Training Sensors & Measurement Math Sensor Transfer Functions
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Sensor Transfer Functions

24 min Sensors & Measurement Math

Sensor Transfer Functions

Every sensor converts a physical quantity (temperature, pressure, light) into an electrical signal. The mathematical relationship between input and output is the transfer function.

Linear Transfer Function

$$V_{\text{out}} = mX + b$$

where $X$ is the measured quantity, $m$ is the sensitivity (slope), and $b$ is the offset.

Common Sensor Types
  • Thermocouple: Voltage proportional to temperature (approximately linear)
  • Strain gauge: Resistance changes linearly with strain
  • Thermistor: Resistance changes exponentially with temperature (nonlinear)
  • Photodiode: Current proportional to light intensity (linear)
Example 1

A temperature sensor outputs 0.5 V at 0°C and 2.5 V at 100°C. Find the transfer function and predict the output at 37°C.

Slope: $m = (2.5 - 0.5)/(100 - 0) = 0.02$ V/°C.

Intercept: $b = 0.5$ V.

$V = 0.02T + 0.5$.

At 37°C: $V = 0.02(37) + 0.5 = 1.24$ V.

Example 2

A sensor reads 3.1 V. Using $V = 0.02T + 0.5$, find the temperature.

$T = (V - 0.5)/0.02 = (3.1 - 0.5)/0.02 = 130$°C.

Example 3

A thermistor has $R = R_0 e^{\beta(1/T - 1/T_0)}$ with $R_0 = 10{,}000$ Ω at $T_0 = 298$ K and $\beta = 3950$. Find $R$ at $T = 373$ K.

$R = 10000 \cdot e^{3950(1/373 - 1/298)} = 10000 \cdot e^{3950(-0.000673)}$

$= 10000 \cdot e^{-2.66} \approx 10000 \times 0.070 = 700$ Ω.

Practice Problems

1. A pressure sensor: 1 V at 0 psi, 5 V at 100 psi. Find the transfer function.
2. Using that function, what pressure gives 3.4 V?
3. A sensor has sensitivity 0.05 V/°C and offset 0.2 V. Output at 60°C?
4. Is a thermistor transfer function linear or nonlinear?
5. A strain gauge has $R = 120(1 + 2\epsilon)$ Ω. Find $R$ when $\epsilon = 0.001$.
6. What is sensitivity in the context of a sensor transfer function?
Show Answer Key

1. $V = 0.04P + 1$

2. $P = (3.4 - 1)/0.04 = 60$ psi

3. $V = 0.05(60) + 0.2 = 3.2$ V

4. Nonlinear (exponential)

5. $R = 120(1 + 0.002) = 120.24$ Ω

6. The slope $m$ — how much the output changes per unit change in input.

The Decibel Scale and Signal Levels