Near-field Antenna Measurement Theory - Planar · 2017-04-18 · Unit vectors = “hat” or lower case e Vector components shown as subscript K, torK r r, ... xyz z xyz yz kkxkykz

Copyright Feb 2002 Page 1NEARFIELD SYSTEMS, INC.

AMTA EDUCATIONAL SEMINAR 2002

Near-field Antenna Measurement Theory -Planar


Overview

l Development Of Plane Wave Theoryl Development Using Measurement Approachl Understanding Working Equationsl Planar Transmission Equationsl Solution Using FFTl Sampling And Data Point Spacingl Planar Probe Correction


Notation Summary

Time Convention Phase increases with +z

Consistent with published papers on Near-field Theory

Time convention in instrumentation and NSI software is Phase decreases with +z.

i te ω−

j te ω+

Only affects complex quantities such as conversion from linear to circular components

-iωt convention +jωt convention

2x y

R

E i EE

−=

2x y

R

E i EE

+=


Notation Summary Vectors

Vectors = arrow over symbol or underline

Unit vectors = “hat” or lower case e

Vector components shown as subscript

,K t or Kr r

,ˆ ˆˆ , , x Ax A or e eθ

, ,x Ak t sφ


Theoretical Basis for Planar Near-field Measurements

l Scattering Matrix Theory Developed by Dr. D. M. Kerns in 1960’s

l Does Not Require:l Ideal Probe, measuring one component of field at a pointl Reciprocal antennasl AUT have ideal polarization, field separability or symmetry l Separation between AUT and probe within a special range

l Only Two Approximations Necessaryl Multiple reflections small enough to neglectl Measurements made over a finite plane

l Fast and Efficient Data Processing (Primarily FFT)


Angular Spectrum Of Plane Waves

Each vector component is a complex number

( , , )

( , , )

( , )

Electric Field E r

E r Superposition of Plane Waves

b Angular Spectrum Vectors

θ φ

θ φ

θ φ

=

=

=

r

r

r


Propagation Vector

Propagation vectorˆ ˆ ˆ

= Defines direction of plane wave

x y x

kk x k y k z

== + +

r


Propagation Vector

ˆ ˆ ˆPropagation vector

= Defines direction of plane wave

b(k) = Angular Spectrum Vector, Function of k

b(K) = Angular Spectrum Vector, Function of K

ˆ ˆ Transverse (x-y) Part

x y x

x y

k k x k y k z

K k x k y

= = + +

= + =

r

r r r

r r r

r of k

r


Propagation Vector

22 2 2 2

2 2 2

2

z

ˆ ˆ ˆ Propagation vector

ˆ

Since For losless medium

Any two components define the third.

= k

Therefore and both define directions.

x y z

z

x y z

y z

k k x k y k z

K k z

k k k k k k

k k k

k K

πλ

γ

= + + =

= +

≡ ⋅ = + + =

± = − −

r

r

r r

r r


Single Plane Wave


Two Plane Waves Different Directions


Evanescent Plane Waves

Propagating in the x-y plane

Exponentially Attenuated in the z-direction

2 2 2x yk k k+ >

γ is Pure Imaginary

Sometimes called Imaginary Space,

θ and φ not defined


Scattering Matrix for Waveguide

1 11 1 12 2

2 21 1 22 2

b S a S a

b S a S a

= +

= +

1 2

a1

b1

a2

b2


Planar Scattering Matrix Schematic

0S′

1S 2S′Y

Z

AUTProbe

0S

0b

0a

0b′

0a′

( )b Kr r

( )a Krr

( )b K′r r

( )a K′rr


Transmission Equation Notation

O = AUT coordinate system originO’ = Probe coordinate system at (x0,y0,z0) in O

S0 = AUT portS’

0 = Probe porta0 = Complex Amplitude incident on AUT Transmission Line portb0 = Complex Amplitude emerging from AUT Transmission Line porta’

0 = Complex Amplitude incident on Probe Transmission Line portb’

0 = Complex Amplitude emerging from Probe Transmission Line porta(K)= Complex Amplitude incident on AUT Space side apertureb(K)= Complex Amplitude emerging from AUT Space side aperturea’(K)= Complex Amplitude incident on Probe Space side apertureb’(K)= Complex Amplitude emerging from Probe Space side aperture


Single, Linearly Polarized Plane Wave

Equations for AUT

0 10( 0) ( 0)y yb K a t K= = =r r

Equations for Probe

0 02

( 0) ( 0)

( 0) ( 0)

ikdy y

y y

a K b K e

b a K s K

′ = = =

′ ′= = =

r r

r r

0 0 10 02( 0) ( 0)ikdy yb F a t K e s K′ ′ ′= = =

r r


Single Plane Wave With Two Polarizations

Equations for AUT

0 10(0) (0)b a t=r r

Equations for Probe

0 02

(0) (0)

