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

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. 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

6. 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.
(4) Accelerometers.
(5) Strain gauges.

7. 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.

8. 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 account 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.

9. Dynamic Measurements
The dynamic response 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.

10. Zero order systems Zero order:
Systems response instantly to measurands, no instrument is trully zero order.
e.g. Linear potentiometer used as position sensor. 10

11. 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.

12. First order and second order systems12

13. First order systems
Mechanical analogs to capacitance – springs or devices that stored thermal energy – e.g. Common thermometer.
First order equation : (Response A)
[y(t) – y(0)]/[(ye) – y(0)] = 1 – e(-t/)
where y(t) – response at time t,
y(0) – initial response.
ye - steady state response.
 - time constant (s) – determine the curve – Find by t =  , y(t)/ye = 13

14. First-Order Systems: Step Response
Here, we define the term error fraction as: (KA = ye)
Non-dimensional step response of first-order instrument 14 14

15. Determination of Time Constant – KA = ye
0.368
Slope = -1/ τ t 15 15

16. Second order systems
Sping/mass systems – Load cells, pressure transducers, accelerometers.
Have damping – dissipate energy.
Inf only :
Low damping – underdamped – oscillation.
Overdamped – No oscillation.
Critical damping – No oscillation. 16

17. 1st and 2nd order systems’ representation
Response time – time to achieve a fraction of steady response from initail condition – 95% response time = 0.95 of y/ye. (Ensure overshoot passed for 2nd order system.)
Rise time – time requied y/ye to increase from 0.1 to 0.9 .
Settling time – time until the oscillation is less than some fraction of steady state amplitude, say 5% = 95% of steady state amplitude.
Minimal dynamic error – small settling time (response time). 17

18. 1st and 2nd order systems
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.
Speed of response - indicates how fast the sensor (measurement system) reacts to changes in the input variable. (Step input). 18

19. 2nd orders - Dynamic Characteristics
overshoot
100%  5%
settling
time
rise time
Typical response of the 2nd order system 19 19

20. First order system example
To calculate rise time (10% y/ye to 90% ) and 90% response time.
Initial thermometer temperature =20C.
Thermometer immersed into water, its temperature = 80C.
Thermometer time constant = 4 seconds.
Solution:
y/ye = (T – Ti)/(T – Ti) = 1 – et/
y/ye = 0.1 = 1 – et1/4 ; t1 = 0.42s.
y/ye = 0.9 = 1 – et2/4 ; t2 = 9.21s.
Rise ∆T = t2 – t1 = 8.79s.
90% Response time = 9.21s. 20

21. First order system example21

22. Frequency response Useful dynamic response.
Output determined by pure sine wave input. Repeat for a range of frequencies. See figure.
Bandwidth – constant ratio of output and input amplitude – good area.
Outside – large systematic errors.
Ensure dynamic response frequency much less than the device natural frequency (transducer), say 0.2 to 0.4n, but n influence by mounting. 22

23. Frequency response 23

25. 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()  (output/input) (20log = -3 dB in decibel unit)
Cutoff frequency: the frequency at which the system response has fallen to (-3 dB) of the stable low frequency.