1. MECH 373 Instrumentation and Measurement
Lecture 3
(Course Website: Access from your “My Concordia” portal)
Contents of today’s lecture:
• Dynamic measurements
– Zero order, first order, second order systems
– Time constant, response time, rise time, settling time
– Frequency response
• Experimental design

2. Dimensions and Units

3. Calibration • Calibration:
A test in which known values of the input are applied to a
measurement system (or sensor) for the purpose of observing the system (or sensor) output.
• Static calibration:
A calibration procedure in which the values of the
variable involved remain constant (do not change with time).
• Dynamic calibration:
When the variables of interest are time dependent and
time-based information is need. The dynamic calibration
determines the relationship between an input of known dynamic behavior and the measurement system output.

4. Static Calibration Curve
Sensitivity is defined as the ratio of the change in the magnitude of the output to the change in the magnitude of measurand. That is:
In other words, the slope of a calibration curve provides the sensitivity of the measurement system
Static Sensitivity: The slope of a static calibration curve. It relates changes in the indicated output to changes in the input

5. Least-Squares Regression
Given a set of N data points of the form (x,y), the least-squares regression yields the mth order polynomial
that minimizes the sum of the squares of the deviations
For example, the best straight line fit in the least squares sense is obtained for the case m=1, for which

6. Least-Squares Regression
To find the values of that minimize we compute the derivatives and set them equal to zero

7. Dynamic Calibration
We have seen so far how to obtain the input-output relationship of an instrument when the input is held constant, neglecting the measurement system dynamics. This process is called static calibration
However, when the input variables of interest are time dependent and time-based information is sought, we need to consider the system dynamics
A dynamic calibration determines the relationship between an input of known dynamic behavior and the measurement system output
Usually such calibrations involve either a sinusoidal signal or a step change as the known input signal
To perform a “static calibration” we use a piecewise-constant function as input. The measurements can only be performed when the system reaches steady state

8. Static vs. Dynamic Calibration
Example: What is the difference between static and dynamic calibration? What type of calibration would you recommend for
(1) an oral thermometer
(2) a pressure gage used in a water line, and
(3) a car speedometer? Explain your reasoning.

9. Dynamic Measurements
If a measurand is unchanging in time and if the measurement system instantaneously shows an equilibrium response to the measurand, the measurement system is siad to be static.
However, in the general case, when the measurand is changing in time and the measuring system does not show instantaneous response, the measurement process is said to be dynamic.
In making dynamic measurements, we must acount for the dynamic charactristics of the measuring system, the dynamic interaction between the measuring system and the test system, and the dynamic charateristics of the test system.

10. Dynamic Measurements
The dynamic respnse of a measurment system can usually be placed into one of three categories: zero order, first order and second order.
In the following slides, the dynamic chracteristics of different measurement systems are discussed.

11. General Model For A Measurement System
nth Order ordinary linear differential equation with constant coefficient
F(t) = forcing function
Where m ≤ n
y(t) = output from the system
x(t) = input to the system
t = time
a’s and b’s = system physical parameters, assumed constant
Measurement
system
x(t)
y(t)
y(0)
The solution
Where yocf = complementary-function part of solution
yopi = particular-integral part of solution

12. Complementary-Function Solution
The solution yocf is obtained by calculating the n roots of the algebraic characteristic equation
Characteristic equation
Roots of the characteristic equation:
Complementary-function solution:
1. Real roots, unrepeated:
2. Real roots, repeated:
each root s which appear p times
3. Complex roots, unrepeated:
the complex form: a  ib
4. Complex roots, repeated:
each pair of complex root which appear p times

13. -- undetermined coefficients, variation of parameters etc.
Particular Solution
-- undetermined coefficients, variation of parameters etc.
Method of undetermined coefficients:
Where f(t) = the function that describes input quantity
A, B, C = constant which can be found by substituting yopi into ODEs
Important Notes 
After a certain-order derivative, all higher derivatives are zero.
After a certain-order derivative, all higher derivatives have the same functional form as some lower-order derivatives.
Upon repeated differentiation, new functional forms continue to arise.  

14. Zero-Order Systems All the a’s and b’s other than a0 and b0 are zero.
where K = static sensitivity = b0/a0
The behavior is characterized by its static sensitivity, K and remains constant regardless of input frequency (ideal dynamic characteristic). xm Vr +
Where 0  x  xm and Vr is a reference voltage
y = V
x = 0 -
A linear potentiometer used as position sensor is a zero-order sensor.

