A Resonant Printed Monopole Antenna With an Embedded Non-Foster Matching Network

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 11, NOVEMBER 2013 5363

A Resonant Printed Monopole Antenna With anEmbedded Non-Foster Matching NetworkHassan Mirzaei, Student Member, IEEE, and George V. Eleftheriades, Fellow, IEEE

Abstract—Non-Foster reactive elements are embedded in-side compact resonant antennas, rather than being employedin matching networks located at the antenna terminals. Theseembedded non-Foster elements interact with the inherent Fosterreactances of the antenna, resulting in a broadband antennawith a high radiation resistance. The class of suitable antennasfor this application is discussed, a design procedure is presentedand a sensitivity and a stability analysis are performed. Finally,experimental results for a representative non-Foster antenna arepresented and discussed.

Index Terms—Active antennas, electrically small antennas,impedance matching, negative resistance circuits, non-Fostermatching.

I. INTRODUCTION

A FUNDAMENTAL limit, due to Wheeler [1], Chu [2] andHarrington [3], inversely relates the minimum achiev-

able of an antenna to its physical size; therefore, a small an-tenna assumes a high quality factor. In turn, to match a high-Qsmall antenna with a lossless passive matching network, theachievable matching bandwidth is limited by another funda-mental limit due to Bode [4], Fano [5] and Youla [6]. This limitinversely relates the matching bandwidth to the of the load tobe matched (here, the antenna).The inclusion of active components in the antenna, or in the

matching network, is key to overcome either of these funda-mental limits. On this basis, the employment of non-Foster re-active elements in the matching network of antennas has beenproposed [7]. These elements do not obey the Foster’s reac-tance theorem for passive reactive elements [8]. Negative ca-pacitors and negative inductors are representative examples ofnon-Foster reactive elements which can be implemented usingtwo groups of active networks called “negative impedance con-verters” (NICs) and “negative impedance inverters” (NIIs). Acollection of NIC circuits can be found in [9] and [10], whereasa NII circuit has been presented in [11].In antenna-matching applications, the reported works mostly

include the employment of non-Foster components at theterminals of dipole or monopole antenna families [7], [10],[12]–[14]. The loop antenna has also been considered for this

Manuscript received May 23, 2012; revised June 11, 2013; accepted July 30,2013. Date of publication August 06, 2013; date of current version October 28,2013.The authors are with the Edward S. Rogers Sr. Department of Electrical and

Computer Engineering, University of Toronto, Toronto, ON M5S 2E4, Canada(e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2013.2276912

application [15]. Ideally, an antenna with a small and fre-quency-dependent radiation resistance (e.g., a small monopole,a dipole and a loop antenna) can be matched to the characteristicimpedance of the system using four non-Foster reactances [16],[15]. These four elements include one negative inductor andone negative capacitor in series with the antenna terminal, plusa non-Foster T-transformer containing two negative inductorsin its series branches. This non-Foster transformer, in additionto stepping up the small antenna resistance, is employed tocancel out the inverse-quadratic frequency dependence of theradiation resistance. In a practical implementation however, thegrounded non-Foster elements are generally easier to imple-ment than the floating ones. In addition, the stability of everysingle non-Foster element has to be ensured in a given design.Considering these factors and the underlying complexity, otherschemes with a smaller number of non-Foster elements areusually used in practice.One popular scheme only uses a single parallel negative ca-

pacitor at the terminals of a monopole antenna. Using this singlenegative capacitor the reactive part of the input impedancecan be effectively eliminated, over a broad frequency band,but the remaining resistive part is highly frequency dependent,assumes very large values and is not matched to the charac-teristic impedance of the system. Nevertheless, improvementsin gain [7], [12] and signal-to-noise ratio (SNR) [14] over awide frequency range have been reported using such a simplescheme. The reasoning behind this method has been outlinedby Harris et al. [7] and recited in [14], [18].In this paper, we take another approach. We work on passive

antennas which are preferably small, but unlike the smallmonopoles and dipoles, provide a reasonable radiation resis-tance at their resonant frequency, comparable to the systemcharacteristic impedance. These antennas are preferable asthey eliminate the external non-Foster transformer requirement[16] for the applications where the antenna has to be matchedto the characteristic impedance of the system. Suitable smallantennas, which are resonant and exhibit large radiation resis-tance, are exemplified by [19], [20] and [21]; however, due tothe large for small antennas, they provide a small bandwidth.Moreover, rather than using the non-Foster reactive elements

at the terminals of the antenna, we investigate embedding theseelements inside the antenna to obtain a broadband matched an-tenna. In fact, in antenna applications, reactive loading with pas-sive Foster reactances is frequently employed to shape the cur-rent distribution on the antenna, and hence, control the resonantmodes, input impedance behavior and radiation pattern. How-ever, the possibility of loading an antenna with non-Foster re-active elements, in addition to Foster ones, can potentially lead

