LISAT2007

LISAT2007

Harold A. Wheeler’s Antenna Design Legacy

Alfred R. Lopez

Greenlawn, NY USAal.lopez@baesystems.com

May 4, 2007

LISAT2007Outline of Presentation

• Harold A Wheeler (Some Highlights)• Harold A. Wheeler’s Antenna Design Legacy

– Matching to a transmission line– Electrically small antenna– Planar array

• Electrically Small Antenna– “Small antenna behaves as a lumped element with some radiation”

• Matching To A Transmission Line– Art of impedance matching using the reflection chart as the

primary tool• Closing Remarks

LISAT2007Harold A Wheeler

LISAT2007Harold A. Wheeler (Some Highlights)• May 10, 1903 - April 25, 1996• 1925 Invented automatic volume control (AVC)• 1939 IRE Morris Liebmann Memorial Prize (paper on TV

amplifiers)• 1941 Development of mine detector used by WWII allied forces• 1947 Started Wheeler Laboratories, inc.• 1948 Chairman of IEEE (IRE) Long Island Section• 1964 IEEE Medal of Honor (highest IEEE award)• 1965-1968, Hazeltine, Chairman of the Board• 180 US patents• More than 90 publications over a span of 57 years

– 1928 Automatic volume control for radio receiving sets– 1947 Fundamental limitations of small antennas– 1975 Small antennas– 1985 Antenna topics in my experience

LISAT2007Harold A. Wheeler’s Antenna Design Legacy

• 1985 Paper, Antenna Topics in My Experience– Wideband Matching To A Transmission Line (1930s to 1980s)

• The union of antenna impedance and circuit theory• Horizontal biconical dipole (6-18 MHz, 1936)• Vertical Monopole on a mast (157-187 MHz, 1943)• Double tuning on reflection chart

– Small Antennas (1940s to 1980s)• 1947 “Behaves as a capacitor or an inductor with some

radiation resistance” (lumped-element small antenna concept)• 1975 Effective volume concept

– Planar Arrays (1960s)• Concept of infinite array (1948)• Relations from current sheet, scan angle (1964)• The impedance crater (1965)• The grating-lobe series (1965)• Array simulators in waveguide (1963)

LISAT2007

Small Antenna Legacy

LISAT2007Small Antennas, 1947 Paper( )Ab

2/6Q3πλπ

<

Original figure appearing in Wheeler’s 1947 paper

LISAT2007Some Definitions (Wheeler)

Radianlength

The radianlength is the wavelength divided by 2π (λ/2π)

Radiansphere

The radiansphere is a sphere whose radius is the radianlength. It is the boundary between the near field and far field of a small antenna. Volume of radiansphere = VRS = 4/3 π (λ/2π)3

Small Antenna

“The small antenna to be considered is one whose maximum dimension is less than the radianlength.”

“An antenna within this limit of size can be made to behave as lumped capacitance or inductance, so this property is assumed.”

LISAT2007Wheeler and Chu

VolumeOccupied WheelerV Factor VolumeEffectivek

VolumeEffectiveVkVV

29

VV

29Q

Papers) 1975 and (1947 Wheeler

OC

E

OC

RS

E

RS

===

==

antenna small theof dimension maximum

the is diameter whosesphere of Volume

VolumeChu V

V Vfor VVQ

Q ON BOUND LOWERQPaper) (1948 Chu

Chu

RSChuChu

RSChu

Chu

==

<<=

=

Wheeler’s formulas yield accurate values, and are invaluable for the design of small antennas.

Chu’s lower bound assumes no stored energy within the sphere and can only be realized in theory. It is a theoretical reference point.

For Cylindrical Antenna: VOC = Ab Where A = area of cylinder base and b = height of cylinder

LISAT2007Wheeler’s Formula for Capacitor Antenna

material (core) fill of ypermitivit Relative

2 ab For

1kkk

factor Fill k ab41k

1 factor Shape k factor volume Effective kk k

length dipole discs, between Distance b radius Disc a Where

k1k

VV

29

VV

29

k1

ba26Q

r

rSC

SCFC

FC

SC

SC

FCSCa

2SC

rSC

OC

RS

E

RS

a2

3

)(CapacitorWheeler

=ε

<−ε+

≈

=π

+=

>===

==

−ε+==

π

πλ

π=

LISAT2007Validation of Wheeler’s Formula for Capacitor

1 .10 3 0.01 0.1100

1 .103

1 .104

1 .105

1 .106

Rad

iatio

n Q

.

