A Wideband Metamaterial-Inspired Compact Antenna Using Embedded Non-Foster Matching
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A Wideband Metamaterial-Inspired CompactAntenna Using Embedded Non-Foster Matching
Hassan Mirzaei, Student Member, IEEE, and George V. Eleftheriades, Fellow, IEEEInvited Paper
Abstract—Passive electrically small antennas have a smallradiation resistance and a narrow bandwidth. In this paper,a new type of active antenna is reported in which an activecircuit generating a non-Foster impedance is embedded in ametamaterial-inspired small antenna with an inherent sizableradiation resistance. This circuit interacts with the reactiveelements of the antenna. The end result is a compact, broadbandantenna which is well matched and has the potential for a highefficiency. In this process, the need for an active (or passive)step-up transformer for the radiation resistance is eliminated.This work opens up the possibility of utilizing non-Fostercomponents embedded in metamaterial-based antenna structuresto obtain efficient antennas by manipulating the current and fielddistribution in and around the antenna.
Index Terms—Metamaterials, active antennas, negative resis-tance circuits, electrically small antennas, impedance matching.
I. INTRODUCTION
The input impedance of an antenna Za has a resistive Ra
and a reactive part Xa. The resistive part consists of a radiation
resistance Rr corresponding to the radiated power and a loss
resistance Rl associated with the dissipated power:
Za = Ra + jXa = (Rr +Rl) + jXa (1)
The input impedance of a small antenna is highly reactive
(i.e. |Xa| large) and the radiation resistance is small (in the
series model implied by Eq. (1)) which, in turn, implies a large
quality factor, Q =∣∣∣Xa
Ra
∣∣∣. Indeed, the minimum achievable Q
from an antenna is inversely related to its physical size by a
fundamental limit theorem referred to as the Chu-Harrington
limit [1], [2]. If one decides to match a high-Q small antenna
with a lossless passive matching network, the achievable
matching bandwidth is also limited by another fundamental
limit called the Bode-Fano-Youla limit [3]–[5]. This limit
inversely relates the matching bandwidth to the Q of the load
to be matched (here the antenna). To overcome this limit,
the employment of active non-Foster components has been
proposed in the matching network of antennas (e.g. see [6]–
[8]). Non-Foster components are simply negative reactances
(or susceptances) that do not follow the Foster’s reactance
theorem. This theorem is based on the conservation of energy
for lossless passive two-terminal devices and states that the
derivative of the reactance (and susceptance) with respect to
frequency must be positive. The non-Foster components are
implemented using active circuits called ‘negative impedance
converters’ (NICs). The function of a NIC is shown in Fig. 1.a.
As this figure suggests, a NIC is a two-port network for
which the input impedance is the negative of the impedance
Fig. 1. The input impedance looking into one port of a two-port NIC is thenegative and scaled value of the impedance connected to the other port.
Fig. 2. (a) A series RC as the simplified equivalent circuit of a shortmonopole. (b) Because of the large Q, the series RC can be approximatedwith a parallel RC. (c) Non-Foster impedance matching for a short monopoleusing a single non-Foster capacitor. (d) A more complete theoretical schemefor perfect matching of a short monopole.
connected to the other port. In general, the impedance can be
scaled by a k factor as well.
In most of the reported applications of the non-Foster
components in matching networks, an antenna from the dipole
or monopole family has been used. At low frequencies where
the antenna is electrically small, the input reactance of the
antenna is highly capacitive (Xa negative) and Za in Eq. (1)
can be approximated with a series RC network as shown in
Fig. 2.a [6]. Then, because of the high Q for small antennas,
the series RC can be approximated with a parallel RC in
Fig. 2.b. Using this simplified model it can be seen that placing
a parallel negative capacitance at the input terminal of the
small antenna cancels out the antenna’s reactance (Fig. 2.c).
However, in theory, the perfect matching of an antenna
using non-Foster components requires four ideal non-Foster
components as shown in Fig. 2.d. The series RLC model
for the antenna is based on the fact that wire antennas show
a series resonance behavior up to their first resonance. The
radiation resistance of the antenna is small and frequency
dependent (proportional to f2). After canceling out the series
LC with non-Foster components, to match this resistance
to the characteristic impedance of the system over a broad
1950978-1-4244-9561-0/11/$26.00 ©2011 IEEE AP-S/URSI 2011
frequency range, a step-up ‘T’ transformer is used to cancel out
the frequency dependence of Rr as well. This scheme is too
complicated and considering all practical trade offs, usually the
former approach with only a negative capacitor is used. For
this scheme, in practice, the loss resistance or conductance that
usually accompanies the non-Foster components can become
comparable or even greater than the radiation resistance,
hence, degrade the radiation efficiency to a great extent.In this paper we use another approach. We start with
a compact metamaterial-inspired resonant antenna which is
capable of achieving a high radiation resistance close to the
characteristic impedance of the system. However due to the
high Q of small antennas, such an antenna exhibits a small
bandwidth (e.g. [9], [10]). Subsequently, non-Foster compo-
nents are employed inside the antenna structure in a fashion
that will be explained later in this paper. These non-Foster
reactances interact with the inherent Foster reactances of the
antenna, resulting in a broadband small antenna with a high
radiation resistance. This approach is remarkable from several
points of view; while keeping the size of the antenna intact,
it eliminates the need for any external matching network, thus
saving cost and space. The end result is a compact, wideband
antenna that can be a good candidate for wireless applications.
