EE3901/EE5901 Sensor Technologies

Chapter 1 Notes

Introduction to Sensors and Measurement Uncertainty

Introduction to sensors

A sensor is a device that measures some property of its environment. Examples of properties that we can measure include:

• Force
• Distance
• Speed
• Sound pressure level (loudness of audio)
• Chemical composition
• … and many more.

Discussion Question: What are other types of sensors can you think of?

All sensors are energy converters. Every measurement involves a transfer of energy. For example, a photodetector absorbs light energy and converts it into electric current. In this example, there is a continual transfer of energy from the light source to the sensor. However, in other cases, there may be an equilibrium condition where the net energy flow is zero. For example, consider a temperature sensor that is initially at room temperature and is then placed into a hot furnace. The temperature sensor must initially absorb heat from the furnace, which is a transfer of energy. Eventually the sensor comes to the same temperature as its new surroundings, at which point an equilibrium has been reached and there is no net flow of energy.

In this subject, we will focus on sensors that respond with an electrical signal as opposed to any other kind of response (e.g. optical, chemical, thermal, etc). For our purposes, an electrical signal is a voltage, current, charge, resistance, capacitance, or inductance. We will learn how to design interface circuits to make these electrical signals accessible to downstream electronics such as an analog-to-digital converter. The goal of these interface circuits is to amplify, convert, transmit, and ultimately make this electrical signal available in a more convenient and robust manner for use by another circuit.

Measurements

When discussing measurement, it is important to distinguish between two related quantities:

• The measurand is the physical property that we seek to observe, for example, temperature, pressure, flow rate, etc. The measurand is the “true value” that exists in the world. When we perform a measurement, we are trying to learn some information about this true value.

• The measurement is the result of using a sensor to observe the measurand. The measurement is always subject to some uncertainty. Hence we should always think of the measurement as merely being an estimate of the measurand.

If the measurand is denoted then the measurement is written , where the ‘hat’ indicates that it is an estimate of the true value.

The transfer function

The transfer function the mathematical relationship between the sensor stimulus and the sensor output , written as . For example, the sensor stimulus might be the temperature and the response might be a voltage.

The transfer function is used to convert the sensor response into the actual measurement, e.g. . Recall that the ‘hat’ on the indicates that it is an estimate of the true measurand .

Example of a transfer function

A thermocouple is a type of temperature sensor which produces a voltage

where is the temperature, and the coefficients , and are constants that depend upon the types of metals used in the thermocouple.

You could solve this equation (using the quadratic formula) to obtain the measured temperature .

Digital vs analog sensors

An analog sensor outputs a continuous electrical quantity such as voltage, current, resistance or capacitance. An digital sensor always chooses one output state at a time from a fixed set of possibilities. Very often this will be a signed or unsigned integer of a given number of bits. Recall that an n-bit integer has possible values.

Sensor characteristics

Sensitivity

Sensitivity is how large the response is for a given change in the measurand. It is the slope of the transfer function, as shown in Figure 1. Precisely, given a transfer function , the sensitivity is defined as

where is the derivative of the transfer function. Notice that the sensitivity is evaluated at a specific value . If the transfer function is non-linear then the sensitivity will vary depending upon the value of the measurand.

The units of sensitivity will be (electrical quantity) per (physical quantity). For example:

• A displacement sensor may have a sensitivity of 10 V/mm.
• A pressure sensor may have a sensitivity of 80 mV/kPa.

For digital sensors, the sensitivity relates to “counts” per physical quantity or “least significant bits” (LSB) per physical quantity e.g. a light sensor with 50 counts/lux. You will also see this written as 50 digits/lux or 50 LSB/lux. This means the digital sensor’s output integer rises by 50 for every increase of 1 lux.

Resolution

The resolution is the smallest change in measurand stimulus that can be accurately detected.

For analog sensors, this is often limited by noise. A small change may be undetectable if it is indistinguishable from noise.

For digital sensors, assuming no problems of noise, the resolution will be defined by the sensitivity. Specifically,

The sensitivity is how many LSBs there are per unit of measurand; the resolution is much measurand there is per LSB.

Span or range

The full span is the range of measurands that can be accepted by the sensor. The full span is also called the range of a sensor. This property is illustrated in Figure 2.