Disc DipoleComputer Model

Q Versus Disc Dipole LengthDisc Radius / λ = 0.005 top, 0.01, and 0.05

(Circles are computed points)1000000

100000

Q

10000

1000

100 0.001 0.01 0.1Disc Dipole Length / Wavelength

LISAT2007Wheeler’s Formula for Inductor Antenna

material (core) fill of typermeabili Relative

2 ab For

1/1kkk

factor Fill k

a b if value this than lessSomewhat ba9.01k

1 factor Shape k factor volume Effective kk k

length axial Loop b radius Loop a Where

k1/1k

VV

29

VV

29

k1

ba26Q

r

rSL

SLFL

FL

SL

SL

FLSLb

2SL

rSL

OC

RS

E

RS

b2

3

)Inductor(Wheeler

=µ

>−µ+

≈

=

<+≈

>===

==

−µ+==

π

πλ

π=

LISAT2007Validation of Wheeler’s Formula for Inductor

1 .10 3 0.01 0.110

100

1 .103

1 .104

1 .105

1 .106

Rad

iatio

n Q

.

Loop InductorComputer Model

Q Versus Loop Axial LengthLoop Radius / λ = 0.005 top, 0.01, 0.025 and 0.05

(Circles are computed points)1000000

100000Q

10000

1000

100

100.001 0.01 0.1

Loop Axial Length / Wavelength

LISAT2007Q Dependence on εr and μr

tyPermeabili Relative k

1/1kVV

29Q

: AntennaInductor

yPermitivit Relative k

1kVV

29Q

: AntennaCapacitor

r

2SL

rSL

OC

RSWheeler

r

2SC

rSC

OC

RSWheeler

=µ

−µ+=

=ε

−ε+=

kSC > 1

Q increases with increasing εr

kSL > 1

Q decreases withincreasing μr

LISAT2007Wheeler’s Implementation of Chu’s Lower Bound on Q

mk/211

29

+

Vr = VS

Original figure first appearing in Wheeler’s 1975 paper

Two Typos

LISAT2007Optimum Shape for Cylindrical Antennas

Q Ratio = Q / QChu(Ratio of Total-

To-External Stored Energy)

0 0.5 1 1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

9

10

Cylinder Diameter to Length Ratio

.

b

D = 0.84b

Capacitor

D = Cylinder Diameterb = Cylinder Axial Length

b

Inductor

D = 2.24b

0.84 2.24Cylinder Diameter to Length Ratio

LISAT2007Optimum Spherical-Cap Dipole

Wheeler Prediction 1985: Minimum Q when cap area about ½ sphere area

Lower Bound for Q of Capacitor (Electric) Small Antennas

Area Ratio

QRatio

1.75

0.440 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

.

.

LISAT2007Small Antenna Q Ratios

Antenna Type Q Ratio = Q/QChu

Spherical Inductor, µr = ∞ 1.0

Spherical Inductor, µr = 1 3.0

Cylindrical Inductorµr = 1, Diameter / Length = 2.24

4.4

Disc DipoleDiameter / Length = 0.84

2.4

Spherical-Cap Dipole 1.75

Q Ratio = Antenna Q / Chu Lower-Bound Q

It is believed that the Q for the spherical-cap dipole is the lower bound for the Q of the capacitor (electric) antenna

LISAT2007“The Largest Antenna in the World is a Small Antenna”

Wheeler, 1985

Antenna located at North West Cape, Australia

Operates at 15.5 KHz

Diameter of outer tower circle is 1.7 miles (2.7Km)

Tower height is 1300 Ft. (390m)

Radiation Q = 435

Radiation Bandwidth = 36 Hz

Resonance Bandwidth = 134 Hz

LISAT2007

Antenna Impedance Matching Legacy

LISAT2007Impedance Matching

• 1935 Antenna perceived as a circuit element. Circuit theory applied to impedance matching antenna to transmission line.

