Near-Field Antenna Measurement Theory II Cylindrical · abTa bRa ¢ == ¢= ¢¢ 0 1 1 l F¢...

Copyright Feb. 2002 Page 1NEARFIELD SYSTEMS, INC.

AMTA EDUCATIONAL SEMINAR 2002

Near-Field Antenna Measurement Theory II

Cylindrical


Overview

l Cylindrical coordinate systemsl Brief summary of rigorous derivation of transmission

equationl Development of transmission equation using

measurement approachl Comparison to planar transmission equationl Translation of centers for probe receiving coefficientsl Far-field quantitiesl Probe correctionl Probe coefficients from far-field patternl Sample measurements and probe correction data


Cylindrical Coordinates

θ

φ

ρ

r

x

y

( , , )( , , )

Cylindrical Coordinates zSpherical Coordinates r

ρ φθ φ


AUT And AUT-Centered Probe Coordinate Systems


Schematic Of AUT, Probe And Cylinder

C0 = AUT-Centered and Measurement Coordinate System

C0’ = AUT-centered Probe Coordinate System

C1 = Probe-Centered Coordinate System


Near-Field Cylindrical Range


Modal Expansion of Electric Field

( ) ( )1 (1) 2 (1)

(1) (1) (1)

(1) (1) (1) 2 (1)

( , , )

ˆˆ( ) ( )

1 ˆˆ ˆ( ) ( ) ( )

n n n nn

in i zn n n

in i zn n n n

E z B M B N d

inM H H e e

nN i H H H z e e

k

γ γ

φ γγ

φ γγ

ρ φ γ γ γ

κρ ρ κ κρ φρ

γγκ κρ ρ κρ φ κ κρ

ρ

∞∞

=−∞ −∞

= +

′= −

′= − +

∑ ∫r r r

r

r


To Calculate Electric Field Anywhere

l Find the Cylindrical mode coefficients

l Evaluate the Hankel Functions and derivatives at the radius and z specified by the product

l Sum over all the modal indices and integrate over all

values of γ

2 2kκρ ρ γ= −

( ) ( )1 (1) 2 (1)( , , ) n n n nn

E z B M B N dγ γρ φ γ γ γ∞∞

=−∞ −∞

= + ∑ ∫r r r

( )snB γ


Far Electric Field

( )( )

( )

1

2

ˆcos( )2 sin, , ( )

ˆcos( )

ikrnn in

n n

B kk eE r i e

r i B kφ

θ φθφ θ

θ θ

∞

=−∞

− = − −

∑r

Does not require calculation of Hankel functions

Requires only the cylindrical mode coefficients ( )snB γ


Cylindrical Mode Coefficients

Since Far-field, Near-field, Gain and Polarization ratios can be found from the cylindrical mode coefficients, determining for a given antenna is the goal of the near-field measurements.

( )snB γ


Cylindrical Waves And Notation

Amplitude of each cylindrical wave is specified by the coefficient and does not vary with any of the cylindrical coordinates (ρ, φ, z).

( )snB γ

θ

φ

ρ

r

x

y


Derivation of the Transmission Equation

l Express the antenna fields in cylindrical coordinates using vector mode functions.

l Write the scattering matrix for the antenna and probe in their own coordinate systems.

l Using field expressions in each coordinate system, derive the joining equations.

l Use joining equations and scattering matrix for each antenna to derive the transmission equation.

l Solve the transmission equation for unknown antenna cylindrical mode coefficients.



γ specifies a “direction of propagation” since the phase of the cylindrical wave in the z-direction is given by .i ze γ

X

Z

kγ



The s-index in specifies the polarization of the wave since s=1 modes produce φ-component (horizontal) far-fields and s=2 modes produce θ-component (vertical) far-fields.