Often the maximum of the range is caused by physical limits of the underlying sensor, e.g. a pressure sensor will have a maximum rated pressure that it can withstand. The minimum value is often limited by the sensor’s resolution, but could also be affected by physical limits, for example, a temperature sensor may not operate at extremely low temperatures.

Error, bias, precision and accuracy

A measurement error is when the measured value differs from the measurand. In practice there is always some degree of measurement error because no sensor system can ever be perfect.

There are several ways to define error. In simplest terms, the error is the raw difference between the measured and true value:

The error has the same units as the measurement. A positive error means that the measurement is too large, and a negative error means that the measurement is too small.

The absolute error is the absolute value of the error:

The relative error is scaled in proportion to the magnitude of the true value:

The relative error is dimensionless, and is often represented as a percentage. In this case it can be called a percentage error. Some authors will take the absolute value of the relative error so that it is always a positive number.

The error is a property of a single specific measurement. However, it is often useful to “zoom out” and discuss the statistical properties of measurement errors that occur over multiple uses of the sensor. Let us define measurement as a probabilistic process. For instance, let

represent the probability of measuring given a particular true measurand . In practical systems, we will often assume that follows a Gaussian distribution, which allows for convenient mathematical analysis of measurement uncertainty (as we will see in subsequent weeks). A sketch of this measurement probability is given in Figure 3.

We define the key properties as follows.

Bias, also called trueness, is a constant offset in the sensor response. In the probabilistic interpretation of measurement,

where is the expected value of . Notice that bias can be corrected for because it is a simple constant offset. The bias can be estimated (e.g. through a calibration procedure where the same measurand is measured repeatedly), and the correction can be included in the transfer function.

Suppose you have a sensor with a simple linear transfer function of the form where is the sensitivity. For example, a differential pressure sensor may have a sensitivity of 1 mV/kPa, giving a transfer function in SI units of where is voltage and is pressure. Any detected bias could be incorporated by modifying the transfer function to read where is the bias.

Precision is how much information is gained from a single measurement. If there is a lot of measurement noise then a single measurement would have a wide range of uncertainty, and would be less precise. A formal definition is

The key point is that a precise sensor always gives the same result, regardless of whether that result is true.

However, the word ‘precision’ is also used to refer to the resolution of a measurement, especially when specifying the number of significant figures. It is important to be aware of the difference between limited resolution and random measurement noise.

The term accuracy is often used when discussing measurement. Unfortunately, there are several different definitions of accuracy, and its meaning is not always clear. Probably the most common definition of accuracy is that it means the same thing as ‘bias’. In this definition, an accurate sensor is one where the expected value of the measurement is close to the measurand. Notably, the sensor can be accurate (have low bias) even if it is not precise (i.e. not repeatable).

Another common meaning of ‘accuracy’ is as a qualitative label of how ‘good’ a sensor is. In this case, both bias and precision are important.

Finally, ‘accuracy’ is also sometimes used discuss a specific measurement (as opposed the entire sensor). In this case accuracy can be synonymous with ‘error’.

Noise

Noise is an unwanted signal that interferes with a measurement. The most common mathematical model of noise that we will consider is additive white Gaussian noise (AWGN).

Define to be the measurement obtained from a sensor system. In the presence of AWGN, this measurement is given by

where is a hypothetical noiseless measurement, and is a random variable drawn from a Gaussian distribution. We can consider as the “true value” that would have been measured at that instant if there were no noise. In practical scenarios, and cannot be determined independently; all we can do is calculate statistical properties of the measurement and its noise.

Importantly, the distribution of the noise has zero mean. If you find this objectional, consider the following. Suppose that there were a physical process that resulted in the additive noise term having a non-zero mean. Such a process could be treated as a bias and simply subtracted away. Once the bias is subtracted off, the remaining component necessarily has zero mean. Hence we can always apply a calibration process such that the only remaining noise is that with zero mean. Overall, the purpose of the noise model is to account for physical processes that cannot be removed by calibration.

The terminology “additive white Gaussian noise” indicates several essential properties of this type of noise:

1. It is “additive”, meaning that the noise is mathematically added to the underlying signal, and hence linear relationships are preserved.
2. It is “white”, meaning that it has no frequency dependence. In other words, it has a flat power spectrum across the entire measurement bandwidth. This implies that the noise cannot be easily filtered out with a suitable high-pass or low-pass filter.
3. It is “Gaussian”, meaning that in the time domain, its values are sampled from a Gaussian (aka normal) distribution.

