Chapter 1 Measurement basics

1.1 Why do we measure things?

Measurements form the basis of empirical research. The purpose of measurements are usually divided in the following three categories:

• Observe a process : Home temperature, electricity consumption, water consumption, CO level in garage, medical diagnostics. Not actually research and not (at least automatic) control.
• Research : Test theories with measurements (testing a model for plant growth), create experimental models for a relationship between factors for phenomena with no existing theory (predict cattle lameness based on weight distribution during milking).
• Control : we want to automatically control a process given certain parameters. e.g. control the fan speed and inlet openings of a barn based on temperature and \( CO_2 \) measurements.

Fault diagnostics is one popular application of measurements based on observing a process. A machine, animal or a plant is monitored with a sensor and model predicts when there is a failure or disease. E.g the lameness in cattle can be predicted using force sensors, pig cough can be measured using sound analysis and bearing failure in machinery can predicted based on resonance frequency. The models used to predict problems can be very simple based on limit values or more complex models that are fitted to data based on sample dataset(This is called machine learning).

The objectives can also overlap e.g. we may want to also research the process we are controlling. The purpose of the measurement affects its requirements for accuracy and reliability.

1.2 The phases of a successful measurement

1.3 Measurement system components

Figure 1.1: The components of a measurement system

Sensor is a transducer that convert a physical quantity to a measurable analog signal.

ADC (analog-to-digital converter) converts the analog signal to digital

A computer/datalogger/microcontroller is used to store and display the information.

The wires from sensor to ADC should be as short as possible to minimize noise and level changes. Voltage signals suffer from voltage drop when long wires are used due to the resistance of the circuit. Current signals however are a lot less sensitive to interference and don’t have a problem with level changes.

Digital signals can be transmitted without a loss of information with efficient compression (e.g. using the Internet) and are therefore the preferred method for transmitting data.

1.4 Signals

A signal is a description of how one parameter relates to another. Usually signals are measured as a function of time e.g. how temperature changes over time, however they can be also measured as function of e.g. distance or light intensity.

1.5 Analog to digital conversion

Analog-to-digital converterss ( ADC ) work by taking a sample of the original signal and comparing it to a fixed number of known voltage levels. The signal is then rounded to the closest reference level. The resolution of the ADC is given in bits which means that the number of values the digitize signal can get, for \( n \) bit ADC is \( 2^n \) . Each ADC works for a certain voltage range e.g. 0 – 5 V.

• Taking a sample of analog signal for digitization is called the sample-and-hold method.
• Analog signal is rounded to the nearest possible value. The difference between the original value and the digitized sample is called quantization error .
• The difference between adjacent conversion levels is called least significant bit (LSB)
• The maximum quantization error is therefore \( \pm\frac{1}{2} \) LSB.

Suppose we have an 8bit ADC with the range of 0-5V. It can produce \( 2^8 = 256 \) distinct numbers (0-255) so with the range of 5V its voltage resolution is 5/256 \( \approx \) 20 mV. This value is also called LSB .

This means in practice that when digitized analog values 10 – 30 mV are rounded to 20mV and analog values 30 – 50 mV to 40mV etc. The quantization error for analog value 10mV is 20mV - 10mV = \( \frac{1}{2} \) LSB.

Now if you have a temperature sensor with the sensitivity of \( \frac{10mV}{^{\circ}C} \) . then digitization introduces an error of \( \pm 1^{\circ}C \) to the measurement. If you want to increase the precision of your measurement you need to use an ADC with higher resolution . This still doesn’t guarantee that the measurements will be accurate , for that you need calibration .

The voltage resolution of the ADC can be calculated with:

So the voltage resolution for 12bit ADC with -10 – 10V range is:

The number of values we get an ADC determines how accurately we can measure things in a certain range. We can get the best accuracy when use the whole voltage range of the instrument. Usually we need to scale the signal either by amplifying or reduce the original signal to achieve this.