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5364 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 61, NO. 11, NOVEMBER 2013

to broadband applications [22]. The embedding technique, pre-sented in this paper, requires that the resonant frequency of theunderlying passive antenna can be reconfigured by a tunablepassive lumped element inside the antenna. A combination ofnon-Foster elements, then, replaces the passive tuning elementand make the antenna wideband. This approach, while keepingthe size of the antenna intact, eliminates the need for any ex-ternal matching network, as the antenna is intrinsically matchedto the characteristic impedance of the system.Related approaches have been theoretically proposed in [23],

[24], for a PIFA antenna, and in [25], whereas the authors of thepresent paper have demonstrated the first experimental resultsin [26]. In this paper, we extend the theory, perform sensitivityand stability analysis and present further experimental resultsbased on the usage of the non-Foster antenna as a receiver in atransmit/receive test setup.The paper is organized as follows. In Section II, the idea

of embedding non-Foster matching networks is described byadopting simple resonant models for the underlying passiveantennas. A practical design of embedded non-Foster matchingnetworks, comprising three suggested schemes, are presentedand applied to a sample antenna in Section III. Sensitivityanalysis is performed for these embedded matching networkschemes in Section IV. The fabrication details are presented inSection V, followed by experimental results in Section VI. Asuitable stability analysis is presented in Appendix A.

II. RESONANT ANTENNAS WITH EMBEDDED NON-FOSTERMATCHING NETWORKS

In this section, simple series and parallel resonant modelsare employed to explain the performance of an embeddednon-Foster matching network. It should be noted that thesesimple resonant models are only valid in a limited bandwidtharound the resonant frequency. However, they are used here tointuitively explain the performance of an embedded matchingnetwork. For the actual design of an embedded non-Fostermatching network, as explained in Section III, the model of theantenna is not used.Suitable antennas for embedding non-Foster matching net-

works include frequency-reconfigurable resonant antennas withan embedded passive tuning element or . Depending onthe type of the tuning element ( or ) and the resonant modelsuited for the antenna (series or parallel), four situations can beidentified, which are tabulated in Table I. In these models, theloading reactive elements and , in the underlying passiveantennas, appear, respectively, in parallel and in series with theantenna self-capacitance and self-inductance .Subsequently, in the non-Foster version, is replaced with

a parallel combination of - and - , whereas is replacedwith a series combination of these two elements, as shown inTable I. By such replacements, the self-reactances of the an-tenna are totally canceled out, and ideally, a broadband antennais obtained. In other words, in a frequency-reconfigurable pas-sive antenna, each resonant frequency corresponds to a spe-cific value of the tuning susceptance or reactance

. The idea behind the embedded matching networkis to use a two-terminal network, instead of or , to satisfythe resonance condition over a certain finite bandwidth rather

TABLE ISIMPLE RESONANT MODELS FOR PASSIVE ANTENNAS AND THEIR ACTIVECOUNTERPARTS WITH EMBEDDED NON-FOSTER MATCHING NETWORKS

than at a single frequency point. In reality however, this goalcannot be achieved by using only passive reactive components,and non-Foster reactive components are required, as inferredusing the simple resonant models in Table I, and as will be ex-plained in detail in Section III.The correspondence between the passive and non-Foster ver-

sions of antennas, shown in the last column of Table I, can befurther explained by evaluating the resonant frequency equa-tions for two cases where and are the loading element:

when loaded by (1a)

when loaded by (1b)

Calculating and from these equationsfor all resonant frequencies in the desired frequency band yields

for (2a)

for (2b)

which highlights the correspondence of and with aparallel and series combination of and , respectively.However, this correspondence is not unique and, in practice,other options are available, as explained in the followingsection.

III. PRACTICAL DESIGN OF EMBEDDED NON-FOSTERMATCHING NETWORKS

In practice, the detailed antenna model is not required for thedesign of an embedded non-Foster matching network. The de-sign can be performed totally based on the frequency-reconfig-urability of the underlying passive antenna. The design processleads to the synthesis of a two-terminal network which can re-place the tuning elements or in the passive antenna to

MIRZAEI AND ELEFTHERIADES: RESONANT PRINTED MONOPOLE ANTENNA 5365

Fig. 1. Selected antenna for embedding a non-Foster matching network alongwith the antenna dimensions.

achieve a broadband operation. In this section, we divide thedesign process into three steps. With the explanation of eachstep, we apply the design methodology to a selected antenna.