II. THEORY: ANTENNAS WITH EMBEDDED NON-FOSTER
COMPONENTS.
The employment of lumped components inside the structure
of antennas to tune their resonant frequency is a well known
method. As an example, top-hat or inductor-loaded monopoles
can be considered in this category. There are also some
metamaterial-inspired small antennas with internal lumped
components [11], [12]. The operating frequency of these
antennas can be tuned by changing the value of the internal
lumped elements. These metamaterial-inspired antennas have a
large radiation resistance but a limited bandwidth. In Fig. 3.a,
one of these antennas with an internal interdigital capacitor
is shown [11]. This dual-band antenna acts like a printed
monopole in its higher band where the interdigital capacitor
is effectively a short circuit. At its lower band, which is the
focus of this paper, the structure of the antenna resembles one
unit cell of a metamaterial transmission line. At this band, the
currents in the two sides of the vertical slot are in opposite
directions and the slot radiates. Although, the original antenna
is intended for WiFi applications with the lower band around
2.44 GHz [11], the antenna can be easily scaled to operate at
any frequency band.The antenna scaled to 300 MHz is depicted in Fig. 3.b.
Apart from some modifications required to embed an active
circuit that will be explained in Section III, the antenna is
quite similar to the original one in Fig. 3.a. By replacing the
interdigital capacitor with a tunable lumped capacitor Ct , the
resonant frequency fr of the antenna can be tuned. This is
shown in Fig. 3.c (results from Ansoft HFSS). In the inset of
this figure, the tuning susceptance Bt = ωCt vs. fr is depicted
which follows a non-Foster behavior, because dBt
dω < 0. [12],
[13].This means that by replacing the tuning capacitor with
an appropriate combination of non-Foster components, all
Fig. 3. (a) A dual-band metamaterial-inspired antenna with a lumpedinterdigital capacitor in which the interdigital capacitor can be replaced by alumped tunable capacitor Ct. (b) Antenna scaled to the UHF band is modifiedin order to provide the DC bias path for the embedded active circuit. (c)Antenna’s resonant frequency fr can be tuned by a tuning capacitor Ct in abroad frequency range (results from Ansoft HFSS). The tuning susceptanceBt vs. resonant frequency fr shows a non-Foster behavior. At resonance, thede-embedded S11 of the antenna on the Smith chart shows a series-resonancebehavior.
required tuning susceptances can be realized at once resulting
in satisfying the resonance condition over a wide bandwidth;
hence a wideband antenna with a large radiation resistance can
be obtained. The quantification for Ct starts with the lumped-
element equivalent circuit for the antenna. If one can assume a
simple series or parallel RLC model for the antenna then the
following approach can be used similar to the one presented in
[12] for a tuning inductor. For the antenna under consideration
the de-embedded S11 is depicted in the inset of Fig. 3.c on the
Smith chart. From this figure, the resonant frequency of the
antenna is associated with a resonance (as opposed to an anti-
resonance) on the Smith chart, hence, a series RLC model as
in Fig. 4.a can be assumed. Using this simplified model, the
resonant frequency can be expressed as:
fr =1
2π√La(Ca + Ct)
(2)
In this equation La and Ca are the antenna’s self capacitance
and inductance and are attributed to the energy storage in the
antenna’s near field. From this equation Ct can be calculated
as:
Ct =A
f2r
+B
A =1
4π2La;B = −Ca (3)
1951
Fig. 4. (a) A simplified model for the antenna around the resonant frequency.(b) For wideband operation, the tuning capacitor Ct can be replaced with aparallel combination of non-Foster components Lnf and Cnf .
This equation shows that Ct is inversely proportional to the
square of fr. The associated tuning suscptance can be found
in the corresponding frequency band:
jBt = jωCt =1
jω( −14π2A )
+ jωB
=1
jωLnf+ jωCnf (4)
From this equation, as depicted in Fig. 4.b., Ct is equivalent to
the parallel combination of Lnf and Cnf where Lnf and Cnf
are negative and denote the equivalent non-Foster element that
can replace Ct to obtain a broadband antenna. It should come
as no surprise that in fact Lnf and Cnf are the negatives of
the antenna’s self-inductance and self-capacitance respectively,
because:
Lnf =−1
4π2A= −La
Cnf = B = −Ca (5)
In practice, fr can be simulated or measured in some
discrete tuning element values. As suggested in [12] by fitting
a curve of the kind shown in Eq. (3) to the discrete data
available, the unknowns A and B in this equation can be
determined. Then Eq. (5) can be used to calculate the non-
Foster inductance and capacitance.
