A New Electro-Optical-Thermal Modelling for Non-Dispersive IR Sensing Technique of Gas Concentration

Abstract

In this research, a new electro-optical-thermal modeling is proposed and built by simulation program with integrated circuit emphasis (SPICE). In particular, it is constructed for use in the non-dispersive infrared (NDIR) sensing technique of gas concentration. This model, based on the theory of circuitry and the Beer-Lambert law, includes various equivalent elements for the optics, sensor, and circuits. To build and investigate the validity of the proposed model, an NDIR for measurement of CO₂ is built with the hybrid combination of a thermopile sensor with a specific wavelength filter, an infrared micro electro mechanical systems (MEMS) heater, an optical tube, amplification circuits with a chopper amplifier, advanced RISC machine (ARM)-based micro processing unit and discrete electronic devices. The thermal properties of the light source with periodic modulation have been studied from the output signal of a thermopile within the limit of modulation frequency. Based on the thorough measurements of output signals and transient responses, the thermal and optical parameters of the sensor and optical components for this model are extracted. The comparison of the simulation and experimental data of the NDIR measurement for different CO₂ concentrations shows a great agreement with a maximum error of 0.27% at 3500 ppm. This approach allows for the development of a high-level sensor and circuit integrated simulation based on the most fundamental principles and multiple variables.

1. Introduction

NDIR is a reliable gas sensing technology that is used for measuring gas concentration [1]. The theory behind it is when IR strikes a gas, the energy states of atoms vibrating in molecules will transform into separate steps if the wavelength meets the requirements. Therefore, we can infer gas concentrations by measuring IR absorption at the wavelengths ranging from 3 to 10 μm.

The most important component of an NDIR instrument is the IR sensor. The heating light source emits IR and the gas in the chamber absorbs certain wavelengths of lights. Then the IR sensor detects the attenuation of these wavelengths and refers to the gas concentration based on the Beer-Lambert law.

A downside of light source operation is the searing heat of their performance over 1500 K under direct modulation that further enables massive power radiation to be released in a short-wave band. Moreover, we need to calculate complex equations to obtain the thermal behavior of the light source. Special surface structure modeling reveals a substantial dependence of emissivity on radiation wavelength. This allows the radiation spectra of the source to be adjusted to the absorption band of the gas whose concentration is being evaluated [2]. Furthermore, those studies provide an alternative technique to research modulation techniques and various sorts of light sources to sustain the operating temperature within an appropriate range [3].

The development of simulation analysis of electronic circuits is based on the improvement of algorithms, component models, and physical characteristics. It plays an important role in studying and analyzing electrical properties in circuit design. However, there is still a lack of integrated analysis and simulation technology due to the complexity of optical, mechanical, and electronic circuit integration systems.

Besides electric properties, the thermal effects on various semiconductor devices are considered and equivalent circuit models were developed for the analysis using SPICE circuit simulation software [4,5,6,7,8]. Comparisons of theoretical formulations have been made and electro-thermal models of temperature sensors were built by various researchers [9,10,11].

In this article, we use SPICE to build a highly integrated optical-electrical-thermal modeling for NDIR. This model is based on the extraction of component parameters and the derivation of the characteristics of thermal radiation. Therefore, the proposed model can calculate and analyze the transient response of NDIR photoelectric thermal behavior under different modulation conditions. The actual experimental results [12,13] are also in good agreement with the simulation results.

2. Modeling and Theory

2.1. Application of Beer-Lambert Law and Thermal Modeling to NDIR Response

By combining a narrow bandpass filter, a detector, an IR illumination, and a gas chamber, we can create an optical system for NDIR gas sensors; also, these components impact the output signal of IR.

The corresponding relations between the electrical model and thermal model can be revealed by the following heat equation. Changes in temperature over time for IR light sources are described with a simple heat transfer equation including the electrical heating power and heat dissipations.

$$\frac{dT}{dt}=\frac{1}{c\rho s}{p}_{el}(t)-\frac{1}{c\rho s}k(T-T_0)-\frac{1}{c\rho s}\epsilon \sigma (T^4-T_a^4)$$

(1)

where c is the specific heat, ρ it’s the density, s is the surface area of the light source wire which is also described in Figure 6, k is the heat transfer coefficient, ε is the emissivity, σ is the Stefan-Boltzmann constant. Electrical heating power P_el(t), dissipated for light source, is described as a function of the supplying voltage U(t) and resistance R(T) of the light source changed with the temperature T, the initial temperature as T₀, the room temperature as T_a.

$$p_{el}(t)=\frac{U^2(t)}{R(T)}$$

(2)

