Instrumentation_ Measurment_course_Chapter_2_.pdf

Instrumentation and Measurement
Chapter Two
Performance Characteristics of Measuring
instruments
Academic Year: 2023/24
Engineering College
Electromechanical Engineering Dep’t
4/2/2024 Prepared by Bereket Walle
1

Out lines
1 Introduction
2 Static Calibrations
3 Static Characteristic
4 Dynamic characteristics of instruments
Zero-order instruments
first-order instruments
Example of first-order instruments
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2.0 Introduction
Measurement Model & Terminologies Transducer
A device that converts a signal from one physical form to a
corresponding signal having a different physical form Physical form:
Such as; mechanical, thermal, magnetic, electric, optical, chemical
Transducers are ENERGY CONVERTERS or MODIFIERS
Sensor is a device that receives and responds to a signal or stimulus.
This is a broader concept that includes the extension of our perception
capabilities to acquire information about physical quantities.
Transducers: sensors and actuators
Sensor: an input transducer (i.e., a microphone)
Actuator: an output transducer (i.e., a loudspeaker)
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2.0 Introduction
The sensor generates a signal variable that can be manipulated:
Processed, transmitted or displayed
Sensor response: As stated above, the sensor is the ocean, scientist
and engineer’s interface with the real world.
Any sensor acts as a filter on the data it processes. The observer
must understand these filter effects in order to be able to remove
them and truly estimate the statistics of the environment.
The overall performance of an instrument is based on its static and
dynamic characteristics.
These effects of instrument can be solved with system calibrations.
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2.1 Static Calibrations
Static Calibrations: Sensors must be calibrated against some
standard in order that the results can be relied upon.
Static calibration refers to the input-output relations obtained when
only one input of the instrument is varied at a time, all other inputs
being kept constant.
It indicates how well the instrument measures the desired input and
rejects the spurious (or undesired) inputs.
If the sensor is made to measure constant or slowly varying quantities,
its performance can be evaluated with only the static characteristics.
The dynamic description is necessary if we need to measure rapidly
varying quantities.
This involves the study of the instrument input-output relations and
the use of differential equations.
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Figure: Calibration curve example
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Calibration
The relationship between the physical measurement variable input
and the signal variable (output) for a specific sensor is known as the
calibration of the sensor.
For sensor or any measuring instrument calibration is done by
providing a known physical input to the system and recording the
output.
The data are plotted on a calibration curve. In this example, the
sensor has a linear response for values of the physical input less than
X0.
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For values of the physical input greater than X0, the calibration curve
becomes less sensitive until it reaches a limiting value of the output signal.
This behavior is referred to as saturation, and the sensor cannot be
used for measurements greater than its saturation value.
In some cases, the sensor will not respond to very small values of the
physical input variable. The difference between the smallest and
largest physical inputs that can reliably be measured by an instrument
determines the dynamic range of the device.
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Modifying and Interfering Inputs
Figure: Interfering and modifying inputs
In some cases, the sensor output will be influenced by physical
variables other than the intended measurand.
In above figure X is the intended measurand, Y is an interfering input,
and Z is a modifying input.
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Interfering input Y causes the sensor to respond in the same manner
as the linear superposition of Y and the intended measurand X.
The measured signal output is therefore a combination of X and Y
with Y interfering with the intended measurand X.
An example of an interfering input would be a structural vibration
within a force measurement system.
Modifying inputs changes the behavior of the sensor or
measurement system, thereby modifying the input/output relationship
and calibration of the device. For various values of Z(modifying
input), the slope of the calibration curve changes. Consequently,
changing Z will result in a change of the apparent measurement even
if the physical input variable X remains constant.
A common example of a modifying input is temperature; it is for this
reason that many devices are calibrated at specified temperatures.
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Steps for calibration:
1. Identify all the possible inputs of the instrument.
2. Decide which of the inputs will be significant in your application.
3. Determine the apparatus and methods to control (vary or maintain
constant) all significant inputs over the desired range.
4. By varying one input and holding the other inputs constant, develop
the sensor input-output relations.
Example: Calibration of the pressure gage sensor
