Research ArticleInfluence Analysis of Transmission Lines on a StableNon-Foster-Loaded Electrically Small Dipole

Zhixun Huang , Hu Yang, and Taolin Liu

National University of Defense Technology, DeYa Street No. 109, P.O. Box 410005, Changsha, China

Correspondence should be addressed to Taolin Liu; zhaosenliu@163.com

Received 10 July 2018; Revised 6 November 2018; Accepted 19 November 2018; Published 28 January 2019

Academic Editor: Angelo Liseno

Copyright © 2019 Zhixun Huang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Non-Foster-loaded antennas have the advantages of compact size and large bandwidth. Meanwhile, they suffer from two issues:internal instability and simulation inaccuracy resulting from distribution parameters. The most commonly used stability analysismethod for microwave circuits, Rollett’s criteria, is not suitable for negative impedance circuits. This paper has explained thereason and proposed an effective method for stability analysis. Transmission lines between lumped components are found to bea main reason of inaccurate simulations, which is analyzed in this paper, and it is concluded that their influence also exists athundreds of megahertz. In order to solve this problem and improve simulation accuracy, circuit and electromagneticcosimulation is conducted. Finally, a 320mm dipole loading with a negative capacitor is fabricated to verify the analysis.Simulated and measured results indicate that the proposed stability analysis is effective and the simulation accuracy issignificantly improved. The matched dipole achieves less than −10 dB reflection coefficient from 30MHz to 580MHz.Furthermore, a 14 dB gain improvement is obtained in electrically small condition.

1. Introduction

Non-Foster circuits (NFCs) are circuits whose reactancedecreases with frequency increases. So the circuits can actas negative inductors and capacitors which are consequentlycalled non-Foster elements [1]. These non-Foster elementshave been used in RF and microwave devices to achieve bet-ter performance. NFCs can be used in broad impedancematching circuits as they violate the Bode-Fano limitation.NFCs are also employed in electrically small antennas (ESAs)to reduce their Q value and broaden their bandwidth [4–6].In addition, NFCs are loaded in a waveguide and microstripfor broadband fast-wave propagation and squint-free beam-forming of antenna arrays [7–10].

The first use of NFCs can be traced back to the 1920swhennegative impedance converters (NICs) were designed withvacuum tubes to realize non-Foster elements for loss compen-sation in a telephone cable [2]. As semiconductor technologydeveloped, the performance of transistors got better, andtransistor-based NICs were introduced by Linvill in 1953 [3].After that, different kinds of grounded and floating-pointNICs became known. FETs are voltage-controlled current

sources which enable transformation of inductance intocapacitance. So FET-based non-Foster circuits can transferinductors into negative capacitors, namely, negative imped-ance inverters (NIIs). Operational amplifiers can alsoimplement NFCs. This kind of NFCs is simpler thantransistor-based NICs, and it is easier to achieve the desiredperformance, but due to the limited bandwidth gain productof operational amplifiers, their bandwidths are narrower. Ofall the NFCs mentioned above, the transistor type is usedmost commonly.

Even though NFC enables circuits to cancel the reactance,thus broadening the bandwidth, the circuit plays a less thanexpected role in the previous measurement. Possible reasonsare as follows: mismatch and instability. Firstly, mismatchmay be due to the performance of NFCs changing with fre-quency. The negative reactance varies with frequency, and itis not likely to cancel the reactance in a wide frequency band.What is more, the interface transmission lines also have animpact on negative impedance which is often overlooked,resulting in simulation inaccuracy. Instability is one ofthe most important issues with NFCs because they arepotentially unstable two-port networks [11]. Scarification of

HindawiInternational Journal of Antennas and PropagationVolume 2019, Article ID 1273574, 8 pageshttps://doi.org/10.1155/2019/1273574

the performance of NFC is often unavoidable to stabilizethe circuit.

This paper compares different methods of NFC stabil-ity analysis and proposes a feasible method. The influenceof the interface transmission line is analyzed and verifiedby simulation. Finally, an electrically small dipole loadedwith a transistor-based NIC for broadband operation wasdesigned and fabricated.

