Harold A. Wheeler’s Antenna Design Legacy

Alfred R. Lopez

Greenlawn, NY USA

al.lopez@baesystems.com

October 24, 2007

Outline of Presentation

• Harold A Wheeler (Some Highlights)

• Harold A. Wheeler’s Antenna Design Legacy

– 1984 AP Symposium IEEE Centennial Session

“Antenna Topics in My Experience”

• Impedance matching to a transmission line

• Electrically small antenna

• Planar arrays

• Electrically Small Antenna

• Matching To A Transmission Line

• Closing Remarks

Harold A Wheeler

Harold 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

Harold A. Wheeler’s Antenna Design Legacy

• 1985 AP Trans. Paper, “Antenna Topics in My Experience”(1984 AP Symposium paper not published in the Proceedings)

– Wideband Matching To A Transmission Line

• The union of antenna impedance and circuit theory

• Horizontal biconical dipole (6-18 MHz, 1936)

• Vertical Monopole on a mast (157-187 MHz, 1943)

• Impedance matching using reflection chart

– Small Antennas

• C and L of equal effective volume (1947)

• Radiation power factor proportional to volume (1947)

• Utilization of space (1975)

– Planar Arrays

• 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)

Small Antenna Legacy

1947 Paper: “Fundamental Limitations of Small Antennas”

(((( ))))Ab

2/6Q

3ππππλλλλππππ<<<<

Original figure appearing in Wheeler’s 1947 paper

Some Wheeler Definitions

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.”

Definition of Antenna Radiation Q

Antenna Q

The ratio of 2π times the energy stored in the fields excited by the

antenna to the energy radiated and dissipated per cycle (IEEE 100)

Antenna Radiation Q

The ratio of 2π times the energy stored in the fields excited by the

antenna to the energy radiated per cycle

The antenna radiation Q is a fundamental property of an electrically

small antenna, which is inversely related to the bandwidth capability of

the antenna.

Wheeler (and) Chu Small Antenna Limitations

VolumeOccupied WheelerV

Factor VolumeEffectivek

VolumeEffectiveV

kV

V

2

9

V

V

2

9Q

Papers) 1975 and (1947 Wheeler

OC

E

OC

RS

E

RS

====

====

====

========

antenna small the

of dimension maximum

the is diameter whose

sphere of Volume

VolumeChu V

V Vfor V

VQ

Q ON BOUND LOWERQ

Paper) (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 very useful

theoretical reference point.

For Cylindrical Antenna: VOC = Ab

Where A = area of cylinder base and

b = height of cylinder

Wheeler’s Formula for Capacitor Antenna

material (core) fill of ypermitivit Relative

2 a

b For

1k

kk

factor Fill k

a

b41k

1 factor Shape k

factor volume Effective kk k

length dipole discs, between Distance b

radius Disc a Where

k

1k

V

V

2

9

V

V

2

9

k

1

ba

26Q

r

rSC

SCFC

FC

SC

SC

FCSCa

2SC

rSC

OC

RS

E

RS

a2

3

)(CapacitorWheeler

====εεεε

<<<<−−−−εεεε++++

≈≈≈≈

====ππππ

++++====

>>>>====

========

====

====

−−−−εεεε++++========

ππππ

ππππλλλλ

ππππ====

Validation of Wheeler’s Formula for Capacitor

1 .103

0.01 0.1100

1 .103

1 .104

1 .105

1 .106

Radiation Q

.