We’ll learn some options for reducing the signals amplitude in the exercise. If you need to amplify a signal you can use operational amplifiers, we are not doing that in the course but you can find the needed circuits from basic electronics books. There are also special amplifiers for common small signals e.g. for load cells and thermocouples.

1.6 Sampling

In addition to getting accurate samples of the voltage we also need to consider the rate of change of the signal. Here are two definitions that help us decide how often we need to sample:

The sampling theorem A continuous signal can be properly sampled, only if it does not contain frequency components above one-half of the sampling rate. (Nyquist criterion, Nyquist-Shannon theorem) .

Proper sampling

If you can reconstruct the analog signal from samples, you must have done the sampling correctly. (Smith 1997).

Note that the highest frequency of interest in a signal is easy to define for periodic signals ( waveforms with recurring frequencies. ). For other signals we need to consider how can we reliably measure the fastest peaks e.g for force measurements and ECG (electrocardiograph) signals.

In practice we usually do not know what is the highest frequency is, and we need to find out:

• A practical way is to sample as fast as you can and determine the required sample rate from the data.

Choosing an appropriate sample rate is also important because using too high sample rate increases the need for data storage, power consumption (especially a concern with wireless devices), analysis time etc.

Signals are often shown in figures as continuous lines because when there are many samples in a figure using a symbol for each sample makes the plot look messy like in Figure 1.2 B. It is however important to remember that lines are not actual measured values and just used to connect the dots (if we give each point in a line a value it is called linear interpolation).

If you are plotting the data to inspect if you have sampled often enough it is a good idea to use a symbol to represent each point and leave the lines out. Also make sure you zoom close enough to the closest peaks so you can really tell how far apart successive samples are.

Figure 1.2 shows a the vertical force influencing a force plate while a University lecturer jumps of it. The data was sampled at 200Hz and then decimated to 40Hz using a computer. The lines in subplots A and C look deceptive similar, but subplots B and C suggest that 200Hz is enough to measure the maximum force, but 40Hz is not quite enough and it would be wise to increase the sample rate to be sure. (The difference between maximum forces is 27N).

Figure 1.2: The force measured when a person jumps of a force sensor represented at at two different sample rates and plot types.

1.6.1 Random noise and the Gaussian distribution

In electronics noise is a random fluctuation in an electrical signal. Random noise has a flat spectral density and a Gaussian distribution . Noise exists in all electronic circuits. Figure 1.3 shows what random noise in a static signal looks like.

Figure 1.3: Time series plot and histogram of random noise on static signal with "true" value at 0 .

Measurements systems generally have noise, fluctuation around the true value. If you only take one sample every 15 minutes the noise level will go unnoticed, however when you take several samples from a sensor at moderately high sample (e.g. 10Hz) there is likely some variability in the values. Although if the noise level in the system is lower than the resolution of your ADC it will be hidden. You can analyze the amount of noise in your system using statistics.

Quantization error is also random noise with a standard deviation of \( \frac{1}{\sqrt{12}} \) LSB. This is usually a small addition to the noise already present in the measured analog signal.

The fact that random noise has a Gaussian distribution means that the the most probable value of the signal is the average value and the standard deviation gives us a meaningful estimate of the variability.

If the signal is stationary during repeated measurements you can make the measured value more precise by taking several samples and averaging. If the signal is not stationary, but the noise is random you can use a moving average filter.

Note that you should check whether or not the noise is Gaussian before you do this. If the signal is not Gaussian then you should check whether the variability in the signal is due to interference and whether or not you can get rid off it by improving the measurement set up or filtering the signal.

The normal distribution (Gaussian) defined as \( N(\mu, \sigma) \) is a bell shaped distribution that has mean \( \mu \) and standard deviation \( \sigma \) .

\[ N(\mu, \sigma) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]

The parameters can be calculated from sample \( x \) with \( N \) numbers using:

\[ \mu = \frac{1}{N} \sum_{i=1}^{N} x_i \\ \sigma = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (x_i - \mu)^2} \]

Flat spectral density is another way of saying that the noise doesn’t have a specific frequency that causes more variability than others. You can obtain the spectrogram of the signal using a discrete Fourier transform (DFT), commonly calculated using the FFT algorithm.