2. Transistor-Based NIC Design

2.1. Circuit Analysis. Transistor-based NICs generally workby maintaining the current of the load and inversing the volt-age at the output port. Consider the floating NIC shown inFigure 1. Port 2 is terminated with the device to be matchedwhose impedance is Za while port 1 acts as the output, con-necting with a 50-ohm transmission line. The simplifiedequivalent circuit of the floating NIC is given in Figure 2.Implementing Kirchhoff’s law, assuming a current source I1flowing in port 1, the following equations can be obtained:

1reV1 −

1reV4 = −I1,

−1Za

+ 1re

V2 +1reV3 = 0,

1re

− gm V1 + gmV2 +1ZL

− gm V3 + gm −1re

−1ZL

V4 = 0,

gmV1 +1re

− gm V2 + gm −1re

−1ZL

V3 +1ZL

− gm V4 = 0

1

The solution to the equations are

V1 = 2gmZLre − Za − ZL − 2re I1,V2 = −ZaI1,V3 = − Za + re I1,V4 = 2gmZLre − ZL − Za − re I1

2

Thus, the input impedance Zin seen from port 1 is

Zin =V1I1

= 2gmZLre − Za − ZL − 2re, 3

where re is the emitter resistance of the transistor itself andgm is the transconductance, affected by the operating pointof the transistor. Therefore, by selecting a suitable transistorand employing it at an appropriate operating point, theimpedance ZL can be used to cancel the impedance Za orthe unwanted part of Za. It needs to be pointed out thatthe DC bias network of the transistor is not shown inFigures 1 and 2. Moreover, the distribution parameters ofthe circuit are not considered in the previous analysis. Sowhen (3) is used to design NICs, more work is needed forbetter performance which will be introduced later in thefollowing sections.

A negative capacitor is designed and loaded on a dipole toverify the effect of the NIC, as shown in Figure 3. The dipoleoperates at 450MHz with a 40MHz bandwidth. It is wellknown that the dipole is capacitive in an electrically smallcase, so the load of NIC should be a capacitor. In addition,the used NIC is an unbalanced structure while the dipole isbalanced. So a wideband balun is placed between the dipoleand circuit. NPN epitaxial silicon transistor NE68139e ischosen for the circuit as it has advantages such as low noiseand up to 2GHz cut-off frequency. Simulation result showsthat the NIC loading dipole has −10dB reflection coefficientfrom 24MHz to 828MHz. The imaginary part of thematched dipole is between −10 and 17 ohms, indicating thatthe reactance of the dipole is mostly cancelled out. Theseresults indicate that NICs have the ability to match imped-ance in broadband.

2.2. Stability Analysis. The most commonly used stabilitydetermination method for microwave circuits is Rollett’s cri-teria. Rollett’s conditions for 2-port unconditional stabilityare [12]

K = 1 − S112 − S22

2 + Δ 2

2 S12S21> 1,

Δ = S11S22 − S12S21 < 14

An additional condition must also be satisfied whereinthe ports must be either open-circuit stable (OCS) orshort-circuit stable (SCS). However, this criterion may givean incorrect result when applied to negative impedance cir-cuits [13]. The K factor only analyzes the S-parameters ofthe circuit, but does not consider the unstable hidden mode

ZL

1 2

Figure 1: Schematic of the floating NIC without DC bias network.

ZL 43

gmVbe1 gmVbe2

Vbe

+

_Vbe

re1 re2

2Za

1I1

+

_

Figure 2: Simplified equivalent circuit of the floating NIC with acurrent source I1 at port 1 and an impedance Za at port 2.

2 International Journal of Antennas and Propagation

inside the circuit. The circuit with two non-Foster elementsand two positive resistors shown in Figure 4 is a simple exam-ple to demonstrate our opinion. In this circuit, if R1, R2, L,and C satisfy the following conditions:

R1 = R2 = R,L = R2C < 0,

5

then

Z11 = Z22 =R1 + Ls R2 + 1/sCR1 + Ls + R2 + 1/sC = R 6

Using Rollett’s criteria, the circuit is unconditionally sta-ble as its input and output impedance are constant R in thewhole frequency band. But the series RL and RC will oscillate,so the circuit is practically unstable. Actually, the K factoronly considers the input and output port stability of the cir-cuit, and it is enough for most active microwave circuits likeRF amplifiers. For the non-Foster case, the circuits are

internal unstable, so other ways for stability analysis shouldbe employed.