A. Step I: Selecting a Suitable Passive Antenna

Suitable antennas for the purpose of embedding non-Fostermatching networks, include tunable reactively loaded antennas(also known as frequency-reconfigurable or frequency-agile an-tennas). The resonant frequency of these antennas can be sweptover a broad range using a variable capacitor or an inductor. It should be noted that reconfigurability can also be achieved

using embedded RF switches to redirect the current into dif-ferent routes on the antenna surface. However, this form of re-configurability is not of interest for the current application.An example of a suitable antenna, selected to develop an em-

bedded non-Foster matching network in this paper, is shown inFig. 1. This antenna is a monopole patch loaded with a capacitorand fed by a coplanar waveguide (CPW) [19], [27]. To facili-tate the implementation of an embedded non-Foster matchingnetwork, the antenna is designed to operate at the lower UHFband. An FR4 substrate with relative permittivity equal to4.35 and thickness equal to 0.79 mm is used for this design.Although an antenna for the purpose of embedding a non-

Foster matching network can be of any size, we work on a de-sign that can be regarded as an electrically small antenna. Weare motivated by the fact that broadband electrically small an-tennas are appealing for handheld communication devices. Theantenna in Fig. 1 can be regarded as a small antenna in the fre-quency range of interest in this paper, according to the condi-tions stated by Wheeler in [28].

B. Step II: Frequency-Reconfigurability and Tuning Behavior

The resonant frequency of a selected antenna can be tuned bythe tuning element loading the antenna ( or ). The tuningbehavior is a graph of or versus resonantfrequencies , which can be measured or simulated over thedesired bandwidth in some discrete frequencies for the passiveantenna.The tuning behavior of the antenna under investigation in

Fig. 1 is simulated using HFSS for some discrete values of theloading capacitor. The results are plotted in Fig. 2(a), showing

Fig. 2. (a) Simulation data using HFSS demonstrates that the resonant fre-quency of the antenna can be tuned by tuning the loading capacitor . Fromthese simulation results, one can obtain (b) the graph of the tuning susceptance

versus frequency.

a large tuning bandwidth. The results in this graph are trans-lated to a versus graph as shown in Fig. 2(b). Thisgraph shows a negative slope in accordance with the expectednon-Foster behavior.

C. Step III: Fitting a Synthesizable Reactance Function to theTuning Data and Calculating the Parameters of the EmbeddedMatching Network

In this step, an appropriate function has to be fitted to thegraph representing the antenna tuning behavior. This functionshould represent a synthesizable reactance (susceptance func-tion or inductance function ) using a combina-tion of Foster and non-Foster elements. The idea here is to re-place the tuning element or with a network which satis-fies the resonance condition in a broad frequency range. As thesample graph in Fig. 2(b) illustrates, such a network synthesizesa non-Foster reactance at its terminals; hence, it can be imple-mented by the addition of and elements to the set of theavailable reactive elements (i.e., capacitors and inductors).In this section, we present three synthesizable functions that

can be fitted to the antenna tuning data. The associated net-works, representing these fitted functions, have at most two el-ements; at least one of which is a non-Foster reactive element( or ). The fact that the fitted curve to the tuning data isnot unique, implies that different networks can be employed forthe embedded non-Foster matching network. However, all thesecurves share the property that they exhibit a negative slope inthe frequency band of interest. These three options, which areshown in different rows of Table II, include:1) A function representing a combination of two non-Fosterelements.

2) A function representing only one non-Foster element.3) A function representing combination of one Foster and onenon-Foster element.

In each row of Table II, the type of the tuning element (or ), the reactance or susceptance function to be fitted to thetuning data ( or ), a graph representing the generalform of the fitted curve, and finally the non-Foster network re-placing the tuning elements with the values related to the coef-ficients of the fitted curve are shown in different columns.For the antenna under investigation, these three reactance

functions are fitted to the antenna tuning data of Fig. 2(b). Thefitted curves, along with the corresponding networks, are shownin Fig. 3. The results show that different functions approximatethe tuning behavior with a different level of accuracy. The best


TABLE IITUNING ELEMENTS CAN BE REPLACED WITH A NON-FOSTER NETWORK FOR ACHIEVING A BROAD IMPEDANCE BANDWIDTH. DIFFERENT EMBEDDED

NON-FOSTER NETWORKS ARE DESIGNED THROUGH THE FITTING OF SYNTHESIZABLE REACTANCE OR SUSCEPTANCE FUNCTIONS TO THE ANTENNA TUNING DATA

FIg. 3. Antenna tuning data with three fitted curves, representing three possibleembedded non-Foster matching networks.

choice depends on a compromise between the following dif-ferent factors:i) Error function: that is, how closely the fitted curve repre-sents the tuning data.

ii) Complexity: that is, how difficult the implementation ofthe network is. In particular, implementations with fewernon-Foster components are preferred.

iii) The values for the network elements: that is, very large orvery small values for the capacitors and inductors are notdesirable.

iv) The sensitivity of the fitted curve to the value of the net-work elements: that is, how the antenna response changesif there exists some variation in the implementation of thenetwork elements. This variation is more pronounced forthe non-Foster elements, because of dispersion and lim-ited quality factor of these elements when implementedusing active circuits. A sensitivity analysis for the threesuggested functions is presented in the following section.