Again if there are marked peaks in the spectrum you should try find the origins of those peaks. Power line interference at 50Hz is quite common in measurements, but you can remove it by using a lowpass or band-reject filter. Generally frequency interference can be removed using digital filters (or analog, but digital filters are better).

1.7 Properties of measurements

This section contains brief explanations about the properties of measurements and general terminology. It deals with both static and dynamic measurements.

Static measurement is a measurement of quantity that doesn’t change during sampling e.g. the weight of a Cow at this instant, moisture of grains from a sample from the dryer.

Dynamic measurement is a measurement of constantly chancing quantity and needs additional consideration compared to static measurements. Force measurements of moving objects, speed and acceleration etc.

1.7.3 Dynamic properties of measurements

Step response : is the measurement systems response to a step input. It is used to determine the dynamic properties of the system.

Rise time

is the that takes for the output (measured value) to rise from 10% change to 90% change after a step input.

Time constant \( \tau \)

is the time that the system takes to reach 0.63 \( \frac{1}{e} \) its final value after a step input.

Figure 1.4: Step response and rise time. Step input and the corresponding change in the measured response (output).

1.8 Calibration

In calibration the measured value is compared to the true value (or more accurate value obtained from better instrument or lab analysis).

• Results are used to calculate calibration coefficients .
• Is used to correct systematic error in the measurements.
• It’s important to have enough calibration points, especially if the sensors response curve is not linear.

It usually helps if you know physics you e.g. use a known mass to calibrate force sensors, rotary movement with known radius to calibrate accelerometers, boiling water and ice to calibrate temperature sensors etc. Calibration also needs to be checked during long running measurements because the properties of the sensors or system can change over time. Sensors in tough environments such as cattle buildings or field are especially prone to changes in calibration.

Some instruments have calibration certificates from the manufacturer. There are specialized calibration labs to do the calibration for you.

1.9 More terms

Span : This is the measurement range of an instrument e.g. 0-5 V, -10 – 10V.

Drift: is the change of the zero level of a measurement over time. Drifting can occur e.g. due to temperature when the properties of the amplifiers or sensors change.

Sensitivity is the change in output of the sensor relative to the input: 10 \( \frac{mV}{^o C} \) , \( \frac{mV}{V} \) …

Resolution tells how accurately a quantity can be measured with a sensor or a system. For ADCs the resolution is expressed in bits and you can calculate the voltage resolution. For sensors this is usually indicated in the measured unit e.g. \( \pm 0.1 ^{\circ} \) C or \( \pm 2cm \) .

Hysteresis means that there is a difference in readings depending whether the measured value is approached from above or below (e.g. rising or falling temperature).

1.10 Exercises

1. LM35A is integrated circuit temperature sensor. Answer following questions based on the datasheet:( http://www.ti.com/lit/ds/symlink/lm35.pdf )

• Accuracy
• Measurement range
• Rise time from 0-100% in still air?
• Write the equation for converting LM35 voltage into temperature in \( ^{\circ} \) C.
• What would be a good agricultural application for the sensor?

1. The resolution of an ADC is 12bits and the span is 0-5V. What is the voltage resolution?
2. Explain the following terms:
   1. Hysteresis
   2. Accuracy
   3. Sensitivity
   4. Precision
   5. Rise time
   6. Least significant bit
3. How do you choose a sampling rate for a dynamic measurement?
4. You want to measure weight with 1kg resolution starting from 0kg. What is the maximum weight you can measure with a 12bit ADC?
5. How do you measure the rise time of a system?
6. What are the advantages of digital signals over analog signals?
7. How would you measure the signal-to-noise ratio of a system?
8. Define random noise.