Many researchers implement time-domain approach forNFC stability analysis, applying a Gaussian pulse excitationand observing the time-domain response of the signal. Thecircuit is stable if the energy of the signal decays as time goesby. The problem is that transients can be very long in NICcircuits. Additionally, the width of the Gaussian pulse shouldbe designed for operating frequency based on the Fouriertransform which is complicated.

Another stability test that has been widely recognized forNICs is the normalized determinant function (NDF) test. Inthis way, all the possible feedback effects, including thoseresulting from parasitic elements inside the device models,are taken into account [14]. NDF calculates the open-loopimpedance and closed-loop impedance of the circuit. The cir-cuit is stable only if there are no zeros of NDF in the right halfof the complex frequency s-plane (RHP). To simplify the cal-culation, a Nyquist plot is usually used to check the numberof RHP zeros which can avoid solving the complex formula.The number of clockwise encirclements of the origin indi-cates the number of RHP zeros that were produced by the cir-cuit. However, the NDF of the NFCs may be inaccuratebecause the parasitic parameters and transmission lines havean impact on NFCs which cannot be accurately evaluated.The NDF will also become complicated after consideringthese factors.

In this paper, we test the loop gain of the circuit for stabil-ity analysis. This method determines whether the transmis-sion function will be infinite in some frequencies, which isessentially consistent with the NDF method, but it is notas complicated as the NDF method. For a NFC system,

V_DC

Vdc = 10 V {t}SRC6

V_DC

Vdc = 10 V {t}SRC5

C

C = 22 𝜇FC37

C

C = 1 𝜇FC36

C

C = 10 nFC35

C

C = 1 𝜇FC35C

C = 1 𝜇FC52

C

C = 22 𝜇FC47

C

C = 1 𝜇FC46

C

C = 10 nFC48

C

C = 50 pF {t}C45

R

R = 4.7 kohmR15

R

R = 200 ohmR17

R

R = 1.1 kohmR16

R

R = 1.1 kohmR12 R

R = 200 ohmR13

Term

Z = file{DAC1, “z[1, 1]”}ohmTerm2

SNP2

Vo

S4P

Ref

3

2

4

1

BJT_NPN

Model = BJTM1Mode = nonlinear

BJT4 BJT_NPN

Model = BJTM1Mode = nonlinear

BJT3

Term

Num=1Z = 50 ohm

Term1

R

R = 4.7 kohmR11

R

R = 200 ohmR10R

R = 200 ohmR14

C49GRM18

C32GRM18 C51

GRM18

C50GRM18

Figure 3: The schematic of the transistor-based NIC with DC bias network and coupling capacitors, in which the impedance of the dipole isimported in Term2 and the S-parameters of the balun are imported in the S4P component.

1 2L < 0 C < 0

R1 R2

Figure 4: An unstable circuit which would be predicted to be stableby Rollett’s criteria.

3International Journal of Antennas and Propagation

assuming that the open-loop gain of the system is A(s)and the function of the feedback network is β(s), thenthe transfer function of the system is.

F s = A s1 − β s A s 7

The product β s A s is the loop gain of the system.Note that the denominator of (7) is not a sum, becausethe NFCs are positive-feedback networks. The circuit isunstable if there are points where the magnitude of theloop gain is greater than 1, the phase is 0, and the phaseis decreasing with increasing frequency. Because the transferfunction of the system tends to be infinite, the zero input mayresult in an infinite output in such a case [15]. A commercialsimulator, ADS is used to calculate the loop gain and analyzecircuit stability. A device in ADS called “OscTest” is insertedin the circuit. This device can evaluate the closed loop, smallsignal gain of the circuit, which is exactly needed for ourstability analysis.