(0) (0)

ikda b e

b a s

′ =

′ ′ ′= •

rr

r r

0 0 10 02 10 02

0 10 02

(0) (0) (0) (0)

(0) (0)

ikdx x y y

ikd

b F a t s t s e

F a t s e

′ ′ ′ ′ = + ′ ′ = •

r r


Angular Spectrum Of Plane Waves With Arbitrary Polarization

Equations for AUT

0 10( ) ( )b K a t K=r r rr

Equations for Probe

0 02

( ) ( )

( ) ( )

i da K b K e

b a K s K

γ′ =

′ ′ ′= •

rr rr

r rr r

0 0 10 02( ) ( ) ( ) i dx yb P F a t K s K e dk dkγ′ ′ ′= •∫∫

r r rr r


Translation Of Probe In Plane

Equations for AUT

0 10( ) ( )b K a t K=r r rr

Equations for Probe

0 02

( ) ( )

( ) ( ) ( )

i d iK Pa K b K e e

b P a K s K

γ •′ =

′ ′ ′= •

r rrr rr

r rr r

0 0 10 02( ) ( ) ( ) i d iK Px yb P F a t K s K e e dk dkγ •′ ′ ′= •∫∫

r rr r rr r


( )0 0 10 02( , , ) ( ) ( ) x yi k x k yi d

x yb x y d F a t K s K e e dk dkγ +′ ′ ′= •∫∫r rr r

Planar Transmission Equation

Probe Amp and Phase Output

Probe Position

AUT Plane Wave Transmitting Function

Probe Plane Wave Receiving Function

Factors due to Probe Translation in Plane

AUT Input Amp and Phase


Planar Transmission Equation

( )0 0 10 02( , , ) ( ) ( ) x yi k x k yi d

x yb x y d F a t K s K e e dk dk

multiple reflection term

γ +′ ′ ′= •

+∫∫

r rr r

Neglecting multiple reflection term is only approximation in derivation of Planar Transmission Equation


Plane Wave Transmitting Spectrum And Electric Field

The Electric Field at a large distance from the AUT in the direction specified by K

r

010( , ) ( )cos

ikri k a eE r K t K

rθ=

r r rr

-80 -60 -40 -20 0 20 40 60 80

Azimuth in Degrees

-80

-60

-40

-20

0

Rel

ativ

e A

mpl

itude

in d

B

Transmitting Spectrum and E-Field ComparisonMain Comp.Transmitting SpectrumMain Comp. E-Field


Plane Wave Receiving Spectrum And Electric Field

Receiving Spectrum pattern and Far Electric Field pattern are identical

01

01

( , ) ( , )(0,0)(0,0)

E A E t A EtE

=r rr r

-80 -60 -40 -20 0 20 40 60 80

Azimuth in Degrees

-80

-60

-40

-20

0

Rel

ativ

e A

mpl

itude

in d

B

Receiving Spectrum and E-Field ComparisonMain Comp. Receiving SpectrumMain Comp. E-Field


Gain Equation

( )22

0 10

20

0

0

4 ( )( )

1

Characteristic Admittance of Free Space

Characteristic Admittance of Transmission Line

= Antenna Reflection Coefficient

a

a

Y t KG K

Y

πγ

η

η

=− Γ

==

Γ

rrr


Planar Near-field Measurement

MEASUREMENT AND DATA PROCESSING

tA

tE

s'A

s'E

Probe

Receiver

NF Data

Computer

-100 0 100Azimuthal Angle in Degrees

-80

-40

0

Re

lativ

e A

mpl

itude

in d

B

AUT Spectrum Patten AUTSpectrum -1 .0 -0.8 -0 .6 -0.4 -0.2 0.0 0.2 0.4 0 . 6 0.8 1 . 0

Relative x-component propagation vector kx/k

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Re

lativ

e y

-com

pon

ent

pro

pa

gat

ion

vec

tor

ky/

k

Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Amplitude F=12.0 GHz

Probe Data


Sample Planar Near-field Data

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Sample Near-Field for Slot Array AntennaNear-Field Amplitude along y=0 centerline

Am

plit

ud

e (

dB

)

X (in)

-55-50-45-40-35-30-25-20-15-10-5-303510Sample Near-Field Data

Main Component Amplitude



-150

-100

-50

0

50

100

150

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Sample Near-Field for Slot Array AntennaNear-field Phase along y=0 centerline

Ph

ase

(d

eg

)

X (in)

-180

-150

-120

-90

-60

-30

0

30

60

90

120

150

180

Sample Near-Field DataMain Component Phase



-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Sample Near-Field for Slot Array AntennaCross Pol near-field Amplitude along y=0 centerline

