Farr

Fields, LC

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A Power Wave Theory of Antennas

Everett G. Farr Farr Fields, LC

City University of Hong Kong August 26, 2014

Hong Kong

Farr

Fields, LC

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Overview

Part 1: Some UWB Antennas We’ve Worked On

Part 2: The Power Wave Theory of Antennas

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Fields, LC

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Example of UWB Antenna: IRA-3Q

• Diameter: 18 in. (46 cm)

• Radiates a Clean Impulse, with FWHM = 38 ps.

• Frequency range 250 MHz – 20 GHz.

• Excellent impedance match across entire frequency range.

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Data for the IRA-3Q

Gain Impulse Response

5 10 15 20-10

-5

0

5

10

15

20

25

30

Frequency (GHz)

dB

i

2 3 4 5 6 7-1

0

1

2

3

4

5

6

Time (ns)

Impuls

e r

esponse (

m/n

s)

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Applications of IRAs

• Broadband EMC/EMI or RCS testing with single antenna

• Intentional EMI

• Impulse Radar to locate weapons, tanks under trees, mines, or unexploded ordnance

• Broadband communications or surveillance

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Normalized Antenna Pattern IRA-3M

E-Plane H-Plane

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Radome on IRA

IRA-3

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Collapsible IRA

• Compact, Lightweight, rapidly deployable design

• Metallized nylon and resistive fabric

• When collapsed: Length=81 cm, Diam=10 cm

• Suitable for broadband communications in field

• Impulse response FWHM = 70 ps

• Peak Gr = 23 dB at 4 GHz

• Useful from 150 MHz to 8 GHz

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CIRA-2 Data

Impulse Response Gain

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Para-IRA Concept

• Parachute Delivered

• Impulse Radiating Antenna

• Goal is to Illuminate 100-Meter Radius Area with a Wideband Pulse

• Parachute Allows Rapid and Flexible Deployment

Parachute

Parabolic ReflectorConducting MeshFabric

TerminatingResistor

UnzipperBalun

Battery

Feed ArmsConductiveRipstopNylon

Transmitter

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Phase I Antenna Mounted Onto Frame for Testing

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Phase I Tow Test Results

• Measure force on scale to correlate terminal velocity with weight

• Descent Rate Results: A 20 pound package falls at 58 kph

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Folded Horn Antenna

• Useful for medium bandwidth (3-5 GHz) at high power

• Could be scaled X10 to reach 300-500 MHz, and mounted onto truck.

• Nearly flat phase front in aperture

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Feed Point Modifications in FH-1E

• Add dielectric disk: Simulates oil tank near feed, and shifts the

dip in S11 to lower frequency

• Add cone: to maintain 50-Ω impedance

( )2/cotln2

hoZ θπ

η=

• We needed both!

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Fields, LC

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Data on the Optimized Feed Horn, FH-1E

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Time Domain Antenna Measurement System

• With the PATAR® system one person can set up an antenna range, take and process data, then tear down and store the equipment all within 4 hours.

• Equipment fits into a shed

• No anechoic chamber needed due to time gating and temperature stability of scopes

• Bandwidth of 900 MHz to 20 GHz for arbitrary antennas

• For impulse antennas, bandwidth reaches as low as 200 MHz

• Works as well for narrowband antennas as for UWB antennas

• Introduces concept of “Personal Antenna Range”

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Measurement Setup

PSPL 4015D Step Generator

Tektronix TDS 8300 Series Digital Sampling

Oscilloscope with 80E04 Sampling Head

and 2m Extender

Remote Pulser Head

Antenna Under Test

Trigger Line

Received Signal

AZ–EL Positioner

Computer Controller

Transmit Antenna

RS232 Link

Ethernet Link

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Custom Elevation / Azimuth Positioner

• Easy setup, teardown, stowage

• Mast and legs removable

• Easy leveling, aiming

• Precision better than ±0.2 degrees in both azimuth and elevation

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Source End of Range

Includes Pulser, mounting Bracket, and

TEM sensor on fixed tripod

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Parameters Calculated

Impulse Response Return Loss (S11)

Gain, Realized Gain Antenna Pattern

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Part 2: The Power Wave Theory of Antennas

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The Antenna Equation and the Generalized Antenna Scattering Matrix (GASM)

Dominant Polarization on Boresight

r

o

rad

rad

eZ

Er γ

Υ

2

~

~=

1

~~

o

recrec

V

ΖΠ =

1

~~

o

srcsrc

V

ΖΠ =

2

~~

o

incinc

Z

E=Σ

1oZ

inZ~

2oZ

Γ=

inc

src

rad

rec

vhs

h

Σ

Π

πΥ

Π~

~

~)2/(

~

~~

~

~

l

GASM completely specifies response of any antenna, including those with waveguide feeds.

