A Comparison of Active and Passive Metamaterials from Equivalent Lumped Elements Modes

doi:10.4236/opj.2013.32B061

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Optics and Photonics Journal, 2013, 3, 260-264

doi:10.4236/opj.2013.32B061 Published Online June 2013 (http://www.scirp.org/journal/opj)

A Comparison of Active and Passive Metamaterials from

Equivalent Lumped Elements Modes

Ya-Fen Ge1, Hui Huang1,2, Yong Liu1, H ou -J un Sun1, X in Lv1, Li- Mi ng Si 1*

1Beijing Key Laboratory of Millimeter Wave and Terahertz Technology, Department of Electronic Engineering,

Beijing Institute of Technology, Beijing, P. R. China

2Microwave Laboratory, Division of Electronics and Information Technology, National Institute of Metrology,

Beijing, P. R. China

Email: *[email protected]

Received 2013

ABSTRACT

With ever-increasing operating frequencies and complicated artificial structures, loss effects become more and more

important in applications of metamaterials. Based on circuit theory and transmission line principle, the design equations

for effective electromagnetic (EM) parameters (attenuation constant α, phase constant β, characteristic impedance

Z0) of general active and passive metamaterial are compared and derived from the equivalent lumped circuit parameters

(R, G, LL, CL, LR, CR). To verify the design equations, theα, βand Z0 in different cases, including balanced, unbal-

anced, lossless, passive and active, are shown by numerical simulations. The results show that using the active method

can diminish the loss effects. Meantime, it also has influence on phase constant and real part of characteristic imped-

ance.

Keywords: Metamaterials; Equivalent Lumped Circuit elements Modes; Loss Compensation

1. Introduction

The electromagnetic (EM) material parameters, such as

propagation constant (including attenuation constant and

phase constant), characteristic impedance, permittivity,

permeability, refractive index, phase velocity and con-

ductivity, of material always used to describe the EM

response in certain frequency whenever EM wave exist

in or propagation (transmit/evanescent) across media [1].

Based on transmission line theory, where EM wave

propagation is considered in media the EM behaviors

could be entirely represented with equivalent lumped

circuit parameters, i.e., resistance, conductance, induc-

tance and capacitance, which are also frequency de-

pendent parameters like EM parameters of material [1]. It

provided the relationship between the equivalent lumped

circuit parameters and the EM material parameters.

These equivalent lumped circuit parameters also serve as

bridges between complicated EM field calculations and

straightforward circuit analysis. Hence, EM material pa-

rameters can be expressed in terms of the equivalent

lumped circuit parameters. Similarly, the equivalent

lumped circuit parameters can be determined by EM ma-

terial parameters [1,2].

Metamaterial, which control the electromagnetic waves

with their structures rather than the compositions [3-6],

such as zero index materials, negative refractive index

metamaterials, plamonics, have received much attention

owing to their extraordinary properties not readily achiev-

able in nature [7-21]. Left-handed material with negative

refractive index was first investigated theoretically by

Veselogo in 1968 [3], which can exhibits lots of unusual

physical properties, like reversed Doppler shift, inversted

Snell’s low (negative refraction) and reversed Cerenkov

radiation, compared with conventional righthanded ma-

terial (RHM), tremendous attention has been focused on

theory and engineering applications of this novel type of

artificial material [7-21]. There are two main artificial

methods to realize metamaterials: first, based on the

resonant-type structures; second, using quasi-lumped

elements developed on transmission lines. With the first

method, Shelby et al. constructed the first LHM with

periodical array of copper split-ring resonators (SRRs)

and thin copper wires in the microwave regime [4]. By

using the second method which usually termed compos-

ite right/left-handed transmission line (CRLH-TL) method

and utilizes transmission line (TL) principle [15-17], the

EM behavior can be easily controlled by equivalent

lumped circuit parameters. It is not easy to acquire purely

LHM with the second method since unavoidable RH

parasitic effects in nature, but the CRLH-TLs may have

*Corresponding author.

Y.-F. GE ET AL. 261

advantages of planar structure, broader bandwidth, ad-

justable phase response and compatible with microwave

integrated circuits. Many applications of CRLH-TL to

RF/ Microwave/Terahertz (THz)/ Optical components

have been proposed, such as filter, antenna, power di-

vider, phase shifter, coupler, balun and some other active

components [2]. However, most of them are based on

ideal lossless CRLH-TL method which ignores the losses

effects. With ever-increasing operating frequencies and

complicated artificial structures, losses effects must have

paid much attention for engineering application of meta-

materials [18-21].

In this paper, we introduce the design equations of ef-

fective EM material parameters of general active and

passive metamaterial from equivalent lumped elements

modes. And then detailed analysis of ten examples, in-

cluding lossless/passive/active metamaterial cases, based

on these equations will presented.

