1. Chapter 2: Performance Characteristics of Instruments
Introduction to Instrumentation Engineering
Chapter 2: Performance Characteristics of Instruments
By Sintayehu Challa

2. 11/17/2018
Goals for this Chapter
Differentiate the difference between steady state and transient behaviors of instruments
They describe performance characteristics of an instruments
Identify terminologies used to define static and dynamic performance
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

3. Overview Static characteristics Dynamic characteristics
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

4. Performance Characteristics
Type of instrument to be used is decided on the characteristics required
E.g., a  0.5oC accuracy instrument is allowable for human body (feeling) while it may be useless for an instrument in a control system
So for selection, the performance characteristics of measuring instruments must be known
Various terminologies are used to define the performance
Instrument performance characteristics is generally broken down into two, namely
Static characteristics
Dynamic characteristics
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

5. Static Characteristics
Static characteristics describes performance of instruments under constant or slowly-varying input
Concerned with steady-state reading, i.e., when the instrument settles down
Since static characteristics affects the dynamic behavior, the overall performance is then judged by a semi-quantitative superposition of the static and dynamic characteristics
Terminologies used to define static characteristics
Range, span, linearity, sensitivity, threshold, resolution, error, etc . . .
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

6. Range and Span Range: Is the limits between which the input can vary
Range of an instrument defines the minimum and maximum values of a quantity that the instrument is designed to measure, i.e.,
Input range – Imin , Imax
Output range – Omin , Omax
E.g., a resistance thermometer sensor might be quoted as having a range of 00C to +8000C
Span: Is the maximum variation
Input span – Imax – Imin
Output span – Omax – Omin
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

7. Linearity
The output reading of an instrument is linearly proportional to the quantity being measured
Input output relation is governed by a straight line
One desirable property of instruments
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

8. Sensitivity
Ratio of the change in output to the corresponding change in input under steady-state conditions
Is the slope of the input-output curve
Indicates by how much the output of an instrument changes when the quantity being measured changes by a given amount
The sensitivity can be linear or non-linear
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

9. Threshold and Resolution
If the instrument’s input is increased very gradually from zero, there will be some minimum value below which no output change can be detected
This minimum value defines the threshold of the instrument
Manufacturers specify it as an absolute value or percentage of full scale reading
Resolution
Smallest possible increment discernible between measured values
Or the minimum input change that can be detected by the system
As the term is used, higher resolution means smaller increments
An instrument with a five-digit display (say, to ) is said to have higher resolution than an identical instrument with a three-digit display (say, 0.00 to 9.99)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

10. Hysteresis
This is an effect of producing different readings when the measured quantity is approached from above or below
Instrument will not have the same output for the same input in repeated trials
It may be the result of mechanical friction, or thermal effects
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

11. Accuracy
The maximum expected difference in magnitude between measured and true values
Accuracy of an instrument is a measure of how close the output reading of the instrument is to the true/correct value
Equivalently, accuracy is the extent to which the value indicated by a measurement system or element might be wrong
Often expressed as a percentage of full scale reading
E.g., A 1% accuracy over a full scale pressure reading of 100 kPa will be accurate within 1 kPa
Plus or minus inaccuracies are also termed as measurement uncertainties
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

12. Precision
Indicates the ability of an instrument to reproduce a certain reading of a constant input with a given accuracy
If no reproducibility, then the instrument is said to have a drift
E.g., for a true value of 100 V, if measured values are 104, 103, 105, 100, 105, then the accuracy is 5 V
Precision is also defined as the maximum deviation from mean
In the above example, mean = so that max. deviation = 1.6 V
Hence precision is 1.6%
High precision measurement instrument gives a small spread of readings and vice versa
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

13. Accuracy vs. Precision
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

14. System Disturbances (or Environmental Effects)
Zero drift or bias or interfering input
Describes the effect where the zero reading (or intercept) of an instrument is modified by a change in the ambient conditions
Causes a constant error that exists over the full range of measurement of the instrument
Sensitivity drift or scale factor drift or modifying input
Defines the amount by which an instrument’s sensitivity varies as ambient conditions change
It is quantified by sensitivity drift coefficient
Introduction to Instrumntaion Eng‘g – Ch.2 Performance Char. of Instruments