( )snB γ

Z

X

AUT Probe10 (0)B

20 (0)B


Cylindrical Mode Phase Patterns

0

30

60

90

120

150

180

210

240

270

300

330

3603002401801206000

30

60

90

120

150

180

210

240

270

300

330

360300240180120600

Phase for n = 1 Phase for n = 5

The n-index in specifies the phase variation in phi through the factor in definition of M and Nine φ

( )snB γ


Features Of Scattering Matrix Approach

l Does not require evaluation of Hankel functions

l Transmission equation valid in near and far-fieldl Only approximation is multiple reflections neglected and

finite scan dimension in Z.

l Provides for probe correction of arbitrary probe

l Results in efficient data processing using the FFT


Cylindrical Scattering Matrix Schematic

Y

X

AUTProbe

0b

0a

0b′

0a′

( )snb γ

( )snb γ′

( )sna γ

( )sna γ′


Single Cylindrical Wave, γ = 0, n = 0

Z

X

AUT Probe

0a ( )0 0,0b′

20 (0)b

Linear Polarization

0 0Probe at 0, 0z φ= =


Transmission Equation Development

Single cylindrical wave, γ = 0, n = 0, linear polarization

2 20 0 0(0) (0)b T a=

AUT Equations Probe Equations

2 20 0 0 0(0,0) (0) (0)b F a R T′ ′ ′=

2 2 20 0 0 0

2 20 0 0

(0) (0) (0)

(0,0) (0) (0)

a b T a

b R a

′ = =

′ ′ ′=

0

11 l

F ′ =′ ′− Γ Γ


Single Cylindrical Wave, γ = 0, n = 0,

Z

X

AUT Probe10 (0)b

20 (0)b

0a ( )0 0,0b′

Two Polarizations



Single cylindrical wave, γ = 0, n = 0, two polarizations

2 20 0 0

1 10 0 0

(0) (0)

(0) (0)

b T a

b T a

=

=


2

0 0 0 01

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

s

b F a R T=

′ ′ ′= ∑

0 0 0 0

2

0 0 01

(0) (0) (0)

(0,0) (0) (0)

s s s

s s

s

a b T a

b R a=

′ = =

′ ′ ′= ∑


Two cylindrical waves, γ = 0

Z

AUT Probe

X

11 (0)b

14 (0)b

21 (0)b

24 (0)b

0a ( )0 0,0b′

Two polarizations, two values of n



Two cylindrical waves, γ = 0, two polarizations, two values of n

1 1 0

4 4 0

(0) (0)

(0) (0)

s s

s s

b T a

b T a

=

=


0

2

01,4 1

(0) (0) (0)

(0,0) (0) (0)

s s sn n n

s sn n

n s

a b T a

b R a= =

′ = =

′ ′ ′= ∑ ∑

2

0 01,4 1

(0,0) (0) (0)s sn n

n s

b F a R T= =

′ ′ ′= ∑ ∑


Spectrum Of Cylindrical Waves

Z

AUT Probe

X

23 1( )b γ

24 (0)b

21 (0)b

25 2( )b γ

13 1( )b γ

11 (0)b

14 (0)b

0a ( )0 0,0b′

Different values of γ, s and n



Spectrum of cylindrical waves, different values of γ, s and n

0( ) ( )s sn nb T aγ γ=


0

2

01

( ) ( ) ( )

(0,0) ( ) ( )

s s sn n n

s sn n

n s

a b T a

b R a d

γ γ γ

γ γ γ∞ ∞

=−∞ =−∞

′ = =

′ ′ ′= ∑ ∑∫

2

0 01

(0,0) ( ) ( )s sn n

n s

b F a R T dγ γ γ∞ ∞

=−∞ =−∞

′ ′ ′= ∑ ∑∫


Probe Moved On Cylinder

X

Z

AUT

Probe

0a

( )0 0 0,b zφ′



Spectrum of cylindrical waves, different values of γ, s and n probe moved to

0( ) ( )s sn nb T aγ γ=

0 0

0

2

0 0 01

( ) ( ) ( )

( , ) ( ) ( )

s s sn n n

in i zs sn n

n s

a b T a

b z R a e e dφ γ

γ γ γ

φ γ γ γ∞ ∞

=−∞ =−∞

′ = =

′ ′ ′= ∑ ∑∫

0 0( , )zφ

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∞ ∞

=−∞ =−∞

′ ′ ′= ∑ ∑∫


Measurements With Two Probes

0 0

2

0 0 0 01

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

n s


=−∞ =−∞

′ ′ ′= ∑ ∑∫

0 0

2

0 0 0 01

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

n s


=−∞ =−∞

′′ ′ ′′= ∑ ∑∫

First probe usually has same polarization as AUT

Second probe usually is cross polarized to AUT


Inversion Of Transmission Equations

Using Fourier series for n and Fourier integral for γ, for first probe data

0 0

2

1

2

0 0 0 0 020 0

( ) ( ) ( )

1( , )

