ENT 350 Biosensor & Transducer: Topic 4 - Transfer Function Notes

Learning Objectives

• Define the concept of a transfer function and explain its critical importance within sensor systems.

• Describe the various methodologies and formats used to represent a transfer function.

• Analyze the fundamental components of a linear transfer function, specifically focusing on sensitivity and offset.

• Identify and define key static performance characteristics of sensors, including range, resolution, saturation, and hysteresis.

• Distinguish the differences between static and dynamic sensor responses.

• Define the time constant ($\tau$) as it pertains to a first-order system.

• Explain the underlying purpose and the step-by-step basic process of sensor calibration.

Fundamental Objective of Sensing

• The primary role of a sensor is the transduction of a physical, chemical, or biological stimulus into a measurable electrical output.

• Inputs (stimuli) can include variables such as temperature, pressure, or glucose concentration.

• Outputs are typically recorded as electrical signals such as voltage ($V$) or current ($I$).

• The central challenge in sensor engineering is reliably determining the magnitude or state of an unknown stimulus based on a known electrical output.

• The solution to this challenge is establishing the mathematical relationship between the input and output, known as the Transfer Function.

Defining the Transfer Function

• Static Transfer Function: This is denoted by the mathematical relationship $E = f(s)$. It describes how the output electrical signal ($E$) relates to the input stimulus ($s$) when the system is in a steady state (unchanging over time).

• Representation: This relationship can be expressed through several formats, including mathematical equations, graphs, or look-up tables.

• Inverse Transfer Function: In practical applications, the output ($E$) is measured to deduce the unknown stimulus ($s$). This is calculated using the inverse function:     $s = f^{-1}(E) = F(E)$

Ideal Case: Linear Transfer Function

• The linear transfer function is the simplest and most desirable model for a sensor, represented by a straight-line equation:     $E = A + Bs$

• Intercept or Offset ($A$): This represents the electrical output of the sensor when the input stimulus is zero ($s = 0$).

• Slope or Sensitivity ($B$): This represents the degree of change in the output signal relative to a specific change in the input stimulus. The sensitivity is calculated as:     $B = \frac{\Delta s}{\Delta E}$

• Sensitivity Relationship: A sensor characterized by a higher slope (a steeper line on a graph) is considered more sensitive.

A More Practical Linear Model

• In many real-world scenarios, it is impractical or impossible to measure a sensor's response at a true zero input (e.g., reaching absolute zero temperature).

• Practical Calibration: Sensors are often calibrated using a known, non-zero reference point, denoted as ($s_0, E_0$).

• The practical linear transfer function equation is:     $E = E_0 + B(s - s_0)$

• The corresponding inverse function used to determine the stimulus from a measured output is:     $s = \frac{E - E_0}{B} + s_0$

Key Static Performance Characteristics

• Range (or Span): These are the minimum and maximum stimulus values the sensor is designed to measure. Operating outside this defined range means the accuracy of the output is no longer guaranteed.

• Resolution: This is the smallest increment of change in the input stimulus ($s$) that results in a detectable change in the output signal ($E$). It is frequently limited by the resolution of the processing electronics, such as the number of bits in an Analog-to-Digital Converter (ADC).

• Saturation: This occurs when the sensor's output reaches its limit and no longer changes, even as the input stimulus continues to increase. This typically happens when the stimulus exceeds the intended physical or electrical operating range of the sensor.

Hysteresis: An Imperfection in Response

• Definition: Hysteresis is a phenomenon where the sensor's output for a specific input stimulus depends on the direction of the change (whether the stimulus is increasing or decreasing).

• Hysteresis Loop: Because the transfer function for an increasing input differs from that of a decreasing input, a loop is formed when plotted.

• Pressure Sensor Example: A pressure sensor may output $2.1\,V$ for a pressure of $50\,kPa$ when the pressure is rising, but if measurements are taken while the pressure is falling, the same $50\,kPa$ might result in an output of $2.0\,V$.

• This characteristic is a frequent source of error in both mechanical and chemical sensing systems.

Nonlinear Transfer Functions

• Many sensors are inherently nonlinear, particularly when they operate over a broad range. These relationships are often modeled using standard mathematical functions:

  • Logarithmic: $E = A + B\ln(s)$

  • Exponential: $E = Ae^{ks}$

  • Power: $E = A + Bs^k$

Strategies for Dealing with Nonlinearity: Linearization

• In nonlinear sensors, sensitivity is not a constant value; it fluctuates depending on the current input value. Three techniques are commonly used to manage this:

  1. Direct Calculation: This involves using the full nonlinear inverse function. For example, if the function is known, one might solve for stimulus as $s = k\frac{E - A}{B}$. While accurate, this is computationally intensive for the processor.

  2. Linear Piecewise Approximation: This method involves breaking the nonlinear curve into several smaller segments. Each individual segment is then approximated with a straight line. This approach provides a balance between accuracy and computational simplicity.

  3. Look-up Table (LUT): A table of corresponding input ($s$) and output ($E$) values is stored in the device's memory. When the device measures an output ($E$), it searches the table to find the corresponding stimulus ($s$).

Static vs. Dynamic Response

• Static Response: This refers to the sensor's output when the input stimulus is held constant at a steady state.

• Dynamic Response: This describes how the sensor output changes over time specifically in response to a change in the input stimulus.

• Clinical Relevance:

  • A clinical thermometer must have a dynamic response fast enough to provide a reading in seconds rather than minutes.

  • An ECG (Electrocardiogram) electrode must exhibit a rapid response to accurately capture the high-frequency electrical signals generated by the heart.

Characterizing Dynamic Response: The Time Constant

• Many sensors exhibit behavior that can be modeled as a first-order system. When subjected to a sudden "step-like" change in input, their output follows a characteristic exponential curve.

• The Time Constant ($\tau$): This is the fundamental parameter used to define the speed of the dynamic response.

• Definition of $\tau$: The time required for the sensor's output to reach $63.2\%$ of its final steady-state value following a step change in input.

• Response Speed: A smaller $\tau$ indicates a faster-responding sensor.

• Steady-State Rule: The output of the sensor is considered to have reached its final steady-state value (specifically >99\% of the final value) after a duration of five time constants ($5\tau$).

The Process of Sensor Calibration

• Calibration is the experimental procedure used to establish the specific transfer function of a sensor.

• The Calibration Process:

  1. Apply a series of known and highly precise inputs ($s_1, s_2, s_3, \dots$) utilizing a trusted reference standard.

  2. Record each corresponding steady-state output ($E_1, E_2, E_3, \dots$) generated by the sensor.

  3. Plot the resulting data points on a graph of $E$ versus $s$.

  4. Fit a mathematical model (such as a linear equation or a specific curve) to these data points.

• The resulting equation from this curve-fitting process becomes the established transfer function for that sensor.

Summary

• The Transfer Function ($E = f(s)$) is the core mathematical relationship defining the link between input and output.

• The Inverse Transfer Function ($s = F(E)$) is the tool used in practice to calculate an unknown stimulus from a measured electrical signal.

• Linear Sensors are the ideal standard, defined by a constant sensitivity ($B$) and an offset ($A$).

• Real-world Sensors possess specific static characteristics such as range, resolution, saturation, and hysteresis that limit performance.

• Dynamic Response determines the speed at which a sensor reacts to environmental changes, characterized by the time constant ($\tau$).

• Calibration is the essential experimental method used to discover and define a sensor's unique transfer function.