non-foster reactances for electrically- small antennas ... · pdf filenon-foster reactances...

Non-Foster Reactances for Electrically-Small Antennas, High-Impedance

Surfaces, and Engineered Materials

James T. AberleAssociate Professor of Electrical Engineering

School of Electrical, Computer and Energy Engineering

Arizona State University

Outline of Presentation

• What Does “Non-Foster” Mean?• Possible Applications of Non-Foster

Reactances– Electrically Small Antennas– High-Impedance Surfaces

– Artificial High-Permeability Materials

• Realization of Non-Foster Reactances

2IEEE Waves and Devices 19 Feb 2010Copyright © 2010James T. Aberle

Foster’s Reactance Theorem

• The theorem is a consequence of conservation of energy.

• The slope of the input reactance (susceptance) of a lossless passive one-port is always positive.

• All zeros and poles of the impedance (admittance) function are simple, and a zero must lie between any two poles, and a pole between any two zeros.


Consequences of Foster’s Reactance Theorem

• Impedances (admittances) of passive one-port networks rotate clockwise on the Smith Chart as frequency increases.

• There is no such thing as a negative capacitor or a negative inductor (for passive circuits).


Foster Network

CC1C=30 pF

LL1

R=L=40 nH

LL2

R=L=40 nH

TermTerm1

Z=50 OhmNum=1

RR1R=50 Ohm

50 100 150 200 2500 300

0

20

40

-20

60

freq, MHz

rea

l(Zin

1)

ima

g(Z

in1

)

freq (10.00MHz to 300.0MHz)

S(1

,1)


Non-Foster Network

LL3

R=L=-40 nH

LL1

R=L=40 nH

LL2

R=L=40 nH

TermTerm1

Z=50 OhmNum=1

RR1R=50 Ohm

50 100 150 200 2500 300

0

20

40

60

80

100

-20

120

f req, MHz

real

(Zin

1)im

ag(Z

in1)

f req (10.00MHz to 300.0MHz)

S(1

,1)



• What Does “Non-Foster” Mean?

• Possible Applications of Non-Foster Reactances–Electrically Small Antennas– High-Impedance Surfaces




VHF Whip: A Canonical ESA

•Geometry of monopole antenna as modeled in Antenna Model software. The monopole is a copper cylinder 0.6 meters in length and 0.010 meters in diameter, mounted on an infinite perfect ground plane.•Frequency range is 30 to 90 MHz.


Input Impedance of VHF Whip From Simulation


Two Tools to Help With Analysis

• Exact two-port representation of antenna in frequency domain in terms of s-parameters.

• Approximate lumped equivalent circuit model of antenna over frequency range of interest.


ajXlR

1:N0Z

0Z

Two-Port Representation of Antenna

The quantities in this box are re-evaluated at every frequency for which we have data.



/

radiation resistance

(1 ) dissipative loss resistance

antenna reactance

a a a r l a

r cd a

cd a

a

Z R jX R R jX

R e R

R e R

X

= + = + += == − ==

0Z

RN r=



• Antenna impedance and radiation efficiency are used to produce a Touchstone *.s2p file for use in circuit simulation – the exact two-port representation of the antenna at each frequency for which we have data.

• Allows concepts like transducer power gain and stability measures to be applied to antennas. The latter being particularly important for considering the use of non-Foster reactances in antenna matching networks.


Approximate Equivalent Circuit of the Antenna

• To model the antenna, we assume that the real part of the antenna impedance varies as the square of frequency, and the imaginary part behaves as a series LC.

−+

=

aaa C

LjRZω

ωωω 1

2

00

Impedance produced by equivalent circuit



• Evaluation of the model parameters (R0, La and Ca):

( )

( )

( )

ℑ

ℑ=

−

−

ℜ=

2

1

22

11

00

11

1

ω

ω

ωω

ωω

ω

a

a

a

a

a

Z

Z

C

L

ZR



pF 69.8

nH 194

30.40

==

Ω=

a

a

C

L

RLLa

R=L=194 nH

Z1P_EqnZ1P1Z[1,1]=4.30*(freq/52e6)**2

CCaC=8.69 pF

TermTerm1

Z=50 OhmNum=1

40 50 60 70 8030 90

-500

-400

-300

-200

-100

-600

0

freq, MHz

imag

(Zin

1)im

ag(V

HF

_Whi

p_S

2P..Z

in1)

40 50 60 70 8030 90

5

10

15

0

20

freq, MHz

real

(Zin

1)re

al(V

HF

_Whi

p_S

2P..Z

in1)


LZ

0ZΓ

Matching Network

Matching Network Concept


Points to Consider

• A real passive matching network can only approach and never exceed the performance predicted by the Bode-Fano criterion.