IV. SENSITIVITY ANALYSIS

As outlined in Section III, a reactance function can be closelyfitted to the tuning data and this function can be synthesizedusing ideal and elements. Ideally, by replacingthe passive tuning element with the synthesized network a wide-band operation, as shown with the dashed line in Fig. 2(a), canbe obtained. However, several practical issues exist that limitthe achievable bandwidth. This section is intended to show thata reasonable error in the element values of the non-Foster net-work, used as an embedded non-Foster matching network, will


Fig. 4. Percentage error in the three susceptance functions, synthesized as embedded matching networks, due to a % error in the value of the correspondingnetwork elements. (a) The maximum relative error in (only) the susceptance functions. (b) The cumulative maximum relative error comprising the relative error inthe synthesized functions plus the relative error in the curve fittings (c) The significant effect of a 10% deviation from the ideal values of the embedded non-Fosterelements on the antenna’s input reflection coefficient, suggest that the improvement in the bandwidth using an embedded matching network will be limited (theresults are obtained from the Agilent ADS EM/Circuit co-simualtion).

inherently limit the achievable bandwidth. In order to keep theanalysis simple, we only consider the error in the value of theideal non-Foster components. Other practical factors includingthe quality factor of the implemented non-Foster elements, sta-bility considerations and dispersion in an actual implementationof the non-Foster elements will further limit the bandwidth im-provement. However, including all these factors will make thesensitivity analysis difficult.The sensitivity analysis is performed in two-steps. First, we

assume a reasonable relative error in the value of the elementsof the non-Foster networks in the right column of Table II. Thenwe calculate the cumulative error for each implementation forthe antenna presented in this paper.The sensitivity of the three synthesized susceptance and in-

ductance functions, in Section III-C, Table II, to a relative errorin the values of the elements, is calculated and summarized inTable III. To perform a fair sensitivity comparison, the relativeerrors, rather than the absolute errors, for the reactance functionsare calculated. These relative errors are then plotted in Fig. 4(a),for the three embedded matching networks of Fig. 3, designedfor the antenna under investigation. This plot shows the fre-quency response of the maximum relative error for the threesusceptance functions, assuming a % relative error in thevalue of the network elements. Although this graph suggests alarger maximum relative error for the non-Foster network com-posed of two non-Foster components, the cumulative error cal-culated below suggests otherwise.To calculate the cumulative error, it should be noted that dif-

ferent susceptance functions in Table III can be fitted to the an-tenna tuning data with different levels of accuracy, as shown inFig. 3. Indeed, the susceptancewith the larger maximum relativeerror in Fig. 4(a), can be fittedmore closely to the antenna tuningdata. The cumulative maximum relative errors, for the three de-signed embedded non-Foster matching networks, are calculatedfrom the following relation and plotted in Fig. 4(b):

cumulative relative error

(3)

TABLE IIIRELATIVE SENSITIVITY FOR THE THREE TYPES OF EMBEDDED NON-FOSTER

MATCHING NETWORKS, PRESENTED IN TABLE II

where is due to the relative error in the value of the net-work elements, which is calculated from the third column ofTable III, and represents the exact value of the an-tenna tuning data (or its best approximation from the discretedata points in Fig. 3). The plot in Fig. 4(b) shows that the cumu-lative errors of all designs are comparable and can attain largevalues. However, the design with a parallel combination of twonon-Foster elements results to a smaller overall cumulative error


Fig. 5. Photo of the fabricated antenna from the top and bottom views, showingthe DC biasing of the embedded non-Foster circuit.

in the bandwidth of interest and thus is selected for implementa-tion of an embedded non-Foster matching network in this paper.To observe the effect of varying the non-Foster element

values on the bandwidth, the antenna in Fig. 1 is simulatedusing the ideal non-Foster loading elements, and subsequentlywith the ideal non-Foster elements with a 10% deviation. The

simulation results, shown in Fig. 4(c), demonstrate thatthe effect of variations in the element values is significant onthe matching. In practice, these errors can increase even morewith the limited quality factor of the implemented non-Fosterelements. These results indicate that, due to these errors whichare unavoidable in a practical implementation, an extremelybroad bandwidth, similar to the dashed-line graph in Fig. 2(a),cannot be achieved. However, as shown in the experimentalresults in Section VI, significant improvement in the antennabandwidth can still be obtained.

