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ME 370 Engineering Measurements and Instrumentation
Example : RMS Value-Typical Wave
Calculate the RMS value for signal
Signal: physical information being transmitted
Waveform: shape and form of signal
Amplitude (or magnitude): measure of the signal strength
Range: lower and upper measurement limits
Frequency: the signal changes over time
Cyclic frequency: cycles/sec (Hz), f
Circular frequency: rad/sec, w = 2pf
Period: T= 2p/w = 1/f
Analog: continuous in time
Discrete (digital): information available at discrete points in time (quantized)
(a) Analog signal representation
(b) Analog display
Measurement of a continuous signal at discrete time intervals
Magnitude can be any value
(a) Discrete time signal
(b) Discrete time waveform
Exist at discrete values in time
Magnitude is discrete
(a) Digital signal
(b) Digital waveform
Root-Mean-Square (RMS) Value
Sampling: discrete representation of an analog signal
The discrete recreation of the amplitude and the frequency of an analog signal depends on
Time interval between each measurement (t)
Total sample period T = Nt, N is the number of measurements
Sampling rate: fs = 1/ t
For a given frequency signal
How fast should sample?
Sampling Theorem (Nyquist)
To reconstruct a signal, the sampling rate (fs) must be greater than twice the highest frequency (fm) component of interest
All frequencies above Nyquist frequency appears aliasing
Definition: describes an effect that causes different continuous signals to become indistinguishable when sampled
Alias frequencies (m integer)
Minimum Aliasing Frequency
Choose integer m to minimize the value
Only consider the positive frequencies
There is no way to distinguish negative and positive frequency
To reconstruct the signal, the lowest frequency signal is usually more important
fs = 1 Hz, signals f = 0.1 Hz and f = 0.9 Hz are indistinguishable
Example: Alias Frequencies
Consider signals at f = 25Hz, 70Hz, 160Hz, 510 Hz. If the sampling frequency fs = 100 Hz, compute the minimum alias frequencies
Set fs to ~5 times fmax
Require pre-knowledge about the signal
Filter out signals beyond fN before sampling
Completely avoid aliasing
For a periodical function (period T)
High Frequencies Give Details
Fourier Series Properties
a0 represents the average signal
an represents the even components
bn represents the odd components
For an odd function, f(-t) = -f(t), an= 0
For an even function, f(-t) = f(t), bn= 0
Frequency Spectrum: Fourier Series
For each frequency component
Circular frequency: wn= 2pn/T =2pfn(rad/sec)
Regular frequency: fn = n/T (Hz)
Each coefficient amplitude at frequency fn
Plot RMS amplitude of Fourier series term
as a function of fn (Harmonics)
Even and Odd Functions
Example: Fourier Series Slope
Example: Fourier Series Square Wave
Discrete Fourier Series
Discrete Fourier Series
Discrete Fourier transform (DFT)
… That is, instead of integrating over a continuous time, values of the function for discrete times are summed
Amplitude Ambiguity (Frequency leakage)
If period of sampling (Tr) is NOT and integer multiple of the fundamental period (T1), amplitude ambiguity occurs.
No Amplitude Ambiguity occurs if:
Tr = m T1
for m = 1,2,3,…
Tr = N δt = N/fs
N/fs = m 1/f1
m = Nf1/fs
If m is not an integer, leakage occurs
Frequency resolution of DFT (f)
When Nt ≠ kT, k = 0, 1, 2,…, and T signal period
Leakage occurs at adjacent frequencies
No Leakage Example
Tr = N δt = 256 x 0.000313 = 0.08
T1 = 1/f1 = 1 / 100 = 0.01
Tr = m T1 m = Tr / T1 = 0.08 / 0.01 = 8
Integer, thus no leakage occurs
Sampling on period <=> No leakage
Signal: y(t) = 10 cos (2p100t)
Tr = N δt = 1024 x 0.000100 = 0.1024
Tr = m T1 m = Tr / T1 = 0.1024 / 0.01
NOT Integer, thus leakage occurs
Not sampling on period <=> Leakage
Options to Control Leakage
Sample on period
Requires pre-knowledge of the signal: which is usually not available
Total sampling time T = Nt as long as possible: but how long and at what cost?
Windowing: multiply sample signal by a window function
Then take DFT/FFT
A DFT / FFT always windows an infinite signal
Then take DFT/FFT: A DFT / FFT always windows an infinite signal
Hamming Window Reduces Leakage
A microphone is used to receive the sound, that is digitally sampled by a sound card. The digital sound is played back by a speaker. The specifications are listed below.