Stability analysis of the NIC shown in Figure 3 is con-ducted, and it is found that the circuit is unstable below100MHz. The magnitude of the loop gain is close to oneand the phase is 0 at 80MHz. To stabilize the NIC, a parallel15-ohm resistor and 20pf capacitor network was added inthe emitter of the transistor connected to the output port,wherein the resistor improves the stability of the circuit,and the capacitor shorts the resistor at high frequencies,reducing signal loss. Simulated loop gain of the circuit isshown in Figure 5. It can be observed that the magnitude ofloop gain after stabilizing becomes further away from 1 andthe phase of which is less than 0 from 15MHz to 1GHz, indi-cating that the circuit is stable in this frequency range.

2.3. Transmission Line Analysis. Though the VHF bandwavelength is much longer than the distance of the device

in NFCs, simulations and experiments indicate that the influ-ence of the transmission line (TL) between devices cannot beignored. Specifically, the interface TL of the antenna andNFC has great impact on the output performance. Figure 6compares the reflection coefficient of the circuit in Figure 3at different interface TL lengths. The result shows that evenin tens of megahertz, TL will have an impact on the overallreflection coefficient, not to mention hundreds of megahertzor gigahertz. Generally, the interface TL of the antenna andNFC is two SMA connectors, which will introduce a transi-tion from coaxial to microstrip. The impedance of theantenna with interface TL is different from the original one,resulting in simulation inaccuracy and mismatch. A good

Mag

1.02

1.00

0.98

Freq (MHz)

0.96

0.94

0.92

0.90

0.880 200 400 600 800 1000

AfterBefore

(a)

2

0

Phas

e (de

g)

Freq (MHz)

−2

−4

−6

−80 200 400 600 800 1000

AfterBefore

(b)

Figure 5: Loop gain of the NIC with dipole terminated before and after adding the stabilization network: (a) magnitude of the loop gain;(b) phase of the loop gain.

0

−5

−10

S11

(dB)

Freq (MHz)

Without line1 cm2 cm

−15

−20

−250 200 400 600 800 1000

Figure 6: Reflection coefficient of the NIC-loaded dipole withdifferent lengths of the interface transmission line.

4 International Journal of Antennas and Propagation

way to minimize the influence of the interface TL is integrat-ing the antenna and NFC on a printed circuit board to elim-inate the use of SMA connectors and connecting the dipoledirectly to the balun without TLs.

For the unavoidable TLs between lumped components(TLBLCs) in NFC, their impact on the circuit must be ana-lyzed and minimized. In order to reduce the transmissionmismatch, the TLs of the signal path in the circuit shouldbe a 50-ohm microstrip line. The characteristic impedanceZ0 of the microstrip line is determined by the equivalent per-mittivity εe, thickness of the printed circuit board h, and linewidth w:

Z0 =

60εe

ln 8hw

+ w4h , w

h≤ 1,

120πεe w/h + 1 393 + 0 667 ln w/h + 1 444 , w

h≥ 1

8

If the line width is wider, the difficulty of layout designincreases, and the size of the printed circuit board alsobecomes larger. After comprehensive consideration, a

0.5mm thick FR4 board with a relative permittivity of 4.4 isused. The 0.95mm wide 50-ohm microstrip line is conve-nient for layout design. With the layout of the NFC attained,the influence of the TLBLCs can be analyzed. In this paper,only the TLBLCs of the signal path is discussed, which is alsothe part that has the greatest impact on the circuit. The NICwith its signal path layout is shown in Figure 7. A sinusoidalsignal source was added at port 1 to see the influence of theTLBLCs on NIC. The voltages at nodes 1, 2, and 3 in the cir-cuit were tested, which would be the same without TLBLCs.The result shown in Figure 8 reveals that the magnitudes ofthe voltages at these three nodes are almost the same, butthe voltage at node 1 produces a significant phase shift with500MHz input signal. It indicates that the TLBLCs of the sig-nal path with the characteristic impedance of 50 ohms willnot cause circuit mismatch but will obviously affect the phaseof the signal. Comparing the phase of reflection coefficient ofthe NFC with and without TLBLCs, the maximum phasedeviation exceeds 30 degrees in the range of 50–500MHz.It indicates that the effect of the TLBLCs on the phase ofthe NFC is not only determined by its length. For example,the phase of the NFC with TLBLCs at 100MHz is 34 degreeshigher than the one without, and the phase difference isequivalent to a line length of 283.3mm, which is much more

