a Bloch impedance

ZBlochx ≈ 60 Ω, are shown in figures 7, 8 and 9. In re-gions where an isotropic behaviour is exhibited, the wave vector must have an absolute value which is independent of the direction. In terms of figure 9 this is represented by cone shaped characteristics or relating to the isofrequency plots this results in circles around the pointβxp=βyp= 0which

can be observed for8.5 GHz≤ f ≤12 GHz. It should be mentioned that the gap in figure 9 is an artefact of the visu-alization and does not represent a stop band as validated by the dispersion diagram depicted in figure 10. In the region Γ−X andM −X the index of refraction is determined and presented in figure 11 which also reveals the isotropy of the unit cell. A prototype, seen in figure 1, was fab-ricated and measured. The results of the measurements are presented in figure 12 and the corresponding dispersion dia-gram in figure 13. As seen from the increasing attenuation constant between9.5 GHz≤f ≤10.5 GHzthe fabricated prototype is not balanced correctly. Further analysis reveals that the series resonant frequency isfrs = 10.57 GHzand the parallel resonant frequency isfrp= 9.67 GHz. This can be explained by tolerances within the fabrication process.

7.996 7.996 7.996 7.996 8.5 8.5 9.004 9.496

β_x p

βy

p

−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Fig. 7.Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in thexandy

directions. 10.996 12.004 13 13 13.996 13.996 15.004 15.004 16 16 16 16 β

x p

βy

p

−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3

Fig. 8. Isofrequency contour plot (in GHz) of the right-handed mode with respect to two-dimensional wave propagation in thex

andydirections.

5 Conclusions

A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between8.5 GHz ≤ f ≤

12 GHz. Within this frequency region a frequency dependent index of refraction ranging from−1.5≤0.8is achieved. A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.

References

AWR Microwave Office, http://www.awrcorp.com/. CST Computer Simulation Technology, http://www.cst.com/. Caloz, C. and Itoh, T.: Electromagnetic Metamaterials, Wiley,

2006. Fig. 6. Simulated scattering parameters of the structure seen in

fig-ure 1:S11red line,S21green line,S31orange line,S41blue line.

Hence, a first order Taylor series expansion inφmay be per-formed aroundφ=0 resulting in

Z≈jZ0φ (7)

and

Y≈j 2

Z0

φ. (8)

Composite Right/Left-Handed behaviour is achieved if at least one frequency band with∂ωφ (ω) <0 and another with ∂ωφ (ω) >0 exists. This can be accomplished by choosing

appropriate network elements forY andZ, respectively. For example, a parallel resonant circuit is inserted forY and a series resonant circuit forZ.

M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

5 troduced in the previous section. The S-parameters of the

resulting unit cell including the feeding network are shown

in figure 6. Although this is not the response of the unit cell

-7.5 -5.0 -2.5 0

9 10 11 12

M

a

g

n

it

u

d

e

/

d

B

f GHz -135 -90 -45 0 45 90 135 180

9 10 11 12

P

h

a

se

/

D

eg

re

e

f GHz

Fig. 6.

Simulated scattering parameters of the structure seen in

fig-ure 1:

S

11

red line,

S

21

green line,

S

31

orange line,

S

41

blue line.

itself, the isotropic behaviour and the power balance must be

conserved anyway. As seen, the result is very close to the

the-oretical expectations of an isotropic structure which requires

magnitudes of -6 dB, identical phase shifts of all

transmis-sion paths as well as a phase difference of 180

◦

between

transmitted and reflected waves. The waveguiding properties

of the de-embedded unit cell, possessing a Bloch impedance

Z

Blochx

≈

60 Ω

, are shown in figures 7, 8 and 9. In

re-gions where an isotropic behaviour is exhibited, the wave

vector must have an absolute value which is independent of

the direction. In terms of figure 9 this is represented by cone

shaped characteristics or relating to the isofrequency plots

this results in circles around the point

β

x

p

=

β

y

p

= 0

which

can be observed for

8 .

5 GHz

≤

f

≤

12 GHz

. It should be

mentioned that the gap in figure 9 is an artefact of the

visu-alization and does not represent a stop band as validated by

the dispersion diagram depicted in figure 10. In the region

Γ

−

X

and

M

−

X

the index of refraction is determined

and presented in figure 11 which also reveals the isotropy

of the unit cell.