The objective in this example is to determine the relationship between
the desired input (pressure) and the output (scale reading).
The first step of the calibration process requires identifying the
desired, interfering and modifying inputs of the pressure gage.
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In the second step you must determine how, or in what conditions,
you are going to use the sensor.
what will be the surrounding temperature?
If the temperature (which is an interfering input for this sensor) varies
over a large range during the normal use of the sensor, maybe you will
have to do the (desired-input / output) calibration for different value of
the temperature.
Then you must ensure that, by choosing the appropriate experimental
conditions, all the inputs of the pressure gage, except the fluid
pressure, are kept constant.
The fluid pressure (true value) must be varied with another
instrument, in increments, over some range, causing the measured
value also to vary:
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Figure: Pressure Gage Calibration
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The average calibration curve is taken as a straight line of equation:
q0 = mqi + b
m and b can be obtained with the least-squares criterion. In this
example, m = 1.08 and b = −0.85kPa
The reading qo allow us to write an estimate of the true pressure as
qi = (qo + 0.85)/1.08
However, this value, obtained from the least-squares line, must have
some plus-or-minus error, given by the standard deviation sqi,
S(qi )2
= (1/N)
X (qo − b)
m
− qi
2
Sqi = 0.18kPa
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Thus if the reading from the gage is 4.32 kPa, our estimate of the
true pressure would be
qi ± 3Sqi = (4.32 + 0.85)/1.08 ± (3x0.18) = 4.79 ± 0.54kPa
(with a ±3Sqi limits)
Note: By taking an error of three times the standard deviation, we
ensure that the true pressure will be in the defined
range[4.79 − 0.54; 4.79 + 0.54] with a probability of 99.7%.
The total error of measurement is in two parts:
- the bias, 0.47 kPa (=4.79 - 4.32)
- the imprecision, 0.54 kPa
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2.2 Static Characteristics
1. accuracy is defined as the difference between the true value of the
measurand and the measured value indicated by the instrument.
The accuracy (lack of error) of an instrument can be evaluated in
terms of the concepts of precision and bias.
In practice, every device will produce an error on the measured
quantities.
Knowing the accuracy of that device will allow us to put bounds on
the error.
For absolute measurement accuracy is better
Typically, the true value is defined in reference to some absolute or
agreed upon standard.
For any particular measurement there will be some error due to
systematic (bias) and random (noise) error sources.
The combination of systematic and random error can be visualized by
considering the analogy of the target.
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2. Precision of a measuring instrument is the degree to which it
produces similar results for the same inputs on a number of occasions.
For relative measurement precision is better than accuracy
3. Systematic Error Sources (Bias): of a measurement is the part of
the error that can be removed by calibration. It has the same value
for all measurements.
4. The imprecision (or random error) is a type of error that is not
known precisely for a given measurement, just bounded.
It is different for every reading and we cannot remove it.
The total inaccuracy of the measuring instrument is defined by the
combination of bias and imprecision.
One class of cause factors are those that change the input output
response of a sensor resulting in miscalibration.
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The modifying inputs and interfering inputs discussed above can
result in sensor miscalibration.
Figure: Measurement accuracy and precision
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For example, if temperature is a modifying input, using the sensor at a
temperature other than the calibrated temperature will result in a
systematic error.
Such systematic error can be corrected by using compensation
methods.
There are other factors in sensor calibration resulting in systematic errors.
In some sensors, aging of the components will change the sensor
response and hence the calibration.
Damage or abuse of the sensor can also change the calibration. In
order to prevent these systematic errors, sensors should be
periodically re calibrated.
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Invasiveness: is interaction between measurement and measuring
device which causes systematic errors; however, in many cases, it can
be reduced to an insignificant level.
For example; Invasiveness would be to use a large warm
thermometer to measure the temperature of a small volume of cold
fluid. Heat would be transferred from the thermometer and would
warm the fluid, resulting in an inaccurate measurement.
Systematic errors can also be introduced in the signal path of the
measurement process.
Finally, systematic errors or bias can be introduced by human
observers when reading the measurement.
A common example of observer bias error is parallax error.
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Note: Invasiveness is a disturbance caused by the measurement process itself
It can be Physical ( e.g Temperature, Electrical (e.g change in resistance), Chemical change in
chemical reaction)