Signal to noise ratio (SNR) and dynamic range (DR)

The signal to noise ratio (SNR) is a measurement of the power in the signal to the power in the noise. Generally if there is a large SNR then obtaining high precision measurements is straightforward, whereas if there is a low SNR then the measurement becomes more challenging and more subject to uncertainty.

The SNR is often expressed in decibels:

where refers to average power and refers to RMS magnitude (e.g. voltage or current). The factor of 20 arises because so regardless of whether is voltage or current.

It is common to plot the transfer function on log-log axes when analysing signal to noise ratios. This is because of the algebraic identity , which means that on logarithmic axes the SNR is proportional to the distance between a measurement point and the noise floor, as illustrated below in Figure 4.

The dynamic range (DR) is the ratio between the largest and smallest values of the measurement:

again where P is power and M is RMS magnitude.

In cases where the minimum is 0 (e.g. absolute scales like light intensity, sound pressure, etc), the lower value is the noise floor. In this case the DR is simply the SNR for the largest possible measurand:

For reference, human hearing has a DR of roughly 140 dB, and human eyesight has a DR of roughly 90 dB.

Hysteresis

Hysteresis is a history dependence, meaning that different measurement results can be obtained despite the measurand being the same. Specifically the measurement depends upon the recent history of the sensor (as shown in Figure 5).

There are several reasons for hysteresis, for instance:

• Slow response time, so that the output is a weighted average over its recent history.
• Temperature changes in a sensing element.
• Backlash in gears, for instance in a rotation sensor that uses gears to couple to the axle being monitored. When the direction of rotation changes, there will be a small amount of movement in the new direction before the gear teeth ‘bite.’ This is called backlash, and will cause the rotation sensor to display hysteresis because the position of the sensor will be slightly offset depending upon the direction of rotation.
• Other chemical or physical properties changing in the sensing element in response to the environment being measured.

Hysteresis is sometimes deliberately introduced to avoid rapid switching near a threshold. A circuit called a Schmidt trigger is sometimes used for this purpose.

Response time

Sensors do not respond immediately to an input stimulus.

The response time or rise time is the time required to reach a given threshold, typically 90% of the final value. If the sensor’s underlying physics results in an exponential response () then the rise time is often defined to be the time constant , which is the time required for the response to rise by . These types of exponential responses are common because they arise from first order differential equations, including for example RC and RL type circuits.

Bandwidth

If the measurand is time-varying, then it is important to consider whether the sensor can respond quickly enough to keep up with the system being measured. The ability of a sensor to track a changing measurand is determined by its bandwidth.

To formally define bandwidth, we consider the frequency response of the sensor system. Much like you can analyse the frequency response of a circuit such as a low-pass filter, so too can you define the frequency response of a sensor. The frequency response of a sensor gives the measurement response when the measurand input is a sinusoid of a given frequency. The frequency response of a typical sensor system is sketched in Figure 6.

Sensors typically have low-pass characteristics, i.e. there is some maximum frequency at which the output voltage or output current starts to drop. This defines the bandwidth of the sensor. Precisely, we define the bandwidth to be point at which the output power has dropped by half. Recall that half power corresponds to a drop of 3 dB.

Measurement is a statistical estimation problem

Recall that the central problem of sensing is to use measurements to obtain information about the measurand. This is fundamentally a question of statistical estimation. If the measurand is not changing, then each measurement is a random variable drawn from the same distribution. The randomness represents the measurement noise. (We will consider the case of a time-varying measurand in the next chapter.)

Review of probability and statistics

We will typically use a Bayesian interpretation of probability, i.e. probability represents a degree of belief. The probability reflects the degree to which a statement is supported by the available evidence.

Expected value

The expected value is the most likely outcome. In the context of measurement, it is the value that we believe best represents the true measurand based upon the available evidence.

Mathematically, it is defined as follows. Let be a random variable defined by its probability density . The expected value of is

Adjust the limits of integration if the probability density is defined over a different range. We will often write the expected value with a bar, e.g. .

Sometimes we want the expected value of some calculated result instead of the expected value of the measurement itself. In this case the expected value of some function of X is given by:

For a finite sample of measurements, the expected value is just the average computed in the usual way:

Variance and standard deviation

The variance represents how far samples are from the mean. There are two related quantities: is called the variance, and has units of the measurement squared. The standard deviation (with the same units as the measurement) is called . The standard deviation is the square root of the variance.