Furthermore, the corresponding relationships between the electrical model and the optical model must be developed on the basis of the Beer-Lambert law, which characterizes the absorptivity of IR by a concentration of gas (C) and absorption length (d), as illustrated in Figure 1. A broadband infrared source creates IR, the gas in an optical tube’s test chamber absorbs IR at gas-specific wavelengths, and an IR detector at the cell’s end monitors residual radiation strength. A narrow bandpass gas-specific filter is positioned on the front of the sensor to measure the concentration of the target gas. In consequence, as the received IR light intensity (I) declines, so does the signal level of the IR sensor. While we take a look at the Stefan-Boltzmann law, we can find out that the fourth power of the temperature (T) is in direct proportion to the total power of the radiation received on the IR sensor per time (t), and the formulation for the sensor’s net flux is given as (3). d(θ) is the effective length of the well-controlled gas chamber matching to the angle θ, and λ is the gas absorption coefficient. A(θ) and B are the geometrical parameter factors for the IR illumination and detector, respectively. T_b is called the working temperature, which is the temperature of the infrared light source after heating. A(θ) is the geometrical parameter factor including the effective luminous surface area of the infrared light source, optical path of the gas chamber and the filter of the sensor. B is the geometrical parameter factor detector which keeps at the T_a. Because T_b⁴ is much larger than T_a⁴, the product value of T_a⁴ term is ignored. Since the radiation from the source far outweighs the sensor, (3) may be rewritten and simplified in (4). The angle from 0 degrees counterclockwise to the optical is θ degrees.

$$I(t)={\displaystyle \int A(\theta ){\epsilon }_b\sigma T_b^4(t)\times e^{-\lambda Cd(\theta )}d\theta -B{\epsilon }_a\sigma T_a{}^4}$$

(3)

$$I(t)=A{\epsilon }_b\sigma T_b^4(t)\times e^{-\lambda Cd}$$

(4)

2.2. Periodic Modulation of Irradiation Architecture

To achieve the effective operation of NDIR measurement, the modulated heating of the light source and the corresponding signal acquisition are arranged. Based on the periodic modulation of heating and the average working temperature of the source, it is necessary to derive and build a new theoretical model and transient response for NDIR gas detection. The characteristic behaviors of the system are described and illustrated in Figure 2.

The heating of IR light source produces IR passing the optical detection tube and part of radiation impinges on the thermopile through filter. At the heating phase, the temperature rise for the light source reaches a stable working temperature, and at the subsequent cooling phase, the temperature fall will drop to the ambient temperature. For each cycle of the modulation, the time for the cooling and heating is denoted as t_c and t_h are the durations for the cooling and heating phase, respectively, as shown in Figure 2. The modulation of heating is conducted by an ARM-based microprocessor which controls a N-type Power MOS as a switch with on and off settings, as shown in Figure 3. An NDIR CO₂ measurement system was developed and set up in a CO₂ gas chamber to evaluate the feasibility of our proposed NDIR electrothermal modeling. The measurement module consists of a light source at one end of a 5 cm long optical tube and an IR sensor at the other end of the optical tube, as shown in Figure 3. An NDIR CO₂ measurement system was developed and set up in a CO₂ gas chamber to evaluate the feasibility of our proposed NDIR electrothermal modeling. The measurement module consists of a light source at one end of a 5 cm long optical tube and an IR sensor at the other end of the optical tube, as shown in Figure 3. At the other end of the optical tube, there is a thermopile infrared detector. The sensed voltage is input to the electrical module for calculation and comparison. The result is input into the microcontroller for feedback control of the heating of the infrared light source.

2.3. Working Principle Based on Periodic Modulation

The thermal conductivity and temperature of the infrared light source can be derived as G and T_b, respectively. G is the thermal conductivity, and the thermal energy of the infrared light source is represented by the product of the temperature difference and the thermal conductivity. H is the heat capacity, and the hot wire obtains the heat energy to generate the capacity of heating up. As shown in (5).

$$H\frac{d{T}_b}{dt}+\mathrm{G}(T_b-T_a)+A_b{\epsilon }_b\sigma T_b^t=P_e$$

(5)

According to Fourier’s law, we can state the temperature variations in the heating and cooling processes as ΔΤ_r and ΔΤ_f and the heating and a cooling period can be derived as t_h, t_c in (6) and (7), separately.

$$\Delta T_r=\frac{P_e{t}_h}{H}-\frac{\mathrm{G}\left(T_b-T_a\right)t_h}{H}-\frac{A_b{\epsilon }_b\sigma T_b{}^4{t}_h}{H}$$

(6)

$$\Delta T_f=-\frac{\mathrm{G}\left(T_b-T_a\right)t_c}{H}-\frac{A_b{\epsilon }_b\sigma T_b{}^4{t}_c}{H}$$

(7)

After m times of heating within N times cycles, the working temperature of IR light source can be expected to reach a stable value followed by $m\cdot \Delta T_r+(N-m)\cdot \Delta T_f=0$. The working temperature can be approximated by the following expression in (8).

$$T_b\cong {\left(\frac{m}{N}\frac{P_e}{A_b{\epsilon }_b\sigma }\right)}^{1/4}$$

(8)

The Beer-Lambert law serves as the foundation for the NDIR working concept. The light intensity received by the IR detector, as well as the radiation of the light, would be transmitted to an electronic output, which can be equated with V_th in (9). That means the output signal will be a complex function of the concentration of the gas, which is denoted as C. There are two coefficients related to the amplification factor and offset of the circuit, denoted as X and Y. S is the Seebeck coefficient which describes the transfer of temperature rise to voltage.

$$V_{th}=XS\left(A\frac{m}{N}\frac{P_e}{A_b}\times {\mathrm{e}}^{-\lambda Cd}-B{\epsilon }_a\sigma T_a{}^4\right)+Y$$

(9)

3. NDIR Design Optical-Electrical-Thermal Modeling

A new IR optical-electronic model is built on the NDIR sensing concept, and the novel method is validated by simulated and measured results. The corresponding elements between the thermal and electric characteristics explain this SPICE simulator-based model. The flow chart for establishing the electro-optical-thermal model is shown in Figure 4. Essentially,

Table 1. Correspondence between the characteristics of the electrothermal model and the electrical equivalent parameters.

Electrical Parameters	R2(V/A)	C1(AS/V)	V(t)(V)	I(t)(A)
Thermal parameters	Zt (K/W)	H(J/K)	T(t)(K)	P(t)(W)
Extracted Values	271.5 kΩ	22.182 μF	1700 V	80 mA

presents the characteristics of the known electro-thermal prototype as well as the matching association among the electrical equivalent parameters.

In the simulation, for instance, the heating power is substituted by electricity, implying that current is injected or discharged to the branching with heating or heat ventilation behavior identical to the IR illumination. Introducing multiple ABM elements, including GVALUE and EVALUE, plays a vital role in radiation heat dissipation, simulating inner radiation, and the input voltage variation of the temperature sensing, which is supplied by the radiological transfer function to the voltage signal.

In Figure 5, the IR illumination is represented by R15, in which the resistance at ambient temperature is 5.1 and the resistance at 1700 K is 35.8. In addition, a resistor R5 linked in series with the resistance, 5.1. The IR illumination is powered by an N-type metal-oxide-semiconductor, and the driving circumstances are set up to mimic, with the findings in comparison to experimental data. The current produced via a particular zero-bias DC voltage source will be delivered to the following phase in the construction of an IR light source with an electrical heating effect.

Figure 5 depicts the prototype of an IR illumination used in the modeling of electro-optical-thermal evaluation based on (5). GVALUE G2 calculates and converts the working temperature to heating power, and the conduction heat losses and radiation losses are shown by R2 and G1, respectively. The heating power that results from the Joule-Thomson effect is determined by the input voltage difference of G2, which represents the increase in temperature of the IR illumination. It is significant to mention that the sense of the injection current indicates the addition of heating power. The thermal resistance of the IR light source is substituted by capacitor C1, and the values of C1 and R2 are approximated using material and geometrical variables.

A GVALUE G1 component causes the dissipation of radiation heat loss. Furthermore, it is defined by Stefan-Boltzmann’s equation and takes the ground as a reference, which indicates that the temperature is absolute zero. The GVALUE G1 is utilized to compute the infrared wave diversity ratio to the corresponding characteristic equation. As illustrated in Figure 5, the secular equation of G1 considers numerous factors of heat radiation loss, which includes the active area of emissivity and tungsten wire, with emissivity $\epsilon$ set to 1.0.

The formulation of El derives from a series of signal conversions that represents the receiving IR strength out from the IR into the ocular pipe, and the sensor’s thermoelectric conversion generates an output signal for the measurement of the thermopile output signal. The surface area of an IR light source is measured by utilizing an optical microscope after measuring and calculating the geometry of the light source. As indicated in Figure 6, 50% of the area is considered as the effective radiation area for outward radiation, and the effective surface area of tungsten is measured and assessed as 9.65⁻⁷ m².

On the basis of the Beer-Lambert law, the absorption coefficient, the gas concentration and the length of the optical tube can all be stated in the formulation of E1.

We can create a Vref by utilizing a voltage divider circuit, which serves as an analogue signal reference, and the output is stated as Vref. The output voltage of the thermopile is generated by the output of EVALUE and magnified by a unity gain buffer AD8552 with reference to Vref voltage, as illustrated in Figure 7.

4. NDIR Sensing Module and Experimental Configuration

As illustrated in Figure 7, the IR light source, OIR-715, was used, with a wavelength range of visible to 4.4 μm, particularly for hydrocarbon detection at 3~3.5 μm, CO₂ detection at 4154.4 μm, and a cut-off wavelength of 5 μm. At the opposite end of the optical tube, there is a thermopile IR detector, HTS-E21-F3.91/F4.26 with a resistance of roughly 30 k. The gas chamber is developed by a high reflectance inner layer to exclude extraneous IR from the wall.

An ultra-low offset chopper amplifier AD8552 amplifies the sensor output signal. The amplified signal is transformed to digital code which uses a 24 bit ADC AD7799 after amplification, and an ARM-based CPU NUC120 is constructed to execute and make computation and calibration under control. The ARM controls a p-type PMOS to turn the current on and off through the heater on and off in order to increase or decrease the temperature to achieve the aim of controlling the IR light source. ARM, which is based on calibration methods and a fitting curve, measures the concentration in this experiment.

5. Simulations and Measurements

Several conditions are set to research the thermal and electrical characteristics of NDIR for CO₂ concentration. In the thermoelectric model, the capacitance component C1 indicates the heat capacity as well as the series resistaence of the heating source using renewable electricity. Hence, we attempt to examine the temporal reaction of the IR piezoelectric component’s output signal throughout the heating process of the hot filament to research the distinguishing features of the hot filament’s increasing temperature. Figure 8 depicts the simulated temperature rise value when the CO₂ concentration is modified to 3000 ppm.

In the modeled transient response, the thermal time constant matching to 63.2% of the rising curve in the research for the heating of an IR light source is 272 ms. The time constant for the IR light source OIR-715 T11 CC6BI (IR Lamp) is 290 ms, according to the technical specifications. The simulation findings demonstrate that the filament heating time constant is similar to the solid IR bulb heating time constant.

The thermoelectric model was used to simulate the thermal response of the IR source under different heating powers, and the simulated data was plotted as a curve as shown in Figure 9. The higher the heating power of the hot wire, the shorter the time to reach thermal equilibrium and the higher the temperature of the heat balance. The temperature here is converted from the output voltage of the analog model. The thermoelectric model is then used to simulate the operating temperature at different heating powers, and the simulated data is plotted as a curve as shown in Figure 10. The results show that the heating power of the hot wire and the working temperature show a polynomial relationship of the nonlinear phenomenon, which reveals a thermal equilibrium including the balance of heating and dissipation including radiation loss.

There are several unknown geometrical and material parameters along the optical path which can be derived effectively by comparing the output voltages from the measurement and simulation. From the curve in Figure 9, we choose the IR light source to establish a proper working temperature at 1700 K, and the output voltage from the thermopile is amplified and recorded in Figure 11. The parameter extracted in EVALUE E1 is achieved from the curve fitting for the measurement of CO₂ concentration at 4000 ppm in Figure 11. The output signal is altered from the analog reference V_ref 1.816 V to 2.4 V. The power consumption and current of the infrared light source simulated at different temperatures are recorded in

Table 2. Power and current values at different simulated temperatures.

Heating Current (mA)	505.24	362.67	236.87	130.54	48.55
Heating power (mW)	0.012	0.087	0.286	0.671	1.302
Equilibrium temperature (K)	1000	2000	3000	4000	5000

. The thermoelectric model was used to simulate the thermal response of the infrared source at different gas concentrations, and the simulated data were plotted as a curve, as shown in Figure 12.

6. Discussion

The simulation results with the actual experimental data are compared with other research studies [12,13]. The measurement of CO₂ concentration is verified by a calibration procedure to build a transfer function between the gas concentration and output voltage, which is shown in Figure 13. It is clear to see that the calibration curve shows good linear approximation behavior. It also reveals that the comparison of simulation and experiment of NDIR measurement for different CO₂ concentrations shows a great agreement with a maximum error of 0.27% at 3500 ppm. The above research results show that the electro-optical-thermal model we set up from the CO₂ measurement parameter extraction can provide high reliability and efficient simulation and analysis of NDIR technology.

In this research, the analysis becomes even more complicated when the response of the heating light source along with its IR sensor and circuits is presented. With this model, we expect to provide a more theoretical, practical, fast, and accurate simulation analysis for different NDIR gas measurement theoretical studies.

7. Conclusions

In this article, we have built a thorough optical-electro-thermal integrated model for NDIR using SPICE. The simulation model is very versatile and with the knowledge of process parameters and operating conditions, and various emitter characteristics which can be understood and optimized. In this paper, the optical-electro-thermal model of an NDIR gas sensor is presented, as well as a simulation approach that computes the transient response of optical-electro-thermal behavior of NDIR under different modulation conditions. It was found that the simulated results are in close agreement with that of the experimental results. The model provides insight and understanding of many physical and dynamic properties and is cost-effective as electrical, optical, and thermal simulations can be performed in the same design environment.