# Input-output Transfer Function Analysis of a Photometer Circuit Based on an Operational Amplifier

## Abstract

**:**

## 1. Introduction

## 2. The photo effect and photodiode model

- 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

_{P}is light generated photocurrent, I

_{N}is noise current, R

_{SH}is the shunt resistance, R

_{S}is the series resistance, C

_{J}is the junction capacitance, and R

_{L}is an external load resistance connected to the photodiode.

## 3. The photometer circuits

_{f}is the negative feedback resistor used to convert the photocurrent into an output voltage linearly related to the light energy. This optoelectronic circuit is used in both integrated circuits containing a photodiode and a transimpedance amplifier on a single chip, and in discrete designs.

## 4. Frequency response analysis by using an input-output transfer function approach

- Performance, good disturbance rejection: loop-transfer function, L(s), large
- Performance, good command following: L(s) large
- Stabilization of unstable plant: L(s) large
- Mitigation of measurement noise on plant outputs: L(s) small
- Small magnitude of input signals: controller transfer function, K(s), small and L(s) small
- Physical controller must be strictly proper: K(s) → 0 and L(s) → 0 at high frequencies
- Nominal stability: L(s) small
- Robust stability: L(s) small

_{P}(s) to V

_{o}(s) is given by

_{SH}‖ R

_{S}is the parallel equivalent of R

_{SH}and R

_{S}. And the transfer function of the circuit shown in Fig. 3 from I

_{P}(s) to V

_{o}(s) is given by

_{S}= 10 Ω and R

_{SH}= 100 kΩ and C

_{J}equal to 580 pF for a reverse voltage of 0 V applied across the BPW21.

_{f}was chosen to be equal to 24.783 kΩ, and R

_{1}, R

_{2}, R

_{3}and R

_{4}were chosen to be equal to 1 kΩ, 100 Ω, 100 kΩ and 22.08 kΩ, respectively.

_{r}(s) never enters the unit circle, which means that the closed-loop system with the transfer function given by (12) is robust. For this system, the stability margins are the following: GM = ∞ and PM = 90°.

_{1}(t) x

_{2}(t)]

^{T}is equal to x(0)=[0 0]

^{T}and the input u(t) is a sinusoid of varying frequency from 1 Hz on, the trajectories of the state variables of this state-space representation is the superposition of many ellipses, one ellipse for each frequency under consideration, with variable length axes. Fig. 8 shows the trajectories of the state variables of the above system for an input of 3 mW of incident light at 10 Hz, 100 Hz and 1 kHz, respectively. Note the fast, satisfactory convergence of the phase variables (i.e., x

_{1}(t) and x

_{2}(t)) of system. If the initial condition of the state vector of the system is not zero, the trajectory described by the state variables will converge to one similar to the ones shown in Fig. 8 (MATLAB simulation).

## 5. Experimental results

## 6. Conclusions

## Acknowledgments

## References

- Graeme, J. Photodiode amplifiers: op amp solutions; McGraw-Hill: New York, 1996. [Google Scholar]
- Hernandez, W. A survey on optimal signal processing techniques applied to improve the performance of mechanical sensors in automotive applications. Sensors
**2007**, 7, 84–102. [Google Scholar] - Zhou, K.; Doyle, J. C.; Glover, K. Robust and Optimal Control; Prentice-Hall, Upper Saddle River: New Jersey, 1996. [Google Scholar]
- Skogestad, S.; Posthlethwaite, I. Multivariable Feedback Control; John Wiley and Sons: England, 1996. [Google Scholar]
- Hernandez, W. Improving the response of an accelerometer by using optimal filtering. Sensors and Actuators A
**2001**, 88, 198–208. [Google Scholar] - Hernandez, W. Improving the response of several accelerometers used in a car under performance tests by using Kalman filtering. Sensors
**2001**, 1, 38–52. [Google Scholar] - Hernandez, W. Improving the response of a wheel speed sensor using an adaptive line enhancer. Measurement
**2003**, 33, 229–240. [Google Scholar] - Hernandez, W. Improving the response of a wheel speed sensor by using frequency-domain adaptive filtering. IEEE Sensors Journal
**2003**, 3, 404–413. [Google Scholar] - Hernandez, W. Robust multivariable estimation of the relevant information coming from a wheel speed sensor and an accelerometer embedded in a car under performance tests. Sensors
**2005**, 5, 488–508. [Google Scholar] - Hernandez, W. Improving the response of a rollover sensor placed in a car under performance tests by using a RLS lattice algorithm. Sensors
**2005**, 5, 613–632. [Google Scholar] - Hernandez, W. Improving the response of a wheel speed sensor by using a RLS lattice algorithm. Sensors
**2006**, 6, 64–79. [Google Scholar] - Hernandez, W. Improving the response of a load cell by using optimal filtering. Sensors
**2006**, 6, 697–711. [Google Scholar] - Hernandez, W. Wheel speed sensors. In Encyclopedia of Sensor; Grimes, C. A., Dickey, E. C., Pishko, M. V., Eds.; American Scientific Publishers, 2006; vol. 10, pp. 461–472. [Google Scholar]
- Hernandez, W. Optimal estimation of the acceleration of a car under performance tests. IEEE Sensors Journal
**2007**, 7, 392–400. [Google Scholar] - Hernandez, W. Optimal estimation of the relevant information coming from a variable reluctance proximity sensor placed in a car undergoing performance tests. Mechanical Systems and Signal Processing
**2007**, 21, 2732–2739. [Google Scholar] - Johnson, M. Photodetection and measurement: maximizing performance in optical systems; McGraw-Hill: New York, 2003. [Google Scholar]
- Pallàs-Areny, R.; Webster, J. G. Sensors and signal conditioning; John Wiley & Sons: New Yok, 2001. [Google Scholar]
- Burr-Brown Application Bulletin: Photodiode Monitoring with Op Amps, SBOA035 (AB-075). Available from: < http://focus.ti.com/lit/an/sboa035/sboa035.pdf>.
- Hernandez, W. Photometer circuit based on positive and negative feedback compensations. Sensor Letters
**2007**, 5, 612–614. [Google Scholar] - Hernandez, W. Linear robust photometer circuit. Sensors and Actuators A
**2007**. Accepted for publication. [Google Scholar] - Hernandez, W. Performance analysis of a robust photometer circuit. IEEE Transactions on Circuits and Systems-II
**2007**. Accepted for publication. [Google Scholar] - Hernandez, W. Robustness and noise voltage analysis in two photometer circuits. IEEE Sensors Journal
**2007**, 7, 1668–1674. [Google Scholar]

**Figure 8.**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.

**Figure 9.**Response of the robust photometer circuit to a step disturbance at the output of the photodiode.

© 2008 by MDPI Reproduction is permitted for noncommercial purposes.

## Share and Cite

**MDPI and ACS Style**

Hernandez, W.
Input-output Transfer Function Analysis of a Photometer Circuit Based on an Operational Amplifier. *Sensors* **2008**, *8*, 35-50.
https://doi.org/10.3390/s8010035

**AMA Style**

Hernandez W.
Input-output Transfer Function Analysis of a Photometer Circuit Based on an Operational Amplifier. *Sensors*. 2008; 8(1):35-50.
https://doi.org/10.3390/s8010035

**Chicago/Turabian Style**

Hernandez, Wilmar.
2008. "Input-output Transfer Function Analysis of a Photometer Circuit Based on an Operational Amplifier" *Sensors* 8, no. 1: 35-50.
https://doi.org/10.3390/s8010035