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In this paper an input-output transfer function analysis based on the frequency response of a photometer circuit based on operational amplifier (op amp) is carried out. Op amps are universally used in monitoring photodetectors and there are a variety of amplifier connections for this purpose. However, the electronic circuits that are usually used to carry out the signal treatment in photometer circuits introduce some limitations in the performance of the photometers that influence the selection of the op amps and other electronic devices. For example, the bandwidth, slew-rate, noise, input impedance and gain, among other characteristics of the op amp, are often the performance limiting factors of photometer circuits. For this reason, in this paper a comparative analysis between two photodiode amplifier circuits is carried out. One circuit is based on a conventional current-to-voltage converter connection and the other circuit is based on a robust current-to-voltage converter connection. The results are satisfactory and show that the photodiode amplifier performance can be improved by using robust control techniques.

Semiconductor junctions convert the photon energy of light into an electrical signal by releasing and accelerating current-conducting carriers within the semiconductor [

Some of the most important characteristics and features of photodiodes are their low cost, excellent linearity, good quality of the time domain performance, the speed of the their response, their low noise, they are compact and lightweight devices, and so on.

Also, photodiodes are used in safety equipment, blood particle analyzers, pulse oximeters, X ray detection, photographic flash control, light meters, automotive applications, optical communications, fiber optic links, and so on.

However, depending on the kind of application, the electronic circuits used to monitoring photodiodes should meet some specific requirements. For example, in laboratory applications, where the temperature is held constant, the responsivity, the shunt resistance, the shunt capacitance and the dark current of the photodiode, which are all temperature dependent photodiode parameters, do not vary from their expected values, and the response of the photodiode is satisfactory if the users guarantee that the power supply is stable and that there are not any noise or disturbance sources corrupting the response of the photodiode. In this kind of applications, conventional photometer circuits perform satisfactorily and designers do not have to work hard in order to meet the final user requirements.

On the other hand, in industrial applications, where photometer circuits have to work under severe working conditions such as the endurance of high temperatures, high humidity, dangerous chemical attacks, undesirably strong vibrations, pollution, and so on [

Generally speaking, in practice the signals produced by sensors are frequently corrupted by noise that cause sensor operations to deviate from their true value, which causes an undesirable degree of uncertainty in the measurements carried out by the sensors. Therefore, in order to cancel the noise that corrupts the relevant information coming from sensors, robust and optimal signal treatment stages are needed [

The present paper is aimed at showing the importance of the application of the above-mentioned techniques to the design of photometer circuits, and it also shows the importance of the frequency response analysis in order to provide insights into the benefits and trade-offs of the current-to-voltage converter connections usually used to build linear photometer circuits.

According to Graeme [

Carriers released within the depletion region of a semiconductor junction produce the majority of this current due to the electric field of this region. That region contains ionized or depleted atoms that support a voltage differential across the junction. The associated electric field accelerates the carriers toward the terminals of the diode, adding conduction energy to the carriers and reducing the probability of recombination. Applying reverse bias to the junction expands the depletion region to encompass more of the material of the diode within the accelerating field [

Forward bias narrows the depletion region and lowers the barrier to carrier injection. Diffusion component of current greatly increases and drift component decreases. Carrier density is large (exponential), making the junction conductive and allowing a large forward current.

In photodiodes, semiconductor doping levels and the junction depth are two of the most important parameters. The depth and extent of the junction determines the location of the depletion region and the light wavelengths that produce an efficient response. Photons generate carriers at a range of depths with a given range proportional to the photon wavelength [

The most important characteristics of a photodiode are the following [

Spectral response

Radiometric sensitivity

Responsitivity

Quantum efficiency

Sensitivity

Linearity

Dark current

Shunt resistance

Junction capacitance

Reverse breakdown voltage

Open circuit voltage

Response time

Noise current

Angular response

Package style

Also, according to [_{P}_{N}_{SH}_{S}_{J}_{L}

Due to their excellent linearity, gain accuracy, high input resistance, high open-loop gain, low noise, low offsets and wide bandwidth, among other characteristics, operational amplifiers are commonly used in monitoring photodiodes [

In addition, the energy transmitted by light to a photodiode can be measured as either a voltage or a current output, and when the photodiode output is monitored as a voltage, the response has a logarithmic relationship to the light energy received since the sensitivity of the diode varies with its voltage. Nevertheless, when a linear relationship between the photodiode response and the light energy received is desired, the voltage drop across the photodiode is held constant for a fixed sensitivity and the photocurrent is linearly related to the incident light energy [

However, according to [_{f}

According to [

On the other hand, in the field, outside the laboratory or in applications in which the temperature is not constant, as the responsivity, the shunt resistance, the junction capacitance and the dark current of the photodiode are temperature dependent, and the noise is dependent upon the characteristics of the photodiode and the operating conditions, the uncertainty of measurement of linear photometer circuits based on conventional CVC connections, as the one shown in

From the Control Engineering point of view, in

In [

The analysis of the relative error between the theoretical sensitivity and the experiemental sensitivity of the circuit shown in

In addition, in [

Finally, in [

However, there is al least one more type of analysis that should be carried out to these circuits. This is the frequency response analysis by using an input-output transfer function approach. This is the aim of the next section.

