Wednesday, January 30, 23:55 PM
Fourier Analysis Note if a function is odd we only need to use the sines and if it is even we only need to use the cosines.
Finite Statstics Outlier Analysis
Required Measurements Design-Stage Uncertainty
Zero Order Uncertainty
Temperature Measurement Wheatstone (RTD)
If F(t) is a unit step then:
Radiative Measurement Strain Single
Multiple (assume all GFs are the same)
Rosette Measurement System Behaviour
First Order Second Order Mechatronics Sensors
Seismic Proving Ring
Pressure Transducers Pressure Transducers
Fluid Velocity Measurement Flow Rate Measurement
Above 2 lead to: 15˚ and 7˚ 2/3 - Static and Dynamic Characteristics of Signals
Tuesday, January 29, 2013 6:21 PM
An input signal often comes into a measurement device as a step function. Due to the time required to read the input the output signal
is often of a lagging form. This is an important factor to be aware of when selecting a measurement device/system.
A static signal is something like a temperature or pressure of a tire.
A dynamic signal is something like an AC current or pressure in an internal combustion engine.
• Can be a periodic, step, etc. function in time
There are numerous ways that we can analyze input and output signals to learn about the variable we are measuring. Techniques
include: classifying the signal and frequency analysis (histograms with a distribution).
Types of Waveforms:
• Analog (Describes a signal that is continuous in time)
• Discrete Time (Information about the magnitude of the signal is only available at discrete time points)
• Digital Signal (Available only at discrete time intervals with quantized data levels as well)
MECH 215 Page 16 Direct current offset can be used to highlight the AC component. To do this we subtract the average from a set of dynamic voltages to
see the highlighted AC signal. This works because it shrinks the data down to where the fluctuations are more obvious.
We often use the mean of a data set to find a representative value. The root-mean-square calculation is also used fairly often in signal
We can approximate very complex signals by using Fourier Analysis. A Fourier Series involves using an
infinite series of sine and cosine functions.
A periodic function y(t) having an arbitrary period T can be represented by:
MECH 215 Page 17 Where n is a constant multiplier of t.
An even function is one with the same y-value on either side of the y axis. An odd function is the
opposite sign of the y-value:
If a function y(t) is even, we can represent it using a series of Cosines (cos is symmetric about the y-axis)
If the function y(t) is odd, we can represent is with a series of Sines (sin is opposite sign on either side of
The fundamental frequency of a Fourier series is the n*w term in the first Fourier term. All of the other
MECH 215 Page 18 The fundamental frequency of a Fourier series is the n*w term in the first Fourier term. All of the other
terms are an n multiple of this term and are therefore called harmonics.
We can plot the absolute values of the amplitudes of the Fourier series terms against their
corresponding frequency to produce the frequency content or amplitude-frequency spectrum:
MECH 215 Page 19 4 - Data Acquisition, Sampling, and Digital Devices
Wednesday, February 06, 2013 7:27 PM
Advantages to digital data acquisition are that collecting and processing large amounts of data is efficient and fast.
Sampling is a process by which an analog signal is made discrete.
Important data from an analog signal such as frequency and amplitude can be represented by a discrete signal if the
proper conditions are met.
• Time increment (δt) between each discrete number
• Total sample period (Nδt)
• Frequency content of the analog signal
The sampling rate (based on δt) is very important when measuring an analog signal: When sampling, the sampling rate must be more than twice the highest frequency found in the measured signal.
Although in practice it is often a good idea to ensure the sampling frequency is MUCH greater than twice the input
frequency. The criterion are:
The Nyquist Frequency (F )Nis the maximum frequency that can be represented in a signal based on the sample rate. All
frequencies in the signal above F Nake on an alias frequency between 0 and F . N
Aliasing is an effect that makes different signals becomethe same or aliases of each other when sampled.
We can use a folding diagram to convertthe signal frequency with respect to the nyquist frequency into the aliasing
frequency with respect to the nyquist frequency.
The out of phase section of the folding diagram means that the output signal will be a negative compared to the input
Analog-to-digital converterconvertsanalog voltagevalues to binary number using quantization. Signal exits at discrete
time intervals at discrete magnitudes.
