Uncertainty Considerations In Spherical Near-field Antenna

Measurements Phil Miller

National Physical LaboratoryIndustry & Innovation DivisionTeddington, United Kingdom

Outline

• Introduction and Spherical Configuration

• Typical NIST 18 Point Uncertainty Budget

• Consideration of the Terms

• Identification of the Uncertainties

• Deriving the Uncertainty Budget

• Conclusion

Introduction

• The Ability to be able State the Uncertainty to which a Quantity is Measured is as Important as the Measurement itself

• Performing an Uncertainty Analysis for a Validated Measurement Facility can involve a Lot of Work. Reducing the Confidence in the Uncertainty can Reduce Effort

• Need to perform Measurement Validation to avoid Large Systematic Errors

Spherical Configuration

θ

Phi Axis

ThetaAxis

Test Antenna

ChiAxis

Probeφ

θ

‘Model Tower’ Spherical Range Configuration

18 Point Error Budget

Aliasing Uncertainty 7Truncation Uncertainty8θ, φ Position Uncertainty 9

Impedance Mismatch Uncertainty6Normalisation Constant5Probe Alignment Uncertainty 4Probe Relative Pattern3Probe Polarisation Uncertainty2Gain Calibration Uncertainty1

UNCERTAINTY CONTRIBUTIONNo.

18 Point Error Budget (Cont.)

Cable Errors/Rotary Joints16Temperature Effects17Receiver Dynamic Range18

Receiver Phase Uncertainty15

System Phase Errors due to• Receiver Phase Uncertainty• Cable Errors/Rotary Joints• Temperature Effects

14Receiver Amplitude Non-linearity13Probe-AUT Multiple Reflections12Positioner and AUT Misalignment Uncertainty11‘R’ Position Uncertainty10

SGH and Probe Calibration Uncertainty

• Determined when SGH and Probe are Calibrated

• SGH and/or Probe Gain Uncertainty is a Direct Contribution to Gain Uncertainty Budget

• 0.2 dB Probe Pattern Error at Edge of Reflector produces a– 0.03 dB Gain Uncertainty – 0.32 dB Uncertainty on a –26.8 dB First Sidelobe (-55

dB error level signal)

Probe Alignment Uncertainty

• Horizontal Probe Positioning Uncertainty can produce Large Measurement Uncertainty in Polar Mode, Smaller in Equatorial Mode

• Chi Axis Misalignments causes Polarisation Uncertainty proportional to the Sine of the Angle of Misalignment

• The Effect of Range Length Uncertainty can be calculated by varying the Inputted Range Length in the Transform

Normalisation Constant and Mismatch Uncertainty

• Normalisation Constant Uncertainty can be determined by Repeatedly Disconnecting and Reconnecting the Relative/Direct Connection and Noting the Variation

• Mismatch Uncertainty is Determined by applying the Measurement Uncertainty in the Reflection Coefficient Measurement in turn to the Real and Imaginary Part of each Reflection Coefficient in :-

( )( )22

2

11

1

GL

GLcM

Γ−Γ−

ΓΓ−=

Aliasing Error

• Caused by Under-Sampling the Data

• Use the TICRA result to Estimate Uncertainty

• where Ptr is the Power Level of the Neglected Modes

NM ≤nkrN 10 +=12M

2πΔφ+

=12N

2πΔθ+

≤

)(045.0 301 trPkrn −=

Truncation Error and θand φ Position Errors

• Truncation Error is caused by Acquiring Too Small an Area of Data

• Estimate the Error due to Truncation by Acquiring a Larger Area of Data and then Perform Transforms on Progressively Reduced Area Data Sets

• θ and φ Position Errors can cause Pointing Errors if Systematic. Otherwise of Second Order Importance unless Antenna is Offset in the Minimum Sphere

• When Antenna is Offset use ‘R’ Position Uncertainty Techniques

‘R’ Position Errors

• Primarily Phase Errors caused by Run-Out Errors and Tilt Errors in the Positioner System

