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7 Strain, Force, Pressure and Flow Measurements

7 Strain, Force, Pressure and Flow Measurements

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24.21
A bonded strain gauge consists of a thin wire or conducting film arranged in a
coplanar pattern and cemented to a base or carrier. The basic form of this type
of gauge is shown below:
A bonded strain
gauge

FORCE

Small surface area
Low leakage
High isolation

FORCE

Figure 24.16 – A Bonded Strain Gauge
The strain gauge is normally mounted so that as much as possible of the length
of the conductor is aligned in the direction of the stress that is being measured.
Lead wires are attached to the base and brought out for interconnection.
Semiconductor strain gauges have a greater sensitivity and higher-level output
than wire strain gauges. They can also be produced to have either positive or
negative changes when strained. However, they are temperature sensitive, are
difficult to compensate, and the change in resistance is nonlinear.

PMcL
2015

Strain, Force, Pressure and Flow Measurements

Index

24 - Sensor Signal Conditioning

24.22
24.8 High Impedance Sensors
Many popular sensors have output impedances greater than several megohms,
and thus the associated signal-conditioning circuitry must be carefully designed
to meet the challenges of low bias current, low noise, and high gain. A few
examples of high impedance sensors are:
High impedances
sensors…

 Photodiode preamplifiers
 Piezoelectric sensors
 Humidity monitors
 pH monitors
 Chemical sensors
 Smoke detectors
Very high gain is usually required to convert the output signal of these sensors
into a usable voltage. For example, a photodiode application typically needs to
detect outputs down to 30 pA of current, and even a gain of 106 will only yield
30 mV. To accurately measure photodiode currents in this range, the bias
current of the op-amp should be no more than a few picoamps. A high
performance JFET-input op-amp is normally used to achieve this specification.
Special circuit layout techniques are required for the signal conditioning

…require special
interfacing circuits

circuitry. For example, circuit layouts on a printed circuit board (PCB)
typically need very short connections to minimise leakage and parasitic
elements. Inputs tend to be “guarded” with ground tracks to isolate sensitive
amplifier inputs from voltages appearing across the PCB.

Index
24 - Sensor Signal Conditioning

High Impedance Sensors

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2015

24.23
24.9 Temperature Sensors
Temperature measurement is critical in many electronic devices, especially
expensive laptop computers and other portable devices – their densely packed Temperature is an
extremely important

circuitry dissipates considerable power in the form of heat. Knowledge of physical property to
system temperature can also be used to control battery charging, as well as to measure
prevent damage to expensive microprocessors.
Accurate temperature measurements are required in many other measurement
systems, for example within process control and instrumentation applications.
Some popular types of temperature sensors and their characteristics are
indicated in the table below:
Sensor

Range

Accuracy

Excitation

Feature

Thermocouple

-184°C to
+2300°C

High accuracy
and
repeatability

Needs cold
Lowjunction
voltage
compensation

RTD

-200°C to
+850°C

Fair linearity

Requires
excitation

Low cost

Thermistor

0°C to
+100°C

Poor linearity

Requires
excitation

High
sensitivity

Semiconductor

-55°C to
+150°C

Linearity: 1°C

Requires
excitation

10 mV/K,
20 mV/K
or 1A/K
typical
output

Accuracy: 1°C

Popular types of
temperature sensor
and their
characteristics

Table 24.1 – Popular Types of Temperature Sensors
In most cases, because of low-level and/or nonlinear outputs, the sensor output
must be properly conditioned and amplified before further processing can
occur. Sensor outputs may be digitized directly by high resolution ADCs –
linearization and calibration can then be performed in software, reducing cost
and complexity.

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2015

Temperature Sensors

Index
24 - Sensor Signal Conditioning

24.24
Resistance Temperature Devices (RTDs) are accurate, but require excitation
current and are generally used within bridge circuits.
Thermistors have the most sensitivity, but are also the most nonlinear. They are
popular in portable applications for measurement of battery and other critical
system temperatures.
Modern semiconductor temperature sensors offer both high accuracy and
linearity over about a -55°C to +150°C operating range. Internal amplifiers can
scale output to convenient values, such as 10 mV/°C.

Index
24 - Sensor Signal Conditioning

Temperature Sensors

PMcL
2015

24.25
24.10 Summary


A sensor is a device that receives a signal or stimulus and responds with an
electrical signal. The full-scale outputs of most sensors are relatively small
voltages, currents, or resistance changes, and therefore their outputs must
be properly conditioned before further analog or digital processing can
occur.