• The analog input side of the converteris specified over the signal's entire range E FSR
• The digital side is specified in the number of bits in the converter'sregister (M). This determinesthe number of
different binary values. This resolution is given by:
Since digital data is quantized, any converted value that falls between two quanta would result in error. This error is
called quantization error and is given by half the size of the resolution.
Saturation error occurs because the voltage range is limited. If the range is exceeded by the signal then the data
saturates or becomes constant at the max or min quanta value. There are four types of error that can occur during the conversion process:
• Zero Shift
Main componenentsof a data acquisition system are:
• Analog filters to control the frequency content of the sampled signal
• Amplifiers for selectable gains
Signal amplitude attenuation using a voltage divider circuit to reduce the input voltage.
Shunt resistor circuits can be used to convert and input signal as a current to a voltage.
A multiplexer can take multiple input signals and convert them into a single A/D convertor. 5 - Data Reduction, Probability and Statistics
Friday, February 08, 2013 10:44 PM
Sources that contribute to variations in data can come from the following:
• Measurement Systems
• Measurement Procedure and Technique
• Measured Variables
○ Temporal Variation
Based on the distribution of the data we can perform many analytical tools to understand it. There is no
way to be able to determine a true value but using the following terminology we can make a very good
Where the true value is the average plus or minus the uncertainty at a certain confidence level.
The most basic tool for data analysis is the histogram and it measures the frequency of the data in a
measured range. The number of ranges to be used can be determined by:
Where N is the number of samples taken.
The histogram can be extended to look at frequencies of each interval which leads to distribution curves
and we will be looking at normal (or Gaussian) distributions:
The probability density function for this type of distribution is very complicated but by defining a few
terms we can take advantage of data tables.
The Z variable tells us how far away from the mean (or centre line) the point of interest is:
By looking up P(Z1) in data tables we can find the probability that the next measured point will be within
the range of that Z value.
Note: this is only for one side of the middle so if we want within 1 Z for example on both sides we would
multiply this by 2. 6 - Finite Data Sets
Friday, February 08, 2013 11:07 PM
Values derived from a finite set are only estimatesof the true value.
We can calculate the accuracy or closeness of the data using the following equations:
For a normal distribution of x about the mean:
Where the circled variable replaces Z. It is the t-estimatorand is a function of N-1 and the desired
precision interval to come up with a value from the tables.
The T-estimatortimes the sample standard deviation gives us the precision interval for the given
If we have a large number of finite data sets we can calculate the standard deviation of the means which
can help improve our estimateof the true value. 7 - Outlier Analysis
Wednesday, February 13, 2013 1:46 AM
An outlier is a data point that deviates markedly from other membersof the sample.
Chauvenet's Criterion identifies outliers having less than a 1/2N probability of occurrence.
Where P(Z0) is the probability of the data point being measured as recorded on the Z tables where Z is:
We also have the ability to calculate the number of required measurementsto obtain a certain
Where t has subscripts of the DOF (N-1) and the desired confidence level.
The number of measurementscan be calculated by:
Where d is half of the confidence interval.
This method can be difficult to use howeversince the DOF of t depends on N so we must estimate Sx. An
alternative approach is to make an initial number of measurementsto obtain the sample variance then:
And the number of additional measurementsrequired for the confidence level is simply N -TN 1
ALWAYS ROUND UP THE NUMBER OF MEASUREMENTS!! 8 - Regression Analysis
Wednesday, February 13, 2013 2:07 AM
A regression is an analysis technique to fit a curve to set of data. We often have "noise" in an output
signal that also has to be removed.
The least squares fit analysis tries to fit a line to minimize the squares of the error of each point with
respect to the line.
For y = f(x) a polynomial fit can be written as:
Where the goal is to minimize D so we set the partial derivativesto 0. 9 - Uncertainty Analysis
Wednesday, February 13, 2013 7:10 AM
Every measurementwe make has an error, or the difference between what we measure and the true
Measurementerrors can contain systematicerror as well as random error.
