The lecture discusses measurement system behavior and modeling. It aims to understand how systems respond to different input signals and how instruments modify measurements. Models of zero-order, first-order, and second-order systems are presented. A first-order model is developed to simulate a thermometer's temperature response over time. Second-order systems can oscillate at their natural frequency depending on damping. Characterizing system order, time constants, damping, and frequency response allows predicting instrument behavior.

1. Lecture 3
Measurement System Behavior
Goals of the lecture
• Understand how the system responds to a variety of
input signals
• Understand how the instrument being used modifies
the measurand
• Characterize the behavior of 0th, 1st and 2nd order
systems
• Use the response of first and second order systems to
predict the behavior of simple instruments and
transducers
• Use transfer functions to describe system behavior

2. Example
Your boss asks you to find the temperature of the
liquid in a tank which is part of a manufacturing
process. Using a mercury thermometer, you record
the temperatures over several minutes – what value
do you report to her?

3. The Problem
For every sensor (as well as every part of the sensor system),
we need to describe how that portion of the system responds
to the imposed input

4. General Model for a Measurement System
Most measurement system dynamic behavior can be characterized by a linear
ordinary differential equation of order n:
where:

5. • System response is determined by mechanical elements (mass, stiffness, and damping)
and electrical elements (resistance, inductance, and capacitance) that form the components
of most measurement system.
– Sensors may use mechanical elements (e.g., a spring of known stiffness can be used to
determine static force by measuring spring deflection) or electrical elements (change of
resistance to measure strain as in a strain gage), or most commonly, a combination of both.
– Electrical filters, amplifiers, and other electronics use electrical elements to eliminate
noise, remove “DC” signals, or to boost signal level.
• No element is “pure” - our spring also has mass that will restrict its ability to determine
dynamic forces (i.e., the spring’s motion creates an additional force due to its
acceleration).
• Elements interact - for example, the spring’s stiffness and mass create its natural
frequency
•All systems possess damping (from friction, viscosity, inductance, etc.) that is usually
helpful

6. Zero-Order System
The simplest model of a system is the zero-order system, for which all the derivatives
drop out:
y = K F(t)
K is measured by static calibration.
Used for static measurements only - cannot be used dynamically
A tire gauge determines unknown pressure by measuring the deflection of a spring.

7. First-Order Systems
Example
Suppose a bulb thermometer originally indicating 20ºC is suddenly exposed to a fluid
temperature of 37 ºC. Develop a simple model to simulate the thermometer output
response.
The rate at which energy is exchanged between the sensor and the
environment through convection, , must be balanced by the
storage of energy within the thermometer, dE/dt.
For a constant mass temperature sensor,
This can be written in the form
dividing by hAs
This equation can be written in the
form first order differential equation:
or

8. First-Order Systems (Cont.)
The solution to the first ordinary differential equation:
The output response of the system to unit step input
τ is the time required for the output to achieve 63% of the difference between the initial
and final values
or
Rise time, settling time – these are instrumentation specs (e.g., rise time = 15 ms, 5%
settling time = 30 ms)

9. Harmonically Excited First–Order System
First-Order Systems (Cont.)
The angular frequency ω(Ω) of the input and output are the same, but the amplitude
(and phase lag) of the output depend on ωτ.
We can therefore describe the entire frequency response characteristics in terms of a
magnitude ratio and a phase shift
Note: τ is the only system characteristic which affects the frequency response

10. The amplitude is usually expressed
in terms of the decibel dB = 20 log10
M(ω).
First-Order Systems (Cont.)
The frequency bandwidth of an
instrument is defined as the frequency
below which M(ω)=0.707, or dB = -3 ("3
dB down").
First order systems act as low pass
filters, in other words they
attenuate high frequencies.
A useful measure of the phase shift is the
time delay of the signal:
Zero phase lag is ideal, but a linear phase
lag is often acceptable

11. Second-Order Systems
Example of a second order system
where ωn is the natural frequency and ζ (zeta)
is the damping ratio.
or

12. Homogeneous Solution
The form of the homogeneous solution depends on the roots of the characteristic
equation
The particular solution will depend on the forcing
function F(t).

13. Unit Step Function Input
For an underdamped system, 0 < ζ < 1,
F(t) = AU(t),
For a critically damped system
(ζ =1) the solution is:
For an over-damped system (ζ >1)
the solution is:

14. For under-damped systems, the output oscillates at the ringing frequency ωd
and and
Rise Time
By definition it is the time required for the system to achieve a value of 90% of the step
input. The rise time is decreased by decreasing the damping, Obviously there is a tradeoff
between fast response and ringing in a second order system.
Settling Time
The settling time is defined as the time required for the system to settle to within ±10% of
the steady state value.

15. Frequency Response
If F(t) = A sin ωt, the solution is given by
The first term is a transient which will
eventually die out
the steady-state
response
where
and
Resonance Frequency
Under-damped second order systems may
resonate or oscillate at a greater magnitude
than the input, M(ω) > 1.
Systems with a damping ratio greater than ζ >
0.707 do not resonate.