Assume no response is outside the bandwidth of the device. Determine…
Frequency Response: 70-16k Hz
Frequency Response: 0-15k Hz
Sampling rate: 22k Hz
Frequency Response 20-20k Hz
Example 7 (cont’d)
(2a) What range of sound frequencies can be accurately sampled by this microphone and sound card combination?
(2b) Suppose a sound has a frequency of 15,500 Hz and it is recorded using the microphone and sound card. What will the frequency of the tone be if played back through the speaker?
(2c) Within the range of frequencies you can accurately sample, what range of frequencies can be played back through the speakers?
(2d) Suppose you perform an FFT on a sound data file and notice a large amplitude spike at 50 Hz. Did this sound likely exist when the microphone sensed the sound waves? If not, state one possible source of this frequency component in the data file.
(2e) If you were to place a filter between the microphone and soundcard, what frequency range would you allow to pass and why?
Analog-Digital Conversion (ADC)
Analog signals are:
Converted to yes/no or on/off pulses
Coded in a meaningful form (e.g., binary words)
Readouts are precise and require no interpolation
Easily coupled to other devices and a computer
Signals are easily reproduced
Voltages are generally low (5, 10, and 12 V are typical)
Data Acquisition Systems
Data Acquisition Board
Convert analog signal to digital signal (A/D)
M-bit A/D converter output 2M different binary numbers
Things to consider when selecting an A/D converter
Number of inputs
Resolution (# of bits)
Data Acquisition Board: Inputs
(a) Single ended
(b) differential ended
Single-ended Vs. differential-end connections
Single Ended Vs Differential Ended
# of outputs (N/2)
Smaller noise (cancel out common noise)
Input requires separate ground-reference point or return signal
# of outputs (N)
Input can share a common reference point with other signals
A/D Considerations - Resolution
Quantization error (resolution error)
ADC Considerations – Resolution
Resolution in bits:
M-bit ADC has2M quantized discrete values
8 bit 28 = 256
12 bit 212 = 4,096
16 bit 216 = 65,536
Resolution in volts:
Voltage range (full scale voltage range) (EFSR) divided by the resolution in bits:
Q = EFSR / 2M(Note that smaller Q is better)
To select a high-resolution ADC it is better to look for higher bits than lower voltage range, for practical reasons.
ADC Considerations – Error
Reaching upper/lower limit of the voltage range.
Error due to the size of each quantized discrete value.
Resolution (quantization) error: |eq |≈ ±Q/2
Error % = (Resolution error / sensor range )%
The voltage output from a J-type thermocouple referenced to 0°C is to be used to measure temperatures of 50-70°C. The output voltages will vary linearly over this range from 2.585 to 3.649 mV.
If the thermocouple voltage is input to a 12-bit A/D converter having a ±5 V range, estimate the percent quantization error in the digital value.
If the analog signal can be first passed through an amplifier circuit, compute the amplifier gain required to reduce the quantization error to 5% or less.
fs= 500 Hz
N = 256
For the f(t), what are the fundamental frequency, fundamental period and amplitude of the wave?
Does aliasing occur? If yes, what is the closest frequency at which it does not occur? (show all calculations)
Does leakage occur? If yes, what is the closest number of samples per period (N) at which leakage does not occur? (Show all calculations)
fs= 1/dt non-aliasing if: fs ≥ 2fm
Tr=N.dt non-leaking if: m is an integer
m = Tr/T1
Operation Amplifier (Op-Amp)
IC (transistors, resistors and capacitors)
DC-coupled voltage amplifier
Input: 2 sources (differential) over a range of frequencies (bandwidth)
Output: 1 single-ended output
Power: Requires power supply
Accepts input from 2 sources over a range of frequencies and responds to a voltage difference between the two inputs
Typical Op-Amp chip
High gain output: A → 104 – 105
High gain (A) output: ~100
Inherently unstable: A is a function of frequency A 1 at high f
Some Op-Amps have low-pass filter built-in
The IC can be used in a circuit in two ways.
If the voltage goes into pin 2 then it is known as an INVERTING AMPLIFIER.
If the voltage goes into pin 3 then the circuit becomes a NON-INVERTING AMPLIFIER.
Infinite open-loop gain
Infinite voltage range available at the output (Vout).
