Measurement Uncertainty Estimation of a Robust Photometer Circuit

In this paper the uncertainty of a robust photometer circuit (RPC) was estimated. Here, the RPC was considered as a measurement system, having input quantities that were inexactly known, and output quantities that consequently were also inexactly known. Input quantities represent information obtained from calibration certificates, specifications of manufacturers, and tabulated data. Output quantities describe the transfer function of the electrical part of the photodiode. Input quantities were the electronic components of the RPC, the parameters of the model of the photodiode and its sensitivity at 670 nm. The output quantities were the coefficients of both numerator and denominator of the closed-loop transfer function of the RPC. As an example, the gain and phase shift of the RPC versus frequency was evaluated from the transfer function, with their uncertainties and correlation coefficient. Results confirm the robustness of photodiode design.


Introduction
In general, there are many parameters that may affect a measurement result. Although it is impossible to identify all of them, the most significant can usually be identified and the magnitude of their respective effects on the measurement result can be estimated. Further, the way they impact the measurement result can, in many cases, be mathematically modeled [1].
In this paper, the uncertainty of measurement of a robust photometer circuit (RPC) based on both positive and negative feedback compensations was estimated. A rapid communication about the performance of the RPC was presented in [2]. Also, a detailed explanation of the electronic design of the RPC was given in [3]. The exact closed-loop transfer function (CLTF) of this complex feedbackcontrolled system was given in [4], a noise voltage analysis of it was carried out in [5] and an input-out transfer function analysis was carried out in [6].
In the above-mentioned references the importance of applying robust control techniques [7,8] to improve the disturbance rejection performance of photometer circuits was demonstrated. In addition, general information about signal conditioning and photodiode monitoring with operational amplifiers (opamps) by using non-robust feedback control techniques can be found in [9][10][11]. Other applications of robust and optimal filtering and control techniques to improve the performance of sensors can be found in [12][13][14][15][16][17][18][19][20][21][22][23].
The knowledge of the photodiode transfer function allows estimation of the RPC input from a measurement of its output. However, without an accompanying statement of the estimated uncertainty of RPC input, results are incomplete and in order to estimate the RPC input uncertainty, some estimation of the transfer function uncertainty is needed. The uncertainty of the measurement is a nonnegative parameter characterizing the dispersion of the quantity values being attributed to the measurands based on the information used [24].
The aim of this paper is to estimate the uncertainty of the RPC transfer function (at a level of confidence of approximately 95% [25]) and show how from this information it is possible to estimate other RPC parameters, such as its gain and phase response, with their respective uncertainties. The description of the RPC transfer function is made through the coefficients of both numerator and denominator of this function.

CLTF of the RPC
In accordance with [2][3][4][5], the RPC is shown in Figure 1. Note that in this figure the photodiode diode has been substituted by its circuit model, which according to [9][10][11], among other references, consists of a current generator (I P ) proportional to the incident light intensity, a junction capacitance (Cj ), a shunt resistance (Rj ), and a series resistance (Rs). Also, in this figure, R1, R2, R3 and R4 are the feedback resistors previously calculated in [3] that guarantee the robust disturbance rejection performance characteristic of the photometer circuit. Therefore, taking into consideration opamp parameters such as the input resistance (R i ), the input capacitance (C i ), the open-loop gain (A o ) and the gain bandwidth product ( ), the CLTF from the current generator is the parallel equivalent of j R and s R , and: ( )  [4]. In [4] the equation that describes the CLTF of the RPC as a function of the above opamp parameters was shown along with the stability analysis of the feedback system and some simulations and experimental results. Here and ω represents angular frequency), is the Laplace transform of the current ) (t i P . Thus, taking into consideration (1), the CLTF from the power of the incident light is the sensitivity of the photodiode at a specific wavelength λ and From the above equations, it can be seen the influence of several aspects that are usually of concern for circuit designers such as operational amplifier parameters. For the problem at hand, the opamp parameters that have been taken into consideration to obtain the above equations are the ones that often limit the performance of photometer circuits based on opamps [4].