15. Sensitivity to Disturbance
Instrument’s specifications are valid only under controlled conditions of temperature, pressure etc…
As variations occur in the ambient temperature etc., certain static instrument characteristics change
E.g., sensitivity can be affected by zero and sensitivity drift as shown in the figure
Sensitivity to disturbance measures the magnitude of this change
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

16. Sensitivity to Disturbance …
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

17. Error Two types of errors, namely systematic and random errors
Systematic error: Cause repeated readings to be in error by the same amount, i.e., consistent error signs
Due to instrument short coming and environmental effects
Related to calibration errors and can be eliminated by correct calibration
Accuracy is related to such type of errors
Random errors: Caused by random electronic fluctuations in instruments, unpredictable behavior of the instrument, influences of friction, etc…
Such errors are related to precision
Characterized by positive and negative errors
Random fluctuations usually follow certain statistical distribution
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

18. Error
Difference between result of measurement and true value of the quantity being measured
E.g., If the measured value is 10.1 when the true value is 10.0, the error is If the measured value is 9.9 when the true value is 10.0, the error is-0.1.
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

19. Tolerance
A term that is closely related to accuracy and defines the maximum error that is to be expected in some value
When used correctly, tolerance describes the maximum deviation of a manufactured component from some specified value
E.g., crankshafts are machined with a diameter tolerance quoted as so many microns (10-6m)
Electric circuit components such as resistors have tolerances of perhaps 5%
One resistor chosen at random from a batch having a nominal value 1000 ohm and tolerance 5% might have an actual value anywhere between 950 ohm and 1050 ohm
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

20. Overview Static characteristics Dynamic characteristics
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

21. Dynamic Characteristics of Instruments
Describe behaviors between the time an input quantity changes its value and the time when the instrument output attains a steady value
Are useful when the input signal is rapidly varying
Used to study performance under transient conditions
In general, both static and dynamic characteristics are important to characterize a given instrument
Ordinary linear differential equation with constant coefficients is the most widely used mathematical model to study dynamic response in the form
Usually linear, time-invariant (LTI) system assumed
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

22. Dynamic Characteristics …
In an LTI system, input and output, for time t > 0, are related as:
where qi is the measured (input) quantity, q0 is the output reading and a an, b bm are constants
Using the method of Laplace-transform, equation (2.1) is Or
(2.1)
(2.2)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

23. Dynamic Characteristics ….
If we limit consideration to step changes in the measured quantity only, then Equation (2.1) reduces to:
So that
Based on the order of Equation (2.3), equipments are classified as zero-, first-, and second-order instruments
(2.3)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

24. Zero-order Instrument
When all the coefficients a an other than a0 and bo are assumed to be zero, Equation (2.3) then degenerates into
Any instrument that closely obeys Equation (2.4) is defined to be a zero-order instrument
The two constants can be combined to give
where K=bo/ao is static sensitivity
From Equation (2.4), no matter how qi might vary with time, the output follows it perfectly with no distortion or time lag
Zero-order instrument represents ideal or perfect dynamic performance
(2.4)
(2.5)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

25. Zero-order Instrument …
Example: Displacement measuring potentiometer
Given linear distribution of resistance along length L, the output voltage eo can be written as
Measurement error em = Kqi - eo = o (ideal)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

26. First-order Instrument
If all the coefficients a an except for a0, a1, and b0 are assumed zero in equation (2.3) then:
Using the Laplace transform and rearranging, we get
Defining K = b0/a0 as the static sensitivity and =a1/a0 as the time constant of the system, equation (2.7) becomes
(2.6)
(2.7)
(2.8)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

27. First-order Instrument …
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First-order Instrument …
Examples of first-order instruments
Temperature measurement system
Amplifiers
Electromechanical and electronic meters
Graphical recorders
CRO (RC and RL network of CRO)
The dynamic behavior can be studied with the response of the system due to standard test inputs
Example: Impulse, step, ramp, sinusoidal inputs
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

28. First-order Instrument …
If equation (2.8) is solved analytically, the output q0 in response to a step change in qi at time t is shown in the figure below
Time constant  is the time taken for the output quantity q0 to reach 63% of its final value
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

29. Performance Parameters
Dynamic characteristics that are useful in characterizing the speed of response of any instrument include
Rise Time: Time required for a response to reach 90% of the step input (final value)
Settling time: Time to reach and stay within a stated  tolerance value around its final value
Knowing fast response requires a small value of 
Need to know which parameters to vary to reduced settling time
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