• The matchable bandwidth is limited by the Q of the load.

• The matchable bandwidth can only be increased by de-Qing the load – that is by intentional introduction of dissipative losses into the matching network – and concomitant reduction in radiation efficiency.


Bode-Fano Criterion

Γ⋅

≤∆

m

Qf

f

1ln0

π

Maximum value of fractional bandwidth that can be achieved with any passive, lossless matching network.

Q of the load

Maximum allowable reflection coefficient

035.03

1 ,9.81

0

≤∆⇒=Γ=

f

fQ m


Single-Tuned Mid-band Match

TermTerm1

Z=4.30 OhmNum=1

LLe

R=L=884 nH

CCaC=8.69 pF

Z1P_EqnZ1P1Z[1,1]=4.30*(freq/52e6)**2

LLa

R=L=194 nH


S(1

,1)

m1

m2

m1freq=S(1,1)=-10.409 / -73.141impedance = Z0 * (0.992 - j0.630)

51.80MHzm2freq=S(1,1)=-10.477 / 71.883impedance = Z0 * (1.008 + j0.630)

52.20MHz

008.052

8.512.52

0

=−≈∆f

f

009.02

1

0

==∆Qf

fAnalytical:


Wheeler-Lopez Double-Tuned Matching

A.R. Lopez, “Wheeler and Fano Impedance Matching”, IEEE Antennas and Propagation Magazine, Vol. 49, No. 4, August 2007


Wheeler-Lopez Double-Tuned Matching with Antenna De-Qing

TermTerm2

Z=50 OhmNum=2

RRDR=200 Ohm

CC2C=53.0 pF

TFTF1T=0.354

LL2

R=L=177 nH

LLe

R=L=884 nH

TermTerm1

Z=50 OhmNum=1

ZinZin1Zin1=zin(S11,PortZ1)

Zin

N

S_ParamSP1

Step=1 MHzStop=90 MHzStart=30 MHz

S-PARAMETERS

S2PSNP1File="VHF_whip.s2p"

21

Re f


S(1

,1)

40 50 60 70 8030 90

-12

-10

-8

-14

-6

freq, MHz

dB(S

(1,1

))

40 50 60 70 8030 90

-22

-20

-18

-16

-14

-24

-12

freq, MHz

dB(S

(2,1

))

Decent match, poor efficiency


Matching Network with Non-Foster Reactances

Z1P_EqnZ1P1Z[1,1]=4.30*(freq/52e6)**2

LLa

R=L=194 nH

CCaC=8.69 pF

LLa1

R=L=-194 nH

CCa1C=-8.69 pF

TermTerm1

Z=50 OhmNum=1

LL2

R=L=-44.9 nH

LL1

R=L=-44.9 nH L

L3

R=L=44.9 nH

AntennaDualizer


S(1

,1) m1

m1freq=S(1,1)=4.843E-4 / -3.517E-5impedance = Z0 * (1.001 - j5.952E-10)

90.00MHzCancels frequency squared dependence of radiation resistance.

Van Der Pol, Proc, IRE, Feb. 1930


0Z

0ZmL

( )ma LL +−

aC−

Two-port model of antenna

Active matching network

Antenna with More Practical Matching Network using Non-Foster Reactances.


Optimized Non-Foster Matching Network

VARVAR1

Cneg=8.72637 oLneg=226.588 oLm=49.5959 o

EqnVar

CCa1C=-Cneg pF

LL2

R=L=-Lneg nHL

L3

R=L=Lm nH

TermTerm1

Z=50 OhmNum=1

TermTerm2

Z=50 OhmNum=2

S2PSNP1File="VHF_whip.s2p"

21

Ref


S(1

,1)

40 50 60 70 8030 90

-26

-24

-22

-20

-18

-28

-16

freq, MHz

dB

(S(1

,1))

40 50 60 70 8030 90

-0.4

-0.3

-0.2

-0.1

-0.5

0.0

freq, MHz

dB(S

(2,1

))27IEEE Waves and Devices 19 Feb 2010

Copyright © 2010James T. Aberle



• Possible Applications of Non-Foster Reactances– Electrically Small Antennas

– High-Impedance Surfaces– Artificial High-Permeability Materials



High-Impedance Ground Plane

Sievenpiper, et. al, IEEE Trans. MTT, Nov. 1999

Capacitive FSS

Low Permittivity

Spacer

Metal BackplaneMetal Vias

10”

Coaxial Feed

Bent WireElement


EM Properties of the Sievenpiper High-Impedance Ground Plane

• Surface impedance is (ideally) an open-circuit (emulating a PMC rather than a PEC like a conventional ground plane).

