Evaluation of expanded uncertainties in luminous intensity and illuminance calibrations

Ferhat Sametoglu

doi:10.1364/AO.47.005829

Applied Optics
Vol. 47,
Issue 31,
pp. 5829-5847
(2008)
•https://doi.org/10.1364/AO.47.005829

Evaluation of expanded uncertainties in luminous intensity and illuminance calibrations

Ferhat Sametoglu

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Ferhat Sametoglu, "Evaluation of expanded uncertainties in luminous intensity and illuminance calibrations," Appl. Opt. 47, 5829-5847 (2008)

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Abstract

Detector-based calibrating methods and expressions for calculation of photometric uncertainties related to uncertainties in the calibrations of luminous intensity of a light source, illuminance responsivity of a photometer head, and calibration factors of an illuminance meter are discussed. These methods permit luminous intensity calibrations of incandescent light sources, luminous responsivity calibrations of photometer heads, and calibration factors of illuminance meters to be carried out with relative expanded uncertainties (with a level of confidence of 95.45%) of 0.4%, 0.4%, and 0.6%, respectively.

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					$u_{i} (y)$	$h (y, x_{i})$
$X_{i}$	$x_{i}$	$u (x_{i})$	pdf	$c_{i}$	(cd)	(%)
$c_{DVM}$	0.999992	0.000004	NTD	$1762.65 cd$	0.00705	0.02
$c_{DVMD}$	0.000001	0.0000004	RTD	$1762.65 cd$	0.00071	0.0002
$U_{L}$	$0.584720 V$	$0.0000036 V$	NTD	$3014.50 cd / V$	0.01085	0.04
$U_{L} (J)$	$5.8472 A$	$0.0006 A$	NTD	$- 301.45 cd / A$	$- 0.18087$	10.3
R	$0.099994 Ω$	$0.0000020 Ω$	NTD	$- 17627.44 cd / Ω$	$- 0.03525$	0.4
$R_{D}$	$- 0.0000010$	0.0000004	RTD	$- 1762.64 cd$	$- 0.00071$	0.0002
$α_{R S} Δ T$	0.0	0.0000023	RTD	$- 1762.64 cd$	$- 0.00407$	0.01
$m_{i}$	6.24	0.14	RTD	$- 0.00662 cd$	0.00096	0.0003
d	$4.000 m$	$0.00018 m$	RTD	$141.16 cd / m$	0.02541	0.2
$d_{P}$	$0.00212 m$	$0.000021 m$	RTD	$141.16 cd / m$	0.00296	0.003
$α_{S R} Δ T$	0.0	0.0000078	RTD	$141.16 cd$	0.00110	0.0004
$P_{c}$	$221.12 nA$	$0.024 nA$	NTD	$1.27 cd / nA$	0.03007	0.3
$c_{PCM}$	1.00033	0.00010	RTD	$1.27 cd$	0.00013	0.00001
$P_{m D}$	$- 0.000030$	0.000012	NTD	$1.27 cd$	0.00002	0.0000001
$s_{v}$	$12.592 nA / lx$	$0.019 nA / lx$	NTD	$- 22.44 cd \cdot lx / nA$	$- 0.42391$	56.3
$s_{v D}$	$- 0.0006$	0.0024	RTD	$- 22.44 cd$	$- 0.05386$	0.9
$ccf$	1.0000	0.0005	RTD	$282.47 cd$	0.14124	6.3
$scf$	0.9992	0.0004	NTD	$282.70 cd$	0.11308	4.0
$M_{R}$	0.9998	0.00006	NTD	$282.54 cd$	0.01783	0.1
$L_{a ε}$	$0.0 °$	$0.00064 °$	RTD	$- 282.47 cd / °$	$- 0.18103$	10.3
$L_{a φ}$	$0.0 °$	$0.00064 °$	RTD	$- 282.47 cd / °$	$- 0.18103$	10.3
$P_{θ}$	$0.0 °$	$0.00012 °$	RTD	$- 282.47 cd / °$	$- 0.03262$	0.3
$P_{N}$	0.0	0.00012	RTD	$- 282.47 cd$	$- 0.03390$	0.4
$I_{x} (T)$	$281.20 cd$	combined uncertainty, $k = 1$			$\pm 0.56 cd$
		effective degree of freedom, $v_{eff}$			$> 100$
		expanded uncertainty, $k = 2$			$\pm 1.13 cd$
		expanded uncertainty, $k = 2$			0.4%