15. First-Order Systems
All the a’s and b’s other than a1, a0 and b0 are zero.
where K = b0/a0 is the static sensitivity
 = a1/a0 is the system’s time constant

16. First-Order Systems: Step Response
Assume for t < 0, y = y0 , at time = 0 the input quantity, x increases instantly by an amount A. Therefore t > 0
The complete solution:
yocf
yopi
Transient response
Steady-state response
Applying the initial condition, we get C = y0-KA, thus gives

17. First-Order Systems: Step Response
Here, we define the term error fraction as
Non-dimensional step response of first-order instrument

18. Determination of Time Constant
0.368
Slope = -1/

19. First-Order Systems: Frequency Response
From the response of first-order system to sinusoidal inputs,
we have
The complete solution:
Transient response
Steady state response =
Frequency response
If we do interest in only steady-state response of the system, we can write the equation in general form
Where B() = amplitude of the steady-state response and () = phase shift

20. First-Order Instrument: Frequency Response
The amplitude ratio
The phase angle is
Dynamic error
-3 dB
0.707
Cutoff frequency
Frequency response of the first-order system
Dynamic error, () = M(): a measure of an inability of a system to adequately reconstruct the amplitude of the input for a particular frequency

21. Dynamic Characteristics
Frequency Response describe how the ratio of output and input changes with the input frequency. (sinusoidal input)
Dynamic error, () = 1- M() a measure of the inability of a system or sensor to adequately reconstruct the amplitude of the input for a particular frequency
Bandwidth the frequency band over which M()  (-3 dB in decibel unit)
Cutoff frequency: the frequency at which the system response has fallen to (-3 dB) of the stable low frequency.

22. Dynamic Characteristics
Example: A first order instrument is to measure signals with frequency content up to 100 Hz with an inaccuracy of 5%. What is the maximum allowable time constant? What will be the phase shift at 50 and 100 Hz?
Solution:
Define
From the condition |Dynamic error| < 5%, it implies that
But for the first order system, the term can not be greater than 1 so that the
constrain becomes
Solve this inequality give the range
The largest allowable time constant for the input frequency 100 Hz is
The phase shift at 50 and 100 Hz can be found from
This give  = -9.33o and = o at 50 and 100 Hz respectively

23. Second-Order Systems = the static sensitivity
In general, a second-order measurement system subjected to arbitrary input, x(t)
The essential parameters
= the static sensitivity
= the damping ratio, dimensionless
= the natural frequency

24. Second-Order Systems Consider the characteristic equation
This quadratic equation has two roots:
Depending on the value of , three forms of complementary solutions are possible
Overdamped ( > 1):
Critically damped ( = 1):
Underdamped (< 1): :

25. Second-Order Systems Case I Underdamped (< 1):
Case 2 Overdamped ( > 1):
Case 3 Critically damped ( = 1):

26. Second-order Systems Example: The force-measuring spring
consider a spring with spring constant Ks under applied force fi and the total mass M. At start, the scale is adjusted so that xo = 0 when fi = 0;
the second-order model:

27. Second-order Systems: Step Response
For a step input x(t)
With the initial conditions: y = 0 at t = 0+, dy/dt = 0 at t = 0+
The complete solution:
Overdamped ( > 1):
Critically damped ( = 1):
Underdamped (< 1): :

28. Second-order Instrument: Step Response
Ringing frequency:
 = 0
Ringing frequency:
0.25
Rise time decreases  with but increases ringing
0.5
Optimum settling time can be obtained from  ~ 0.7
1.0
Practical systems use 0.6<  <0.8
2.0
Non-dimensional step response of second-order instrument

29. Dynamic Characteristics
overshoot
100%  5%
settling
time
rise time
Typical response of the 2nd order system

30. Second-order Instrument: Frequency Response
The response of a second-order to a sinusoidal input of the form x(t) = Asint
where
The steady state response of a second-order to a sinusoidal input
Where B() = amplitude of the steady state response and () = phase shift

31. Second-order Instrument: Frequency Response
The amplitude ratio
The phase angle
 = 0.1
0.3
0.5
1.0
2.0
0 = 0.1
0.3
0.5
1.0
2.0
Magnitude and Phase plot of second-order Instrument

32. Second-order Systems
For overdamped ( >1) or critical damped ( = 1), there is neither overshoot nor steady-state dynamic error in the response.
In an underdameped system ( < 1) the steady-state dynamic error is zero, but the speed and overshoot in the transient are related.
Rise time:
overshoot Td
Maximum overshoot:
Peak time:
peak
time
Resonance
frequency:
settling
time
Resonance
amplitude:
rise time

33. Dynamic Characteristics
Speed of response: indicates how fast the sensor (measurement system) reacts to changes in the input variable. (Step input)
Rise time: the length of time it takes the output to reach 10 to 90% of full response when a step is applied to the input
Time constant: (1st order system) the time for the output to change by 63.2% of its maximum possible change.
Settling time: the time it takes from the application of the input step until the output has settled within a specific band of the final value.
Dead time: the length of time from the application of a step change at the input of the sensor until the output begins to change

34. Experimental Design
Experimental design is the first step in any measurement experiment. It involves developing a measurement test plan following three steps:
Parameter Design Plan – test objective and identification of process variables and parameters and a means for their control. You should ask:
What question am I trying to answer ?
What variables to be measured ?
What variables will affect my results ?
System and Tolerance Design Plan – selection of measurement technique, equipment and test procedure based on some preconceived tolerance limits for error. You should ask:
How will I do the measurement and how good do the results have to be ?