Infinite bandwidth (i.e., the frequency magnitude response is flat everywhere with 0 phase shift).
Infinite input impedance (Zin = ).
Zero input current.
Zero input offset voltage (i.e., when the input terminals are shorted, the output is a virtual ground or Vout = 0).
Infinite slew rate.
Zero output impedance (i.e., Zout = 0, so that output voltage does not vary with output current or load).
Infinite Common-mode rejection ratio (CMRR).
Infinite Power supply rejection ratio for both power supply rails. The power supply sources are called rails.
Op Amps & Negative Feedback
No current flows in
Not affected by load
Negative feedback drives to be equal to (i.e. an imaginary short)
Allow near- ideal characteristics
Usually use a differential amplifier circuit with buffering
Can achieve variable gain
Real Op-Amp: Saturation
Output voltage cannot exceed supply Vcc
For a gain-> Vout > Vcc, Vout clipped to Vcc
Real Op-Amp: Clipping
For a gain Vout > Vcc ⇒ Vout clipped to Vcc
Real Op-Amp: Slew Rate
The maximum rate of change of a signal in a circuit
Distort signal if signal changes exceeds slew rate
Real Op-Amp: Distort Signal
For the instrumentation amplification circuit shown below, both inputs, V1 and V2 can be assumed to sinusoidal and in phase, of the form Asin(2pft), with different amplitudes (A1 and A2) but the same frequency f.
Other known specs: R = 10 kW . R gain is user-defined
Amplifier LM201A: Supply Voltage : ±10V; Slew Rate Limit: 0.7 V/ms
Example 2 (cont’d)
If A1 = 0.6 V, A2 = 1.5 V
(a) What is the maximum gain and corresponding value of Rgain that can be used to prevent clipping?
(b) For the gain established in part (a), what is the corresponding maximum frequency of the input signal that can be detected without slew rate distortion?
Common Mode Rejection
Common Mode Rejection (CMR)
For differential measurement, the voltage difference of two signals is the only desired signal
Common mode voltage: a voltage signal that is common to both signals may exist.
An ideal differential amplifier only amplifies the differential signal, while rejecting the common mode voltage signals
Non-ideal: common voltage signals are also amplified
Common Mode Rejection
Consider two signals
Eliminate noise by subtracting signals
Real Common Mode Rejection
Actual output of a differential instrumentation amplifier
Vd : differential voltage (Vwanted)
Vcm : common mode voltage (Vnoise)
Acm : common mode gain
Ad : differential gain (gain for an instrumentation amplifier)
Ideal differential op-amp
Acm= 0 → Vocm = 0
Real differential op-amp
Acm : close to 0
Ad: typically much larger
Common Mode Rejection Ratio (CMRR)
CMRR defined as:
Indicates how much of most common mode signal is rejected.
Indicates how much of most common mode signal is rejected.
Depends on signal frequency
Attenuate unwanted signal components
Pass desired signal components
Low pass, high pass, band pass, notch
Passive filters: RLC circuits
Active: powered chip (i.e., op-amp)
Typical Ideal Filters
A Simple RC Circuit
eo as a function of ei:
Time constant, τ
Increment of time such that when
t = τ ,
t = 2 τ,
For this circuit,
Response to a Periodic Input
Let the input
The Transfer Function has a magnitude, M and phase shift ϕ
Low Pass Filter (Passive)
Magnitude ratio: attenuation
Cut-off frequency, fc , when
M = 0.707 (or gain = -3 dB)
Low Pass Filter Response
High Pass Filter (Passive)
Circuit diagraph shown here R = 100 KΩ, C = 330 pF, determine
Output for input
Plot |E1/Ei|, |E2/Ei|, and |Eo/Ei| over a range of frequencies when an AC signal is passed through the following circuit.
R=100 kW; Rgain= 200 kW
Strain gauge systems
Strain measurement circuit
Engineering Mechanics Review
L = L0 + dL
Units: 10-6 in/in or 10-6 m/m
Em is Elastic modulus
Is this relationship valid for all stress/strain?
Engineering Mechanics Review (cont’d)
Ratio of lateral strain to axial strain
An ideal strain measurement should:
Have good resolution
Insensitive to ambient conditions (temperature, pressure etc.)
