near-field antenna measurement theory iii spherical

Copyright Feb. 2002 NEARFIELD SYSTEMS, INC.

AMTA EDUCATIONAL SEMINAR 2002

Near-Field Antenna Measurement Theory IIISpherical


Overview

l Spherical Coordinate Systemsl Spherical Modesl Development Of Transmission Equation l Comparison To Planar And Cylindricall Translation Of Centers For Probe Receiving Coefficientsl Far-field Quantitiesl Probe Correctionl Sample Data


Spherical Measurement Schematic

Showing definition of three angular variables θ, φ, χ.


E-Field And Spherical Waves

1 2

1

( , , ) ( ) ( )n

mn nm mn nm im

n m n

E r Q M Q N e φθ φ θ θ∞

= =−

= + ∑ ∑r r r

and are the spherical wave functions which define the amplitude and phase of the spectrum of waves that will represent any field.

nmMr

nmNr


Development of Transmission Equation

l Scattering matrix for AUT using spherical modesl Scattering matrix for probe using AUT coordinate

spherical modesl With probe at θ and φ = 0 develop transmission

equation for:l Single mode, single polarizationl Single mode, dual polarizationl Spectrum of modes

l Rotate probe to (θ,φ,χ)

Copyright Feb. 2002NEARFIELD SYSTEMS, INC.

Spherical Near-field Scanning


Spherical Transmission Equation

( )

( )

2( )

1

0 0

0

, , ( )

( , , , )(1 )where , ,

measured data on sphere of radius a

a s na smn im n im

m n s

a l

W P Q e d e

b aW

a

µ φ µχµ

µ

χ φ θ θ

χ φ θχ φ θ

=

=

′ ′− Γ Γ=

=

∑∑∑ ∑


Definition Of Indexes

s naP µ = Probe spherical mode coefficients with respect to measurement sphere of radius a.

= AUT spherical mode coefficients for position and orientation of AUT in measurement sphere.

smnQ

s = polarization index = 1,2 ; 1= TE modes, 2=TM modes

m = azimuthal (φ) index for AUT,

µ = azimuthal (φ) index for probe.

n = polar angle (θ) index for both probe and AUT


Transmission Equation Comparison

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

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

0 0

2

0 0 0 01

( , ) ( ) ( ) in i zs sn n

n s

b z F a R T e e dφ γφ γ γ γ∞ ∞

=−∞ =−∞

′ ′ ′=

∑ ∑∫

( )2

( )

1

, , ( )a s na smn im n im

m n s

W P Q e d eµ φ µχµ

µ

χ φ θ θ=

=

∑∑∑ ∑


Far-field And Gain Equations

( )

1 1 20lim

1

21 20

2210

( , , ) ( 1) ( ) ( )

4( , ) ( 1) ( ) ( )

1

ikr nn s mn nm mn nm

rn m n

ns n mn nm mn nm

n m n

a eE r Q M Q N

kr

G Q M Q Nk

θ φ θ θ

πηθ φ θ θ

∞+ −

→∞= =−

∞−

= =−

= − +

= − + ′− Γ

∑ ∑

∑ ∑

r r r

r r


Solution Of Transmission Equation

l Inversion of Transmission Equation to solve for the coupling product

l Probe Correction to solve for AUT Transmitting coefficients

l Calculation of Far-field, Gain, Polarization


Inversion Of Transmission Equation

( )

2 2 2 ( )2 0 0 0

, ,

( )

1( )

(2 1) 8

( ) sin

nn mm im n im

m n

im n im

e d en

e d e d d d

π π πµµ φ µχµ

µ

φ µ χµ

δ δ δθ

π

θ θ θ φ χ

′ ′ ′

′ ′ ′−′ ′

= +

×

∑∫ ∫ ∫

Using the Orthogonality Relationship


Inversion Of Transmission Equation

( )2 2 2 ( )2 0 0 0

1 1 2 2

1( , , ) ( )

8sin

,

m n a a im n im

m n a n a m n n a m n

I W e d e

d d d

Where

I P Q P Q

π π πµ φ µ χµ

µ µ µ

χ θ φ θπ

θ θ φ χ

′ ′ ′ ′ ′ ′−′ ′

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′

=

×

= +

∫ ∫ ∫


Problems With Spherical Calculations

l 1- For arbitrary probe, measurements must be taken for small increments of probe rotation , requiring scanning over sphere many times and increasing computations.

l 2- Integration in θ with its evaluation of the complicated function is time consuming and uses approximate numerical integration.

l 3- Calculation of gain and far-field require evaluation of complicated M and N functions.