5. Random Error Sources (Noise):
Random error is sometimes referred to as noise, which is defined as a
signal that carries no useful information. If a measurement with true
random error is repeated a large number of times, it will exhibit a
Gaussian distribution.
The precision of the measurement is normally quantified by the
standard deviation that indicates the width of the Gaussian
distribution
There are a variety of sources of randomness that can degrade the
precision of the measurement starting with the repeatability of the
measurand itself.
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For better accuracy two or more sensors may be used to observe the
same variable in a single physical environment called sensor fusion.
The process of sensor fusion is modeled as shown in figure bellow,
sensors output signals are combined in some manner (typically in a
processor) to provide a single enhanced measurement.
Figure: Sensor fusion model
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Transducers or measurement systems are not perfect systems.
Mechatronics/Electromechanical design engineer must know the
capability and shortcoming of a transducer or measurement system to
properly assess its performance.
There are a number of performance related parameters of a
transducer or measurement system.
These parameters are called as sensor specifications.
Sensor specifications inform the user to the about deviations from the
ideal behavior of the sensors.
Following are the various specifications of a sensor/transducer system.
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6. Range
The range of a sensor/instrument indicates the limits between which
the input can vary. For example, a thermocouple for the measurement
of temperature might have a range of 25 − 225oC.
7. Span
The span is difference between the maximum and minimum values of
the input. Thus, the above-mentioned thermocouple will have a span
of 200oC.
8. Error
Error is the difference between the result of the measurement and the
true value of the quantity being measured.
A sensor might give a displacement reading of 29.8 mm, when the
actual displacement had been 30 mm, then the error is 0.2 mm.
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9. Sensitivity
is defined as the change in output of the instrument per unit change
in the parameter being measured.
The factor may be constant over the range of the sensor (linear), or it
may vary (nonlinear).
The sensitivity of the device is determined by the slope of the
calibration curve.
For example, a general purpose thermocouple may have a sensitivity
of 41V /oC
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10. Sensitivity to disturbance
All calibrations and specifications of an instrument are only valid
under controlled conditions of temperature, pressure etc.
These standard ambient conditions are usually defined in the
instrument specification.
As variations occur in the ambient temperature etc., certain static
instrument characteristics change.
The sensitivity to disturbance is a measure of the magnitude of this
change.
Such environmental changes affect instruments in two main ways,
known as zero drift and sensitivity drift. Zero drift is sometimes
known by the alternative term, bias.
Zero drift or bias describes the effect where the zero reading of an
instrument is modified by a change in ambient conditions.
This causes a constant error that exists over the full range of
measurement of the instrument
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11. Linearity
Linearity is normally desirable that the output reading of an
instrument is linearly proportional to the quantity being measured.
The nonlinearity indicates the maximum deviation of the actual
measured curve of a sensor from the ideal curve.
Figure: None linearity error
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12. Hysteresis
If the input measured quantity to the instrument is steadily increased
from a negative value, the output reading varies in the manner shown
in curve (A). If the input variable is then steadily decreased, the
output varies in the manner shown in curve (B).
The non-coincidence between these loading and unloading curves is
known as hysteresis
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Figure: hysteresis error
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Hysteresis is most commonly found in instruments that contain
springs, such as the passive pressure gauge and the Prony brake (used
for measuring torque).
It is also evident when friction forces in a system have different
magnitudes depending on the direction of movement, such as in the
pendulum-scale mass-measuring device.
Hysteresis can also occur in instruments that contain electrical
windings formed round an iron core, due to magnetic hysteresis in the
iron.
This occurs in devices like the variable inductance displacement
transducer, the LVDT and the rotary differential transformer.
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13. Dead space/time
The dead space of a transducer/measuring instrument is the range of
input values for which there is no output.
Dead space is defined as the range of different input values over
which there is no change in output value.
Any instrument that exhibits hysteresis also displays dead space.
The dead time of a sensor device is the time duration from the
application of an input until the output begins to respond or change.
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14. Resolution
Resolution is the smallest detectable incremental change of input
parameter that can be detected in the output signal.
It is the attitude or capacity of seeing or understanding small change
in input and show or display it on the out put side.
Resolution can be defined as the smallest change that can be
detected by a measuring instrument.
Resolution can be expressed either as a proportion of the full-scale
reading or in absolute terms.
For example, if a LVDT sensor measures a displacement up to 20 mm
and it provides an output as a number between 1 and 100 then the
resolution of the sensor device is 0.2 mm.
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15. Threshold
If the input to an instrument is gradually increased from zero, the
input will have to reach a certain minimum level before the change in
the instrument output reading is of a large enough magnitude to be
detectable.
This minimum level of input is known as the threshold of the
instrument.
Manufacturers vary in the way that they specify threshold for
instruments.
Some quote absolute values, whereas others quote threshold as a
percentage of full-scale readings. As an illustration, a car
speedometer typically has a threshold of about 15 km/h. This means
that, if the vehicle starts from rest and accelerates, no output reading
is observed on the speedometer until the speed reaches 15 km/h.
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16. Stability
Stability is the ability of a sensor device to give same output when
used to measure a constant input over a period of time. The term
drift is used to indicate the change in output that occurs over a
period of time. It is expressed as the percentage of full range output.
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17. Repeatability
Repeatability is the attitude to give an output measured values with
low distance among themselves in following transductions repeated:
- with the same input
- in the same/equivalent (and well defined) experimental conditions
with same instrument and observer, same location and same conditions
- in a short time interva.
18. Reproducibility
describes the closeness of output readings for the same input when
there are changes in the method of measurement, observer, measuring
instrument, location, conditions of use and time of measurement.
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19. Response Time
It is the time taken by a sensor to approach its true output when
subjected to a step input is sometimes referred to as its response time.
20. Tolerance
It is acceptable range of variations of metrologic characteristics
among a population nominally constant.
It is a term that is closely related to accuracy and defines the
maximum error that is to be expected in some value.
It is not, strictly speaking, a static characteristic of measuring
instruments.
It is mentioned here because the accuracy of some instruments is
sometimes quoted as a tolerance figure.
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2.3 Dynamic characteristics of instruments
The sensor response to a variable input is different from that
exhibited when the input signals are constant
The reason for dynamic characteristics is the presence of
energy-storing elements
Inertial: masses, inductances
Capacitances: electrical, thermal
Dynamic characteristics are determined by analyzing the response of
the sensor to a family of variable input wave forms: Impulse, step,
ramp, sinusoidal, white noise
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The dynamic response of the sensor is (typically) assumed to be linear.
For any LTI measuring system, it can be modeled by a
constant-coefficient linear differential equation
ak
dky(t)
dtk
+ ... + a2
d2y(t)
dt2
+ a1
dy(t)
dt
+ a0y(t)
In practice, these models are confined to zero, first and second order.
Higher order models are rarely applied
These dynamic models are typically analyzed with the Laplace
transform, which converts the differential equation into a polynomial
expression
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Think of the Laplace domain as an extension of the Fourier transform
Fourier analysis is restricted to sinusoidal signals
x(t) = sin(ωt) = e − jωt
Laplace analysis can also handle exponential behavior
x(t) = e−σt
sin(ωt) = e−σ+jωt
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2.3.1 The Laplace Transform (review)
The Laplace transform of a time signal y(t) is denoted by
L[y(t)] = Y(s)
The s variable is a complex number s = σ + jω
The real component defines the real exponential behavior
The imaginary component defines the frequency of oscillatory behavior
The fundamental relationship is the one that concerns the
transformation of differentiation
L