2. Theory and Design Equations

The equivalent circuit of the unit-cell of general active

and passive metamaterial is shown in Figure 1. The

unit-cell is modeled with six lumped element circuit pa-

rameters, i.e., R, G, CL, LL, CR, and LR. These parameters

are normalized to length and may be detailed interpreted

as following: a series resistance (R, unit in []) and

a shunt conductance (G, unit in [S/m]) to account for the

effects of losses (conductive loss and dielectric loss, re-

spectively); a series capacitance (CL, unit in [

/m



]) and

a shunt inductance (LL, unit in [

m]) to obtain the

left-handed transmission line (LH-TL) by using the dual

principle of circuit theory; as well as a series inductance

(LR, unit in [/

m]) and a shunt capacitance (CR, unit in

m]) from unavoidable practice effects.

The period, or termed lattice constant, of the unit-cell

is P. For satisfying the principle of ideal effective ho-

mogenous TL, it should be chosen to meet the electri-

cally small condition:

, at least , typically

410

PP P



 (1)

where



is the guided wavelength.

From Figure 1, the series impedance Z and shunt ad-

mittance Y for the unit-cell of general active and passive

metamaterial can be written as:

Figure 1. Equivalent circuit for unit-cell of general active

and passive metamaterial.

ZRjLC





 







(2)

YGj CL





 







(3)

Based on the classical TL principle [1], the propaga-

tion constant



and characteristic impedance Z0 of the

general active and passive metamaterial are given by:

 

 Y (4)

 (5)

where



is attenuation constant (unit in Np/m) and



is phase constant (unit in rad/m).

By inserting (2) and (3) into (4) and separating the real

and imaginary parts, the equations of



and



are

distilled and simplified as (6) and (7), respectively. When

R = G = 0, it become ideal lossless case, which has al-

ready been widely used in former publications and re-

ports for calculation simplicity [2]. From (6) and (7), it is

clear that the attenuation constant 0



 and the

propagation constant j







in this ideal case. Simi-

larly, using (2) and (3) into (5) the characteristic imped-

ance Z0 is given as (8).

As we have stated above, since the electrical size of

the unit-cell of general active and passive metamaterial is

small enough to suppress all diffraction scattering effects,

the general active and passive metamaterial can be char-

acterized with the some other effective constitutive pa-

rameters, such as permittivity, permeability and refrac-

tive index. The effective values of them can be easily

extracted from the EM material parameters (



and Z0)

[13-15].

3. Numerical Simulations and Comparison

Analysis

Ten numerical examples, including lossless (R=G=0),

passive lossy (R > 0, 0G



) and active lossy (R<0,



) cases, are presented in this section to study losses

effects in general active and passive metamaterial. The

equivalent lumped circuit parameters are presented in

Table 1.

Conductance generally represents the dielectric loss

and it has the relationship with dielectric loss tangent

(tan



) is:

0tan



 



(9)

where r



is the relative permittivity (dielectric constant),

and is permittivity in free space.

Take the low cost substrate material Duroid (

8.85410)







2.2





tan 0.0009





) as an example, which has been widely

used for RF/Microwave engineering, the corresponding

Y.-F. GE ET AL.

262

conductance is 0.001 (unit in [S/m]) at 10 GHz.

Table 1 shows balanced case (example 1-5), unbal-

anced case (example 6-10), lossless case (example 1 and

6), passive lossy case (example 2, 3, 7 and 8) and active

lossy case (example 4, 5, 9 and 10).

In the balanced case (example 1-5), from the parame-

ters, it is easy to calculate the transition frequency:

GHz. The attenuation constant curve is shown in

Figure 2. As can be seen in Figure 2, the attenuation

constant is increasing with resistance in passive case

(), but decreasing with the enhanced active (which

can be represented by the absolute value of negative re-

sistance) metamaterial. Hence, included active elements

may be a convenient way to overcome the major draw-

back of losses in metamaterials.

0R

The phase constant curve in the balanced case is shown in

Figure 3. The sign of phase constant ( ) is crucial impor-

tant for metamaterial, because it determines the backward

wave, forward wave dan broadside (in the balanced case) or

stopband (in the unbalanced case) frequencies.

 

224222224222 22

1121 2

LRRLRLL RLLRL LR

CLCLCLGL CLCLRCCL

 





 



1





(6)

 

224222224222 22

1 12121

LRRLRLL RLLRL LR

CLCLCLGL CLCLRC CL

 





 











(7)



1/4

422 2222

0422222 2

exp 2

LLR

LLR LRL

LRLRl LLL R

LjCRjL

LCL CLCR

ZjArg

CCLCL LGCjLGjC















 



























 























(8)

Table 1. Equivalent lumped circuit parameters of active and passive metamaterials.

Example R []

/mLR []

/nH mCL [pF m



] LL [nH m



] CR [] /pF mG [S/m]

1 0 1 1 1 1 0

2 0.1 1 1 1 1 0.001

3 1 1 1 1 1 0.001

4 -0.1 1 1 1 1 0.001

5 -1 1 1 1 1 0.001

6 0 1 2 5.5 1 0

7 0.1 1 2 5.5 1 0.001

8 1 1 2 5.5 1 0.001

9 -0.1 1 2 5.5 1 0.001

10 -1 1 2 5.5 1 0.001

Figure 2. Attenuation constant curve for balanced case

(example 1-5). Figure 3. Phase constant curve for balanced case.