A prototype, seen in figure 1, was

fab-ricated and measured. The results of the measurements are

presented in figure 12 and the corresponding dispersion

dia-gram in figure 13. As seen from the increasing attenuation

constant between

9 .

5 GHz

≤

f

≤

10 .

5 GHz

the fabricated

prototype is not balanced correctly. Further analysis reveals

that the series resonant frequency is

f

rs

= 10

.

57 GHz

and

the parallel resonant frequency is

f

rp

= 9

.

67 GHz

. This can

be explained by tolerances within the fabrication process.

7.996 7.996 7.996 7.996 8.5 8.5 9.004 9.496 β

x p

β y

p

−3 −2 −1 0 1 2 3

Fig. 7.

Isofrequency contour plot (in GHz) of the left-handed mode

with respect to two-dimensional wave propagation in the

x

and

y

directions.

10.996 12.004 13 13 13.996 13.996 15.004 15.004 16 16 16 16 β

x p

β y

p

−3 −2 −1 0 1 2 3

Fig. 8.

Isofrequency contour plot (in GHz) of the right-handed

mode with respect to two-dimensional wave propagation in the

x

and

y

directions.

5 Conclusions

A two-dimensional CLRH unit cell was presented which

possesses an isotropic behaviour between

8 .

5 GHz

≤

f

≤

12 GHz

. Within this frequency region a frequency dependent

index of refraction ranging from

−

1 .

5 ≤

0 .

8 is achieved. A

systematic procedure based on even and odd mode analysis

was presented which allows to determine the element values

of the equivalent circuit model of the proposed structure as

well as its feeding network.

References

AWR Microwave Office, http://www.awrcorp.com/.

CST Computer Simulation Technology, http://www.cst.com/.

Caloz, C. and Itoh, T.: Electromagnetic Metamaterials, Wiley,

2006.

Fig. 7. Isofrequency contour plot (in GHz) of the left-handed mode with respect to two-dimensional wave propagation in thexandy

directions.

M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

5 troduced in the previous section. The S-parameters of the

resulting unit cell including the feeding network are shown

in figure 6. Although this is not the response of the unit cell

-7.5 -5.0 -2.5 0

9 10 11 12

M

a

g

n

it

u

d

e

/

d

B

f GHz -135 -90 -45 0 45 90 135 180

9 10 11 12

P

h

a

se

/

D

eg

re

e

f GHz

Fig. 6.

Simulated scattering parameters of the structure seen in

fig-ure 1:

S

11

red line,

S

21

green line,

S

31

orange line,

S

41

blue line.

itself, the isotropic behaviour and the power balance must be

conserved anyway. As seen, the result is very close to the

the-oretical expectations of an isotropic structure which requires

magnitudes of -6 dB, identical phase shifts of all

transmis-sion paths as well as a phase difference of 180

◦

between

transmitted and reflected waves. The waveguiding properties

of the de-embedded unit cell, possessing a Bloch impedance

Z

Blochx

≈

60 Ω

, are shown in figures 7, 8 and 9. In

re-gions where an isotropic behaviour is exhibited, the wave

vector must have an absolute value which is independent of

the direction. In terms of figure 9 this is represented by cone

shaped characteristics or relating to the isofrequency plots

this results in circles around the point

β

x

p

=

β

y

p

= 0

which

can be observed for

8 .

5 GHz

≤

f

≤

12 GHz

. It should be

mentioned that the gap in figure 9 is an artefact of the

visu-alization and does not represent a stop band as validated by

the dispersion diagram depicted in figure 10. In the region

Γ

−

X

and

M

−

X

the index of refraction is determined

and presented in figure 11 which also reveals the isotropy

of the unit cell.

A prototype, seen in figure 1, was

fab-ricated and measured. The results of the measurements are

presented in figure 12 and the corresponding dispersion

dia-gram in figure 13. As seen from the increasing attenuation

constant between

9 .

5 GHz

≤

f

≤

10 .

