Calibration of a power sensor at a frequency of 18 GHz
Author: EA
This Example is taken from EA 4/02. See EA 4/02 Section S6 for more details.
The measurement involves the calibration of an unknown power sensor with respect to a calibrated power sensor used as a reference by substitution on a stable transfer standard of known small reflection coefficient. The measurement is made in terms of calibration factor, which is defined as the ratio of incident power at the reference frequency of 50 MHz to the incident power at the calibration frequency under the condition that both incident powers give equal power sensor response. At each frequency one determines the (indicated) ratio of the power for the sensor to be calibrated respectively the reference sensor and the internal sensor that forms part of the transfer standard, using a dual power meter with ratio facility.
Model Equation:
K_{X} = (K_{S} + δK_{D}) * (M_{Sr} * M_{Xc}) / (M_{Sc} * M_{Xr}) * p_{Cr} * p_{Cc} * p

List of Quantities:
Quantity  Unit  Definition 

K_{X}  unknown calibration factor  
K_{S}  calibration factor of the reference power sensor  
δK_{D}  drift of calibration factor of the reference power sensor since its last calibration  
M_{Sr}  mismatch factor of the reference power sensor at the reference frequency of 50 MHz  
M_{Xc}  mismatch factor of the unknown power sensor at the calibration frequency of 18 GHz  
M_{Sc}  mismatch factor of the reference power sensor at the calibration frequency of 18 GHz  
M_{Xr}  mismatch factor of the unknown power sensor at the reference frequency of 50 MHz  
p_{Cr}  correction of the observed ratio for nonlinearity and limited resolution of the power meter at power ratio level of the reference frequency  
p_{Cc}  correction of the observed ratio for nonlinearity and limited resolution of the power meter at power ratio level of the calibration frequency  
p  =p_{iX}/p_{iS,} ratio of the output power ratios indicated at the power transfer system in realizing equal response for the unknown and the reference power sensor 
K_{S}: 
Type B normal distribution Value: 0.957 Expanded Uncertainty: 0.011 Coverage Factor: 2 
REFERENCE SENSOR: The reference sensor was calibrated six months before the calibration of the unknown power sensor. The value of the calibration factor, given in the calibration certificate, is (95,7±1,1)% (coverage factor k=2), which also may be expressed as 0,957±0,011
δK_{D}: 
Type B rectangular distribution Value: 0.001 Halfwidth of Limits: 0.002 
DRIFT OF THE STANDARD: The drift of the calibration factor of the reference standard is estimated from annual calibrations to be 0,002 per year with deviations within ±0,004. From these values the drift of the reference sensor which has been calibrated half a year ago is estimated to be 0.001 with deviations within ±0,002.
M_{Sr}: 
Type B Ushaped distribution Value: 1.0 Halfwidth of Limits: 0.0008 
MISMATCH FACTORS: As the transfer standard system is not perfectly matched and the phase of the reflection coefficients of the transfer standard, the unknown and the standard power sensors are not known, there will be an uncertainty due to mismatch for each sensor at the reference frequency and at the calibration frequency. The probability distribution of the contribution is Ushaped and the limits are calculated from the magnitude of the reflection coefficients (see EALR2S1:S6.8).
M_{Xc}: 
Type B Ushaped distribution Value: 1.0 Halfwidth of Limits: 0.0168 
M_{Sc}: 
Type B Ushaped distribution Value: 1.0 Halfwidth of Limits: 0.014 
M_{Xr}: 
Type B Ushaped distribution Value: 1.0 Halfwidth of Limits: 0.0008 
p_{Cr}: 
Type B normal distribution Value: 1.0 Expanded Uncertainty: 0.00142 Coverage Factor: 1.0 
LINEARITY AND RESOLUTION OF THE POWER METER: The expanded uncertainty of 0,002 (coverage factor k = 2.0) is assigned to the power meter readings at the power ratio level of the reference frequency and of 0,0002 (coverage factor k = 2.0) at the power ratio level of calibration frequency due to nonlinearity of the power meter used. These values have been obtained from previous measurements. Since the same power meter has been used to observe both p_{iX} and p_{iS} the uncertainty contributions at the reference as well at the calibration frequency are correlated. Because power ratios at both frequencies are considered, the effect of the correlations is to reduce the uncertainty. Thus only the relative difference in the readings due to systematic effects must be taken into account resulting in a standard uncertainty of 0,00142 associated with the correction factor p_{Cr} and 0,000142 with the correction factor p_{Cc.}
p_{Cc}: 
Type B normal distribution Value: 1.0 Expanded Uncertainty: 0.000142 Coverage Factor: 1.0 
p: 
Type A Method of observation: Direct Number of observation: 3
Arithmetic Mean: 0.97597 
Uncertainty Budgets:
K_{X}: unknown calibration factor
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

K_{S}  0.95700  5.50·10^{3}  normal  0.98  5.4·10^{3}  11.0 % 
δK_{D}  1.00·10^{3}  1.15·10^{3}  rectangular  0.98  1.1·10^{3}  0.5 % 
M_{Sr}  1.000000  566·10^{6}  Udistr.  0.93  530·10^{6}  0.1 % 
M_{Xc}  1.0000  0.0119  Udistr.  0.93  0.011  46.9 % 
M_{Sc}  1.00000  9.90·10^{3}  Udistr.  0.93  9.2·10^{3}  32.6 % 
M_{Xr}  1.000000  566·10^{6}  Udistr.  0.93  530·10^{6}  0.1 % 
p_{Cr}  1.00000  1.42·10^{3}  normal  0.93  1.3·10^{3}  0.7 % 
p_{Cc}  1.000000  142·10^{6}  normal  0.93  130·10^{6}  0.0 % 
p  0.97597  4.80·10^{3}  normal  0.96  4.6·10^{3}  8.1 % 
K_{X}  0.9330  0.0162 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

K_{X}  0.933  0.032  2.00  95% (ttable 95.45%) 