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Relationship to Currently Defined Quantities

Realized Gain 2

2|

~|

4~hGr

λ

π=

h~

transfer function

)(th impulse response

Effective Length h

Z

Z

Z

ZZ

E

Vh

o

o

o

oin

inc

ocV

~~

~

~~

2

1

1

1+==

Impedance Mismatch 2|

~|1 Γ−

Factor Γ~

reflection coefficient

)(tΓ reflection impulse response

RCS 2~

4 lπσ =

l~

scattering coefficient

)(tl scattering impulse response

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New Definitions and Symbols

waveintensity radiation radiated

~~

vedensity waflux power incident

~~

power wave received

~~

power wave source

~~

2

2

1

1

==

==

==

==

r

o

radrad

o

incinc

o

recrec

o

srcsrc

eZ

Er

Z

E

V

V

γΥ

Σ

ΖΠ

ΖΠ

Π, Σ, and Υ are Greek for P, S, and U, which are the commonly used symbols for power, power flux density, and radiation intensity.

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Relationships between Power Expressions and

Power Wave Expressions

Power Power Wave

( )( ) 2*

2*

~~~Re

21~

~~~Re

21~

recrecrecrec

srcsrcsrcsrc

IVP

IVP

Π

Π

==

==

Power Flux Density Power Flux Density Wave

2* ~~~

Re2

1~incincincinc AdHES Σ=•

×= ∫∫

rrr

Radiation Intensity Radiation Intensity Wave

22* ~

ˆ~~

Re2

1~radradradrad rrHEU Υ=•

×=

rr

Power waves add phase to well-known power expressions.

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Antenna Equation and GASM in the Time Domain

• Antenna equation and GASM in the time domain

•

′

Γ=

)(

)(*

)(2/)(

)()(

)(

)(

t

t

tvth

tht

t

t

inc

src

rad

rec

Σ

Π

πΥ

Π

l

where “ ' ” indicates a time derivative and the “•* ” operator is a

matrix-product convolution operator, defined as

∗+∗

∗+∗=

∗

•

)()()()(

)()()()(

)(

)(

)()(

)()(

222121

212111

2

1

2221

1211

tatstats

tatstats

ta

ta

tsts

tsts

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Antenna Equation for Two Polarizations and Arbitrary Angles