5 GHz

the fabricated

prototype is not balanced correctly. Further analysis reveals

that the series resonant frequency is

f

rs

= 10

.

57 GHz

and

the parallel resonant frequency is

f

rp

= 9

.

67 GHz

. This can

be explained by tolerances within the fabrication process.

7.996 7.996 7.996 7.996 8.5 8.5 9.004 9.496 β

x p

β y

p

−3 −2 −1 0 1 2 3

Fig. 7.

Isofrequency contour plot (in GHz) of the left-handed mode

with respect to two-dimensional wave propagation in the

x

and

y

directions.

10.996 12.004 13 13 13.996 13.996 15.004 15.004 16 16 16 16 β

x p

β y

p

−3 −2 −1 0 1 2 3

Fig. 8.

Isofrequency contour plot (in GHz) of the right-handed

mode with respect to two-dimensional wave propagation in the

x

and

y

directions.

5 Conclusions

A two-dimensional CLRH unit cell was presented which

possesses an isotropic behaviour between

8 .

5 GHz

≤

f

≤

12 GHz

. Within this frequency region a frequency dependent

index of refraction ranging from

−

1 .

5 ≤

0 .

8 is achieved. A

systematic procedure based on even and odd mode analysis

was presented which allows to determine the element values

of the equivalent circuit model of the proposed structure as

well as its feeding network.

References

AWR Microwave Office, http://www.awrcorp.com/.

CST Computer Simulation Technology, http://www.cst.com/.

Caloz, C. and Itoh, T.: Electromagnetic Metamaterials, Wiley,

2006.

Fig. 8. Isofrequency contour plot (in GHz) of the right-handed mode with respect to two-dimensional wave propagation in thex

andydirections.

2.2 Dispersion characteristics

As shown in Eqs. (3) and (4) the impedance matrix of the ideal unit cell comprises only two different types of elements. These are on the one hand the input impedancesZI=Znn

forn∈1,2,3,4 and on the other hand all the remaining en-tries denoted by ZT. However, we want to take possible anisotropic behaviour into account for the calculation of the dispersion characteristics. Thus, a third element is intro-duced which allows to distinguish between the impedance

(4)

76 M. Ebersp¨acher et al.: Design and analysis of an isotropic 2-D planar CRLH waveguide structure 6 M. Ebersp¨acher et al.: Design and Analysis of an Isotropic 2D Planar CRLH Waveguide Structure

−4 −2 0 2 4 −4 −2 0 2 4 6 8 10 12 14 16 18

β_x p

β

y p

f /

G

H

z

Fig. 9.Surface plot of the two-dimensional wave vectorβexhibited by the CRLH structure.

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

Fig. 10.Simulated wave propagation characteristics. Dispersion di-agram for the regionΓ−X(left) and the corresponding attenuation constant (right).

Ebersp¨acher, M. A., Eccleston, K., and Eibert, T. F.: Realization of Via-free Microstrip Composite Right/Left-Handed Transmission Lines, GeMiC, 2009.

Eccleston, K.: Beam forming transition based upon a zero-phase-shift metamaterial, IET Microw. Antennas Porpag., 4, 1639– 1646, 2010.

Lay, A., Itoh, T., and Caloz, C.: Composite right/left-handed trans-mission line metamaterials, IEEE Microwave Magazine, 5, 34– 50, 2004.

Shelby, R. A., Smith, D. R., and Schultz, S.: Experimental veri-fication of a negative index of refraction, Science, 292, 77–79, 2001.

-2 -1 0 1

9 10 11 12

n = c0 / vp h f GHz

Fig. 11. Index of refraction obtained by EM simulations for the regionsΓ−X(red line) andM−Γ(blue line).

-7.5 -5.0 -2.5 0

9 10 11 12

Fig. 12.Measured scattering parameters of the fabricated structure seen in figure 1:S11red line,S21green line,S31orange line,S41

blue line. 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

Fig. 13.Measured wave propagation characteristics. Dispersion di-agram for the regionΓ−X(left) and the corresponding attenuation constant (right).

Fig. 9. Surface plot of the two-dimensional wave vectorβexhibited by the CRLH structure.

−4 −2 0 2 4 −4 −2 0 2 4 6 8 10 12 14 16 18

β_x p

β

y p

f /

G

H

z

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

-2 -1 0 1

9 10 11 12

n = c0 / vp h f GHz

-7.5 -5.0 -2.5 0

9 10 11 12

blue line. 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

Fig. 10. Simulated wave propagation characteristics. Dispersion di-agram for the region0−X(left) and the corresponding attenuation constant (right).

of two ports which are arranged oppositely and perpendicu-larly, respectively. Consequently, the impedance matrix for such a four-port network reads

Zr=



  

ZI ZT ZX ZX

ZT ZI ZX ZX

ZXZX ZI ZT

ZXZX ZT ZI



  

. (9)

It is assumed here that the two ports 1, 2 and 3, 4 are arranged along one Cartesian coordinate axis. Performing the Floquet ansatz based on Zryields the dispersion characteristics

[cosh(γyp+γxp)+cosh(γyp−γxp)](Z2T−Z2X)

−[cosh(γxp)+cosh(γyp)](2ZIZT−2Z2X)

−2Z_X2+2Z_I2=0. (10) For wave propagation along a principle axis, i.e. the region

0−Xin the dispersion diagram, this expression simplifies to cosh(γxp)=cosh(γyp)=ZI/ZT. (11) In this case all nodal points of the periodic arrangement be-ing on a line perpendicular to the propagation direction are

−4 −2 0 2 4 −4 −2 0 2 4 6 8 10 12 14 16 18 β

x p

β

y p

f /

G

H

z

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

-2 -1 0 1

9 10 11 12

n = c0 / vp h f GHz

-7.5 -5.0 -2.5 0

9 10 11 12

blue line. 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

Fig. 11. Index of refraction obtained by EM simulations for the regions0−X(red line) andM−0(blue line).

−4 −2 0 2 4 −4 −2 0 2 4 6 8 10 12 14 16 18

β_x p

β

y p

f /

G

H

z

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

-2 -1 0 1

9 10 11 12

n = c0 / vp h f GHz

-7.5 -5.0 -2.5 0

9 10 11 12

blue line. 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz βp π 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 0.5 1.0 f GHz αp

Fig. 12. Measured scattering parameters of the fabricated structure seen in figure 1:S11red line,S21green line,S31orange line,S41

blue line.

equipotential points. Thus, no current is flowing across nodal points in transverse direction and the four-port matrix can be reduced to a two-port impedance matrix given by the up-per left submatrix of Zrcomprising the elementsZIandZT. Consequently, the Bloch impedance reads

ZBlochx=

q

ZI2−ZT2. (12)

3 Design

The goal is to realise an isotropic two-dimensional CRLH unit cell which exhibits a left-handed mode forf <10GHz and a right-handed mode forf >10GHz fulfilling Eq. (3) around the transition frequency offT=10GHz. Due to the given operation frequencies the use of discrete elements, for example surface mounted devices, may lead to difficulties. Hence, the design of the 2d unit cell is based on the 1d struc-ture presented in (Ebersp¨acher et al., 2009) which is fabri-cated in an entirely printed circuit technology based on trans-mission line theory. The original design can be considered as an interdigital capacitor whose arms supporting the individ-ual fingers are extended to open-ended stubs in order to

(5)

M. Ebersp¨acher et al.: Design and analysis of an isotropic 2-D planar CRLH waveguide structure 77

−4 −2

0 2

4

−4 −2 0 2 4 6 8 10 12 14 16 18

β_x p

β

y p

f /

G

H

z

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz

βp π

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

0.5 1.0

f GHz

αp

-2 -1 0 1

9 10 11 12

n

=

c0

/

vp

h

f GHz

-7.5 -5.0 -2.5 0

9 10 11 12

M

a

g

n

it

u

d

e

/

d

B

f GHz

-135 -90 -45 0 45 90 135 180

9 10 11 12

P

h

a

se

/

D

eg

re

e

f GHz

blue line.