Have a high-frequency response for dynamic strain measurements
Resistance strain gauges
Resistance Strain Gauges
Resistance of metallic conductor
re is resistivity (Wm)
L is length (m)
Ac is cross-sectional area (m2)
Change in resistance
Common metallic strain gauge material is a 55% copper, 45% nickel alloy called constantan
re is resistivity = 49 x 10-8 Wm
Typical strain gauge resistance is 120 W
What length of wire of diameter 0.025 mm is required for this resistance?
Gauge Factor (GF)
Piezoresistive effect (change of resistivity due to strain)
Gauge Factor: change in resistance with strain
Provided by manufacturer
Gauge Factors are determined by a calibration process in a biaxial strain field
Most strain gauges used to measure uniaxial strain
Strain Gauge Electrical Circuits
Typically one or more legs of a Wheatstone bridge circuit
Wheatstone Bridge with One Active Strain Gauge
For an initially balanced bridge
R1 = R2 = R3 = R4 = R and Eo = 0
Then if a strain is applied
Wheatstone Bridge to Measure Strain
For Wheatstone bridge with one active strain gauge
Reading typical specification sheet
Compensation for apparent strain
Multiple loading modes
For the strain gauge setup shown below, what is the applied tensile load if the measured bridge output under load is 2.5 mV? The nominal resistance of the strain gauge is 120Ω; the bridge excitation voltage Ei = 10V. The bar is 3 cm wide by 1 cm high and is made of steel (E = 200 GPa). Assume GF of the strain gauge is 2.
Multiple Strain Gauges in a Bridge Circuit
Consider the general case where all 4 resistors in a Wheatstone bridge are active strain gauges
If we choose strain gauges of equal resistances and gauge factors
Bridge Constant, k
Ratio of the actual bridge output to that of a single gauge
Used for multiple strain gauge bridge configurations
Multiple Strain Gages
BRIDGE CONSTANT κ
Single Gage in Uniaxial Strain
κ = 1
Two gages sensing equal and opposite strain
κ = 2
Two Gages in Uniaxial Strain
Four gages with pairs sensing equal and opposite strain
κ = 4
One axial gage and one poison gage
κ = 1 + ν
Four gages with pairs sensing equal and opposite strain – Sensitive to torsion only. Typical Shaft Arrangement
Consider the following Wheatstone Bridge system:
3 fixed and identical resistors with R = 2.5 k Ω
1 gauge resistor @ R1 with GF = 2.
Sensor is placed on a sheet of 0.05 cm x 10 cm (t x w) cross section made of steel (E= 200 GPa)
The sheet is subject to tension of 0 – 70 kN.
An input potential of 15 V is used.
What is the maximum expected strain?
What is the maximum expected δR1 ?
What is the system’s sensitivity in mV/ Ω?
What is the system’s output range?
Using a 12 bit ADC with voltage range of ±5 V to collect and convert data:
What is the quantization error?
What is the % error?
What gain amplifier should be used to reduce % error to 1%?
In A test setup, the Eo is read to be 12.7 mV, what force is acting on the sensor and what is the corresponding strain?
What is system’s theoretical sensitivity in terms of mV/N?
Typical Spec Sheet
Sensitivity to Lateral Strain
Error % of axial strain
Lateral sensitivity, Kt [%]
Caused by gage and bonding material
Typically cycle before final testing
(There are 3 additional cyles to 6000 microstrain)
AR/R0 in micro-ohms per ohm
Applied strain level in microstrain
Example 2: Temperature Compensation Arrangement
What is dEo/Ei for the following strain gauge arrangement?
Example 3: Strain Magnification Arrangement
Error and Uncertainty Analysis
Error and Uncertainty
Important note: Terminology related to measurement uncertainty is not used consistently among experts and texts.
The definitions here are the most commonly used.
Error: recognizable deficiency due to experimental mistakes and/or shortcomings
Uncertainty: potential deficiency due to lack of knowledge
Measurement error: difference between measured and true values
Measurement uncertainty: the estimate of the probable error in a measurement
Any difference in the value assigned to a case (measurement) and the true value is known as the measurement error.
Image source: http://physicslabs.ccnysites.cuny.edu
Example: Some potential sources of error in measuring the length of the pencil:
Spacing marks on the ruler are 0.5 cm apart but length of the pencil is not an integer (n) or half-integer (n + 0.5).
The person conducting the measurement must use their judgment in reading the value off the ruler and may not have perfect vision, stable hands, etc.
Measurement errors can be generally categorized as Systematic or Random errors.
Systematic Errors: Systematic errors in experimental observations usually come from the measuring instruments.