• Propagation of TM and TE surface waves is not supported (thus can be called an electromagnetic bandgap structure).


Model for Surface Impedance of Sievenpiper HIS

Γ

For plane waves at normal incidence, the substrate may be understood as an electrically short length of shorted transmission line in parallel with a shunt capacitance at the reference plane of the outer surface.

Γ

OpenShort

Zstub

Zin

Shunt Capacitance

2ββββd

θθθθ

Pha

se A

ngle

, θθ θθ

Frequency

180o

90o

0o

-90o

-180o

fo

BW ~ 10% to 20% of fo.

Cηηηηo Zo, ββββ Short

Zstub

Zin d


Reflection Phase Bandwidth of Sievenpiper HIS

00

2λ

πµω

ω hr=∆

hµr

Spacer layer

FSS layer

Ground plane


Electrically-Thin Broadband High-Impedance Surface

• In principle, one could realize an electrically-thin broadband HIS by using a high-permeability spacer layer.

• A high-permeability meta-material can be realized using artificial magnetic molecules (AMMs) implement with negative inductance circuits.

• Unfortunately, AMM performance is very sensitive to component tolerances.

• But, there is a better way …33IEEE Waves and Devices 19 Feb 2010

Copyright © 2010James T. Aberle

Electrically-Thin Broadband High-Impedance Surface

Kern, Werner, Wilhelm, APS 2003


Electrically-Thin Conventional HIS

TermTerm1

Z=377 OhmNum=1

CC1C=2.17 pF

TLINPTL1

Sigma=0TanM=0Mur=1TanD=0.0009F=1 GHzA=0.0K=2.2L=1.6 mmZ=377/sqrt(2.2) Ohm

m1freq=phase(S(1,1))=90.015

2.308GHzm2freq=phase(S(1,1))=-90.204

2.502GHz

m3freq=phase(S(1,1))=-0.139

2.403GHz

1.5 2.0 2.5 3.0 3.51.0 4.0

-100

0

100

-200

200

freq, GHz

phas

e(S

(1,1

))

m1

m2

m3

m1freq=phase(S(1,1))=90.015

2.308GHzm2freq=phase(S(1,1))=-90.204

2.502GHz

m3freq=phase(S(1,1))=-0.139

2.403GHz

%80

=∆f

f

0.062 in Rogers Duroid 5880


Electrically-Thin HIS with Negative Inductance

LLneg

R=L=-2.02 nH

TLINPTL1

Sigma=0TanM=0Mur=1TanD=0.0009F=1 GHzA=0.0K=2.2L=1.6 mmZ=377/sqrt(2.2) Ohm

TermTerm1

Z=377 OhmNum=1

1.5 2.0 2.5 3.0 3.51.0 4.0

-5

0

5

10

-10

15

freq, GHzph

ase(

S(1

,1))


Unit Cell of Sievenpiper HIGP


Reflection Phase Response of Sievenpiper HIGP

12 13 14 15 16 17 18-140

-120

-100

-80

-60

-40

-20

0

20

40

Frequency (GHz)

Ref

lect

ion

Coe

ffic

ient

Pha

se (

deg)


Unit Cell of Sievenpiper HIGP with Reactive Loading


Equivalent Circuit of Loaded HIGP for Normally Incident Plane Wave

E

Zload

Zload


Reflection Phase of Capacitively Loaded HIGP

0.5 1 1.5 2 2.5 3 3.5 4-200

-150

-100

-50

0

50

100

150

200

Frequency (GHz)

Ref

lect

ion

Coe

ffic

ient

Pha

se (

deg)


Reflection Phase of Negative-Inductor Loaded HIGP

0.5 1 1.5 2 2.5 3 3.5 4-40

-30

-20

-10

0

10

20

30

40

Frequency (GHz)

Ref

lect

ion

Coe

ffic

ient

Pha

se (

deg) Lneg = -2.2nH




• Possible Applications of Non-Foster Reactances– Electrically Small Antennas

– High-Impedance Surfaces

– Artificial High-Permeability Materials• Realization of Non-Foster Reactances


What is an Artificial Material?• An artificial material is a large-scale emulation of an actual

material, obtained by embedding a large number of electrically small inclusions (“artificial molecules”) within a host medium.