According to [

According to [

Performance, good disturbance rejection: loop-transfer function,

Performance, good command following:

Stabilization of unstable plant:

Mitigation of measurement noise on plant outputs:

Small magnitude of input signals: controller transfer function,

Physical controller must be strictly proper:

Nominal stability:

Robust stability:

Fortunately, the above-mentioned conflicting design objectives are generally in different frequency ranges, and most of the objectives can be met by using a large loop gain (|

Furthermore, in accordance with the loop-shaping approach to controller design [

However, robust stability must not be confused with robust performance [

According to [_{P}_{o}_{SH}_{S}_{SH}_{S}_{P}_{o}

As can be seen from the above

Here, in order to carry out a practical analysis of the reason why the above statement is true taking into consideration the GM and the PM of (1) and (2), the photometer circuits were implemented by using the Siemens silicon photodiode BPW21, with _{S}_{SH}_{J}

In addition, in the laboratory experiments the incident light came from the 3 mW RS Modulated Laser Diode Module 194-004, whose nominal wavelength is 670 nm, and the experimental sensitivity of the BPW21 at 670 nm was 0.1345 A/W. Furthermore, in order to have a 10 V output for a 3 mW input to the circuits shown in _{f}_{1}_{2}_{3}_{4}

For the above-given values,

The loop transfer function of the robust photometer circuit [

Thus, the sensitivity function of the robust circuit is

Moreover, the Bode plots of (11) and (12) are shown in

The Nyquist diagram of the loop transfer function of the robust photometer (13) is shown in _{r}

The state-space representation of the SISO system given by (12) is the one given by

Thus, the relationship between (12) and (15)-(16) is given by

For the system given by (15)-(16), if the initial condition of the state vector _{1}_{2}^{T} is equal to ^{T} and the input _{1}_{2}

The response (MATLAB simulation) of the robust closed-loop system to a step disturbance in the sensitivity function (14) is shown in

The photometer circuits presented in this paper (

Furthermore, in order to test the noise rejection performance of both circuits, a 3 mW incident light input signal at 0 Hz and wavelength equal to 670 nm was apply to the above-mentioned circuits, and the power spectrum of the output voltage of these circuits was displayed by using the YOKOGAWA Digital Oscilloscope DL9040.

The result of the experiment was that for the non-robust circuit the peak value of the power spectral density (PSD) of the noise of the output voltage was equal to -50.54 dBV. However, the peak value of the PSD of the noise of the output voltage was equal to -62.00 dBV. Therefore, a signal-to-noise ratio improvement of 10 dBV was achieved by using the robust compensation.

Moreover, the temperature coefficient (TC) of the output voltage of the robust photometer circuit was equal to -21.83 μV/°C, while the TC of the output voltage of the non-robust photometer circuit was equal to -40.16 μV/°C, which shows the importance of the application of robust control techniques to improve the performance of photometer circuits.

In this paper, an input-output transfer function analysis of a photometer circuit based on an op amp has been carried out. The results of the analysis showed the importance of the application of robust control technique to improve the performance of photometer circuits, and showed some of the most important points to be taken into consideration when designing these circuits for applications in which they have to work under severe working conditions, as it happens in most industrial applications.

Moreover, here two photometer circuits, one based on a conventional CVC connection and another based on a robust CVC connection, have been compared with each other. The results of such a comparison showed that the performance of photometer circuits based on robust CVC connections is much better that the performance of photometers circuits based on conventional CVC connections.

The use of robust control techniques to the design of complex sensor systems can bridge the gap between advanced signal conditioning techniques and the design of sensors for a wide range of applications. The reality is that only by the fusion of these concepts can the designer find the way clear to build the sensors that today's industry needs.

This work was supported by the Universidad Politecnica de Madrid, Spain.

Equivalent circuit for a photodiode connected to a load resistance

Conventional current-to-voltage converter connection.

Robust photometer circuit.

Nyquist plot of

Bode plot of the closed-loop transfer function of the non-robust photometer (11).

Bode plot of the closed-loop transfer function of the robust photometer (12).

Nyquist diagram of the loop transfer function of the robust photometer (13).

Trajectories of the state variables of the system given by (15)-(16) for an input of 3 mW of incident light at 10 Hz, 100 Hz and 1 kHz, respectively.

Response of the robust photometer circuit to a step disturbance at the output of the photodiode.

Practical implementation of the circuits shown in