^{1}

^{*}

^{2}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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.

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 [

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 [

In the above-mentioned references the importance of applying robust control techniques [

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 non-negative parameter characterizing the dispersion of the quantity values being attributed to the measurands based on the information used [

The aim of this paper is to estimate the uncertainty of the RPC transfer function (at a level of confidence of approximately 95% [

In accordance with [_{P}

Therefore, taking into consideration opamp parameters such as the input resistance (_{i}_{i}_{o}_{T}_{T}_{P}_{o}_{j}_{s}_{j}_{s}_{1}(_{1}(

Here _{o}_{o}_{P}_{P}

Thus, taking into consideration (1), the CLTF from the power of the incident light _{o}

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 [

The law of propagation of uncertainty given in [_{1}, x_{2},⋯, x_{m})). The measurement function is given by:

However, if there are

Furthermore, the uncertainty matrix of the vector x is given by:
_{i}_{i}_{i}_{j}_{j}_{i}_{i}_{j}_{i}_{j}_{i}, x_{j}_{j}, x_{i}_{i}, x_{j}_{i}_{j}_{i}, x_{j}

In addition, the function y = f(x) is linearized at x = x_{0} and:
_{0},
_{0} and J is the Jacobian matrix of f(x):

Thus, the uncertainty matrix of the vector y is given by U_{y} = J · U_{x} · J^{T}

The elements _{i}_{j}_{ij}_{i}_{j}

According to [

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 _{1} – _{4} were the nominal ones, the series resistance and the shunt resistance of the BPW21 were measured experimentally by using the KEITHLEY Semiconductor Characterization System 4200-SGS, and the sensitivity of the BPW21 was measured experimentally by using the 3 mW RS Modulated Laser Diode Module 194-004 at 0 Hz and nominal wavelength 670 nm. A photograph of the prototype of the RPC with the 3 mW Modulated Laser Diode Module was shown in [

In accordance with [_{i}

When C_{i} = 0, coefficients a_{3}, d_{4} and d_{5} of (1) are equal to zero (see [_{1}(s) is a second order polynomial and d_{1}(s) is a third order polynomial. Thus, in (3) the first term of the numerator, n_{2}(s), is equal to zero and the first two terms of the denominator, d_{2}(s), are equal to zero as well.

In order to have dimensionless parameter when possible the following change in polynomial _{2}(_{2}(_{0} is a conventional value _{0} = 1.5 _{0} has been chosen to be equal to the nominal value of the gain bandwidth product _{T}

Please note that a conventional value has no uncertainty. Working in this way, the parameters _{2} to _{7} are dimensionless and the parameter _{1} (the RPC gain at DC) is expressed in V/W. The expressions that relate the parameters _{i}_{j}_{k}_{2}(_{2}(

The first parameter _{1} can be easily determined by direct calibration: a power stabilized laser, whose power _{C}_{C}_{1} at DC would be _{DC}_{C}_{C}_{DC}

At this point it should be pointed out that as we are carrying out a direct calibration procedure, due to the fact that _{DC}_{2}, _{3}, ⋯, _{7}, the covariance _{DC}_{i}

Thus, the RPC transfer function is described by using the output quantities _{DC}_{1}, _{2}, _{3}, ⋯, _{7}. This transfer function allows us to carry out the estimation of the power of the optical signal _{0}(

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 _{0}cos(_{0}(_{0}cos(

The amplitude _{0} of the output voltage is determined by:

The gain _{DC}

And the phase shift φ is determined by:

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 _{2}, _{3}, ⋯, _{7}, are shown in

The matrix of the estimated correlation coefficients among elements of parameters _{2}, _{3}, ⋯, _{7} is:
_{i}_{j}_{i}_{j}

At first glance, _{2,3} seem to be no correlated with _{4,5,6,7} because they depend on different variables, _{2,3} depend on _{1,2,3} while _{4,5,6,7} depend on _{1,2,3,4,5}. However, as _{1,2,3} are correlated with _{1,2,3,4,5}, _{2,3} and _{4,5,6,7} are correlated as well.

The uncertainty matrix U_{y}_{2}, _{3}, ⋯, _{7} is:

As described previously, the transfer function can be used to determine the gain _{DC}

Again, the uncertainty propagation from the transfer function parameters _{DC}_{1}, _{2}, _{3}, ⋯, _{7} to _{DC}_{2}, _{3}, ⋯, _{7}:

The uncertainty matrix of the vector [gφ] is:
_{2} is the Jacobian matrix of the functions _{2}, _{3}, ⋯, _{7}) and φ = φ (_{2}, _{3}, ⋯, _{7}), and is given by:
_{2} are evaluated numerically.

For example, at frequency

Another important parameter is the cut-off frequency _{c}_{c}_{2}(

In this paper, the cut-off frequency _{c}_{4}, y_{5}, ⋯, y_{7} were determined numerically as well.

In order to be consistent with the above statements, for the analysis, the partial derivatives of _{c}_{2} and _{3} were assumed to be equal to zero. Therefore, the standard uncertainty of the cut-off frequency, _{c}

Finally, the results are the following:
_{95%}(_{c}

In this paper, the uncertainty of the transfer function of a RPC has been estimated in accordance with the Guide to the Expression of Uncertainty in Measurement of the Organization for Standardization. The RPC transfer function has been described through seven parameters and the uncertainty and correlation coefficients of these parameters have been estimated as well. Also, it has been shown that other parameters such as the gain, phase margin and the cut-off frequency can be estimated along with their respective uncertainties taking into consideration the information given by the RPC transfer function.

This work has been partially supported by the Ministry of Science and Innovation (MICINN) of Spain under the research project TEC2007-63121, and the Universidad Politécnica de Madrid.

Robust photometer circuit.

Gain

Phase shift φ(°) vs. frequency

Minimum, typical, maximum value and standard uncertainty of the input quantities.

C_{j} |
522 pF | 580 pF | 638 pF | 24 pF |

R_{j} |
374 MΩ | 416 MΩ | 457 MΩ | 17 MΩ |

R_{s} |
5.31 Ω | 5.90 Ω | 6.49 Ω | 0.24 Ω |

R_{1} |
900 Ω | 1000 Ω | 1100 Ω | 41 Ω |

R_{2} |
90.0 Ω | 100.0 Ω | 110.0 Ω | 4.1 Ω |

R_{3} |
90.0 kΩ | 100.0 kΩ | 110.1 kΩ | 4.1 Ω |

R_{4} |
19.87 kΩ | 22.08 kΩ | 24.29 kΩ | 0.90 kΩ |

R_{i} |
15.0 MΩ | 50.0 MΩ | 55.0 MΩ | 2.0 MΩ |

_{i} |
0 pF | 0 pF | 0 pF | 0 pF |

A_{0} |
106 dB | 114 dB | 125.4 dB | 1.46 dB |

ω_{T} |
0.80π Mrad/s | 1.20π Mrad/s | 1.30π Mrad/s | 0.15 Mrad/s |

σ | 121.1 mA/W | 134.5 mA/W | 148.0 mA/W | 5.5 mA/W |

Parameter estimation, standard uncertainty and relative standard uncertainty.

y_{2} |
0.0053 | 0.0016 | 0.30 |

y_{3} |
0.42 | 0.12 | 0.29 |

y_{4} |
0.148 | 0.029 | 0.20 |

y_{5} |
12.2 | 2.0 | 0.17 |

y_{6} |
55.9 | 8.0 | 0.14 |

y_{7} |
50.3 | 2.9 | 0.057 |