• Reflection Chart (Tool for Impedance Matching)– 1936 Paper “Doublet antennas and transmission lines” (Reflection

chart on resistance and reactance coordinates)– 1940 Reflection Charts (Smith, Carter and Wheeler)– Wheeler developed the art of impedance matching using the

reflection chart as the primary tool (Smith and Carter charts were used as under-lays for the Wheeler reflection chart in graphical solutions to impedance matching problems)

• Optimum impedance matching of single-tuned and double-tuned antennas

LISAT2007Antenna Bandwidth DefinitionsRadiation Bandwidth (Ratio) = 1/Q = (fH – fL)/f0

= Antenna 3 dB Bandwidth= Antenna Resonance Bandwidth

For an antenna tuned to resonance, fH and fL are the high and low frequencies where the magnitude of the reactance is equal to the resistance (f0 is the resonant frequency)

(Impedance) Matching Bandwidth (Ratio) = B = (fH – fL)/f0

fH and fL are the high and low frequencies of a frequency band over which a specified magnitude of reflection coefficient (or VSWR) is not exceeded

B(Γ) = f(Γ)/Q where Γ is the maximum reflection magnitudeMatching Factor = QB(Γ) = f(Γ)

LH0 fff =

Matching Bandwidth is equal to the Radiation Bandwidthmultiplied by the Matching Factor

LISAT2007Optimum Impedance Matching: Wheeler and Fano

2 and 1 n1

2QBn2

n1

=Γ−

Γ=

∞=

Γ=

=

−

π

=

........ 3 ,2 ,1n)nacosh()nbcosh(

)bcosh()nbtanh(

)acosh()natanh(

)bsinh()asinh(n2

sin2QB

Wheeler Fano

It is remarkable that the Wheeler and Fano equations are in exact agreement (n = 1, n = 2), considering their radical difference in form.

Single Tuned n = 1Double Tuned n = 2Infinite Tuned n = ∞

LISAT2007Multiple Tuning Impedance Matching Network

Ideal Transformer(Adjust Impedance Level)

Tuning (n)

(3) (2) (1)

R

C1

L1

L2 C2

L3 C3

Generator Antenna

R0

3322110

0

CL21

CL21

CL21f

Frequency Resonant f

π=

π=

π=

=

LISAT2007Wheeler Single Tuning

L1

C1

R

Ideal ImpedanceTransformer

R0

Generator Antenna

LISAT2007

.

Single Tuning Reflection Chart

Mid-Band Match

WheelerEdge-BandMinimum-ΓMatching

fH

fL

Minimum Γ

OCSC

C

L

R0R

Q = 10

B = 0.2

LISAT2007Single Tuning Equation

( )

1 n 1

21

2QB

)2/tan(1)2/tan(2)tan(

)2/tan()sin()1)(cos()sin()1)(cos(

1)f(z1)z(f

QB)tan( )QB(1RR 1)jQB1(

RR

)sin(j)cos()jexp(jQB1RR

R)f(Z)f(z

ff

ff

RLj1R

ff

ffLjR)f(Z

n2

n1

2

2

22

22

H

H

20

0

00

HH

0

L

0

H10

H

0

0

H10H

=

Γ−

Γ=

Γ−Γ

=

φ−φ

=φ

φ=φ++φφ+−φ

=+−

=Γ

=φ+==+

φ+φ=φ=+==

−

ω+=

−ω+=

LISAT2007Wheeler Double Tuning

L1

C1

R

Ideal ImpedanceTransformer(Set for optimum single-tuned edge-band matching)

R0

Generator Antenna

L2 C2

LISAT2007Double Tuning Reflection Chart

.

Parallel resonant circuit susceptance

WheelerSingle-tunedEdge-BandMinimum-ΓMatching

fH

fL

OCSC

C

L

fH

fL

R0R

LISAT2007Double Tuning Reflection Chart (continued)

.Wheeler Double-Tuned Matching

WheelerSingle-tunedEdge-BandMinimum-ΓMatching

OCSC

C

L

Γ2

Γ1

fH

fL

fH & fLf0

212

1

1

2

1

Γ=Γ

Γ=

ΓΓ

Γ1

Γ2 1

Similar Triangles

LISAT2007Double Tuning Reflection Chart (continued)

.