• Like natural molecules, the electrically small inclusions exhibit electric and/or magnetic dipole moments.

• As a result of these dipole moments, the macroscopic electromagnetic constitutive parameters (εr and µr) are altered with respect to the host medium.

HB

ED

0

0

µµ

εε

•=

•=


Why Create an Artificial Magnetic Material?

• “Naturally” occurring magnetic materials (ferrites) are heavy, fragile and expensive, and they also exhibit relatively high magnetic losses and dielectric constant.

• Available ferrite materials provide a limited selection of relative permeabilities.

• The permeability tensor of the ferrite is controlled by applying a static magnetic field –permanent magnets and/or electromagnets are required.


Artificial Magnetic Metamaterial

•A 3-dimensional lattice of artificial molecules. •Electrically small loop with a load impedance

dx

dy

dz

dz

dy

a

W

ZL

y

z


Simple Circuit Model for Artificial Magnetic Molecule (AMM)

Llooploss

xx

ZLjR

aj

H

m

N

++−==

+=

ωωµα

αµ

40

1

VH

Rrad L loop

+-

ZL

I

Relative permeability

Number of AMMs per unit volume

Magnetic polarizability of

each AMM


Broadband, High Permeability Requires Negative Inductance

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16

18

Frequency, GHz

Re

lativ

e P

erm

eabi

lity

imag

real

a

W

ZL= -jωLd

0 1 2 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16

18

Frequency, GHz

Re

lativ

e P

erm

eabi

lity

imag

real

a

W

ZL= -jωLd

Circuit Theory Model


How to Extract Material Properties (TEM waveguide containing material sample)

wave port 1 wave port 2

material sample

PMC walls


How to Extract Material Properties

• HFSS-Two port S parameters calculation of TEM

waveguide containing material sample. • MATLAB

-Shifting of the reference planes.-Conversion of S-parameters to ABCD

parameters.-Calculation of propagation constant and

characteristic impedance of equivalent transmission line.-Evaluation of the material properties.


How to Extract Material Properties (Loop Configuration)

lumped element

copper loop


Negative Inductance is Modeled as a Frequency-Dependent Capacitance

( ) neg

equivLf

C 22

1

π=


Cancellation of Parasitic Capacitance Using NIC

To remove the resonance, the parasitic capacitance of the loop should be compensated by a negative capacitance.


Remedy for Snoek-Like Phenomenon: Add Negative Capacitance in Shunt with Negative Inductance


Negative Impedance Converter (NIC)

• An ideal NIC is a two-port network such that when a load impedance is attached to the output terminal, the input impedance is the (possibly scaled) negative value of the load impedance.

IdealNIC

ZL

Zin=-kZL (k>0)

IdealNIC

ZL

Zin=-kZL (k>0)

Canonical NIC

R

RZ

+

-ZZ in −=


S_ParamSP1


S-PARAMETERS

OpAmpAMP2

BW=500 MHzGain=100 dB

RR4R=100 OhmTerm

Term1

Z=50 OhmNum=1

RR3R=50 Ohm

RR2R=1 kOhm

RR1R=1 kOhm

Simple Op-Amp Test Circuit

Can specify DC gain and unity gain BW.FOM: RL > 15 dB


1E7 1E8 1E91E6 4E9

-30

-20

-10

-40

0

freq, Hz

dB(S

(1,1

))

m1

m1freq=dB(S(1,1))=-14.883

7.000MHz

Unity-gain BW = 500 MHz


NIC Return Loss BW vs. Op-Amp Unity Gain BW

500 1000 1500 20005

10

15

20

25

30

Op-Amp Unity-Gain BW (MHz)

NIC

Ret

urn

Loss

BW

(MH

z)

Op-amp based NICs contra-indicated for applications above 30 MHz


Fabricated OPA690 NIC Evaluation Board


NIC ZL

Rin

Signal Generator

Vin Vneg

Iin

-ZL

Vg

Rg

Zin

Circuit for evaluating the performance of a grounded negative impedance

Lin ZR >Stability requires that


V g V in V n e g

V tS in eV g

P h a s e = 0D a m p in g = 0D e la y= 0 n s e cF re q = 0 .5 M H zA m p litu d e = 1 0 0 m VV d c = 0 m V