Random Errors: Random errors in experimental measurements are caused by unknown and unpredictable changes in the experiment. These changes may occur in the environmental conditions or depend on the observer.
Two types of systematic error can occur with instruments having a linear response:
Offset or zero setting error in which the instrument does not read zero when the quantity to be measured is zero.
Multiplier or scale factor error in which the instrument consistently reads changes in the quantity to be measured greater or less than the actual changes.
Systematic Error may occur because:
There is something wrong with the instrument or its data handling system.
The instrument has measurement limitations.
The instrument is not calibrated.
A/D Considerations – Error
Systematic errors in a linear instrument (full line). Broken line shows response of an ideal instrument without error
Accuracy and Systematic Error
The accuracy of a measurement is how close the measurement is to the true value of the quantity being measured. The accuracy of measurements is often reduced by systematic errors, which are difficult to detect even for experienced researchers, due to lack of knowledge.
Random Error may occur because:
The experimenter cannot operate the instrument properly (e.g. reading values off a ruler)
The environment influences the experiment (e.g. cooling from wind or heating from sun).
Other variations in the measurement process (e.g. delay in pushing the timer’s buttons)
Random errors often have a Gaussian normal distribution. In such cases statistical methods may be used to analyze the data. The mean m of a number of measurements of the same quantity is the best estimate of that quantity.
The Gaussian normal distribution.
m = mean of measurements.
s = standard deviation of measurements.
68% of the measurements lie in the interval m - s < x < m + s
95% of the measurements lie within m - 2s < x < m + 2s
99.7% of the measurements lie within m - 3s < x < m + 3s
Precision and Random Error
The precision of a measurement is how close a number of measurements of the same quantity agree with each other. The precision is limited by the random errors. It may be determined by repeating the measurements.
Accuracy and Precision
Mean: Average value of each data point, and is the best estimate of the true value (when n is large).
Standard deviation: Shows the accuracy of the estimate (how far the data points are from the average).
Variance: square of the standard deviation
Standard error of the mean: is the standard deviation of the sample-mean’s estimate from the population’s mean.
In other words, if one takes random samples (same size) from the same population, the standard deviation of those samples’ means is approximately equal to SEM.
SEM is used to determine (estimate) how precisely the sample mean estimates the population mean.
Lower SEM values indicate more precise estimation of the population mean.
Large standard deviation results in large SEM values.
Large sample size results in small SEM values.
Dr. M’s Party
Number of guests
Average age (years)
Dr. M's Party
Standard deviation of “age”
Calculate mean, STDev, and SEM.
Data set 1: 0,46,3,19
Data set 2: 10,20,15,11,23,20,20
Following data were collected by students measuring the acceleration of earths gravity , calculate standard deviation from the mean.
Root-sum-squares (RSS) method:
Rule of thumb:
Use the 95% probability level (P% = 95%) for all uncertainty calculations
Uncertainty: estimate of the probable error in a measurement
Uncertainty analysis: the process of identifying and quantifying errors
Test objects are known
Measurement process is clearly defined and all known calibration corrections for bias error have been applied
Data are obtained under fixed conditions
User has experience with the system
When to Perform Uncertainty Analysis?
Before any measurements are taken
NOT from your data
Advanced stage and/or single measurement
When a large data set is not available
Only have limited data
When deciding on measurement procedure(s)
Multiple measurement stage
When sufficient repetitions and replications of the measured data are available
You have enough data to do statistics
Design Stage Error
Estimate total error associated with instrument(s)
Error due to instrument resolution, u0
Resolution: the minimum readable output
Calibration error, uc
Instrument Resolution Error, u0
A/D converter (M-bit, Full scale range EFSR)
Instrument Error Types
Instrument Error Types (cont’d)
Uncertainty From Instrument Specs uc
Estimate of the bias error(s) caused by the instrument (evaluated from calibration)
For example: Omega PX541 series pressure transducer
Estimate suitability of an instrument for a measurement
Guide for selecting equipment and procedures
Estimate minimum uncertainty in a measurement system
Use RSS method to combine estimate from each sensor and instrument
An Omega PX541 series 0-30 psi pressure transducer (specifications on the right) is used in a laboratory where the room temperature may vary ±10°C. What is the total error that is introduced into the system using this pressure transducer? (Design stage uncertainty)
An Omega PX541 series 0-30 psi pressure transducer (specifications provided below) is used in a laboratory where the room temperature may vary ±10°C. A 200 W resistor is used to convert the 4-20 mA pressure transducer output to a voltage signal that is subsequently measured by a 12-bit 0-5V voltmeter. The voltmeter has an accuracy of 0.01% of the reading. What is the pressure transducer resolution and the design stage uncertainty?
