Non-Foster Reactances for Electrically- Small Antennas, High-Impedance Surfaces, and Engineered Materials

James T. Aberle Associate 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

• 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. IEEE Waves and Devices 19 Feb 2010 4 Copyright © 2010James T. Aberle 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).

L L Term L2 C L1 Term1 C1 R L=40 nH L=40 nH R1 Num=1 R= C=30 pF R= Z=50 Ohm R=50 Ohm

20 real(Zin1) imag(Zin1) 0 S(1,1)

-20 0 50 100 150 200 250 300 freq, MHz

freq (10.00MHz to 300.0MHz)

L L Term L2 L L1 Term1 L3 R L=40 nH L=40 nH R1 Num=1 R= L=-40 nH R= Z=50 Ohm R= R=50 Ohm

120

100

40 S(1,1) real(Zin1) imag(Zin1) 20

-20 0 50 100 150 200 250 300 f req, MHz

f req (10.00MHz to 300.0MHz)

• 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

•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.

• 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.

jX a Rl Z0 N 1:

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

= + =++ Zaa R jX arl R R jX a = = Rr e cd R a radiation resistance = − = R/ (1 ecd ) R a dissipative loss resistance = X a antenna reactance R N = r Z 0

• 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.

• 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.

2  ω   1  Z = R   + jωL −  a 0  ω   a ω   0   Ca 

Impedance produced by equivalent circuit

• Evaluation of the model parameters ( R0, La and Ca): = ℜ{ (ω )} R0 Za 0

 −1 L  ω a ℑ{}Z ()ω   1 ω    a 1   1   =    −1 1 ω   ℑ{}Z ()ω   2 ω    a 2   2 Ca 

= Ω R0 30.4 L Term Z1P_Eqn La C = L=194 nH La 194 nH Term1 Z1P1 Ca Num=1 Z[1,1]=4.30*(freq/52e6)**2 R= C=8.69 pF = Z=50 Ohm Ca 69.8 pF

20 0

-100 15 -200

10 -300

real(Zin1) -400 5 imag(Zin1) -500

real(VHF_Whip_S2P..Zin1) 0 imag(VHF_Whip_S2P..Zin1) -600 30 40 50 60 70 80 90 30 40 50 60 70 80 90 freq, MHz freq, MHz

Matching Network Z L

• 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.

Q of the load Maximum value of fractional bandwidth that can be achieved with any passive, lossless matching network. ∆ = Γ = 1 ⇒ f ≤ Q 81 ,9. m .0 035 3 f0

L L Term Le Z1P_Eqn La C Term1 L=884 nH Z1P1 L=194 nH Ca Num=1 R= Z[1,1]=4.30*(freq/52e6)**2 R= C=8.69 pF Z=4.30 Ohm

m1 m2 freq= 51.80MHz freq= 52.20MHz S(1,1)=-10.409 / -73.141 S(1,1)=-10.477 / 71.883 impedance = Z0 * (0.992 - j0.630) impedance = Z0 * (1.008 + j0.630)

m2 ∆ − f ≈ 52 2. 51 8. =

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

S-PARAMETERS N Zin S_Param SP1 Zin Start=30 MHz Zin1 Stop=90 MHz Zin1=zin(S11,PortZ1) Step=1 MHz

1 2 C R L Ref Term C2 RD Le Term L Term2 Term1 L2 C=53.0 pF R=200 Ohm L=884 nH S2P R= Num=2 Num=1 L=177 nH SNP1 Z=50 Ohm Z=50 Ohm R= File="VHF_whip.s2p" TF TF1 T=0.354

-6 -12

-14 -8 -16

-10 -18 dB(S(1,1)) dB(S(2,1)) -20 -12

S(1,1) -22

-14 -24 30 40 50 60 70 80 90 30 40 50 60 70 80 90 freq, MHz freq, MHz Decent match, poor efficiency

L L C L L L1 L2 La1 La Term Ca1 Z1P_Eqn C L Term1 L=-44.9 nH L=-44.9 nH L=-194 nH Z1P1 L=194 nH Ca L3 C=-8.69 pF Num=1 R= R= R= Z[1,1]=4.30*(freq/52e6)**2 R= C=8.69 pF L=44.9 nH Z=50 Ohm R=