V_DCSRCVdc = 10 V {t}

V_DCSRCVdc = 10 V {t}

CC37C = 22 𝜇F

CC47C = 22 𝜇F

CC55C = 1 𝜇FCC54C = 10 nF

CC36C = 1 𝜇F

CC35C = 10 nF

CC45

V2

V1

GRM18C49

C = 50 pF {t}

RR15R = 4.7 kohm

RR18R = 4.7 kohm

RR10R = 200 ohm

RR14R = 200 ohm

RR9R = 15 ohm

CC52C = 1 𝜇F

Cl__1C41C = 20 pF

RR17R = 200 ohm

GRM18C46PartNumber = GRM188R71C104KA01

RR12R = 1.1 kohm

V3

GRM18C50

RR13R = 200 ohm

CC56C = 1 𝜇F

S4P

Vo1

43

2Ref

SNP2 TermTerm2Z = file{DAC1, “z[1, 1]”}ohm

GRM18C48PartNumber = GRM188R71C104KA01

R16R

R = 1.1 kohm

BJT_NPNBJT4Model = BJTM1Mode = nonlinear

BJT_NPNBJT5Model = BJTM1Mode = nonlinear

print_dipole_balun_newemModelTerm

Term1Num = 1Z = 50 ohm

Figure 7: Cosimulation schematic of the NIC, where TLBLCs of the signal path are the EM model and other components are thelumped model.

5International Journal of Antennas and Propagation

than the length of the signal path. Furthermore, it can befound that the influence of the phase is less when the lengthof each branch is equal, because phase shift can be compen-sated to some extent in cross-coupling. So, when designingNFCs, not only should the distance of the components beas small as possible to make the length of the entire signalpath short enough, but the length of the TLBLC of eachbranch should also be kept as uniform as possible.

TLs also influence the stability of the NFCs as they mayshift the poles of NDF to RHP, resulting in instability [16].From the previous analysis, TLs have a great influence onthe phase of the NFCs, so when implementing loop gain sta-bility analysis, the TLs should not be ignored too. The actualcircuit layout needs to be included in the simulation to obtaina reliable result. In this paper, we use the EM model ofTLBLCs and the schematic model of the other lumped com-ponents for cosimulation. Simulation result indicates that thecircuit with the dipole terminated is stable in the frequenciesfrom 30MHz to 1000MHz.

3. Experimental Verification

A prototype of the non-Foster circuit terminated with adipole is fabricated as shown in Figure 9. The circuit part is35× 35× 0.5mm3 in size, integrated with a 320mm dipole.No oscillations are observed in the spectrum analyzer whenthe DC bias is turned on. The NIC-loaded dipole is stableas predicted by previous simulation. It is worth mentioningthat the method of circuit stability analysis we use is for aknown terminal load, so the terminal load should be clearedbefore the analysis is conducted.

In order to verify the performance of the negative capaci-tance generated by the NIC, S-parameters are measured usinga vector network analyzer. Figure 10 compares the reflectioncoefficient of measured, simulated NIC-loaded dipole andthe unmatched one. It can be observed that the NIC greatlybroadens the bandwidth of the antenna, especially at low

frequencies. The simulation results especially the cosimula-tion considering the influence of TLs are consistent with themeasured result. At the frequencies above 400MHz, theadvantages of cosimulation are pronounced. Although thetrend of reflection coefficient changes with frequency is simi-lar, the simulated one without TLs is 5 dB lower than themea-sured one while the cosimulated one with TLs is almost thesame as the measured one. This means that NICs may beapplied to a higher frequency using cosimulation. The mea-sured result shows that the operating frequency of thematched dipole ranges from 30MHz to 580MHz, 14 times

V1V2V3

5.85.65.45.25.04.84.6

Nod

e vol

tage

(V)

4.44.24.03.8

0 2 4 6Time (nsec)

8 10

Figure 8: Voltages of nodes 1, 2, and 3 with 500MHz sinusoidalsignal source added in port 1.