dy(t)
dt

= sY 9s) + y(0)
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sY(s) + Y(0)
sY(s) + Y(0)

Other useful relationships are
G(s) is called the transfer function of the sensor/instrument
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The position of the poles of G(s) -zeros of the denominator - in the
s-plane determines the dynamic behavior of the measuring
instrument. such as:
Oscillating components
Exponential decays
Instability
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Pole location and dynamic behavior
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2.3.2 Zero-order instruments
Input and output are related by an equation of the type
L[y(t) = k.x(t)] = Y (s) = kX(s) =
Y (s)
X(s)
= k
Zero-order is the desirable response of a sensor/instrument
No delays
Infinite bandwidth
The sensor only changes the amplitude of the input signal
Zero-order systems do not include energy-storing elements
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2.3.2 Zero-order instruments
Example: A potentiometer used to measure linear and rotary
displacements
This model would not work for fast-varying displacements
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2.3.2 First-order instruments
Inputs and outputs related by a first-order differential equation
L

a1
dy(t)
d(t)
+ a0y(t) = x(t)

=
Y (s)
X(s)
=
1
a1s + a0
=
k
τs + 1
First-order sensors have one element that stores energy and one that
dissipates it
Step response
y(t) = Ak(1 − e−t/τ )
A is the amplitude of the step
k(= 1/a0) is the static gain, which determines the static response
τ(= a1/a0) is the time constant, which determines the dynamic
response
Ramp response
y(t) = Akt − Akτu(t) + Akτe−t/τ )
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2.3.2 First-order Instruments
Frequency response is described by the amplitude and phase shift
plots
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Example of a first-order sensor
What type of input was applied to the sensor?
Parameters
C: Thermal capacitance of the mercury
R: thermal resistance of the glass to heat transfer
θF : temperature of the fluid
θ(t): temperature of the thermometer
The system equivalent circuit is an RC network
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Example of a first-order sensor
Heat flow through the glass (θF − θ(t)/R
Temperature of the thermometer rises as dθ(t)
dt = θF −θ(t)
RC
sΘ(s) =
ΘF (s) − Θ(s)
RC
= (RCs + 1)Θ(s) = ΘF (s)
Θ(s) =
ΘF (s)
(RCs + 1)
= θ(t) = ΘF

1 − e−t/RC

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2.3.3 Second-order Instruments
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Second-order step response
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Second-order ramp response
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Second-order Frequency response
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Example of second-order sensors
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