Y.-F. GE ET AL. 263

Figure 3 shows the zero phase constant 0



 at the

transition frequency 0

of 5 GHz, usually termed Ze-

roth-Order Resonator (ZOR, the physical dimensions can

be arbitrary but not limited by the conventional wave-

length [2]). It also can be observed the continuous leak-

age/fast frequency band (between two air lines c







)

from LH (phase constant 0



, 0.5 < f < 5 GHz) to RH



, 5 GHz < f < 20 GHz) state through the transition

frequency point in all of the numerical calculations in-

cluding lossless, passive and active states.

In Figure 4(a) and (b), the real part and imaginary

part of the characteristic impedance are plotted respec-

tively for balanced case.

Like the balanced case analysis, the attenuation con-

stant, phase constant, real part and imaginary part of

characteristic impedance as function of frequency are

calculated and plotted in Figures 5-7(a) and (b) for un-

balanced case (Example 6-10), respectively.

In the unbalanced case (Example 6-10), the stopband

can be calculated from 2.33 GHz to 3.56 GHz by using

the design equations of Section II and the parameters

presented in the Table 1 . The results show that using the

active method can diminish the loss effects. Meantime, it

also has influence on phase constant and real part of

characteristic impedance.

(a) Real part

(b) Imaginary part

Figure 4. Real and imaginary parts of Z0 as function of fre-

quency for balanc e d case.

Figure 5. Attenuation constant curve for unbalanced case

(example 6-10).

Figure 6. Phase constant curv e for unbalance d case .

(a) Real part

(b) Imaginary part

Figure 7. Real and imaginary part of Z0 as function of fre-

quency for unbalanc e d case.

Y.-F. GE ET AL.

264

4. Conclusions [9] L. M. Si and X. Lv, “CPW-Fed Multi-Band

Omni-Directional Planar Microstrip Antenna Using Com-

posite Metamaterial Resonators for Wireless Communi-

cations,” Progress in Electromagnetics Research, Vol. 83,

2008, pp. 133-146. doi:10.2528/PI ER08050404

The EM material parameters equations for general active

and passive metamaterial are calculated and compared

from equivalent lumped elements modes. In the com-

parison analysis of balanced and unbalanced lossless (R

= G = 0), passive lossy (R > 0, ) and active lossy

(R < 0, ) cases, different losses effects (from the

changes of resistance in this paper) have some effects on

phase constant and the real part of characteristic imped-

ance, as well as a big influence on attenuation constant

and the imaginary part of characteristic impedance. One

can use included active element to metamaterial to im-

prove the attenuation characteristics and then to over-

come the major drawback of losses effects in metamate-

rials.

0G

[10] L. M. Si, W. Zhu and X. Lv, “Determination of the Ef-

fective Constitutive Parameters of Active Transmission

Line Metamaterials,” 2012 International Workshop on

Metamaterials, Nanjing, pp. 1-4, 2012.

doi:10.1109/META.20 12.6464919

[11] L. M. Si, H. J. Sun, Y. Yuan, and X. Lv, “CPW-fed

Compact Planar UWB Antenna with Circular Disc and

Spiral Split Ring Resonators,” Progess in Electromag-

netics Research Symposium, 2009, pp. 502-505.

[12] L. M. Si, H. J. Sun and X. Lv, “Numerical Simulations of

Backward-to-Forward Leaky-Wave Antenna with Com-

posite Right/Left-Handed Coplanar Waveguide,” Chinese

Physics Letters, Vol. 27, 2010.

doi:10.1088 /0256-307X /27/3/034106

5. Acknowledgements [13] L. M. Si and X. Lv, “Terahertz Waves Hairpin Microstrip

Band-Pass Filter and Its Application to Overlaid Dielec-

tric Material Detection,” Modern Physics Letters B, Vol.

22, pp. 2843-2848, 2008.

doi:10.1142 /S02179849080 17412

This work has been supported by the National High

Technology Research and Development Program of

China (Grant No. 2012AA8123012), the Basic Research

Foundation of Beijing Institute of Technology (Grant No.

20120542015), and the National Natural Science Foun-

dation of China (Grant No. 61275107).

[14] Y

. Liu, L. M. Si, M. Wei, P. Yan, P. Yang, H. Lu, C.

Zheng, Y. Yuan, J. Mou, X. Lv and H. Sun, “Some Re-

cent Developments of Microstrip Antenna,” International

Journal of Antennas and Propagation, Vol. 2012, 2012.

doi:10.1155 /2012/428284

[15] A. K. Iyer and G. V. Eleftheriades, “Negative Refractive

Index Metamaterials Supporting 2-D Waves,” 2002 IEEE

MTT-S International Microwave Symposium Digest, Vol.

2, 2002, pp. 1067-1070.

doi:10.1109/MWSYM.200 2.1011823

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