V. FABRICATION

As stated in Section III-A, an FR4 substrate withand mm is used for this design. The schematic of theantenna, along with the dimensions, are shown in Fig. 1. Thefabricated antenna is depicted in Fig. 5, from the top and bottomviews (inset, the bottom view). The NIC circuit is implementedin a space equal to 15 8 mm . Although, the antenna itself isa single layer design, the circuit uses both sides of the boardin order to fit the design into the available space. The bottomlayer of the board is mostly used for running the DC bias linesall across the NIC circuit as well as for hosting trimmer capac-itors which are used for tuning the NIC circuit. The DC bias isprovided through the CPW signal line using a bias-T network,as shown in Fig. 5. This eliminates the need for the extra bi-asing wires or traces which can affect the antenna radiation per-formance. For providing a DC return path, a radio frequency

Fig. 6. Measured results of the non-Foster antenna for four differenttrimmer states.

choke (RFC) connects the monopole patch to the CPW ground,which separates the DC ground from the second terminal of theNIC circuit as well. The DC return path is separated from theCPW feedline by creating a narrow slot in the monopole patch.To maintain the AC signal flow across this DC-blocking slot, acoupling capacitor is placed over this slot. For this design, wearbitrarily use a Linvill’s NIC circuit scheme [29]; a detailedschematic of which for a below-hundred-megahertz design, hasbeen presented in [14]. Based on this design, a new circuit forthe frequency range of interest is designed. The stability of theembedded NIC circuit is verified by the Nyquist test applied tothe loop-gain (see Appendix A).

VI. EXPERIMENTAL RESULTS

A. Input Reflection Coefficient

In the design of Fig. 5, the NIC circuit can be tuned usingtrimmer capacitors to adjust the frequency band of operation.The input reflection coefficient results for four trimmerstates are shown in Fig. 6.The measured results of the antenna with the embedded

non-Foster matching network, in Fig. 6, are then comparedto the simulated results of the underlying passive antenna, inFig. 2(a). This comparison is presented in Table IV, showing asignificant improvement in the bandwidth.

B. Antenna as a Receiver in a Transmit/Receive (Tx/Rx)System

In another experiment, a prototype of the non-Foster activeantenna is employed as a receiver in a Tx/Rx experiment. Theexperiment is performed in an anechoic chamber. The experi-mental setup schematic along with a snapshot of the setup inthe anechoic chamber are shown in Fig. 7. As outlined in theschematic of the test setup in Fig. 7(a), an arbitrary waveformgenerator generates a Gaussian pulse which modulates the RFsignal from a signal generator. The Gaussian shape is used inthis experiment in order to transmit a signal with a controlledbandwidth and with no sidelobes in the frequency domain. The


TABLE IVCOMPARISON BETWEEN THE MEASURED RESULTS OF THE NON-FOSTERANTENNA AND THE SIMULATED RESULTS OF THE ASSOCIATED PASSIVE

ANTENNA FOR DIFFERENT TRIMMER STATES

Fig. 7. (a) Schematic of the Tx/Rx experiment where the non-Foster antenna isused as a receiving antenna. (b) The experimental setup in the anechoic chamber.

full width at half maximum (FWHM) of the Gaussian pulse is450 ns which corresponds to a bandwidth equal to 2 MHz inthe spectral domain. The modulated signal, with a peak powerequal to 20 dBm, is then fed to a horn antenna and transmittedtoward the Rx antenna, located about five meters away acrossan anechoic chamber. A bias-T is used for DC biasing of the Rxantenna and the output is connected to a high-frequency oscillo-scope with a matched input, for time domain observation of thereceived signal. In the receiver side, the non-Foster antenna iscompared to a passive antenna tuned to operate at the same fre-quency by a loading capacitor. Both antennas under test (AUTs)are shown in the inset of Fig. 7(b).These two non-Foster and passive prototypes are tuned to

have an dip around 377 MHz, as shown in Fig. 8, with a2.8% and a 0.5% bandwidth, respectively. This improvement inbandwidth corresponds to a factor of 5.6. The experimental re-sults for three different frequencies, at 368 MHz (close to theedge of the 10 dB for the non-Foster antenna), 375 MHz(close to the center frequency for both antennas) and 388 MHz(outside the band for both antennas), are shown on the samescale in Table V. The results demonstrate that in general, thelevel of the received power by the non-Foster antenna is higher

Fig. 8. Input reflection coefficient of the non-Foster and passive antennas usedin the Tx/Rx experiment are measured using a vector network analyzer. Bothantennas are tuned to operate around the same center frequency.

TABLE VCOMPARISON OF THE NON-FOSTER ANTENNA WITH A PASSIVE ANTENNA INTHE RX MODE. THE OSCILLOSCOPE IS TERMINATED TO A 50-OHM RESISTORTO PROVIDE A MATCHED LOAD FOR THE ANTENNA. THE OSCILLOSCOPEHORIZONTAL AND VERTICAL SETTINGS IN THESE FIGURES ARE 200

ns/DIVISION AND 100 mV/DIVISION, RESPECTIVELY

at these frequencies. The level of the received signal for the pas-sive antenna drops outside of its 10 dB band, while, for thenon-Foster antenna, it is maintained over a significantly largerbandwidth.