Am

plitu

de (

dB)

X (in)

-60-55-50-45-40-35-30-25-20-15-10-5-303510

Sample Near-Field DataCross Component Amplitude


Schematic of Waveguide Probe

Y

X

A (main)

E (Cross)

X-Polarized OpenEnded Waveguide


X-Polarized Probe Az-Component

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e y-

com

pone

nt p

ropa

gatio

n ve

ctor

ky/

k

Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Amplitude F=12.0 GHz

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e y-

com

pone

nt p

ropa

gatio

n ve

ctor

ky/

k

Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, AZ-Component, Phase F=12.0 GHz


X-Polarized Probe El-Component

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e y-

com

pone

nt p

ropa

gatio

n ve

ctor

ky/

k

Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, EL-Component, Amplitude F=12.0 GHz

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0


-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Rel

ativ

e y-

com

pone

nt p

ropa

gatio

n ve

ctor

ky/

k

Open Ended Waveguide Probe PatternAs Measured on Pattern RangeX-Polarized, EL-Component, Phase F=12.0 GHz


Simplified Notation For Probe Correction Discussion

( )0 0( ) i d i K Pb P F a t s e e dKγ •′ ′ ′= •∫∫

r rr r r

Delete subscripts on

Remember that t is Transmitting and s is Receiving in following equations

10 02t and s′r r

Delete Explicit notation showing t and s as functions of K.

Remember that t and s are not constants but are functions of direction of propagation.


Planar Near-field Mathematics

( ) ( ) ( ) ( ) ( )' '

' '

[ ]

[ ]

i d i K PA A A E E

i d i K PA A A E E

B P t K s K t K s K e e dK

B t s t s e e dK

γ

γ

•

•

= +

= +

∫

∫

r v

r v

v v v v v v

v

MAIN COMPONENT TRANSMISSION EQUATION


Planar Near-field Mathematics

( ) ( ) ( ) ( ) ( )" "

" "

[ ]

[ ]

i d i K PE A A E E

i d i K PE A A E E

B P t K s K t K s K e e dK

B t s t s e e dK

γ

γ

•

•

= +

= +

∫

∫

r v

r v

v v v v v v

v

CROSS COMPONENT TRANSMISSION EQUATION


Solving Transmission Equation

( )' '2( )

4

i di K P

A A A E E A

eD K t s t s B P e dP

γ

π

−− •= + = ∫

r vv v v

( )" "2( )

4

i di K P

E A A E E EeD K t s t s B P e dP

γ

π

−− •= + = ∫

r vv v v

MAIN COMPONENT PLANE WAVE SPECTRUM OF MEASURED DATA

CROSS COMPONENT PLANE WAVE SPECTRUM OF MEASURED DATA

Invert equations using Fourier transform (FFT)


Planar Near-field Measurement

Receiver

AUT Spectrum

Computer

-100 0 100Azimuthal Angle in Degrees

-80

-40

0

Rel

ativ

e A

mpl

itud

e in

dB

AUT Spectrum Patten

Synthesized Array

Synthesized Beam

PLANE WAVE SPECTRUM ANALYZER

D Ke

B P e dPA

i d

Ai K P( )

v v vr v

=−

− •zγ

π4 2 c h


Sampling Theorem And Fourier Transform

If the Fourier Transform of the measured data is band limited:

1-Data point spacings are defined .

2-Integration replaced by summation without approximation.

x yandδ δ

2

( )

2

( ) ( , )4

( , )4

x r y s

i di K P

A A

i di k x k yx y

A r sr s

eD K B x y e dx dy

eB x y e

γ

γ

π

δ δπ

−− •

−− +

=

=

∫ ∫

∑∑

r vv

max maxx y

x yk kπ π

δ δ= =


Band Limited Spectrum

-80

-70

-60

-50

-40

-30

-20

-10

0

-1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00

Example of Band Limited Spectrum

Am

plitu

de (

dB)

Kx (cycles/L)

2 2xπλ λ

δπ

= =


Band Limit Greater than k

0.331.5(2 )y

πλδ λ

π= =

-80

-70

-60

-50

-40

-30

-20

-10

0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Far-field Spectrum, No Probe Correction, No Cos(Theta) Factor

Am

plitu

de d

B

Ky/k Cycles/Lambda


Span And Spacing Relations

For near-field data with the number and spacing of data in the

x- and y-directions given by

and the FFT size given by ,

, , ,x y x yN N δ δ

,x yF F

x x x y y yNear Field Span S = (N -1) S =(N -1)