T ra nT ra n 1

M a xTim e S te p = 0 .5 n s e cS to p T im e = 5 u s e c

T R A N S IE N T

RR inR = 1 0 0 O h m

RR 7R = R s c a le

O P A 6 9 0 _ p o rtX 1

RR 1 0R = 5 0 O h m

RR 3R = R s c a le 2

V A RV A R 1

R s c a le 2 = 2 5 0R s c a le = 2 5 0

E q nVa r

RR gR = 5 0 O h m

Schematic captured from Agilent ADS of the circuit for evaluating the performance of the OPA690 NIC


2 4 6 8 10 12 14 16 180 20

-40

-30

-20

-10

-50

0

freq, MHz

dB(R

etur

n_Lo

ss_S

imul

ated

)dB

(Ret

urn_

Loss

_Mea

sure

d)

Simulated and measured return loss for the OPA690 NIC evaluation circuit


• Canonical NIC and most other NIC circuits in the literature produce grounded negative impedance

• But for the applications we are considering here, we need floatingnegative impedance

Ground Negative Impedance Versus Floating Negative Impedance

-Z

-Z


Floating NIC Realized Using Two Op-Amps


S_ParamSP1

Step=0.1 MHzStop=100 MHzStart=0.1 MHz

S-PARAMETERS

MuPrimeMuPrime1MuPrime1=mu_prime(S)

MuPrime

MuMu1Mu1=mu(S)

Mu

TermTerm2

Z=50 OhmNum=2

TermTerm1

Z=50 OhmNum=1

RR2R=1k Ohm

RR1R=1k Ohm

RR4R=1k Ohm

RR3R=1k Ohm

RRLR=50 Ohm

RRnegR=50 Ohm

OpAmpAMP2


OpAmpAMP1


Op-Amp FNIC Test Circuit


m1freq=dB(S(1,1))=-15.048

3.600MHz

1E6 1E71E5 1E8

-40

-30

-20

-10

-50

0

freq, Hz

dB

(S(1

,1))

m1

m1freq=dB(S(1,1))=-15.048

3.600MHz

1E6 1E71E5 1E8

-10

-5

-15

0

freq, Hz

dB

(S(2

,1))

Unity-gain BW = 500 MHz

20 40 60 800 100

1.000

1.002

1.004

1.006

1.008

0.998

1.010

freq, MHz

Mu1

MuP

rime1


Op Amp BW 15 dB RL NIC BW

500 MHz 3.6 MHz

1000 MHz 7.2 MHz

2000 MHz 14.5 MHz

NIC Return Loss BW vs. Op-Amp Unity Gain BW


1i

1v

+

-

LZ

2i

2v

+

-

1Q 2Q

3v 3v′

Floating NIC Circuit Using Two Transistors


NIC “All-Pass” Test Circuit

0Z

LZ0Z LZ−


High-Frequency BJT Device Model

VARVAR1

Cpi=gm/(2*pi*fT)Cc=Cpi/10Rb=7gm=900fT=32Rpi=110

EqnVar

PortP3Num=3

PortP2Num=2

PortP1Num=1

CCcC=Cc pF

RRbR=Rb Ohm

CCpiC=Cpi pF

VCCSSRC1

R2=1e100 OhmR1=Rpi OhmG=gm mS


Test Circuit Results (fT = 32 GHz)

0.2 0.4 0.6 0.80.0 1.0

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.9

-0.2

freq, GHz

dB(S

(2,1

))

0.2 0.4 0.6 0.80.0 1.0

-25

-20

-30

-15

freq, GHz

dB(S

(1,1

))

m1

m1freq=dB(S(1,1))=-19.995

502.0MHz

0.2 0.4 0.6 0.80.0 1.0

1.0000000

1.0000000

1.0000000

1.0000000

1.0000000

freq, GHz

Mu1

MuP

rime1


All-Pass -20 dB RL BW v. fT

0 5 10 15 20 25 30 350

100

200

300

400

500

600

Transistor fT in GHz

All-

pass

ban

dwid

th in

MH

z


Fabricated Two Transistor Floating NIC Using 2N2222 Devices


Measured Results for Two Transistor FNIC


Summary of NIC Developments

• We’ve had some successes in fabricating NICs that work up to about 50 MHz.