Disc Dipole

Computer Model

Q Versus Disc Dipole Length

Disc Radius / λ = 0.005 top, 0.01, and 0.05

(Circles are computed points)1000000

100000

Q

10000

1000

100 0.001 0.01 0.1

Disc Dipole Length / Wavelength

Wheeler’s Formula for Inductor Antenna

material (core) fill of typermeabili Relative

2 a

b For

1/1k

kk

factor Fill k

a b if value this than lessSomewhat b

a9.01k

1 factor Shape k

factor volume Effective kk k

length axial Loop b

radius Loop a Where

k

1/1k

V

V

2

9

V

V

2

9

k

1

ba

26Q

r

rSL

SLFL

FL

SL

SL

FLSLb

2SL

rSL

OC

RS

E

RS

b2

3

)Inductor(Wheeler

====µµµµ

>>>>−−−−µµµµ++++

≈≈≈≈

====

<<<<++++≈≈≈≈

>>>>====

========

====

====

−−−−µµµµ++++========

ππππ

ππππλλλλ

ππππ====

Validation of Wheeler’s Formula for Inductor

1 .103

0.01 0.110

100

1 .103

1 .104

1 .105

1 .106

Radiation Q

.

Loop Inductor

Computer Model

Q Versus Loop Axial Length

Loop Radius / λ = 0.005 top, 0.01, 0.025 and 0.05

(Circles are computed points)1000000

100000

Q

10000

1000

100

10

0.001 0.01 0.1

Loop Axial Length / Wavelength

Q Dependence on εr and µr

tyPermeabili Relative

k

1/1k

V

V

2

9Q

: AntennaInductor

yPermitivit Relative

k

1k

V

V

2

9Q

: AntennaCapacitor

r

2SL

rSL

OC

RSWheeler

r

2SC

rSC

OC

RSWheeler

====µµµµ

−−−−µµµµ++++====

====εεεε

−−−−εεεε++++====

kSC > 1

Q increases with

increasing εr

kSL > 1

Q decreases with

increasing µr

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

mk/21

1

2

9

++++

Vr = VS

Original figure first appearing in Wheeler’s 1975 paper

Two

Typos

Optimum 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 Diameter

b = Cylinder Axial Length

b

Inductor

D = 2.24b

0.84 2.24

Cylinder Diameter to Length Ratio

Optimum 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

Q

Ratio

1.75

0.44

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

.

.

Small Antenna Q Ratios

1.75 Spherical-Cap Dipole

2.4Disc Dipole

Diameter / Length = 0.84

4.4Cylindrical Inductor

µµµµr = 1, Diameter / Length = 2.24

3.0Spherical Inductor, µµµµr = 1

1.0Spherical Inductor, µµµµr = ∞∞∞∞

Q Ratio = Q/QChuAntenna Type

Q Ratio = Radiation 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

“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

Antenna Q = 116

Resonance Bandwidth = 134 Hz

Antenna Impedance

Matching Legacy

Impedance 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 Hemisphere 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)

• Equations for optimum impedance matching of single-tuned

and double-tuned antennas

Antenna Bandwidth Definitions

Antenna 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(ΓΓΓΓ) = f(ΓΓΓΓ)/Q where ΓΓΓΓ is the maximum reflection magnitudeMatching Factor = QB(ΓΓΓΓ) = f(ΓΓΓΓ)

LH0 fff ====

Matching Bandwidth is equal to the Antenna Bandwidth

multiplied by the Matching Factor

Multiple Tuning Impedance Matching Network

Ideal Transformer

(Adjust Impedance Level)

Tuning (n)

(3) (2) (1)

R

C1

L1

L2 C2

L3 C3

Generator Antenna

R0

332211

0

0

CL2

1

CL2

1

CL2

1f

Frequency Resonant f

ππππ====

ππππ====

ππππ====

====

Optimum Impedance Matching: Wheeler and Fano

2 and 1 n

1

2QB

n

2

n

1

====ΓΓΓΓ−−−−

ΓΓΓΓ====

∞∞∞∞====

ΓΓΓΓ====

====

−−−−

ππππ

====

........ 3 ,2 ,1n

)nacosh(

)nbcosh(

)bcosh(

)nbtanh(

)acosh(

)natanh(

)bsinh()asinh(

n2sin2

QB

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 = 1

Double Tuned n = 2

Infinite Tuned n = ∞

Wheeler Single Tuning

L1

C1

R

Ideal

Impedance

Transformer

R0

Generator Antenna

.