III. METAMATERIAL-INSPIRED NON-FOSTER ANTENNA
To replace the passive tuning capacitor in Fig. 3.b with an
active non-Foster circuit, as mentioned before, some modifi-
cations in the structure of the antenna have been made. These
modifications are mainly for providing a DC bias for the active
circuit without affecting the antenna performance. In addition,
the reason for scaling the antenna to operate around 300 MHz
is to relax the design of the non-Foster active circuit using
discrete components.
In the scaled and modified antenna in Fig. 3.b., the bias
voltage is fed through the antenna feeding port. A DC return
path is provided using an RFC placed between the left side of
the antenna and the GND plane and an additional slot separates
this path from the feedline where an AC coupling capacitor is
used to provide an AC short circuit.
To calculate the required non-Foster components to be used
instead of the tuning capacitor, the procedure explained in
Section II is followed. The fr vs. Ct data is extracted from
Fig. 3.c using HFSS simulation results and is shown along
Fig. 5. (a) A curve can be fitted to the Ct vs. fr in which Ct is inverselyproportional to f2
r . (b) By replacing Ct with an ideal parallel Lnf and Cnf
combination, a broadband antenna is obtained.
Fig. 6. Linvill basic NIC two-transistor circuits; (a) floating; (b) single-ended.
with similar data from Agilent ‘Advanced Design System’
(ADS) in Fig. 5.a. From this data, the A and B parameters in
Eq. (3) can be calculated using curve fitting in which Ct is
inversely proportional to f2r . Finally, using Eq. (5), Lnf , Cnf
are calculated and shown in Fig. 5.a
To confirm the function of the non-Foster components in
broadening the bandwidth of the antenna, a simulation is
carried out in which the calculated ideal parallel negative
capacitors and inductors are used instead of the tuning capac-
itor Ct in Fig. 3.b. The result of the antenna input reflection
coefficient S11, can be seen in Fig. 5.b. showing the expected
wideband antenna behavior. We find out, however, that the
response is very sensitive to the values of the non-Foster
components. Since in any practical implementation of non-
Foster components there is some variation with frequency,
this suggests that a smaller bandwidth should be expected and
suitable means for tuning the circuit for optimal performance
should be provided as well.
IV. NIC CIRCUIT
To realize the required non-Foster components the Linvill
two-transistor schemes can be used [7], [14]. The basic floating
and single-ended configurations are shown in Fig. 6. The
complete circuit will include the core part in Fig. 6 plus bias
circuitry and some compensating and stabilizing components
[7]. To have more flexibility in tuning the circuit, high-Q
trimmer capacitors are used in the design. The layout of
the circuit should be designed such that it would fit into
the space provided as can be seen in Fig. 3.b. To account
for parasitics from the layout, the EM/circuit cosimulation
feature of the Agilent ADS is used. The goal of the simulation
1952
Fig. 7. (a) Variation of Lnf with frequency assuming a constant value forCnf . The small series resistance value is required for having a high-Q Lnf .(e) Variation of Cnf with frequency assuming a constant value for Lnf . Thesmall parallel conductance value is required for having a high-Q Cnf .
is to obtain a parallel combination of two non-Foster Lnf
and Cnf components as calculated in Section III. Since the
circuit is frequency dependent, it should be compensated such
that the variation of Lnf and Cnf with frequency would be
in a reasonable range. The simulation results showing the
frequency dependency of the Lnf and Cnf are shown in
Fig. 7.a and b respectively along with the loss elements. The
loss elements of the non-Foster component are namely the
series resistance for Lnf and parallel conductance for Cnf .
The simulation results show that the Lnf and Cnf elements
have a large quality factor which is necessary for a low-loss
operation of the active circuit.
V. EXPERIMENT
The fabricated antenna with the embedded NIC circuit is
shown in Fig. 8.a. which depicts both the top and bottom
layers. In the bottom layer, the trimmer capacitors can be
observed.
As mentioned in Section III, since the antenna performance
is very sensitive to the value of non-Foster components and
due to the variation of the implemented non-Foster compo-
nents with frequency as in Fig. 7.d and e, achieving a very
large bandwidth is difficult. However, it is possible to tune
the NIC circuit for operation at different frequencies with a
reasonable bandwidth. This possibility is illustrated in Fig. 8.b.
As can be observed in this figure, in part of the frequency range
S11 is greater than zero dB. This is due to a small negative
resistance generated by the NIC circuit and translated to the
input port, however, since this negative resistance is well below
the characteristic impedance, the system is stable. It should be
noted that if high-Q tunable inductors with a small size were
accessible the process of tuning would be easier and more
efficient.
VI. CONCLUSION
A metamaterial-inspired antenna with embedded non-Foster
components has been presented. The resulting structure is
a compact wideband antenna without any external matching
network. This opens up the possibility of embedding non-
Foster components in the structure of antennas to manipulate
current and field distribution in and around the antenna in
order to obtain compact and efficient antennas.
Fig. 8. (a) Photos of the fabricated antenna for the top and bottom layers.(b) Experimental results showing the possibility of tuning the operating bandof the antenna.
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