Computation of the overall uncertainty for a measurement system consisting of a chain of components or several segments/instruments.
Let R is a known function of the n independent variables xi1, xi2 , xi3, ..., xin
R = f(xi1, xi2 , xi3, ..., xin)
n is the number of independent variables. Each variable contains some uncertainty (ux1, ux2, ux3,..., uxn) that will affect the result R.
Given a result:
A sensitivity index (weight) is defined:
An uncertainty estimate in the result is determined from:
where is the absolute sensitivity coefficient and should be evaluated at the expected value of
The resistance of a certain size of copper-based resistance thermometer element is given as
where is the initial resistance at 20° C, is the temperature coefficient of resistance for copper, and the temperature of interest is .
What is the element resistance and its uncertainty?
Percent Uncertainty Propagation – “Short Cut” of Product Functions
Good only for product functions
Percent uncertainty is simplified as
A certain obstruction-type flow meter is used to measure the flow of air at low velocities. The relationship describing the flow rate is:
where C is an empirical discharge coefficient, A is the flow area, p1 and p2 are the upstream and downstream pressures, T1 is the upstream temperature, and R is the gas constant for air. Determine the percent uncertainty in the mass flow rate for the given conditions:
C = 0.92 ± 0.005 (from calibration data)
P1 = 25 psi ± 0.5 psi
T1 = 70°F ± 2°F
Δp = p1 – p2 = 1.4 psi ± 0.005 psi (measured directly)
A = 1.0 in2 ± 0.001 in2
Static Response and Calibration
Calibration is the process of quantifying and correcting the error due to the disconnection between input and output.
Apply known input and observe the output.
Applied inputs can be random or sequential.
In sequential input calibration a full cycle of inputs is preferred (min max min).
Calibration Also Determines
Input and output range
The closeness with which a measuring system indicates the actual value
error = true value – indicated value
Precision (random) errors
Measure of random variations found during repeated measurements at a given input value
Bias (systematic) errors
Difference between measured value and true value
The response is not time dependent.
The response (magnitude and/or frequency) is time dependent.
Order of the system (0,1,2,…)
Sensitivity: output change per input unit change
The slope of a static calibration plot is the static sensitivity or static gain
Dynamic (amplitude, frequency, and phase)
Static Response Example
Temperature Sensor 1:
Linear relation between input (temperature) and output (voltage) (constant sensitivity)
Temperature Sensor 2
Non-Linear relation between input (temperature) and output (voltage) (variable sensitivity)
Calibration curve shows inputs and corresponding outputs; they can have different units and/or a gain.
Measurement System Response/Behavior
Measurement output y(t) for input F(t)
Depends upon initial conditions y(0)
General system behavior model:
Zero-order system model: y(t) = KF(t)
K: static sensitivity
F(t): input signal
y(t): output signal
Typical 1st-order systems
Exhibit some form of storage or dissipative capability
Have negligible inertial forces
General first-order system model:
Input Test Signal For System Analysis
Dimensions of a rectangle is given as:
Length = 10 cm
Width = 5 cm
Calculate the area of the rectangle and its uncertainty.
1st Order Response to Step Input
1st-order System Response - Step Input
Steady state response
As t gets infinitely large, solution reaches steady state of (KA)
Dynamic Error, defined as an error fraction
When t = t:
Example 2: Find t From The Graph
Example 3: Find t From The Graph
Error Fraction, G (t)
Example 1: Thermal Response
A certain thermometer has a time constant of 15 s and an initial temperature of 20°C. It is suddenly exposed to a temperature of 100°C.
(1) Determine the rise time (90% )
(2) The corresponding temperature at this time.
Dynamic error in temperature measured using a thermometer after 3 seconds is 70%.
(1) What is magnitude ratio at 3 seconds?
(2) What is the time constant?
(3) what is the magnitude ratio at 1 second?
1st Order Response to Sine Input
A temperature sensor is expected to measure an input having frequency components as high as 50 Hz with an error no greater than 5%. What is the maximum time constant for the temperature sensor that will permit this measurement?
For an input signal at 200 Hz, a 1st order system has a magnitude ratio of 0.707.