Dualizer Antenna

m1 Cancels frequency squared m1 freq= 90.00MHz S(1,1)=4.843E-4 / -3.517E-5 dependence of radiation resistance. S(1,1) impedance = Z0 * (1.001 - j5.952E-10)

Van Der Pol, Proc, IRE, Feb. 1930

freq (30.00MHz to 90.00MHz)

Z0 − ( + ) La Lm

− Ca

Lm Z0

Active matching network Two-port model of antenna

1 2

L Ref C Term Term L2 L Ca1 Term2 Term1 L=-Lneg nH C=-Cneg pF S2P L3 R= Num=2 Num=1 SNP1 Z=50 Ohm Z=50 Ohm L=Lm nH File="VHF_whip.s2p" R= Var Eqn VAR VAR1 Lm=49.5959 {o} Lneg=226.588 {o} Cneg=8.72637 {o}

-16 0.0

-18 -0.1 -20 -0.2 -22

S(1,1) -0.3

dB(S(1,1)) -24 dB(S(2,1))

-26 -0.4

-28 -0.5 30 40 50 60 70 80 90 30 40 50 60 70 80 90 freq, MHz freq, MHz

freq (30.00MHz to 90.00MHz)

Capacitive FSS Low Permittivity Spacer

Metal Vias Metal Backplane

10”

Coaxial Feed

Bent Wire Element

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

• 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).

Z in d

ηηη βββ o C Zo, Short Γ

Zstub

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.

Shunt Capacitance

180 o Zstub θ θ βββ θ θ 2 d 90 o Γ θθθ Zin BW ~ 10% to 20% of f o. 0o Short Open -90 o

FSS layer µ r h

Spacer layer Ground plane ∆ω h = 2πµ ω r λ 0 0

IEEE Waves and Devices 19 Feb 2010 32 Copyright © 2010James T. Aberle 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 … IEEE Waves and Devices 19 Feb 2010 33 Copyright © 2010James T. Aberle Electrically-Thin Broadband High-Impedance Surface

Kern, Werner, Wilhelm, APS 2003

0.062 in Rogers Duroid 5880

m1 m2 freq= 2.308GHz freq= 2.502GHz phase(S(1,1))=90.015 phase(S(1,1))=-90.204

m3 Term freq= 2.403GHz C TLINP Term1 phase(S(1,1))=-0.139 C1 TL1 Num=1 200 C=2.17 pF Z=377/sqrt(2.2) Ohm Z=377 Ohm L=1.6 mm m1 K=2.2 100 A=0.0 m3 F=1 GHz 0 TanD=0.0009 m2

Mur=1 phase(S(1,1)) -100 TanM=0 Sigma=0 -200 1.0 1.5 2.0 2.5 3.0 3.5 4.0 freq, GHz ∆f = 8% f0 IEEE Waves and Devices 19 Feb 2010 35 Copyright © 2010James T. Aberle Electrically-Thin HIS with Negative Inductance

Term L TLINP Term1 Lneg TL1 Num=1 L=-2.02 nH Z=377/sqrt(2.2) Ohm Z=377 Ohm R= L=1.6 mm K=2.2 A=0.0 F=1 GHz TanD=0.0009 15 Mur=1 10 TanM=0 Sigma=0 5

0 phase(S(1,1)) -5

-10 1.0 1.5 2.0 2.5 3.0 3.5 4.0 freq, GHz

-20

-40

-60

-80

Reflection Coefficient Phase (deg) -100

-120

-140 12 13 14 15 16 17 18 Frequency (GHz)

Zload

200

150

100

-50

-100 Reflection Coefficient Phase (deg)

-150

-200 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (GHz)

20 Lneg = -2.2nH

-10

-20 Reflection Coefficient Phase (deg)

-30

-40 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (GHz)

IEEE Waves and Devices 19 Feb 2010 43 Copyright © 2010James T. Aberle 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 ε µ electromagnetic constitutive parameters ( r and r) are altered with respect to the host medium.

