Evalua&ng Luminous Uncertainty given Spectral Power Uncertainty

Transcription

1 Evalua&ng Luminous Uncertainty given Spectral Power Uncertainty by Rolf S. Bergman CORM May

2 Content of Presenta,on Statement of Problem Application of GUM Process De>ining Equation or Model Uncertainty terms Sensitivity Coef>icient Combined Uncertainty including correlation Spreadsheet example calculations Conclusions 14 May 2015 CORM

3 Introduc,on Integrating spheres are normally used to measure lumen output of a lamp Calibration of integrating sphere is done using total spectral >lux standard lamp Standard lamp calibration certi>icate provide values for spectral power >lux at given wavelengths and values for the uncertainty of the >lux at each of those wavelengths. Lumens are calculated from test lamp spectral >lux How does the uncertainty in the spectral 3lux get translated to lumen uncertainty? 14 May 2015 CORM

4 Lumen Equa,on The lumen value is the sum (integral) of the spectral power, S i, over the wavelength region of interest, weighted by the photopic ef>icacy, V i, i.e., 14 May 2015 CORM

5 Lumen Uncertainty The only term on the right- hand- side of the lumen equation that has uncertainty is the spectral >lux values at a given wavelength, λ i. Thus the uncertainty in the lumens is a function of the uncertainty in the spectral >lux, i.e., u(φ) = f[u(s i )] 14 May 2015 CORM

6 Combined Uncertainty The Combined Uncertainty when correlation is de>ined as the square root of the product of the sum of the sensitivity coef>icient, Φ/ x i at that spectral value, x i, times the sensitivity coef>icient at every other spectral value, times the covariance associated with each spectral value pair, u(x i, x j ). This can be written as: where r(s i,s j ) is the correlation coef>icient. 14 May 2015 CORM

7 Sensi,vity coefficients In the last equation we noted that the sensitivity coef>icient was the partial derivative of the lumen value with respect to the variables. There is only one variable with uncertainty in the lumen equation, that of S i. Thus: 14 May 2015 CORM

8 Combined Uncertainty for Lumens Inserting the values of the Sensitivity Coef>icient into the combined uncertainty we obtain: where h i is the fractional uncertainty in the spectral value, S i. 14 May 2015 CORM

9 Evalua,on of Lumen Uncertainty Combined uncertainty equation is evaluated using an EXCEL spreadsheet. Assumptions: A NIST halogen standard lamp with given spectral >lux values, S i, every 5 nm between 380 and 780 nm. The relative spectral uncertainty is accounted for by blue, green and red values; h i = h 1, h 2 and h 3. The correlation coef>icient is not known but we will evaluate the effect of various values of r on the value of the lumen uncertainty, u(φ). 14 May 2015 CORM

10 Lumen Uncertainty 1.0# Lumen&Uncerta2nty&as&a&func2on&of&the&correla2on&coefficient& 0.9# 0.8# Lumen&Uncertainty,&u& 0.7# 0.6# 0.5# 0.4# 0.3# 0.2# 0.1# 0.0# 0.0# 0.1# 0.2# 0.3# 0.4# 0.5# 0.6# 0.7# 0.8# 0.9# 1.0# Correla2on&Coefficient,&r& 14 May 2015 CORM

11 Conclusions Wavelength Integration: Ignoring the spectral uncertainty except in the central visible region where Vi > 0.1 (about 175 nm band) causes negligible errors in the lumen uncertainty Correlation Coef>icient: What is a reasonable correlation coef>icient between spectral >lux values at various wavelengths? r=0.5? If so the reduction in the lumen uncertainty due to multiple values is small, about 21 % As a general rule it is reasonable to assume that the lumen uncertainty is equal to the spectral uncertainty for spectral power in the neighborhood of 560 nm. 14 May 2015 CORM