Design-Stage Uncertainty Analysis
With this error we can report the true value as:
In uncertainty analysis we are looking to quantify u
The zero-order uncertainty is half of the resolution of the device.
Three examples of Instrument error are:
• Linearity • Repeatability
Instrument can be summed up as:
Potential errors in the measurementprocess are:
• Data acquisition
• Data reduction 10 - Error Propagation
Wednesday, February 13, 2013 7:27 AM
Didn't make a note on this as it seemed really complicatedand I don't think we would be expected to
know it. 11 - Temperature Measurement Introduction
Wednesday, February 13, 2013 7:30 AM
Two methods of thermometrybased on thermal expansion:
• Liquid-in-glass thermometer(measuresthe expansion of the liquid)
• Bimetallic thermometer(measuresthe difference in the expansions of the metals)
A resistance temperaturedetector (RTD) calculates the temperaturebased on the changing resistance
of a wire. Usually made of platinum because it has the mostlinear resistance vs. temperatureplot of
The resistance of a wire is given by:
And the resistance can also be measured as a function of the temperatureby:
We can use a Wheatstone bridge to measure the resistance of an RTD. It utilizes a variable resistor so
we can balance the RTD and a galvanometerto measure any current.
The Wheatstonebridge can also be extended using wires to keep the RTD far away from the other
resistances (so the temperaturedoes not affect them). The uncertainty in the resistance of the RTD is related to the uncertainty in each of the other three
resistors and is given by:
This error can then be extended to the error of the temperaturereading by:
Where: 12 - Temperature Measurements - Thermistors and
Tuesday, April 09, 2013 1:49 PM
A thermistoris a thermally sensitive resistor:
• It is made out of a cast chip of a semi-conductor
• Raising temperatureincreases the number of electronsin conduction band
• It has high sensitivity, ruggedness and response time
• Often has a limited temperaturerange (-90 to 130)
The equation for the resistance of a thermistoris:
Where beta is a material constant.
There are two types of circuits that we can use to determinebeta.
• Voltage divider method
• Volt ammetermethod
The dissipation constant defines the amount of electrical power that will raise a body 1 degree above
Beta can be determined with some experimentalvalues and using:
An emf can be produced between a similar metal at two different temperatures.This is the theory that a
thermocoupleworks off of. The junctions between the two different metals are placed into dissimilar
temperaturesand the recorded emf is a function of the temperaturedifference and Seebeck coefficients
of the metals. Three fundamental thermocouplelaws are:
• Law of homogeneousmaterial (a current can not be sustained in a single material with only heat)
• Law of intermediatematerials (sum of thermoelectricforces is zero if circuit is uniform
• Law of successiveor intermediate temperatures(can sum up emf's measured by a thermocouple) 13 - Basic Temperature Measurement with
Tuesday, April 09, 2013 3:20 PM
Selection of thermocouplesis based on:
• Desired uncertainty
The emf output is dependent on the types of metal that make up the thermocoupleas well as the
There are tables for each type of thermocouplethat show the emf output based on the temperature.
Note: All table values are with respect to a reference temperatureof 0˚C.
A thermopileis a device that connects more than one thermocouplein series.
Thermocouplescan be connected in parallel and then the measured voltage is an average.
Using data acquisition hardware with a thermocouplecan be hard due to the low voltagesso to account
for this an amplifier is often used with typical gains between 100 and 500 so the LSB calculation
Potential Problemswith Thermocouples:
• Poor bead construction
• Shunt impedance
• Galvanic action
• Thermal shunting
• Conduction along the thermocouplewire
• Inaccurate ice-point
• Fragile, compared to thermocouples
• External current is supplied which can heat up RTD
• Voltage difference occurs if platinum wire connected to copper connectors
Thermistors- Pros and Cons
• Thermistorsare very sensitive (100 times RTDs, 1000times thermocouple)
• Very small changes are detectable
• Very fast response time
• No standard curves (need one for each batch)
• Low performanceat high T, degrade over time
Linearity of the three different devices: Linearity of the three different devices: 14 - Temperature Measurement - Dynamic Response
Tuesday, April 09, 2013 4:26 PM
When we are making a measurementwit