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-0.5 0 0.5 1.0

f GHz

βp π

8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

0.5 1.0

f GHz

αp

Fig. 13. Measured wave propagation characteristics. Dispersion di-agram for the region0−X(left) and the corresponding attenuation constant (right).

have inductively. Thus, this configuration realises a shunt in-ductor and a series capacitor simultaneously. Both elements are completed by the occurring parasitics to a series resonant circuit and a parallel resonant circuit, respectively. Adjust-ing the resonant frequencies as well as the Bloch impedance can be achieved easily by modifying either the stublength or the fingers of the arrangement allowing to meet given spec-ifications and balancing the structure. Ensuring the artificial structure to behave homogeneously, the length of the unit cell must not exceed the quarter wave length limit (Lay et al., 2004). Dealing with 2d structures, this requirement must be fulfilled in both directions. Along with the desired isotropic characteristic this results in a quadratic footprint of the unit cell possessing a sidelength which is smaller or equal to a quarter of the wavelength. Although the absence of interlayer connections makes the initial design easy to fabricate and hence preferable, it cannot be extended to a two-dimensional unit cell in a straightforward manner, since the open-ended stubs cannot be reduced to fit into the footprint without los-ing its inductive behaviour. Therefore, the open-ended stubs are substituted by short-circuited ones which require to be reduced in length to maintain their electromagnetic proper-ties. This modification influences strongly the field distribu-tion within the interdigital capacitor. Whereas in the original case the maximum of capacitance was realised close to the open ends, in the modified version this appears close to the hostline. In order to reduce the amount of interlayer connec-tions, these are placed at the corners of the unit cell so that they may be used by adjacent cells simultaneously, as seen in Fig. 1.

3.1 Analysis

As seen in Figs. 1 and 2, the boundary of the unit cell is located right between the interdigital capacitor.

In terms of an equivalent circuit model this can be consid-ered as two capacitors connected in series representing to-gether the capacitance of the interdigital capacitor separated

by the cell boundary. However, excitation of the structure may only be performed with a quasi TEM mode bound to the microstrip lines. Consequently, the reference plane for deter-mining the scattering parameters cannot be set correctly. As a result of this, the determinable scattering parameters always contain contributions of adjacent cells, in case of a periodi-cally arranged cascade, or contributions of the feeding net-work, in case of analysing a single section. Thus, to obtain the properties of the unit cell itself the reference plane for the scattering matrix analysis must be placed between two inseparable capacitors which is not possible in practice. This difficulty can be overcome by modelling the unwanted con-tributions with an equivalent circuit model which is used to de-embed the unit cell parameters. For this, the values of the elementsLR/2, 2LL, CR/2, 2RP ,RS/2 and 2CL must be determined based on the simulated or measured network re-sponse which can be done by performing even and odd anal-ysis.

Starting with the impedance matrix representation of the four-port network referred to as extended unit cell Zeuc, which comprises the unit cell and the feeding network, al-lows to calculate

Zeven=Zeuc(1,1)+Zeuc(1,2)+2Zeuc(1,3) (13) and

Yodd=1/[Zeuc(1,1)−Zeuc(1,2)]. (14) It is assumed here that Zeuc can be described by a matrix possessing the same characteristics as the matrix shown in Eg. (9). Zevenis the impedance which appears at the termi-nals if the network is excited simultaneously at all ports with an identical signal. In this case no current is flowing across the nodal point and the equivalent circuit model simplifies to the one seen in Fig. 3. Likewise, exciting the structure at each two ports with signals possessing a phase shift of 180◦ leads to the odd admittance whose equivalent circuit model is seen in Fig. 4. In order to determine the values of the containing elements both equivalent networks are simulated with the circuit simulation software AWR Microwave Office (AWR). The results obtained from the EM structure are then compared with the results obtained by circuit analyses and in an iterative process the elements are modified until both re-sponses match. This fitting sequence can either be performed with an automatic optimizer or by tuning the element values manually by performing the steps as follows: Starting with the observation ofZevenshows that at the resonant frequency of the shunt resonator the impedance of Zeven is expected to peak since the susceptances of the resonator cancel each other out.