Figure 9: Fabricated NIC-loaded dipole.

5

0

−5

−10

S11

(dB)

−15

−20

−250 200 400 600

Freq (MHz)

MeasureCosimulateSimulateWithout match

800 1000

Figure 10: Reflection coefficient of measured, simulatedNIC-loaded dipole and unmatched dipole.

6 International Journal of Antennas and Propagation

the bandwidth of the one without matching. The imaginarypart of the input impedance is shown in Figure 11(b), whichindicates that the capacitive part of the electrically smalldipole has been canceled well. The mismatch problem causedby the capacitive impedance of the electrically small antennahas been greatly improved.

Note that the loss of the NFC may be greater than itsimprovement of the matching. It can be seen inFigure 11(a) that the resistance of the NIC matched dipoleis around 50 ohms while the unmatched one is almost 0 inthe electrically small case. The NFC cannot enhance the radi-ation impedance of the antenna, so the increased resistanceshould be the loss resistance. We found that the NIC in thispaper itself brings approximately 10 dB extra loss, whichweakens the bandwidth benefit of the NIC. The measuredgain of the NIC-loaded dipole at 450MHz is −10dBi whilethe unloaded dipole is 1.78 dBi. However, in the case of100MHz where the wavelength is 0.11λ (λ is the operatingwavelength of the dipole.), the gain of the matched onebecomes −18 dBi and that of the unmatched one is −32dBi.A 14 dB gain improvement is obtained in the electricallysmall case even though the NIC introduces 10 dB loss. Thetotal efficiency (including reflection efficiency and radiationefficiency) of the NIC matched and unmatched dipole isshown in Figure 12. It indicates that the efficiency of thematched dipole increases by 10 percent compared to that ofthe unmatched one below 300MHz. The benefit of matchimprovement is better than the loss of the NIC. However,the efficiency of the NIC matched dipole in 450MHz is 20percent lower than that of the unmatched one, indicating thatthe loss of the NIC is better than the match improvement.Thus, the NIC is more suitable for a serious mismatch caselike electrically small antennas, rather than broadening theoriginal antenna band because of its loss. A reasonableapproach to tackle the issue of loss could be to add alow-noise amplifier after the NIC for compensation.

4. Conclusion

This paper analyzes the impedance matching capability, sta-bility, and TL influence of the NFCs. Circuit characteristics ofLinvill’s negative impedance converter is given by means ofKirchhoff’s law. Difference stability analysis methods includ-ing Rollett’s criteria, NDF, and time-domain analysis arecompared and inapplicability of some of the methods forNFCs is demonstrated. On the other hand, the loop gainmethod used in this paper is proved to be effective in NFCstability analysis. TLs have an impact on the phase of signaland stability of NFCs. In order to reduce uncertainty of inter-face TLs, the NIC and antenna are integrated together toavoid the transition from coaxial to microstrip. A significantmethod to improve simulation accuracy is to add the layout

500

400

300

200

Real

(ohm

)

100

0

−1000 100 200 300

Freq (MHz)

Without matchNIC match

400 500 600

(a)

500

0

–500

Imag

(ohm

)

–1000

–1500

–2000

NIC matchedWithout match

100 200 300Freq (MHz)

400 500 600

(b)

Figure 11: Impedance of the NIC matched and unmatched dipole: (a) Resistance; (b) Reactance.

1.0

0.8

0.6

0.4

0.2

0.0

0 100 200 300Freq (MHz)

Effici

ency

Without matchNIC matched

400 500

Figure 12: Efficiency of the NIC matched and unmatched dipole.