OCSC

C

L

Γ2

LISAT2007Q of Parallel Resonant Circuit

( )

( ) ( )

22

2

2

0

H1

2L

0

0

L

0

20L2

)QB(1

QQ

)QB(1

jQB1ImjQB1jQB1

jQB1)QB(1

Im

jQB1RRIm

fz1Im

BQff

ff

GCfb

+=

+

−=

−−

++

=

+

=

=

−=

−

ω=

Normalized Susceptance of Parallel Resonant Circuit at f = fL

Normalized Susceptance of Series Resonant Circuit at f = fH

LISAT2007Double Tuning Equation

2n 1

2 -1

2QB

:Tuned Doublemagnitude reflection tuned-double Maximum

magnitude reflection tuned-single Maximum -12QB

:case tuned single the hStart wit

n2

n1

2

2

2

212

1

21

1

=

Γ−

Γ=

Γ

Γ=

=ΓΓ=Γ

=ΓΓ

Γ=

LISAT2007

Impedance Matching Circuit

Impedance Matching Factor(QB)

QB forV = VSWR = 2

Single-tuned mid-band match (Non Fano)

QB = 0.707

Single-tuned edge-band matching(Wheeler-Fano, n = 1)

QB = 0.750

Double-tuned matching(Wheeler-Fano, n = 2) QB = 1.732

Triple-tuned matching(Lopez-Fano, n = 3)

a3 = 2.413b3 = 0.678 QB = 2.146

Infinite-tuned matching(Fano-Bode, n = ∞) QB = 2.860

Impedance Matching Formulas - Summary

V1V

1

2QB2

−=

Γ−

Γ=

V21V

12

1lnsinh

1QB2

2−

=Γ−

Γ=

Γ

=

1V12

1ln21sinh

1QB 2 −=Γ−Γ

=

Γ

=

−+

π=

Γ

π=

1V1Vln1ln

QB

Rule of thumb for maximum achievable matching bandwidth (BMaxA)If V > 2 then BMaxA ≈ V / Q

Γ

−+

Γ

=1ln

ab11ln

a1sinhb

1QB

3

3

33

LISAT2007Closing Remarks

• Wheeler’s Small Antenna Legacy– Radianlength– Radiansphere– Small Antenna Lumped-Element Concept– Accurate Formulas for Q of Small Antennas– Relationship to Chu’s Lower Bound on Q of Small Antennas

• Wheeler’s Antenna Impedance Matching Legacy– Antenna perceived as a circuit element. Circuit theory applied to

impedance matching antenna to transmission line.– Reflection Chart (Tool for Impedance Matching)– Simple formulas relating the Q-Bandwidth product to the maximum

reflection magnitude– Wheeler and Fano contributions provide a complete picture of

impedance matching limitations

LISAT2007References

[1] H. A. Wheeler, “Antenna Topics in My Experience,” IEEE Trans. Antennas Propagat.,Vol. AP-33, No. 2, Feb. 1985, pp. 144-151.

[2] A. R. Lopez, “Fundamental Limitations of Small Antennas: Validation of Wheeler’s Formulas,” IEEE Antennas Propagat. Mag., Vol. 48, No. 4, Aug. 2006, pp. 28-36.

[3] A. R. Lopez, “Review of Narrowband Impedance-Matching Limitations,” Antennas Propagat. Mag., Vol. 46, No. 4, Aug. 2004, pp. 88-90.

[4] A. R. Lopez, “Wheeler and Fano Impedance Matching,” IEEE Antennas Propagat. Mag., to be published.

[5] H. A. Wheeler, “Fundamental Limitations of Small Antennas,” Proc. IRE, 35, Dec. 1947, pp. 1479-1484.

[6] L. J. Chu, “Physical Limitations of Omni-Directional Antennas,” J. Appl. Phys., 19, Dec. 1948, pp. 1163-1175.

[7] H. A. Wheeler, “Reflection Charts Relating to Impedance Matching,” IEEE Trans. Microwave Theory Tech., Vol. MTT-32, No. 9, Sep. 1984, pp. 1008-1021.

[8] R. M. Fano, “Theoretical Limitations of the Broadband Matching of Arbitrary Impedances,” J. Franklin Institute, 249, Jan, 1950, pp. 57-83, Feb. 1950, pp. 139-154.

[9] A. R. Lopez, Relevant papers available at http://www.arlassociates.net