(6)

78 M. Ebersp¨acher et al.: Design and analysis of an isotropic 2-D planar CRLH waveguide structure Consequently, the product ofLL andCRis immediately

known. Due to the fact that the unit cell realises a band pass structure with only weak losses, the parallel resistance RP must be highly resistive whereas its series counterpart RS must be of low resistance. Hence, we can conclude that

RP>> RS and for this reason<{Zeven} ≈RP at the paral-lel resonant frequency. Now the quality factor of the shunt resonator can be determined which allows a first estima-tion of LL andCR. As seen in the equivalent circuit, de-picted in Fig. 3, the series resonance frequency occurs at

ωrs=

√

2/√LRCL which is, if a balanced structure is as-sumed, far above the parallel resonant frequency. Thus, the series resonant circuit behaves capacitively, depending on the elementsCL andLR/2, since it is operated below its reso-nance frequency. By modifying these values, the imaginary part ofZevenof the equivalent circuit model may be adjusted until it fits to the desired one.

After these first steps we can evaluate the odd admittance

Yodd at the shunt resonance frequency. Obviously, the par-allel resonator appears as an open-circuit and the remain-ing network is almost the series resonator, as it can be seen in Fig. 4. Again, utilizing the quality factor and the reso-nance frequency allows to estimate the remaining element values. Since both adjustment steps are not decoupled the approximation of the element values can require some fur-ther iterations. Figure 5 shows the comparison of the even impedance and the odd admittance obtained by EM simula-tions and circuit simulasimula-tions after the element values have been determined correctly.

The determined elements can be considered either as the feeding network, in case of analysing a single section, or as a part of adjacent cells in case of a periodically arranged cas-cade. Thus, by finding these element values the equivalent circuit model of the CRLH unit cell is fully determined.

4 Results

A well balanced unit cell is achieved after optimizing the structure supported by EM full-wave simulation using CST Microwave Studio (CST) combined with the procedure in-troduced in the previous section. The S-parameters of the resulting unit cell including the feeding network are shown in Fig. 6.

Although this is not the response of the unit cell itself, the isotropic behaviour and the power balance must be con-served anyway. As seen, the result is very close to the the-oretical expectations of an isotropic structure which requires magnitudes of -6 dB, identical phase shifts of all transmis-sion paths as well as a phase difference of 180◦ between transmitted and reflected waves. The waveguiding properties of the de-embedded unit cell, possessing a Bloch impedance

ZBlochx ≈60, are shown in Figs. 7, 8 and 9. In regions

where an isotropic behaviour is exhibited, the wave vector must have an absolute value which is independent of the

di-rection. In terms of Fig. 9 this is represented by cone shaped characteristics or relating to the isofrequency plots this re-sults in circles around the pointβxp=βyp=0 which can be

observed for 8.5GHz≤f ≤12GHz. It should be mentioned that the gap in Fig. 9 is an artefact of the visualization and does not represent a stop band as validated by the dispersion diagram depicted in Fig. 10. In the region0−XandM−X

the index of refraction is determined and presented in Fig. 11 which also reveals the isotropy of the unit cell.

A prototype, seen in Fig. 1, was fabricated and mea-sured. The results of the measurements are presented in Fig. 12 and the corresponding dispersion diagram in Fig. 13. As seen from the increasing attenuation constant between 9.5GHz≤f≤10.5GHz the fabricated prototype is not bal-anced correctly. Further analysis reveals that the series res-onant frequency isfrs=10.57GHz and the parallel resonant frequency isfrp=9.67GHz. This can be explained by toler-ances within the fabrication process.

5 Conclusions

A two-dimensional CLRH unit cell was presented which possesses an isotropic behaviour between 8.5GHz≤f ≤

12GHz. Within this frequency region a frequency dependent index of refraction ranging from−1.5≤0.8 is achieved. A systematic procedure based on even and odd mode analysis was presented which allows to determine the element values of the equivalent circuit model of the proposed structure as well as its feeding network.

Acknowledgements. The authors gratefully acknowledge the

sup-port of the TUM Graduate School’s Faculty Graduate Center EI at Technische Universit¨at M¨unchen, Germany.

References

AWR Microwave Office, http://www.awrcorp.com/, Last accessed: 15.02.2011.

CST Computer Simulation Technology, http://www.cst.com/, Last accessed: 15.02.2011.

Caloz, C. and Itoh, T.: Electromagnetic Metamaterials, Wiley, 2006.