Measurement №: n	Lamp Voltage $U_{L}$	Photocurrent $P_{c}$	Stray Light $s cf$
	(V)	(nA)
1	0.584718	221.16	0.9987
2	0.584712	221.10	0.9991
3	0.584723	221.09	0.9989
4	0.584720	221.15	0.9992
…	…	…	…
18	0.584719	221.13	0.9993
19	0.584720	221.09	0.9996
20	0.584717	221.11	0.9990
Mean	0.584720	221.12	0.9992
$s (U_{L})$	0.0000036	0.024	0.0004

	Luminous Intensity $I_{x} (T)$
Measurement №: n	(cd)	Luminous Intensity Normalized $I_{xnor} (T)$
1	281.22	0.9997
2	281.25	0.9998
3	281.29	0.9999
4	281.20	0.9996
5	281.31	1.0000
6	281.22	0.9997
Mean value $M_{R} = \frac{1}{n} \sum_{i = 1}^{n} I_{xnori} (T)$		0.9998
Standard deviation $s (M_{R}) = \frac{1}{n - 1} \sqrt{\sum_{k = 1}^{n} (I_{xnori} (T) - M_{R})}$		0.00015

$X_{i}$	$x_{i}$	$u (x_{i})$	pdf	$c_{i}$	$u_{i} (y)$ ( $nA / lx$ )	$h (y, x_{i})$ (%)
$c_{DVM}$	0.999992	0.000004	NTD	$118.95 nA / lx$	0.000476	0.02
$c_{DVMD}$	0.000001	0.0000004	RTD	$118.95 nA / lx$	0.000048	0.0002
$U_{L}$	$0.584720 V$	$0.0000036 V$	NTD	$203.42 nA / lx \cdot V$	0.000732	0.04
$U_{L} (J)$	$5.8472 A$	$0.0006 A$	NTD	$- 20.34 nA / lx \cdot A$	$- 0.012205$	11.7
R	$0.099994 Ω$	$0.0000020 Ω$	NTD	$- 1189.52 nA / lx \cdot Ω$	$- 0.002379$	0.4
$R_{D}$	$- 0.0000010$	0.0000004	RTD	$- 118.94 nA / lx$	$- 0.000048$	0.0002
$α_{R S} Δ T$	0.0	0.0000023	RTD	$- 118.94 nA / lx$	$- 0.000275$	0.01
$m_{i}$	6.24	0.14	RTD	$- 0.000445 nA / lx$	0.000064	0.0003
$s_{v}$	$1.8302 nA / lx$	$0.0027 nA / lx$	NTD	10.42	0.028616	64.4
$s_{v D}$	$- 0.0015$	0.0006	RTD	$10.42 nA / lx$	0.006254	3.1
$P_{T}$	$596.74 nA$	$0.131 nA$	NTD	$0.03 nA / lx \cdot nA$	0.004197	1.4
$P_{R}$	$56.16 nA$	$0.0067 nA$	NTD	$- 0.33 nA / lx \cdot nA$	$- 0.002237$	0.4
$c_{PCM}$	1.00033	0.00010	NTD	$- 0.33 nA / lx$	$- 0.000033$	0.0001
$P_{m D}$	$- 0.000030$	0.000012	RTD	$- 0.33 nA / lx$	$- 0.000004$	0.000001
$ccf$	1.0000	0.0005	RTD	$19.06 nA / lx$	0.009531	7.1
$scf$	0.9992	0.0004	NTD	$19.08 nA / lx$	0.007631	4.6
$ucf$	0.9992	0.00015	NTD	$19.08 nA / lx$	0.002862	0.6
$M_{R}$	0.9997	0.00007	NTD	$19.07 nA / lx$	0.001430	0.2
$P_{θ R}$	$0.0 °$	$0.00012 °$	RTD	$- 19.06 nA / lx \cdot °$	$- 0.002201$	0.4
$P_{θ T}$	$0.0 °$	$0.00012 °$	RTD	$- 19.06 nA / lx \cdot °$	$- 0.002201$	0.4
$R P_{a}$	0.0	0.00044	NTD	$- 19.06 nA / lx$	$- 0.008387$	5.5
$P_{N R}$	0.0	0.00012	RTD	$- 19.06 nA / lx$	$- 0.002287$	0.4
$s_{v T} (T)$	$19.08 nA / lx$	combined uncertainty, $k = 1$			$\pm 0.04 nA / lx$
		effective degree of freedom, $v_{eff}$			$> 100$
		expanded uncertainty, $k = 2$			$\pm 0.07 nA / lx$
		expanded uncertainty, $k = 2$			0.4%