VII. CONCLUSION

Reactively loaded frequency-reconfigurable resonant an-tennas have been suggested as candidates for embeddingnon-Foster matching networks. It has been shown that byreplacing the reactive loading of these antennas with a suitablenon-Foster network, a broad impedance bandwidth can beachieved. The bandwidth increase is obtained by decreasingthe reactive parts of the antenna impedance, rather than de-creasing the antenna quality factor by loading with lossyelements. A design methodology has been proposed in whichan embedded matching network can be designed with theantenna’s frequency-reconfigurability data. Three suitable em-bedded matching networks, for antennas with either capacitiveand inductive loading, have been proposed. These networkscontain at most two reactive elements, at least one of whichis a non-Foster reactive element. The tradeoffs in choosingthe most appropriate network in a practical implementationhave been discussed. A detailed sensitivity analysis has beenperformed, and the cumulative maximum relative errors havebeen calculated for the three proposed networks. The stabilityof the embedded matching network has been verified by the


Nyquist test applied to the loop-gain of the NIC circuit (seeAppendix A). A sample antenna has been fabricated and testedfor its impedance bandwidth and in a Tx/Rx experiment. Theexperimental results show that a significant improvement inthe impedance bandwidth compared to the underlying passiveantenna can be obtained. Moreover, the Tx/Rx experimentshows that the antenna with an embedded non-Foster matchingnetwork can maintain the received signal level over a broaderbandwidth. Nevertheless, the ultimate decision of whetherthese antennas can surpass their passive counterparts dependson practical considerations, including the DC power consump-tion and stability issues. More importantly, perhaps, a detailedsignal-to-noise ratio investigation is still due for non-Fosteractive antennas, but it is out of the scope of the present paper.

APPENDIX ASTABILITY ANALYSIS

A stable terminated NIC circuit should have no poles in theright-half of the complex-frequency plane (no right-half plane(RHP) poles). When employed as an embedded non-Fostermatching network, the NIC circuit is terminated by the antennasurrounding the circuit.To test the existence of the RHP poles, the Nyquist test is

applied to the loop-gain of a feedback system (here the NICcircuit). The standard Nyquist test is performed by plotting theloop-gain in a complex-frequency plane for the complex-fre-quency variable “ ” traveling over a contour around the RHPin the clockwise (CW) direction. The number of the CW encir-clements of the point is equal to the differencebetween the number of RHP zeros and poles of the characteristicequation . Assuming that for a physical system hasno RHP poles, the CW encirclement of the pointis equal to the number of RHP zeros of , that is the number ofthe RHP poles of the terminated NIC circuit.To measure the loop-gain in a NIC circuit, a test point should

be selected around the feedback loop, a signal (voltage orcurrent) should be injected and the gain should be measured inone circulation around the loop. To obtain an accurate value ofthe loop-gain, the signal injection should not alter the actualperformance of the loop. Thus, the test point should be selectedwhere looking into opposite directions the impedance levelis essentially different (high/low). Then, if this condition issatisfied and if looking forward the impedance level is high,a voltage injection and otherwise a current injection methodgives accurate results. However, in the Linvill’s NIC circuitemployed in this implementation, an appropriate point satis-fying the above high/low impedance condition cannot be found.In such a situation, Middlebrook [30] has proposed a methodin which the loop-gain can be obtained by two tests involvingboth the current and voltage injection methods for arbitraryimpedance levels at a test point.To explain the method, the antenna with the embedded NIC

circuit (Linvill’s voltage-inverting NIC [29]) is shown in Fig. 9,where the extra elements including the biasing circuitry and thelayout of the printed circuit board (PCB) for the NIC circuit arenot shown for simplicity. The schematics of the loop-gain mea-surement using the Middlebrooks current and voltage injection

Fig. 9. Antenna with a NIC circuit used as an embedded matching network.In the insets of this figure the current and voltage injection schematics for theloop-gain measurement are shown.

Fig. 10. Nyquist plot of the loop-gain for positive real frequencies shows thata CW encirclement of the point is canceled out with a CCWencirclement. The inset of the figure shows the Nyquist plot for an extendedfrequency range.

methods are shown in the inset of Fig. 9. The voltage loop-gain, and the current loop-gains are calcu-

lated separately. Subsequently, the total loop-gain is obtainedusing [30]:

(4)

The antenna (including the circuit layout) and the embeddedmatching circuit are simulated using the EM/Circuit co-sim-ulation tool of the Agilent ADS. The Nyquist plot of theloop-gain is plotted in Fig. 10 for the frequencies between0.1–2.85 GHz and for the extended frequency range up to5 GHz in the inset of this figure. The frequency limits areselected based on the validity of the models for the circuitcomponents. This plot shows a rather complicated pattern, andfor the frequencies above 2.85 GHz, the Nyquist plot shows noencirclement of the point while tending towardsthe origin as expected. With the help of the plot, it is observedthat a clockwise gain crossover at 262 MHz is canceled outby a counterclockwise gain crossover at 2.841 GHz. A similar


situation happens for the negative real frequencies for whichthe pattern is a mirror image with respect to the real axis. Theseresults indicate that, due to the total zero encirclement of thepoint , the NIC circuit is stable.