= ε •ε D 0 E = µ • µ B 0 H

IEEE Waves and Devices 19 Feb 2010 44 Copyright © 2010James T. Aberle 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.

d z Z z L

dx dz a y dy •A 3-dimensional lattice of artificial molecules. •Electrically small loop with a load impedance IEEE Waves and Devices 19 Feb 2010 46 Copyright © 2010James T. Aberle Simple Circuit Model for Artificial Magnetic Molecule (AMM)

Relative Number of permeability AMMs per µ = + α unit volume xx 1 N Rrad Lloop

I + V Z H - L − ωµ 4 α = m = j 0a + ω + H Rloss j Lloop Z L

Magnetic polarizability of each AMM

16 Circuit Theory Model real 14

10 W 8

6 ω RelativeRelativePermeability Permeability ZL= -j Ld 4

2 a imag 0 0 1 2 3 4 5 6 7 8 9 10 Frequency, GHz

PMC walls

wave port 1 wave port 2

material sample

IEEE Waves and Devices 19 Feb 2010 49 Copyright © 2010James T. Aberle 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.

lumped element

copper loop

1 C = equiv ()π 2 2 f Lneg

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

• 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.

Zin =-kZ L (k>0)

Ideal Z NIC L Canonical NIC

+ = − - Zin Z

R Z

S-PARAMETERS

S_Param R SP1 R1 R Start=1 MHz R=1 kOhm Stop=4000 MHz R3 R=50 Ohm Step=1 MHz

R OpAmp R4 AMP2 Term R=100 Ohm Term1 Gain=100 dB BW=500 MHz Num=1 Z=50 Ohm

R R2 R=1 kOhm Can specify DC gain and unity gain FOM: RL > 15 dB BW.

-10 m1

-20 m1 freq= 7.000MHz

dB(S(1,1)) dB(S(1,1))=-14.883 -30

-40 1E6 1E7 1E8 1E9 4E9 freq, Hz

10 Op-amp based NICs NIC Return Loss BW (MHz) BW Loss Return NIC contra-indicated for applications above 30 MHz 5 500 1000 1500 2000 Op-Amp Unity-Gain BW (MHz)

Rin Rg Vin Vneg

Vg Iin NIC ZL

Signal Generator

Zin -ZL Stability requires that > Rin Z L

T RANSIENT

V a r V A R T ra n E q n R T ra n 1 V A R 1 Rscale=250 R 7 StopTim e=5 usec R=Rscale MaxTim eStep=0.5 nsec Rscale2=250

Vg Vin Vneg R R OPA690_port R g R in X 1 V tS ine R =50 O hm R =10 0 O hm V g V dc=0 m V Am plitude=100 m V Freq=0.5 M Hz Delay=0 nsec R Dam ping=0 R 3 P hase=0 R=Rscale2 R R 1 0 R =50 O hm

-10

-20

-30

-40

dB(Return_Loss_Measured) dB(Return_Loss_Simulated) -50 0 2 4 6 8 10 12 14 16 18 20 freq, MHz

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

-Z

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

-Z

S-PARAMETERS Mu MuPrime S_Param SP1 Mu MuPrime Start=0.1 MHz Mu1 MuPrime1 Stop=100 MHz Mu1=mu(S) MuPrime1=mu_prime(S) Step=0.1 MHz

OpAmp R AMP1 R3 Gain=100 dB R=1k Ohm BW=500 MHz R R4 R=1k Ohm

R R Term Rneg RL Term R Term1 R=50 Ohm R=50 Ohm Term2 R1 Num=1 Num=2 R=1k Ohm Z=50 Ohm Z=50 Ohm R OpAmp R2 AMP2 R=1k Ohm Gain=100 dB BW=500 MHz

0 0

-10 m1

-5 -20

-30

dB(S(1,1)) m1

dB(S(2,1)) -10 -40 freq= 3.600MHz dB(S(1,1))=-15.048

-50 -15 1E5 1E6 1E7 1E8 1E5 1E6 1E7 1E8 freq, Hz freq, Hz

1.010

1.008

1.006

1.004 Mu1

MuPrime1 1.002

1.000

0.998 0 20 40 60 80 100 freq, MHz

Op Amp BW 15 dB RL NIC BW 500 MHz 3.6 MHz 1000 MHz 7.2 MHz 2000 MHz 14.5 MHz

Z L ′ v3 v3

i i 1 Q1 Q2 2

+ +

v1 v2

- -

Z − L Z L Z0

C Cc C=Cc pF

C Port R Cpi Port P1 Rb C=Cpi pF P2 Num=1 R=Rb Ohm Num=2

Var VCCS Eqn VAR VAR1 SRC1 Rpi=110 G=gm mS fT=32 R1=Rpi Ohm gm=900 R2=1e100 Ohm Rb=7 Cc=Cpi/10 Cpi=gm/(2*pi*fT)