( ) ( )nmd µ θ′

′ ′

χ∆


Innovations For Practical Processing

If the probe’s far-field pattern is symmetric for rotation about the axis of the probe, the probe’s spherical mode expansion will only contain coefficients for 1µ = ±

0 1s nAP forµ µ= ≠ ±

This can be accomplished with circular waveguide probe fed with a single mode rectangular waveguide.

Use of Special Probe


Example Of Circular Waveguide

From a design by Stubenrauch, NIST



Use of special probe then only requires data for χ = 0 and χ = 90 degrees. The data from these two measurements are generally referred to as “the φ -component data and the θ-component data” even though the probe is not perfectly polarized.

The “integration” in χ is unnecessary, and two equations result.

Use of Special Probe



2- Integration in θ and evaluation of function. ( )1( )n

md θ′′±

Express the results of the φ-integration and thefunction as Fourier series. The integration in θ then reduces to a matrix multiplication plus an FFT.

The coefficients of the matrix obtained from the Fourier series representation of are obtained efficiently from recursion relations.

( )1( )n

md θ′′±

( )1( )n

md θ′′±



Calculation of Far-field Using Transmission Equation

Far-field is the field an ideal dipole would measure at a large distance from the AUT. Once the AUT spherical coefficients are found they are substituted into the Transmission Equation with ideal dipole coefficients for the probe, and the response calculated.



Calculation of Far-field using Transmission Equation

( )2

( ),

1

, ( )

Ideal dipole coefficients for large sphere

sn smn im nm

m n s

sn

E D Q e d

where D

φθ φ

µ

φ θ θ∞

=

∞

=

=

∑∑∑ ∑


Mode Cutoff

0

0 0

0

For minimum sphere radius MRE

Maximum n value kr NSI uses 10

Minimum angular spacing Radians2r

r

kr

λ

= =

≤ +

≤


Mode Cutoff Example


Translation Of Centers For Probe

,

( )

, , , are indices in AUT coordinate system, , are indices in Probe coordinate system.

s na snP P C a

where s and nand

µ σµνσµν

σ ν

µσ µ ν

= ∑


Steps In Probe Correction

l Measure or calculate the far-field pattern of the probe

l Input this data to the spherical program and calculate the probe spherical coefficients for its coordinate system

l Use the translation of centers equations to obtain the probe coefficients in the measurement coordinates

l Use the translated coefficients in the probe correction


Rotator Schematic


Examples of Spherical Systems

Matra MarconiSpherical NF

Used for satellite antenna testing

NSI Model 700S-50Spherical NF

Antenna Weights to 136 kg; 2m diameter


Spherical Near-field Range


Sample Spherical Near-field Data

-120 -60 0 60 120

Theta in Degrees

-90

-60

-30

0

30

60

90

Phi

in D

egre

esNear-Field Amplitude on Spherical SurfaceX-Band Slot Array, f= 9.338 GHZ, Phi Component



-120 -60 0 60 120

Theta in Degrees

-90

-60

-30

0

30

60

90

Phi

in D

egre

esNear-Field Phase on Spherical SurfaceX-Band Slot Array, f= 9.338 GHZ, Phi Component



-120 -60 0 60 120

Theta in Degrees

-90

-60

-30

0

30

60

90

Phi

in D

egre

esNear-Field Amplitude on Spherical SurfaceX-Band Slot Array, f= 9.338 GHZ, Theta Component



-120 -60 0 60 120

Theta in Degrees

-90

-60

-30

0

30

60

90

Phi

in D

egre

esNear-Field Phase on Spherical SurfaceX-Band Slot Array, f= 9.338 GHZ, Theta Component


Spherical Probe Correction

-80

-70

-60

-50

-40

-30

-20

-10

0

-75 -50 -25 0 25 50 75

Effect of Probe correction on Spherical MeasurementsOpen Ended Waveguide Probe, X-Band

Am

plitu

de (

dB)

Elevation (deg)

Probe Correction No Probe Correction

Difference


Spherical Near-field Data

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

-100 -50 0 50 100

Near Field Amplitude Data for X-Band Slot ArrayA

mpl

itude

(dB

)

Theta (deg)


Spherical Probe Correction

Comparing Planar FF Pattern with Spherical FF Pattern without Probe Correction. OEWG Probe.


Dipole Antenna on Spherical Range


Dipole Far-Field Pattern Results

0345

330

315

300

285

270

255

240

225

210

195180

165

150

135

120

105

90

75

60

45

30

15

-30 -20 -10 dB

Far-field Amplitude for Dipole Measured on Spherical Range

near-field antenna measurement theory iii spherical

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