REFERENCES[1] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc.

IRE, vol. 35, no. 12, pp. 1479–1484, Dec. 1947.[2] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl.

Phys., vol. 19, no. 12, pp. 1163–117, Dec. 1948.[3] R. F. Harrington, “Effect of antenna size on gain, bandwidth and effi-

ciency,” J. Res. Nat. Bur. Stand., vol. 64D, no. 1, pp. 1–12, Jan.–Feb.1960.

[4] H. W. Bode, Network Analysis and Feedback Amplifier Design. NewYork, NY, USA: D. Van Nostrand, 1947.

[5] R.M. Fano, “Theoretical limitations on the broadbandmatching of arbi-trary impedances,” J. Franklin Inst., vol. 249, no. 1, pp. 57–83, 1950.

[6] D. C. Youla, “A new theory of broad-band matching,” IEEE Trans.Circuit Theory, vol. CT-11, no. 1, pp. 30–50, Mar. 1964.

[7] A. D. Harris and G. A. Myers, “An investigation of broadband minia-ture antennas,” Naval Postgraduate School, Monterey, CA, Tech. Rep.AD0677320, Sep. 1968.

[8] R. M. Foster, “A reactance theorem,” Bell Syst. Tech. J., vol. 3, p.259267, 1924.

[9] S. E. Sussman-Fort, “Gyrator-based biquad filters and negativeimpedance converters for microwaves,” Int. J. RF MicrowaveComput. Aided Eng., vol. 8, no. 2, pp. 86–101, 1998.

[10] S. D. Stearns, “Non-Foster circuits and stability theory,” in Proc. IEEEAntennas Propag. Soc. Int. Symp., Jul. 2011, pp. 1942–1945.

[11] S. Kolev, B. Delacressonniere, and J. L. Gautier, “Using a negativecapacitance to increase the tuning range of a varactor diode in MMICtechnology,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp.2425–2430, Dec. 2001.

[12] A. K. Perry, “Broadband antennas systems realized by active circuitconjugate impedance matching,” M.S. thesis, Naval PostgraduateSchool, Monterey, CA, USA, Sep. 1973, Acc. No. AD769800.

[13] A. Bahr, “On the use of active coupling networkswith electrically smallreceiving antennas,” IEEE Trans. Antennas Propag., vol. AP-25, no. 6,pp. 841–845, Nov. 1977.

[14] S. E. Sussman-Fort and R. M. Rudish, “Non-Foster impedancematching of electrically-small antennas,” IEEE Trans. AntennasPropag., vol. 57, no. 8, pp. 2230–2241, Aug. 2009.

[15] T. K. Albee, “Broadband VLF loop antenna system,” U.S. patent3,953,799, Apr. 1976.

[16] G. Skahill, R. M. Rudish, and J. Piero, “Electrically small, efficient,wide-band, low-noise antenna elements,” in Proc. Antenna Applicat.Symp., 1998, pp. 214–231.

[17] G. Skahill, R. M. Rudish, and J. A. Pierro, “Apparatus and method forbroadband matching of electrically small antennas,” U.S. patent 6 121940, Sep. 2000.

[18] H. Mirzaei and G. Eleftheriades, “Antenna applications of non-Fosterelements,” in Proc. IEEE Int. Workshop Antenna Tech., Mar. 2012, pp.281–284.

[19] J. Zhu and G. V. Eleftheriades, “Dual-band metamaterial-inspiredsmall monopole antenna for WiFi applications,” Electron. Lett., vol.45, no. 22, pp. 1104–1106, 22, 2009.

[20] M. A. Antoniades and G. V. Eleftheriades, “A folded-monopole modelfor electrically small NRI-TL metamaterial antennas,” IEEE AntennasWireless Propag. Lett., vol. 7, pp. 425–428, Oct. 2008.

[21] A. Erentok and R. W. Ziolkowski, “Metamaterial-inspired efficientelectrically small antennas,” IEEE Trans. Antennas Propag., vol. 56,no. 3, pp. 691–707, Mar. 2008.

[22] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Application of charac-teristic modes and non-Foster multiport loading to the design of broad-band antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 1, pp.203–207, Jan. 2010.

[23] C. Di Nallo, G. Bit-Babik, and A. Faraone, “Wideband antenna usingnon-Foster loading elements,” in Proc. IEEE Antennas Propag. Soc.Int. Symp., Jun. 2007, pp. 4501–4504.