• We have had many more failures.

• The main issue concerns stability – small and large signal stability.

• We are making progress, albeit very slowly …

• Someday, someone will make a reliable FNIC that works into the 100s of MHz range.


How Best to Use a NIC to Make a Non-Foster Reactance?

• Direct negation:

ZZ− NIC


How Best to Use a NIC to Make a Non-Foster Reactance?

• Using a certain transformation:

Verman, Proc. IRE, Apr. 1931


R R

-R ZL(s)

Zin(s)

( )( ) 2( )( )

( ) ( )L

inL L

R Z s R RZ s R

Z s Z s

+ − −= + =

ZL(s)

R/2 R/2

R/2 R/2

-R

Negative Impedance Transformation

• Can develop an NIC that needs to work for only one value of real impedance• Also suggests the possibility of using negative resistance diodes (Tunnel, Gunn, etc.) for NIC realization

Grounded

Floating


( )( )

1

1

1 11

1 11

L in

neg

neg

negneg

negneg

Z s Z s

sLsC

sLsC

sLsC sC

sL

sLsCsC

sL

−

−

+− −

− −+

2

2

neg

neg

L R C

LCR

=

=

Negative Impedance Transformation


OptimOptim1

Sav eCurrentEF=no

UseAllGoals=y esUseAllOptVars=y esSav eAllIterations=noSav eNominal=noUpdateDataset=y esSav eOptimVars=noSav eGoals=y esSav eSolns=y esSetBestValues=y esNormalizeGoals=noFinalAnaly sis="None"StatusLev el=4DesiredError=0.0MaxIters=250OptimTy pe=Gradient

OPTIM

VARVAR1

Cneg=7.6146 oLneg=291.922 oLm=55.1849 o

EqnVar

GoalOptimGoal2

RangeMax[1]=RangeMin[1]=RangeVar[1]=Weight=Max=10Min=SimInstanceName="SP1"Expr="abs(real(Zin1)-50)"

GOAL

GoalOptimGoal1

RangeMax[1]=RangeMin[1]=RangeVar[1]=Weight=Max=20Min=SimInstanceName="SP1"Expr="abs(imag(Zin1))"

GOAL

sp_nec_NE85630_7_19940401SNP3

Frequency ="0.05 - 3.60 GHz"Bias="Bjt: Vce=10V Ic=30mA"sp_nec_NE85630_7_19940401

SNP2

Frequency ="0.05 - 3.60 GHz"Bias="Bjt: Vce=10V Ic=30mA"

CAPQCneg

Mode=proportional to f reqF=60.0 MHzQ=100.0C=Cneg pF

INDQLm

Rdc=0.0 OhmMode=proportional to f reqF=60.0 MHzQ=100.0L=Lm nH

INDQLneg

Rdc=0.0 OhmMode=proportional to f reqF=60.0 MHzQ=100.0L=Lneg nH

MuPrimeMuPrime1MuPrime1=mu_prime(S)

MuPrime

MuMu1Mu1=mu(S)

Mu

ZinZin1Zin1=zin(S11,PortZ1)

Zin

N

S2PSNP1

21

Ref

TermTerm2

Z=50 OhmNum=2

TermTerm1

Z=50 OhmNum=1

S_ParamSP1


S-PARAMETERS

Schematic captured from Agilent ADS of VHF monopole with active matching network


40 50 60 70 8030 90

-25

-20

-15

-30

-10

freq, MHz

dB(S

(1,1

))Return Loss (dB)

Simulated return loss at input of optimized active matching network and antenna


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75

80

85

90

70

95

freq, MHz

mag

(S(2

,1))

*100

Overall Efficiency (%)

Overall efficiency (in percent) of optimized active matching network and antenna


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0.98

0.99

1.00

0.97

1.01

freq, MHz

Mu1

MuP

rime1

Small-signal geometrically-derived stability factor for the optimized active matching network and antenna


Summary• The use of non-Foster reactances could improve

the performance of ESAs and HIGPs dramatically.

• Some hard-won successes have been achieved in the development of the requisite NICs.

• But an interdisciplinary team with expertise in circuits as well as field theory and sufficient funding is needed to realize reliable high frequency non-Foster reactances and to integrate them into electromagnetic devices.


non-foster reactances for electrically- small antennas ... · pdf filenon-foster reactances...

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