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: *lms@bit.edu.cn
Received 2013
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-
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.
Copyright © 2013 SciRes. OPJ
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 [
]) 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
 (1)
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.
 
 
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)
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
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)
3. Numerical Simulations and Comparison
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
) is:
 
where r
is the relative permittivity (dielectric constant),
and is permittivity in free space.
Take the low cost substrate material Duroid (
tan 0.0009
) as an example, which has been widely
used for RF/Microwave engineering, the corresponding
Copyright © 2013 SciRes. OPJ
Copyright © 2013 SciRes. OPJ
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.
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
 
 
 
224222224222 22
1 12121
 
 
422 2222
0422222 2
exp 2
 
 
Table 1. Equivalent lumped circuit parameters of active and passive metamaterials.
Example R []
/mLR []
/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.
Copyright © 2013 SciRes. OPJ
Copyright © 2013 SciRes. OPJ
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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-
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