(1) Calculate the time constant.
(2) Calculate the phase lag
A 1st order system with 10 ms time constant is subject to a sine wave of 15 V at 30 Hz.
(1) Calculate the amplitude of the steady-state output.
(2) Calculate the magnitude ratio.
(3) y(t) for t = 1 (ms) where y(0) = 0
1st-order System Response - Sine Input
f(w): phase lag
F(t) = Asin(wt)
Magnitude change and phase shift between output signal and input signal are predictable
Magnitude ratio: ratio between output amplitude and input amplitude
What is K?
1st-order System Frequency Response: Magnitude Ratio
1st-order System Phase Response
The dynamic error, is a measure of the inability of a system to adequately reconstruct the amplitude of the input at a particular frequency
Q: How is dynamic error related to attenuation?
2nd Order System Example
Systems that have a storage/dissipative capability AND inertia
Calculate damping ratio, natural frequency, ring frequency.
Homogenous solution determines transient response
Damping ratio () dictates homogenous solution form
Underdamped (0 < < 1)
Critically damped ( = 1)
Over damped ( > 1)
Transient response, yh depends on (See Eqns. 6.57-6.60)
2nd order system response
Standard 2nd-Order System Model
Transient response, yh depends on
Step Response of 2nd-Order System
Example 2: Determine ωn and ζ Experimentally from step input response
Natural Log decrement
Step Response of Underdamped 2nd-Order System
What should the time constant be for a 1st order system to have a dynamic error of 0.01 when input is a sine wave with amplitude of 15 V at 100 Hz?
Step Response of 2nd-Order System: Properties
Rise time: time to first reach 90% of input step value (KA – y0)
Settling/response time: time for oscillations to settle within ±10% of input value
Design damping ratio and natural frequency for desired response
Most systems designed with 0.6 < <0.8
Ringing and Settling Time
2nd Order System Response
Key Points of Last Lecture
2nd order system
Frequency Response of Second-Order System
Input F(t) = Asin(wt)
Total solution: y(t) = ysteady + yh
Transient response, yh depends on z: typically ignore this part!
Magnitude Ratio: 2nd-Order System Frequency Response
Phase Shift: 2nd-Order System Frequency Response
A force transducer behaves as a second-order system. If the undamped natural frequency of the transducer is 1800 Hz and its damping is 30% of critical, estimate the error in the measured force for a harmonic input of 950 Hz. What is the magnitude of the corresponding phase angle shift?
A load (force) cell has an undamped natural frequency of 160 Hz and a damping ratio is 0.707. Estimate the practical frequency range over which it can measure dynamic loads with an inherent error of less than 5%.
Why Measure Acceleration
Motion is system physical characteristic
Characterize system dynamics
Whole body motion, displacement, velocity
Input into control systems
Use measured acceleration to correct changing dynamic condition
Output is a direct indication of either displacement or acceleration
Seismic Transducer – 2nd Order System
Spring constant (k)
Damping coefficient (c)
Critical damping coefficient:
System damping ratio:
Undamped natural frequency:
Steady Solution: Frequency Response
Can be used to track
Output proportional to acceleration
The output magnitude should follow the magnitude of the input acceleration
M(ω) -> 1
Typical ζ= 0.7, why?
ω/ωn < 0.4, why?
Magnitude Ratio: Second-Order System Frequency Response
Typical Accelerometer Characteristics
From: Introduction to Engineering Experimentation, 2nd Ed., by A.J. Wheeler and A.R. Ganji, Pearson, Upper Saddle River, NJ, 2004.
Vibration pickup or vibrometer:
Output is proportional to either displacement or velocity
Condition for w/wn
Damping ratios 0.707
A seismic device can be used as either a vibrometer or an accelerometer
Frequency range below undamped natural frequency
Frequency ranges above undamped natural frequency
An accelerometer has a damping ratio and natural frequency of the vibrometer is 0.7 and 100 Hz, respectively. What is the range of input frequency (bandwidth) for which the error in measured acceleration is less than 1%?
If the sensor is to be used as a vibrometer what frequency range would provide the same error levels?
95% confidence interval: 19 out of 20 (95%) samples from the same population will produce confidence intervals that contain the population parameters.
A confidence interval is a range of values that is likely to contain an unknown population parameter. If you draw a random sample many times, a certain percentage of the confidence intervals will contain the population mean. This percentage is the confidence level.
der System Frequency Response