4

s sn n n

s

in i z

I R T

b z e e d dza

πφ γ

γ γ γ

φ φπ

=

∞− −

−∞

′ ′=

′=

∑

∫ ∫


Inversion Of Transmission Equations

0 0

2

1

2

0 0 0 0 020 0

( ) ( ) ( )

1( , )

4

s sn n n

s

in i z

I R T

b z e e d dza

πφ γ

γ γ γ

φ φπ

=

∞− −

−∞

′′ ′′=

′′=

∑

∫ ∫

For second probe data


Data Point Spacing And Maximum n

Band limits of the AUT pattern in the θ-direction define a data point spacing in z like the planar case

2zλ

δ ≤

Due to the exponential decrease in the reactive cylindrical modes, the maximum n value is

max sin( ),2

MREin NSI Software

n ka and radiansa

a

λθ φ≤ ∆ ≤

≡

Due to these band limits, the integration is replaced by summation without approximation and the FFT is used to calculate the I’s.


Cylindrical Near-field Sampling Criteria

Note that index nmax is therefore driven by the size and mounting offset of the AUT being considered.

The above sampling criteria are valid for the probe exterior to the reactive near-field of the AUT.

AUT linear z-stageProbe

x-Axis

AUT azimuth positioner

Axis of rotation

0 m

AUT support

AUTd

MRE

D

Radius


Probe Correction

21 1 2 2

1

21 1 2 2

1

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

s sn n n n n n n

s

s sn n n n n n n

s

I R T R T R T

I R T R T R T

γ γ γ γ γ γ γ

γ γ γ γ γ γ γ

=

=

′ ′ ′ ′= = +

′′ ′′ ′′ ′′= = +

∑

∑


Probe Correction Equations

1 2 2 11 2

1

1 1

I I I IR R R R

T T

′ ′′ ′′ ′ ′′+ + ℜ′ ′′ ′ ′′ ′ℜ

= =′′ ′′ℜ ℜ− −

′ ′ℜ ℜ

Using concise notation and deleting explicit reference to γ and n


Far Electric Field

( )( )

( )

1

2

ˆcos( )2 sin, , ( )

ˆcos( )

ikrnn in

n n

B kk eE r i e

r i B kφ

θ φθφ θ

θ θ

∞

=−∞

− = − −

∑r

Does not require calculation of Hankel functions

Requires only the cylindrical mode coefficients ( )snB γ


Cylindrical Coefficients And Transmitting Function

0( ) ( ) ( )s s sn n nB b T aγ γ γ= =

Therefore solving for Ts will give far-field, gain, and polarization ratios


Cylindrical And Planar Probe Correction Equations

"E A' " ' " '

E A" "

' '

D Ds s

1 1

A Es

A s EA E

s s

s s

D Ds s

t tρ

ρρ ρρ ρ

− −= =

− −

1 2 2 11 2

1

1 1

I I I IR R R R

T T

′ ′′ ′′ ′ ′′+ + ℜ′ ′′ ′ ′′ ′ℜ

= =′′ ′′ℜ ℜ− −

′ ′ℜ ℜ


Sample Probe Correction Theta Cut

-80

-70

-60

-50

-40

-30

-20

-10

0

-75 -50 -25 0 25 50 75

Cylindrical Near-Field Probe Correction Main Component, Elevation Cut

Am

plitu

de (

dB)

Elevation (deg)

No Probe Correction OEWG Probe Correction

Difference


Sample Probe Correction Phi Cut

-80

-70

-60

-50

-40

-30

-20

-10

0

-75 -50 -25 0 25 50 75

Cylindrical Near-Field Probe CorrectionAzimuth Cut, Main Component

Am

plitu

de (

dB)

Azimuth (deg)

No Probe Correction OEWG Probe Correction

Difference


References

Yaghjian, A.D., "Antenna Measurements on a Cylindrical Surface: A Source Scattering-Matrix Approach," National Bureau of Standards Technical Note 696, 1977, 34 p., Boulder, CO.

Leach, W.M., Jr., and Paris, D.T., "Probe Compensated Near-field Measurements On A Cylinder," IEEE Transactions on Antennas and Propagation, Vol. AP-21, No. 4, pp. 435-445, July 1973.

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Near-Field Antenna Measurement Theory II Cylindrical · abTa bRa ¢ == ¢= ¢¢ 0 1 1 l F¢...

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