30. Exercise From the First order RC and RL SERIES NETWORK
A) Find the input and output Equation?
B) Find the time constant and static sensitivity factor?
C) Find the Transfer and System function?
2. Determine the step response of First order system and sketch the system including all parameter?
3. Repeat Exercise 2 for the Ramp response?
4. Repeat Exercise 2 for the Sinusoidal signal?
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments

31. Second-order Instrument
If all coefficients a an other than a0, a1 and a2 in equation (2.2) are assumed zero, then we get:
Applying Laplace transform and rearranging:
Defining K (static sensitivity), ωn (un-damped natural frequency) and  (damping ratio) as
Equation (2.9), in terms of K, ω and , becomes
(2.9)
(2.10)
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

32. Second-order Instrument …
If equation (2.10) is solved analytically, the shape of the step response depends on the value of 
= 0 is no damping case and constant amplitude oscillations
= 0.2, we is still oscillatory response, but the oscillations gradually die down
When  is increased further, oscillations reduces and overshoot (see curves (C) and (D))
Overdamped response as shown by curve (E)
Output reading creeps up slowly towards the correct reading
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

33. Second-order Instrument …
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Second-order Instrument …
= 1: critically damped
<1: under damped
>1: Over damped
Exercise:
1. Determine the step and sinusoidal response analytically the second order system.
2. Consider a series RLC circuit and find the Transfer function and all constants?
Example of 2nd order system
The d’Arsonval (permanent magnet moving coil) movement
Introduction to Instrumntaion Eng‘g – Ch.2 Performance Char. of Instruments

34. d’Arsonval Movement Used in analogue voltmeter
It consists of a rectangular coil wound round a soft iron core that is suspended in the field of a permanent magnet
The signal being measured is applied to the coil and this produces a radial magnetic field
Interaction between this induced field and the field produced by the permanent magnet causes a torque
Torque results in rotation of the coil
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

35. Dynamic characteristics of Measurement System
Measurement Systems especially in industrial, aerospace and Biological applications are subject to inputs which are not static but dynamic in nature i.e. the inputs vary with time and also the output vary with time.
The Dynamic characteristics of any measurement system described by
Speed of Response
Measuring Lag
Fidelity
Dynamic Error
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments

36. Terms used for dynamic characteristics
Response time: Time elapsed between an input is applied and the time in which the system gives an output corresponding to some specified percentage, e.g. 95%, of its final value
Rise time: Time taken for the output to rise to some specified percentage of the steady-state output. Often the rise time refers to the time taken for the output to rise from 10% of the steady-state value to 90 of the steady-state value.
Settling time: This is the time taken for the output to settle to within some percentage, e.g. 2%, of the steady-state value
Introduction to Instrumntaion Eng‘g - Ch. 2 Performance Char. of Instruments

37. Contd.
4. Measuring Lag:-An instrument does not immediately react a change in output
Measuring Lag is defined as the delay in the response of an instrument to a change in a Measuring quantity.
Two types of Measuring Lag
Retardation type:-In this case the response of the instrument begins immediately after a change in the measured has occurred.
Time Delay type:-In this case the response of the system begins after a “Dead Time” that means after the application of the input.
5. Fidelity of a measurement system is defined as the ability of the system to reproduce the output in the same variation of the input.
In Fidelity measurement system , there is no time lag or Phase shift between the input and output
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments

38. Contd.
6. Dynamic Error is the difference between the measured value of the instrument changing with time and the value indicated by the instrument if no static error is assumed. 7. Standard signals of Instrumentation system Dynamic behavior of measurement systems can be studied with the help of certain standard signals. These standard signals are Step Input, Ramp Input and Sinusoidal input
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments

39. Example
A Step input of 5A is applied to the Analogue current meter. the Analogue current meter pointer swings to 5.18A and finally comes to rest at 5.02A.
Determine the Overshoot of the reading in Ampere and in percentage of final reading
Determine the percentage error of the instrument.
Given
ISTEP= 5 A
I1A= 5.18 A
I2A= 5.02 A
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments

40. Solution The overshoot of the reading
Overshoot Reading= I1A - I2A =5.18 A – 5.02 A =0.16 A
B) The percentage of Errors becomes
Instrum. & Control Eng. for Energy Systems - Ch. 3 Performance Char. of Instruments