[24] G. Bit-Babik, C. Di Nallo, J. Svigelj, and A. Faraone, “Small widebandantenna with non-Foster loading elements,” in Proc. Int. Conf. Electro-magn. Adv. Applicat. (ICEAA), Sep. 2007, pp. 105–107.

[25] P. Jin and R. Ziolkowski, “Broadband, efficient, electrically smallmetamaterial-inspired antennas facilitated by active near-field reso-nant parasitic elements,” IEEE Trans. Antennas Propag., vol. 58, no.2, pp. 318–327, Feb. 2010.

[26] H. Mirzaei and G. V. Eleftheriades, “A wideband metamaterial-in-spired compact antenna using embedded non-Foster matching,”in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2011, pp.1950–1953.

[27] H. Mirzaei and G. V. Eleftheriades, “A compact frequency-recon-figurable metamaterial-inspired antenna,” IEEE Antennas WirelessPropag. Lett., vol. 10, pp. 1154–1157, 2011.

[28] H. Wheeler, “Small antennas,” IEEE Trans. Antennas Propag., vol. 23,no. 4, pp. 462–469, Jul. 1975.

[29] J. Linvill, “Transistor negative-impedance converters,” Proc. IRE, vol.41, no. 6, pp. 725–729, Jun. 1953.

[30] R. D. Middlebrook, “Measurement of loop gain in feedback systems,”Int. J. Electron., vol. 38, no. 4, pp. 485–512, 1975.

Hassan Mirzaei (S’07) received the B.Sc. degree(honors) in electrical and computer engineering fromIsfahan University of Technology, Isfahan, Iran, in1998 and the M.Sc. degree in electrical engineeringfrom Sharif University of Technology, Tehran, Iran,in 2000. He is currently working toward the Ph.D.degree at the University of Toronto, Toronto, ON,Canada.From 2001 to 2007, he was an RF and Microwave

Engineer at several Iranian telecommunication com-panies where he was involved in the design of mi-

crowave circuits and systems. His research interests include dispersion engi-neering, application of wave propagation and interaction theory to circuit de-sign, and circuit-enabled electromagnetic structures.

George V. Eleftheriades (F’06) received theDiploma in electrical engineering degree from theNational Technical University of Athens, Athens,Greece, in 1988, and the M.S.E.E. and Ph.D. degreesin electrical engineering from the University ofMichigan, Ann Arbor, MI, USA, in 1989 and 1993,respectively.From 1994 to 1997, he was with the Swiss Fed-

eral Institute of Technology, Lausanne, Switzerland.He is currently a Professor with the Department ofElectrical and Computer Engineering, University of

Toronto, Toronto, ON, Canada, where he holds the Canada Research Chair/Velma M. Rogers Graham Chair in Engineering. He is a recognized interna-tional authority and pioneer in the area of metamaterials. These are man-mademedia which have electromagnetic properties not found in nature. He introduceda method for synthesizing metamaterials using loaded transmission lines. To-gether with his graduate students, he provided the first experimental evidenceof imaging beyond the diffraction limit with a Veselago–Pendry lens and pio-neered several novel microwave and optical components, and antennas usingthese transmission-line based metamaterials. His current research interests arein the areas of electromagnetic metamaterials, transformation electromagnetics,super-resolution imaging, small antennas and components for broadband wire-less communications, novel antenna beam-steering techniques, plasmonic andnanoscale optical components and fundamental electromagnetic theory.Prof. Eleftheriades was elected a Fellow of the Royal Society of Canada in

2009. He was an IEEE Antennas and Propagation Society (IEEE AP-S) Distin-guished Lecturer (2004–2009) and a member of the IEEE AP-S AdministrativeCommittee (AdCom) (2008–2010). He is a member of the Technical Coor-dination Committee MTT-15 (Microwave Field Theory). He is an associateeditor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He hasbeen the general chair of the 2010 IEEE International Symposium on Antennasand Propagation and CNC/USNC/URSI Radio Science Meeting, Toronto,Canada. He was the recipient of the 2008 IEEE Kiyo Tomiyasu TechnicalField Award, the 2001 Ontario Premiers’ Research Excellence Award, the 2001Gordon Slemon Award presented by the University of Toronto, and the 2004E. W. R. Steacie Fellowship presented by the Natural Sciences and EngineeringResearch Council of Canada (NSERC). He was the coguest editor of theOctober 2011 PROCEEDINGS OF THE IEEE on “Metamaterials: Fundamentalsand Applications in the Microwave and Optical Regimes” and he coedited abook titled “Negative-Refraction Metamaterials: Fundamental Principles andApplications” (Wiley & IEEE Press, 2005). He was the corecipient of theinaugural 2010 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS BestPaper Award, and the 2008 and the 2012 IEEE RWP King Best Paper Awards.

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