The definition of variance is:

Translated into words, this indicates the “average of the squared distance from the mean”.

For an entire population the variance can be calculated using

Note that the above definition applies to an entire population. Most often in statistics we deal with a finite sample drawn from the larger population, in which case the correct (“unbiased”) estimator of variance is

The proof of this estimator is outside the scope of this class, so refer to a statistics book for more details. The idea of the proof is to treat as a random variable (since it depends upon a random sampling of the broader population), then find the expected value of that random variable. The resulting algebra gives the correction factor of .

Visual illustration of the mean and standard deviation

The impact of the mean and standard deviation can be explored using interactive Figure 7. This figure plots the normal distribution (also called the Gaussian distribution), which has probability density function

where is the expected value and is the standard deviation.

Drag the sliders to adjust the mean and standard deviation.

Correlation

Correlation is the tendency of two variables to exhibit a linear relationship. Mathematically:

Correlation is always in the range . If then the variables are said to be uncorrelated.

It is best explained visually (Figure 8). Drag the slider to see different levels of correlation.

Covariance

Covariance is similar to correlation but not normalised. While correlation is a dimensionless number between and , covariance has units and there is no upper or lower limit.

Let be the covariance between variables and . Covariance is also written as . The definition of covariance is as follows:

Equation (25) shows that covariance is equivalent to the correlation multiplied by the standard deviation of each variable. The relationship to correlation provides an intuitive explanation, as shown in Figure 9. When two variables have nonzero covariance, the area of overlap is reduced, overall allowing for better precision in the joint measurement of the entire system.

Note that the covariance of a variable with itself is the variance:

Since the covariance is defined between pairs of variables, it is convenient to list all pairwise combinations in a matrix. For instance, given two variables and , the covariance matrix is of dimensions and is given by:

This matrix is symmetric because .

In the general case of an arbitrary number of random variables, we can form the covariance matrix as follows. Firstly define a vector containing the variables:

and then the covariance matrix is

Visual illustration of the covariance matrix for a 2D Gaussian distribution

Figure 10 shows a two-dimensional Gaussian with the specified correlation coefficient and standard deviations and .

Error propagation

Defining error

There are two ways that we can define error:

1. Absolute error giving a range of possible values, e.g. V, without specifying the likelihood of values within that range. The limits can be thought of as giving worst case scenarios.
2. Error characterised through a probability distribution, for example, as a normal distribution with a given standard deviation. The probability distribution gives the relative likelihood of each amount of error. We will typically use this approach.

Introduction to error propagation

Suppose that you perform a measurement and obtain a result , however, this result is subject to uncertainty. You would like to use that measurement in some arbitrary calculation . The problem of error propagation is to calculate the uncertainty in the calculated result given the uncertainty in .

An intuitive picture is shown in Figure 11. The measurement has error bars shown by the standard deviation . Our task is to calculate .

Derivation of the variance formula

Let be a random variable with expected value and variance . Let be a calculated value based on . Notice that is also a random variable.We seek an expression for the variance .

Suppose we perform a measurement and obtain the result . Take a Taylor series expansion of about .

The notation means take the derivative then substitute . In other words, this quantity is just a number. Let so that .

Our goal is to find . We need an expression for .

Consequently,

Therefore

This is the formula for propagating variance.

Variance propagation with multiple variables

Let be random variables (e.g. from different sensors).

Represent these variables as a column vector .

Let the expected values be .

Let the covariance matrix of these variables be .

Define some calculation .

If is a vector then we seek the covariance matrix . If is a scalar then is a 1x1 (i.e. a scalar) giving the variance .

Derivation

Let be a measurement result.

Define the Jacobian as:

where the measurement result is substituted into the Jacobian, so this is a matrix of numbers. If is a scalar then the Jacobian is a row vector.

Then the Taylor series expansion of about the point is

Notice that the matrix product produces all the first order terms in the multivariate Taylor series.

Proceeding as above, we find

Hence

Hence the covariance matrix is

To summarise, the variance propagation law for a multivariable function is

where is the Jacobian matrix of the calculation evaluated at the measurement result, and is the covariance matrix of the measurements.

Root sum of squares error vs absolute error

There is a simplification to the above method in the case that the variables are uncorrelated. The simplification (which you will prove in the tutorial questions) is as follows:

If then the uncertainty in is given by: