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)

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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

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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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