Applications of the Law of Propagation of Uncertainty
The law of propagation of uncertainty given in [24][25] assumes that the output quantity can be represented by a real number y, so that it can be written as a function that depends on one or more input quantities (i.e. ). The measurement function is given by: However, if there are n output quantities, the relation between the input and output quantities is given by: ( where the superscript T denotes transposition ) and: ( ) , , Furthermore, the uncertainty matrix of the vector x is given by: is the standard uncertainty of the input quantity i = is the estimated covariance of the input quantities i x and j x . The degree of correlation between i x and j x is characterized by the estimated correlation coefficient: , and a change in one does not imply an expected change in the other.
In addition, the function ( ) and: ( ) and J is the Jacobian matrix of ( ) Thus, the uncertainty matrix of the vector y is given by

The elements
j i x f ∂ ∂ / of the Jacobian matrix J are the sensitivity coefficients ij c of the output quantities i y associated to the input quantities j x . In this paper, in order to build matrix J numerical differentiation was used [28].

Uncertainty of the Input Quantities and Typical Value of the CLTF
According to [26], input quantities represent information obtained from sources such as direct measurements, calibration certificates, specifications of manufacturers, and tabulated data. Table 1 shows the minimum, typical and maximum value of the input quantities, and their standard uncertainties as well.
The information of the parameters of the OP07 and the junction capacitance of the BPW21 was taken from their datasheets. The value of the resistors  In accordance with [24,25], taking into consideration the available information concerning the input quantities, in this paper the input quantities were described by triangular a priori distributions. Finally, using the above typical values and taking into consideration that 0 = i C for the OP07, the CLTF of the RPC given by (2) was given by: Where: , coefficients 3 a , 4 d and 5 d of (1) are equal to zero (see [4]). Therefore, ) (s n 1 is a second order polynomial and ) (s d 1 is a third order polynomial. Thus, in (3) the first term of the numerator, ) (s n 2 , is equal to zero and the first two terms of the denominator, ) (s d 2 , are equal to zero as well.
In order to have dimensionless parameter when possible the following change in polynomial ) ( 2 s n and ) ( 2 s d was made:   The first parameter 1 y can be easily determined by direct calibration: a power stabilized laser, whose power C W has been previously measured by a traceable laser power meter, is focused onto the photodiode and the output voltage of the RPC is measured with a traceable voltmeter. The reading provided by the voltmeter is C V , and an estimation of the RPC gain 1 y at DC would be As in this paper the photodiode is operated in the photoconductive mode, the photocurrent is linearly proportional to the incident light energy. Thus, assuming we have no nonlinear distortion in the opamp, the RPC shown in Figure1 is a linear circuit; and for the case in which the optical signal is harmonic, , the output voltage is harmonic as well, ) cos( where ω and α are the angular frequency and phase shift of the optical signal, respectively.
The amplitude 0 V of the output voltage is determined by: And the phase shift ϕ is determined by:

Uncertainty of the Parameters Describing the Transfer Function
The standard uncertainty and the relative standard uncertainty (which is defined as the ratio of the standard uncertainty of the parameter to its typical value) of the parameters 7 3 2 , , , y y y L , are shown in Table 2.
The uncertainty matrix of the vector ] g [ ϕ is: J is the Jacobian matrix of the functions ) , , ,

Cut-Off Frequency
Another important parameter is the cut-off frequency c f . For the case under analysis, from the frequency response shown in Figures. 2 and 3, it can be seen that, as the first zero is located between the second and the third pole and the second zero is located right after the forth pole, the cut-off frequency c f depends mainly on the denominator of ) ( 2 s T . In this paper, the cut-off frequency c f was determined numerically in the frequency range shown in Figures. 2 and 3, and its partial derivatives with respect to 7 5 4 y y y , , , L were determined numerically as well.
In order to be consistent with the above statements, for the analysis, the partial derivatives of c f with respect to 2 y and 3 y were assumed to be equal to zero. Therefore, the standard uncertainty of the cut-off frequency, ) ( c f u , was calculated as follows: