The response of a pellistor sensor to flammable gases is linear at levels of around 60% of the lower explosive limit (LEL). The deviation of this linearity at gas levels close to the LEL is indicated as reduced signal per %LEL. Therefore, sensors are regularly subjected to performance tests for evaluating their linearity over the operating range.
Due to this nonlinearity at higher gas levels, instruments that can be used up to the LEL may experience issues when subjected to performance tests. It is necessary to consider the following effects for linearity corrections to the response of pellistor sensors at high gas levels.
Diffusion Considerations
The pellistors are normally placed in a flameproof enclosure that also contains a sintered metal flame arrester, which is the first diffusion barrier for the target gas. This gas has to subsequently pass through a second barrier at the entrance to the sensor before reaching the proximity of the sensor beads. Once the gas reaches close to the sensor beads, a depletion layer that develops around the detector surface due to the diffusion of combustion products away from the detector acts as a third barrier.
The rate at which the fuel reaches the catalytic surface and the rate at which the gas diffuses to the catalyst’s surface determine the pellistor response. This rate determining step, r_{d} is expressed in the following Equation 1.

(1) 
Where, D_{12 }is the binary diffusion coefficient, A is the crosssectional area through which the gas diffuses, C is the bulk gas concentration, and x is the distance from the diffusion barrier. When there is no change in the sintered metal and the sensor can, there will be an increase in the thickness of the depletion layer with increasing gas concentration at a fixed ambient pressure.
This results in a nonlinear increase in the diffusion time of fuel to the catalyst surface relating to the concentration of fuel. The fuel concentration as a function of distance diffused by the gas through the depletion layer is expressed in Equation 2:

(2) 
Where, C is the concentration at a distance ‘x’ from the diffusion barrier after a time ‘t’ from an initial number of molecules ‘N’ at time ‘zero’. This decreases the diffusive flux of molecules within the vicinity of the catalyst surface with increasing thickness of the depletion layer. When there are increased effects of external diffusion controlling elements, there will be a reduced effect of diffusion through the depletion layer. Consequently, the signal linearity enhances at high gas levels.
Adsorption Considerations
The speed at which the fuel reacts on a pellistor is diffusion controlled under standard operating conditions. Thus, the reaction at the surface of the catalyst is not usually rate determining, following a heterogeneous mechanism based on the reactants’ surface coverage. The rate of reaction on the catalyst surface becomes the rate determining step at low operating temperatures. The steps involved in surface catalysis are as follows:
 Diffusion of reactants to the catalyst surface
 Adsorption of reactants at the catalyst surface
 Chemical reaction on the catalyst surface
 Desorption of products from the catalyst surface
 Diffusion of products from the catalyst surface
The reaction of adsorbed molecules turns out to be the rate determining step for treating the kinetics of these surface reactions. The rate of reaction per unit surface area changes in relation to the part of a surface covered as expressed by a Langmuir isotherm. The associated reaction rate is described in Equation 3 as follows:

(3) 
Where k is the rate constant; b is the adsorption coefficient; P is the partial pressure of fuel. With increasing fuel concentration, the nonlinearity is introduced by the term (1 + bP).
Constant Voltage Responses
The outofbalance voltage acquired from a sensor bridge configuration supplied with a constant voltage in terms of the resistances of the detector and compensator is explained in Equation 4:

(4) 
Where, ÄV is the voltage signal, i is the current through the detector, ÄR_{d} is the change in the detector resistance; R_{d} / R_{C} is the compensator resistance.
NonLinearity Estimation
The nonlinearity is described in the following Equation 5:

(5) 
Where R is the overall response, K is the normal linear slope of response against fuel concentration (F), and K_{2} is the nonlinear slope contribution. The following equation expresses the reciprocal form of Equation 5:

(6) 
In the above equations, a value is introduced for K_{2} for nonlinear correction. When the K_{2} value is zero, there will be a linear sensor response and also minimal impact of the nonlinear contribution at low fuel concentrations. The value of the K_{2} value can possibly be estimated by quantifying the response at two calibration points of the fuel.
For example, while exposing VQ542R sensors to 50% LEL methane and 100% LEL, the response ratio differs between 1.75 and 1.95, with a linear relationship leading to a factor of 2.0. From these values, the mean signal ratio is around 1.85 (R_{100} = 1.85 x R_{50}) and therefore:

(7) 
Equation 7 results in the following expression:

(8) 
After including the relationship between constants in Equation 8, Equation 6 is rearranged as follows:

(9) 
Where, F is the fuel concentration to be read; R is the sensor response signal; and K is the slope for signal calibration at a single gas concentration.
The effect of using Equation 9 for the VQ542R miniature sensing heads is summarized in the following table:
Gas Conc. (%LEL) 
R (1.75) (mV) 
Linear (1.75) (%LEL) 
[F] calc. (%LEL) 
R (1.85) (mV) 
Linear (1.85) (%LEL) 
[F] calc. (%LEL) 
R (1.95) (mV) 
Linear (1.95) (%LEL) 
[F] calc. (%LEL) 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
10 
17 
11 
11 
16 
11 
10 
15 
10 
9 
20 
33 
22 
21 
32 
21 
20 
30 
20 
19 
30 
48 
32 
31 
47 
31 
30 
45 
30 
29 
40 
62 
41 
41 
61 
41 
40 
60 
40 
39 
50 
75 
50 
50 
75 
50 
50 
75 
50 
50 
60 
87 
58 
59 
89 
59 
60 
89 
59 
60 
70 
99 
66 
68 
102 
68 
70 
103 
69 
71 
80 
110 
73 
77 
114 
76 
80 
117 
78 
82 
90 
121 
81 
85 
127 
85 
90 
132 
88 
94 
100 
131 
87 
94 
139 
93 
100 
146 
97 
106 
Conclusion
Based on the range of response ratios Ri00/R50 from 1.75 to 1.95, Equation 9 can be applied to different e2v technologies sensors. However, the effect of the sensor housing on the linearity of the sensor response needs a different K_{2} value as a result of its contribution to smaller response ratios. In the above example of VQ542R sensors, the effects of sensor housing have been included.
This information has been sourced, reviewed and adapted from materials provided by SGX Sensortech (IS) Ltd.
For more information on this source, please visit SGX Sensortech (IS) Ltd.