• If in the Antenna Near-Field can Estimate Uncertainty using Ruze Theory

• `

• Bound using NIST Theory

⎟⎟⎠

⎞exp

!exp

2

errorstheofdistancecorrelationtheisc

2

1

__2____

2

where ,wavelength,the is λ

λπδδ

λπ

( )sinU θ=

⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−

×⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+ ∑

∞

=ncu

nnc( ) ( )exp,,

____2

0 δφθφθ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=GG

n

n

Range and AUT Alignment Uncertainty

• Uncertainty due to Range Alignment Errors best Estimated using J. E. Hansen (ed). “Spherical Near-Field Antenna Measurements”

• AUT Alignment Errors causes Error in the Pointing Measurement

• Uncertainty due to Antenna Flexure can be Estimated using Data Acquisition in the Alternative Sphere

Probe-AUT Interaction Errors

• Caused by Multiple Reflections between the Probe and the AUT

• The Measurement Uncertainty caused by the Probe-AUT Interaction can be estimated by Comparing Two Identical Measurements made with the Range Length Changed between then by λ/4

Receiver Amplitude Non-linearity and System Phase Errors

• Receiver Amplitude Errors cause a Gain Error directly equal to the Receiver Compression Error below 0.1 dB

• System Phase Errors can be caused by RF Cable and AUT instabilities, Rotary Joint Errors and Temperature Effects

• The Uncertainty in the Measurement can be estimated using the same techniques as for the ‘R’ Uncertainty Effects

• Rotary Joint Errors can be removed using a Polar Acquisition

Receiver Dynamic Range

• For a Far-Field Range the Receiver Dynamic Range defines the Noise Floor for the Measurements

• For a Near-Field Measurement the Noise Floor is reduced by the difference between the Near-Field Gain and the Far-Field Gain

• The Amplitude of the Noise Floor has the Largest Effect on the Transformed Pattern

Random Errors

• Caused by any Uncertainty in the Measurements not Previously Considered

• Estimate by Performing Repeat Measurements and Comparing Transformed Data. Use: -

• Where the Quantities have been Normalised to Zero

( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

2,,

log*.20, 21 φθφθφθε EE

The Expression of Uncertainty

• Use UKAS Standard M3003 and Type A Evaluation

• Assume Error in Measurement has Already Been Calculated so that the ‘Sensitivity Coefficient’ is 1

• Calculate Divisor from the Probability P(x) of each Uncertainty Term : -

∫=2

1

)(2x

x

dx`xPxD

Common Divisors

A Coverage Factor k will have been used to obtain an Expanded Uncertainty

1/kNormal from Calibration Certificate

Gaussian Distribution1Normal

Single Multipath Uncertainty√2U ShapedPhase Noise Uncertainty√3RectangularCommentDivisorProbability Distribution

Coverage Factor and Coverage Probability

2.5899%399.73%

295.45%

1.9695%190%

Coverage Factor kCoverage Probability P

Uncertainty Probability

2.5899%399.73%

295.45%

1.9695%190%

Coverage Factor kCoverage Probability P

Gain Uncertainty Analysis - Simplified Example

0.00462Normal k = 2

.08Gain Calibration Error

0.00123Peak Normal

.03Repeatability

0.0008√2U-Shaped.01Multipath Error

Standard Uncertainty Ui(y) Abs

DivisorDistributionValuedB

Source of Uncertainty

0.084 dBExpanded Uncertainty k =2 Giving a Confidence of Approximately 95%

0.0048Standard Uncertainty

Conclusions

• Presented a ‘Cook Book’ Methodology to Performing an Uncertainty Analysis

• Methods may seem Complicated but they Get Easier with Use

• Possessing Independent Calibrated Standards is a Good Way of Validating Your Measurements for Yourself and Your Customer and for Checking your Uncertainty Analysis