Frequency Domain

′′′′

′′′′

′′′′Γ

=

′′

′′

′′

)','(~

)','(~

~

),,,(~

),,,(~

)2/(),(~

),,,(~

),,,(~

)2/(),(~

),(~

),(~~

),,,(~

),,,(~

),(~

,

,

,

,

φθΣ

φθΣ

Π

φθφθφθφθπφθ

φθφθφθφθπφθ

φθφθ

φθφθΥ

φθφθΥ

φθΠ

φ

θ

φφφθφ

θφθθθ

φθ

φ

θ

inc

inc

src

rad

rad

rec

vhs

vhs

hh

ll

ll

),( φθ ′′ source coordinates ),( φθ observation coordinates

Time Domain

′′

′′∗

′′′′′

′′′′′

′′′′Γ

=

′′

′′

′′

•

),,(

),,(

)(

),,,,(),,,,()2/(),,(

),,,,(),,,,()2/(),,(

),,(),,()(

),,,,(

),,,,(

),,(

,

,

,

,

t

t

t

ttvth

ttvth

ththt

t

t

t

inc

inc

src

rad

rad

rec

φθΣ

φθΣ

Π

φθφθφθφθπφθ

φθφθφθφθπφθ

φθφθ

φθφθΥ

φθφθΥ

φθΠ

φ

θ

φφφθφ

θφθθθ

φθ

φ

θ

ll

ll

More Compact Frequency Domain Expression

Γ=

inc

src

rad

rec

vhj

h

Σ

Π

πωΥ

Π~

~

~)2/(

~

~~~

~ T

r

ltr

r

r

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Signal Flow Graphs

Dominant polarization, on boresight

radΥ

~

recΠ~

srcΠ~

incΣ~

Γ~

h~ l

~)2/(

~vhs π 1

1

1

1

Both polarizations, arbitrary angles, vectorized 2-port version

radΥ~r

recΠ~

srcΠ~

incΣ~r

Γ~

T~hr

lr~

)2/(~

vhs πr

1

1

1

1

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Signal Flow Graphs (cont’d)

Both polarizations, arbitrary angles, scalar 3-port version

recΠ~

srcΠ~

Γ~

θh~

θθl~

)2/(~

vhs πθ

1

1

rad,~

θΥ

inc,~

θΣ

1

1

φφl~

rad,~

φΥ

inc,~

φΣ

1

1

φθl~

)2/(~

vhs πφ

φh~

θφl~

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Solve Arbitrary Source with Signal Flow Graph

radΥ

~

recΠ~

srcΠ~

Γ~

)2/(~

vhs π

srcΓ~

1

1

1~

~~

osrc

osrcsrc

ZZ

ZZ

+

−=Γ

Dominant polarization, on boresight

srcsrc

radv

hsΠ

πΥ

~

2

~

~~1

1~

ΓΓ−=

Both polarizations, arbitrary angles

srcsrc

radv

hsΠ

π

φθφθΥ

~

2

),(~

~~1

1),(

~r

r

ΓΓ−=

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Fields, LC

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Scattering from an Antenna with Arbitrary Load

radΥ

~

incΣ~

Γ~

h~ l

~)2/(

~vhs π 1

1

lΓ~

Dominant polarization, on boresight

incradv

hsΣ

πΥ

~~

2

~

~~1

~~

2

+

ΓΓ−

Γ= l

l

l

Both polarizations, arbitrary angles

),(~

),,,(~

),(~

),(~

2~~

1

~~φθΣφθφθφθφθ

πΥ ′′⋅

′′+′′

ΓΓ−

Γ= inc

Trad hh

v

s rl

trrr

l

l

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Array of N Antennas or N Modes in Multimoded Waveguides

Need an h(t) for each array element or mode ,

Γ=

inc

src

rad

rec

vhs

h

Σ

Π

πΥ

Π~

~

~)2/(

~

~~

~

~T

r

r

ltt

tt

r

r

Dimensions are visualized as

=

×

×

××

××

×

×

12

1

222

2

12

1 N

N

NNNN

: mutual coupling coefficient and impulse response

Njih ji ,...,1;2,1,~

==

)(and~

tjiΓΓt

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Proving the Relationship between Transmission and Reception Terms

• Relate power wave expressions to open/short circuit forms using circuit theory.

• Treat two antennas in far field as reciprocal two-port network

2221

1211~~

~~

ZZ

ZZ

2221

1211~~

~~

YY

YY

Port 1 Port 2 Antenna 1 Antenna 2

• Assume Antenna 2 is an electrically small electric dipole, whose open/short circuit characteristics are fully known.

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Impulse Response Example

IRA-3Q

Impulse Response Transfer Function

2 3 4 5 6 7

-1

0

1

2

3

4

5

6

Time (ns)

Impuls

e r

esponse (

m/n

s)

10-1

100

101

10-2

10-1

100

Frequency (GHz)

Tra

nsfe

r fu

nction (

m)

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Review of Waveform Norms (For transient antenna patterns)

Three necessary conditions of norms

)inequality triangle()()()()(

linearity)()()(

otherwise

0)( iff

0

0)(

tgtftgtf

tftf

tftf

+≤+

=

≡

>

=

αα

Commonly used: p–norms

)(sup)(,)()(

/1

tftfdttftft

p

p

p=

=

∞

∞

∞−∫

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Transient Antenna Pattern

• Can consider single polarization or total magnitude

22),,(),,(),,( ththth φθφθφθ φθ +=

r

• Express transient patterns in terms of norms of time domain waveforms

),0,0(

),,(),(

),0,0(

),,(),(,

),0,0(

),,(),(

th

th

th

th

th

th

t r

rφθ

φθ

φθφθ

φθφθ

φ

φφ

θ

θθ

=

==

P

PP

• Normalization to boresight is optional

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Fields, LC

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Radiation from or Coupling into a Complex System

• Complex system looks like a poor antenna

• Antenna parameters should be used o Same in TX and RX o Works in both frequency and time domains

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Fields, LC

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Conclusion: Effects on Standards

None of the terms in the Antenna Equation have been defined

Γ=

inc

src

rad

rec

vhs

h

Σ

Π

πΥ

Π~

~

~)2/(

~

~~

~

~

l

Closest is scalar versions:

Impedance mismatch factor, 2|

~|1 Γ− , instead of Γ

~

Realized gain, rG , instead of h~

RCS, σ , instead of l~

We need to complexify the standards to get to the time domain!

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Fields, LC

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Everett Farr +1(505)410-7722

efarr@farr-research.com www.farr-research.com

This paper is based on

Sensor and Simulation Note 564, Revision 3 Available at our web site

Soon to appear in FERMAT! We welcome your online comments!