7International Journal of Antennas and Propagation

TLBLCs into the schematic and cosimulate the whole circuit.The more consistent the added TLBLCs is with the actual lay-out, the higher the simulation accuracy is, especially in highfrequency. The NIC is applied to match a 320mm electricallysmall dipole. Measured and simulated results show goodagreement with each other. The measurement result showsthat the matched dipole achieves less than −10 dB reflectioncoefficient from 30MHz to 580MHz. A 14dB gain improve-ment is obtained in the electrically small case, verifying thatthe NIC is effective.

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

[1] B. Buyantuev and D. Kholodnyak, “Applications of non-Fosterelements to design advanced RF and microwave devices,” in2018 IEEE Conference of Russian Young Researchers in Electri-cal and Electronic Engineering (EIConRus), pp. 1656–1660,Moscow, January 2018.

[2] J. L. Merrill Jr, “Theory of the negative impedance converter,”Bell System Technical Journal, vol. 30, no. 1, pp. 88–109, 1951.

[3] J. G. Linvill, “Transistor negative impedance converters,” Pro-ceedings of the IRE, vol. 41, no. 6, pp. 725–729, 1953.

[4] R. M. Rudish and S. E. Sussman-Fort, “Non-Foster impedancematching improves S/N of wideband electrically small VHFantennas and arrays,” in IASTED Conference on Antennas,Radar and Wave Propagation, Banff, AB, Canada, 2005.

[5] C. R. White, J. S. Colburn, and R. G. Nagele, “A non-FosterVHF monopole antenna,” IEEE Antennas and Wireless Propa-gation Letters, vol. 11, pp. 584–587, 2012.

[6] K. S. Song, Non-Foster Impedance Matching and Loading Net-works for Electrically Small Antennas, [Ph.D. thesis], Electricaland Computer Engineering - The Ohio State University,Columbus, OH, USA, 2011.

[7] J. Long, M. M. Jacob, and D. F. Sievenpiper, “Broadbandfast-wave propagation in a non-Foster circuit loaded wave-guide,” IEEE Transactions on Microwave Theory and Tech-niques, vol. 62, no. 4, pp. 789–798, 2014.

[8] H. Mirzaei and G. V. Eleftheriades, “Arbitrary-anglesquint-free beamforming in series-fed antenna arrays usingnon-Foster elements synthesized by negative-group-delay net-works,” IEEE Transactions on Antennas and Propagation,vol. 63, no. 5, pp. 1997–2010, 2015.

[9] H. Mirzaei and G. V. Eleftheriades, “Squint-free beamformingin series-fed antenna arrays using synthesized non-Fosterelements,” in 2013 IEEE Antennas and Propagation SocietyInternational Symposium (APSURSI), pp. 2209-2210, Orlando,FL, USA, July 2013.

[10] H. Mirzaei and G. V. Eleftheriades, “An active artificial trans-mission line of squint-free series-fed antenna array applica-tions,” inMicrowave Conference (EuMC), 2011 41st European,pp. 503–506, Manchester, U.K, 2011.

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

[12] J. Rollett, “Stability and power-gain invariants of linear Two-ports,” IRE Transactions on Circuit Theory, vol. 9, no. 1,pp. 29–32, 1962.

[13] S. D. Stearns, “Circuit stability theory for non-Foster circuits,”in 2013 IEEE MTT-S International Microwave SymposiumDigest (MTT), pp. 1–3, Seattle, WA, USA, June 2014.

[14] A. Suárez and F. Ramírez, “Stability and bifurcation analysis ofmulti-element non-Foster networks,” IEEE Transactions onMicrowave Theory and Techniques, vol. 66, no. 4, pp. 1817–1830, 2018.

[15] B. R. Long, Analysis of Stable Negative Impedance LoadedDipole and Canonical Chiral Elements with Application toNovel Active Media, [Ph.D. thesis], The Pennsylvania StateUniversity, 2001.

[16] Q. Tang and H. Xin, “Stability analysis of non-Foster circuitusing normalized determinant function,” IEEE Transactionson Microwave Theory and Techniques, vol. 65, no. 9,pp. 3269–3277, 2017.

8 International Journal of Antennas and Propagation

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International Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Navigation and Observation

International Journal of

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www.hindawi.com Volume 2018

Advances in

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