Far-Field Spacing

( 1)( 1)Far-Field Span

yx

x x y y

yxkx ky

x x y y

kkk F k F

FFS S

F F

δ δ

λ λδ δ

λλδ δ

∆∆= =

−−= =


Comparison Of Data Spectrum And Far-field

-80 -60 -40 -20 0 20 40 60 80Azimuth in Degrees

-100

-80

-60

-40

-20

0

Rel

ativ

e A

mpl

itude

in d

B

Spectrum of Meas. Data and E-Field ComparisonSpectrum of Measured DataMain Comp. E-FieldDifference due to

probe correction and Cos(θ) factor


Probe Correction Equations

E' " '

E"

'

Ds

Main Component Spectrum for A-Polarized AUT1

A

A sA

s

s

Ds

tρ

ρρ

−= =

−

"A" '

A"

'

Ds

Cross Component Spectrum for A-Polarized AUT1

Es

EE

s

s

Ds

tρ

ρρ

−= =

−


Definition Of Polarization Ratios

'' '

' 1 For A Polarized ProbeAs s

E

sand

sρ ρ= > −

"" "

" 1 For E Polarized ProbeAs s

E

sand

sρ ρ= < −


Polarization Ratio For OEWG Probe

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Relative x-component propogation vector, kx/k

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Re

lativ

e y-

com

pon

ent p

ropo

gatio

n ve

ctor

. ky/

k

Amplitude, Polarization Ratio, A/E, forOpen Ended WaveguideProbe Main Component E-Polarized( )20*log ( )s Kρ′′

r


Two Parts Of Probe Correction

l Pattern correction l Due to probe’s main component pattern

l Polarization correctionl Due to probe’s cross component pattern


Approximate Probe Correction

tDsA

A

A

≅ '

MAIN COMPONENT, FOR CASE OF X-MAIN COMPONENT

PATTERN CORRECTION ONLY


Probe Pattern Correction Only

-80 -60 -40 -20 0 20 40 60 80Azimuthal Angle in Degrees

-100

-80

-60

-40

-20

0

Relativ

e A

mplitude

in d

B

Main Component Probe Pattern Correction

Probe Main Comp. Pattern

AUT Probe Corrected Main Comp.

Total Spectrum of "Main" N-F DatatDsA

A

A

≅ '

Polarization correction has almost no effect on main component results


Effect Of Probe Pattern Errors

-80 -60 -40 -20 0 20 40 60 80Azimuthal Angle in Degrees

-100

-80

-60

-40

-20

0

Relative Amplitu

de in

dB

Probe Pattern Errors


AUT Probe Corrected Main Comp.

Total Spectrum of "Main" N-F Data

Probe Cross Component Pattern

Probe pattern errors affect single direction


Theoretical Probe Pattern Error

-20.0

-17.5

-15.0

-12.5

-10.0

-7.5

-5.0

-2.5

0.0

2.5

-75 -50 -25 0 25 50 75

Comparison of Measured and Calculated Probe PatternOEWG, F = 10 GHz, H-Plane Cut

Am

plitu

de (

dB)

Azimuth (deg)

Plot 1 Plot 2 Plot 1 - Plot 2


Approximate Probe Correction

tDs

DsE

E

E

A

As≅ −" '"ρ

CROSS COMPONENT

PATTERN CORRECTION POLARIZATION CORRECTION


Probe Pattern Correction Only

-80 -60 -40 -20 0 20 40 60 80

Azimuthal Angle in Degrees

-100

-80

-60

-40

-20

0

Rela

tive

Am

plitud

e in d

B

Cross Component Probe Pattern Correction


AUT Cross Component (Probe Corrected)

Total Spectrum of "Cross" N-F DatatDsE

E

E

≅ "

Pattern correction has same effect on cross pol as on main pol


Polarization Ratios

-80 -60 -40 -20 0 20 40 60 80


-50

-40

-30

-20

-10

0

10

Relat

ive Am

plitu

de in

dB

Polarization Ratios of AUT and Probe

X-Polarized AUT, Reciprocal of Polarization Ratio Y/X

x-Polarized Probe, Reciprocal of Polarization Ratio, Y/X


Probe Polarization Correction

-80 -60 -40 -20 0 20 40 60 80


-120

-100

-80

-60

-40

-20

Relativ

e A

mpl

itude

in d

B

Cross Component Polarization Correction

Polarization Correction

After Polarization Correction

Spectrum of Data Before Correction


General Conclusions

l Probe correction performed on spectrum not near-field data

l Probe polarization correction rarely if ever needed for main component probe

l Need for accurate cross polarization data on cross component probe depends on

l Relative polarization ratio of AUT and probel Required accuracy

Documents

Near-field Antenna Measurement Theory - Planar · 2017-04-18 · Unit vectors = “hat” or lower case e Vector components shown as subscript K, torK r r, ... xyz z xyz yz kkxkykz