35. Experimental Design (cont’d)
3. Data Reduction Design Plan – Plan ahead on how to analyze, present, and use the anticipated data. You should ask:
How will I interpret the resulting data ?
How will I use the data to answer my question ?
Step 1 needs the following important concepts:
Variables
Parameters
Noise and Interference
Random Tests
Replication and Repetition
Concomitant Methods

36. Variables
One variable in the measurement system is obviously the targeted measured variable, but there might be other variables that affect the outcome
All known process variables should be listed and evaluated for any possible cause and effect relationships
A variable that can be changed independently of other variables is known as an independent variable
A variable that is affected by changes in one or more other variables is known as a dependent variable
Variables that cannot be controlled during measurements but that affect the value of the measured variables are called extraneous variables.
If the values of a variable can be enumerated it is called discrete. Otherwise it is called continuous

37. Variables
Example: For the following calibration system. Identify the independent, dependent, and possible extraneous variables.
Answer:
Independent variables: piston displacement x, temperature T
Dependent variables: gas pressure p
Extraneous variables: noise effects due to room temperature; line voltage variations, connecting wires

38. Parameters A parameter is a function relationship between variables
A parameter that has an effect on the behavior of the measured variable is called a control parameter
A control parameter is completely controlled if it can be set and held at a constant value during a set of measurements
Example (Pendulum):
period
Ratio of l and g is a completely controlled parameter
because l is fixed and g is known

39. Noise and Interference
The way of extraneous variables affects measured data can be classified into noise and interference
Noise is a random variation of the value of the measured signal as a consequence of the variation of the extraneous variables
Interference produces undesirable deterministic trends on the measured value because of extraneous variables
Example:

40. Random Tests
It is important to minimize the effects of extraneous variables in a measurement using random tests
A random test is defined by a measurement matrix that sets a random order in the value of the independent variable applied to measure the dependent variable
Example:
Extraneous
Variable
Pressure
Transducer
+ Voltmeter
Independent
Varables
Dependent
Variable
Apply Random Sequence
(holding temperature constant)

41. Random Tests (cont’d)
Assuming that we can hold the temperature fixed, applying a random sequence of volume values rather than a sequential sequence will allow us to average out the effects of extraneous variables
We will see later that when the measurement system is subject to hysteresis, applying an increasing sequence of values for the independent variable produces a different result as compared to the case of applying the same set of values in a decreasing order
A random sequence enables us to deal with this effect by providing a unique value of the independent variable for each value of the dependent variable in the random sequence
After applying several random sequencies the results are averaged out to find a best-fit curve

42. Random Tests (cont’d)
We have seen that random tests are effective for the local control of extraneous variables that change in a continuous manner
Discrete extraneous variables can also be dealt with by performing a random test
The use of different instruments and different test operators are examples of discrete extraneous variables that can affect the outcome of a measurement
These effects are usually reduced by randomizing a test matrix by using random blocks
A block consists of a data set of the measured variable in which the control variable is varied but the extraneous variable is fixed
The extraneous variable is then varied between blocks

43. Example: Randomized Matrix
The manufacture of a particular composite material requires mixing a percentage by weight of binder with resin to produce a gel. The gel is used in a lay-up procedure to impregnate the fiber. The strength will depend on both the percent binder in the gel and the test operator performing the lay-up.
Formulate a test matrix to find the percent binder influence on strength
Solution: Pick three different bider-gel ratios A, B, C and three typical operators to produce N separate composite test samples for each of the 3 ratios
Create the following test pattern:

44. Replication and Repetition
In general the estimated value of a variable improves with the number of measurements. The mean over measurements is taken as the estimated value
Repeated measurements made during any single test run or on a single batch are called repetitions. It allows for quantifying the variation in a measured variable as it occurs during any one test or batch while the operating conditions are held under nominal control
An independent duplication of a set of measurements using similar operating conditions is referred to as a replication. It allows for quantifying the variation in a measured variable as it occurs between different tests, each having the same nominal values of operating conditions

45. Concomitant Methods
A good strategy is to incorporate the use of concomitant methods in a measurement plan. The goal is to obtain two or more estimates for the result, each based on a different method, which can be compared as a check for agreement
Example: Establish the volume of a cylindrical rod of known material.
Method 1: Measure the diameter and length
Method 2: Measure the weight and compute volume based on specific weight of material