Port P3 Num=3

-15 -0.2 m1 -0.3 -20 -0.4

-0.5

-0.6

dB(S(1,1)) -25

dB(S(2,1)) -0.7

-0.8 -30 -0.9 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 freq, GHz freq, GHz

m1 freq= 502.0MHz 1.0000000 dB(S(1,1))=-19.995

1.0000000

1.0000000 Mu1 MuPrime1 1.0000000

1.0000000 0.0 0.2 0.4 0.6 0.8 1.0 freq, GHz

500

400

300

200 All-pass bandwidth All-pass in MHz bandwidth

100

• 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.

• Direct negation:

− Z NIC Z

• Using a certain transformation:

Verman, Proc. IRE, Apr. 1931

Zin (s) • Can develop an NIC that needs to work for only one -R ZL(s) value of real impedance Grounded • Also suggests the possibility of using negative resistance diodes (Tunnel, Gunn, etc.) for R/2 R/2 NIC realization

Floating -R ZL(s)

R/2 R/2 (RZs+( ) )( − R ) − 2 =L + = R Zin ( s ) R ZsL() Zs L ()

IEEE Waves and Devices 19 Feb 2010 80 Copyright © 2010James T. Aberle Negative Impedance Transformation ( ) ZsL( ) Zs in −1 sL sC neg 1 = 2 −sL Lneg R C sC neg = L 1 1 Cneg 2 sL + R sC − − 1 sC neg sL neg 1 1 −sL − 1 neg sC sC + neg sL

S-PARAMETERS N OPTIM Zin Mu MuPrime S_Param Optim SP1 Zin Mu MuPrime Optim1 Start=30 MHz Zin1 Mu1 MuPrime1 OptimTy pe=Gradient Sav eCurrentEF=no Stop=90 MHz Zin1=zin(S11,PortZ1) Mu1=mu(S) MuPrime1=mu_prime(S) MaxIters=250 Step=1 MHz DesiredError=0.0 StatusLev el=4 FinalAnaly sis="None" NormalizeGoals=no SetBestValues=y es GOAL Sav eSolns=y es Sav eGoals=y es Goal Sav eOptimVars=no OptimGoal1 INDQ CAPQ UpdateDataset=y es Expr="abs(imag(Zin1))" Lneg Cneg Sav eNominal=no SimInstanceName="SP1" L=Lneg nH C=Cneg pF Sav eAllIterations=no Min= Q=100.0 Q=100.0 UseAllOptVars=y es Max=20 F=60.0 MHz F=60.0 MHz UseAllGoals=y es Weight= Mode=proportional to f req Mode=proportional to f req RangeVar[1]= Rdc=0.0 Ohm RangeMin[1]= RangeMax[1]=

sp_nec_NE85630_7_19940401 GOAL SNP3 sp_nec_NE85630_7_19940401 Bias="Bjt: Vce=10V Ic=30mA" Goal SNP2 Frequency ="{0.05 - 3.60} GHz" OptimGoal2 Bias="Bjt: Vce=10V Ic=30mA" Expr="abs(real(Zin1)-50)" Frequency ="{0.05 - 3.60} GHz" SimInstanceName="SP1" Min= Max=10 Var VAR 1 2 Weight= Eqn RangeVar[1]= VAR1 Ref RangeMin[1]= Lm=55.1849 {o} INDQ Lneg=291.922 {o} RangeMax[1]= S2P Term Lm Cneg=7.6146 {o} Term SNP1 Term2 L=Lm nH Term1 Num=2 Num=1 Q=100.0 Z=50 Ohm Z=50 Ohm F=60.0 MHz Mode=proportional to f req Rdc=0.0 Ohm

Return Loss (dB) -10

-15

-20 dB(S(1,1)) -25

-30 30 40 50 60 70 80 90 freq, MHz

Overall Efficiency (%) 95

mag(S(2,1))*100 75

70 30 40 50 60 70 80 90 freq, MHz

1.01

1.00

0.99 Mu1 MuPrime1 0.98

0.97 30 40 50 60 70 80 90 freq, MHz

IEEE Waves and Devices 19 Feb 2010 86 Copyright © 2010James T. Aberle 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.