$X_{i}$	$x_{i}$	$u (x_{i})$	pdf	$c_{i}$	$u_{i} (y)$	$h (y, x_{i})$ (%)
$c_{DVM}$	0.999992	0.000004	NTD	6.32	0.0000253	0.01
$c_{DVMD}$	0.000001	0.0000004	RTD	6.32	0.0000025	0.0001
$U_{L}$	$0.584720 V$	$0.0000036 V$	NTD	$10.81 V^{- 1}$	0.0000389	0.02
$U_{L} (J)$	$5.8472 A$	$0.0006 A$	NTD	$- 1.08 A^{- 1}$	$- 0.0006489$	5.0
R	$0.099994 Ω$	$0.0000020 Ω$	NTD	$- 63.24 Ω^{- 1}$	$- 0.0001265$	0.2
$R_{D}$	$- 0.0000010$	0.0000004	RTD	$- 6.32$	$- 0.0000025$	0.0001
$α_{R S} Δ T$	0.0	0.0000023	RTD	$- 6.32$	$- 0.0000146$	0.003
$m_{i}$	6.24	0.14	RTD	0.000024	0.0000034	0.0001
$P_{R}$	$56.16 nA$	$0.0067 nA$	NTD	$0.018 {nA}^{- 1}$	0.0001189	0.2
$c_{PCM}$	1.00033	0.00010	NTD	0.018	0.0000018	0.00004
$P_{m D}$	$- 0.000030$	0.000012	RTD	0.018	0.0000002	0.000001
$s_{v}$	$1.8302 nA / lx$	$0.0027 nA / lx$	NTD	$- 0.55 lx / nA$	$- 0.0015213$	27.3
$s_{v D}$	$- 0.0015$	0.0006	RTD	$- 0.55$	$- 0.0003325$	1.3
$E_{T}$	$30.79 lx$	$0.067 lx$	NTD	$- 0.03 {lx}^{- 1}$	$- 0.0022082$	57.5
$ccf$	1.0000	0.0005	RTD	1.01	0.0005067	3.0
$scf$	0.9992	0.0004	NTD	1.01	0.0001521	0.3
$ucf$	0.9992	0.00015	NTD	1.01	0.0004057	1.9
$M_{R}$	0.9994	0.00018	NTD	1.01	0.0001801	0.4
$P_{θ R}$	$0.0 °$	$0.00012 °$	RTD	$- 1.01 °^{- 1}$	$- 0.0001170$	0.2
$L X_{θ}$	$0.0 °$	$0.00012 °$	RTD	$- 1.01 °^{- 1}$	$- 0.0001170$	0.2
$R P_{a}$	0.0	0.00048	NTD	$- 1.01$	$- 0.0004864$	2.8
$P_{N R}$	0.0	0.00012	RTD	$- 1.01$	$- 0.0001216$	0.2
$k (T)$	0.9838	combined uncertainty, $k = 1$			$\pm 0.0029$
		effective degree of freedom. $v_{eff}$			$> 55$
		expanded uncertainty, $k = 2$			$\pm 0.0058$
		expanded uncertainty, $k = 2$			0.6%

Measurement №: n	Photometer Head (TH15 C) [ $P_{R} (nA)$ ]	Photometer Head (P30SCT) [ $P_{T} (nA)$ ]	Illuminance Meter (B360) [ $E_{T} (lx)$ ]
1	56.15	596.7	30.8
2	56.17	596.9	30.7
3	56.16	596.8	30.8
4	56.17	596.4	30.8
…	…	…	…
18	56.16	596.8	30.9
19	56.16	596.7	30.8
20	56.15	596.5	30.7
Mean	56.159	596.74	30.79
$s (U_{L})$	0.0067	0.131	0.067

	Photometer Head (PUT)		Illuminance Meter
Measurement №: n	$s_{v T} (T)$	normalized $s_{v T n} (T)$	$k (T)$	normalized $k_{n} (T)$
1	1.2880	0.9996	0.9923	0.9992
2	1.2882	0.9998	0.9919	0.9988
3	1.2885	1.0000	0.9925	0.9994
4	1.2880	0.9996	0.9922	0.9991
5	1.2878	0.9995	0.9928	0.9997
6	1.2881	0.9997	0.9931	1.0000
Mean		0.9997	0.9994
Standard deviation		0.00018	0.00044

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Applied Optics