Single Tuning Reflection Chart

Mid-Band

Match

Wheeler

Edge-Band

Minimum-ΓMatching

fH

fL

Minimum Γ

OCSC

C

L

R0R

Q = 10

B = 0.2

Single Tuning Equation

(((( ))))

1 n

1

2

1

2QB

)2/tan(1

)2/tan(2)tan(

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

)sin()1)(cos(

1)f(z

1)z(f

QB)tan( )QB(1R

R 1)jQB1(

R

R

)sin(j)cos()jexp(jQB1R

R

R

)f(Z)f(z

f

f

f

f

R

Lj1R

f

f

f

fLjR)f(Z

n

2

n

1

2

2

22

22

H

H

20

0

00

HH

0

L

0

H10

H

0

0

H10H

====

ΓΓΓΓ−−−−

ΓΓΓΓ====

ΓΓΓΓ−−−−

ΓΓΓΓ====

φφφφ−−−−

φφφφ====φφφφ

φφφφ====φφφφ++++++++φφφφ

φφφφ++++−−−−φφφφ====

++++

−−−−====ΓΓΓΓ

====φφφφ++++========++++

φφφφ++++φφφφ====φφφφ====++++========

−−−−

ωωωω++++====

−−−−ωωωω++++====

Wheeler Double Tuning

L1

C1

R

Ideal

Impedance

Transformer

(Set for

optimum

single-tuned

edge-band

matching)

R0

Generator Antenna

L2 C2

Double Tuning Reflection Chart

.

Parallel resonant

circuit susceptance

Wheeler

Single-tuned

Edge-Band

Minimum-ΓMatching

fH

fL

OCSC

C

L

fH

fL

R0R

Double Tuning Reflection Chart (continued)

.

Wheeler

Double-Tuned

Matching

Wheeler

Single-tuned

Edge-Band

Minimum-ΓMatching

OCSC

C

L

Γ2

Γ1

fH

fL

fH & fLf0

212

1

1

2

1

Γ=Γ

Γ=

Γ

Γ

ΓΓΓΓ1

ΓΓΓΓ2 1

Similar

Triangles

Double Tuning Reflection Chart (continued)

.

OCSC

C

L

Γ2

Double Tuning Equation

2n

1

2

-1

2QB

:Tuned Double

magnitude reflection tuned-double Maximum

magnitude reflection tuned-single Maximum

-1

2QB

:case tuned single the hStart wit

n

2

n

1

2

2

2

212

1

21

1

====

ΓΓΓΓ−−−−

ΓΓΓΓ====

ΓΓΓΓ

ΓΓΓΓ====

====ΓΓΓΓ

ΓΓΓΓ====ΓΓΓΓ

====ΓΓΓΓ

ΓΓΓΓ

ΓΓΓΓ====

QB = 2.146

a3 = 2.413

b3 = 0.678

Triple-tuned matching

(Lopez-Fano, n = 3)

QB = 2.860

Infinite-tuned matching

(Fano-Bode, n = ∞)

QB = 1.732

Double-tuned matching

(Wheeler-Fano, n = 2)

QB = 0.750

Single-tuned edge-band

matching

(Wheeler-Fano, n = 1)

QB = 0.707

Single-tuned mid-band

match

(Non Fano)

QB for

V = VSWR = 2

Impedance Matching Factor

(QB)

Impedance Matching

Circuit

Impedance Matching Formulas - Summary

V

1V

1

2QB

2

−=

Γ−

Γ=

V2

1V

1

2

1lnsinh

1QB

2

2

−=

Γ−

Γ=

Γ

=

1V1

2

1ln

2

1sinh

1QB 2 −=

Γ−Γ

=

Γ

=

−+π

=

Γ

π=

1V

1Vln

1ln

QB

Rule of thumb for maximum achievable matching bandwidth (BMaxA)

If VSWR > 2 then BMaxA ≈ VSWR / Q

Γ

−+

Γ

=1

lna

b11ln

a

1sinhb

1QB

3

3

33

Closing 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

• Wheeler: “You should work hard to find the easy way”