Amplification,

level

translation,

galvanic

isolation,

impedance

transformation, linearization and filtering are fundamental signalconditioning functions that may be required with sensors.


A resistance bridge, or Wheatstone bridge, is used to measure small
resistance changes accurately. There are a variety of different bridge
circuits, and a variety of amplifying and linearizing techniques to suit each
type.



There are a variety of methods for interfacing to remote bridges. Many
integrated bridge transducers are available as “one-chip” solutions to bridge
driving and measurement.



There are many types of sensors – their use in a certain application requires
an understanding of their physical construction and operation, as well as the
required performance and cost demanded by the overall system.

24.11 References
Jung, W: Op-Amp Applications, Analog Devices, 2002.

PMcL
2015

Summary

Index
24 - Sensor Signal Conditioning

24.26
Exercises
1.
For temperature measurements only one active transducer is used and so it is
not possible to have a linear output if it is placed in a bridge.
(a)

Show that the output from a single-element varying bridge is given by:

vo 

(b)

VB R
4 R  R
2

Since the active transducer resistance change can be rather large (up to
100% or more for RTDs), the nonlinearity of the bridge output
characteristic (the formula above) can become quite significant. It is
therefore desired to linearize the output of a temperature transducer using
the following circuit:

VB

R1

R 2+ R

vo
R1
R2
Derive an equation for the output voltage.

Index
24 - Sensor Signal Conditioning

Exercises

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2015

25.1
25 System Modelling
Contents

Introduction ................................................................................................... 25.2
25.1 Differential Equations of Physical Systems .......................................... 25.3
25.2 Linear Approximations of Physical Systems......................................... 25.5
25.3 The Transfer Function ........................................................................... 25.8
25.4 Block Diagrams ..................................................................................... 25.9
25.5 Feedback .............................................................................................. 25.16
25.6 Summary.............................................................................................. 25.19
25.7 References ........................................................................................... 25.19
Exercises ...................................................................................................... 25.20

PMcL
2015

Contents

Index
25 - System Modelling

25.2
Introduction
In order to understand, analyse and design complex systems, we must obtain
quantitative mathematical models of these systems. Since most systems are
dynamic in nature, the descriptive equations are usually differential equations.
If the system stays “within bounds”, then the equations are usually treated as
linear differential equations, and the method of transfer functions can be used
to simplify the analysis.
In practice, the complexity of systems and the ignorance of all the relevant
factors necessitate the introduction of assumptions concerning the system
operation. Therefore, we find it useful to consider the physical system,
delineate some necessary assumptions, and linearize the system. Then, by
using the physical laws describing the linear equivalent system, we can obtain
a set of linear differential equations. Finally, utilizing mathematical tools, such
as the transfer function, we obtain a solution describing the operation of the
system.
In summary, we:
1. Define the system and its components.
2. List the necessary assumptions and formulate the mathematical model.
3. Write the differential equations describing the model.
4. Solve the equations for the desired output variables.
5. Examine the solutions and the assumptions.
6. Reanalyse or design.

Index
25 - System Modelling

Introduction

PMcL
2015

25.3
25.1 Differential Equations of Physical Systems
The differential equations describing the dynamic performance of a physical
system are obtained by utilizing the physical laws – this approach applies
equally well to electrical, mechanical, fluid and thermodynamic systems.
For mechanical systems, Newton’s laws are applicable.
EXAMPLE 25.1 Spring-Mass-Damper Mechanical System
Consider the simple spring-mass-damper mechanical system shown below:

K
f
Friction

Mass
M

y

r (t )
Force
This is described by Newton’s second law of motion (this system could
represent, for example, a car’s shock absorber). We therefore obtain:

M

d2y
dy
f
 Ky  r
2
dt
dt

where K is the spring constant of the ideal spring and f is the friction constant.

PMcL
2015

Differential Equations of Physical Systems

Index
25 - System Modelling

25.4
EXAMPLE 25.2 Parallel RLC Circuit
Consider the electrical RLC circuit below:

r (t )

R

L

C

v( t )

This is described by Kirchhoff’s current law. We therefore obtain:
C

dv v 1
 
vdt  r
dt R L 

In order to reveal the close similarity between the differential equations for the
mechanical and electrical systems, we can rewrite the mechanical equation in
terms of velocity:
v

dy
dt

Then we have:
M

dv
 fv  K  vdt  r
dt

The equivalence is immediately obvious where velocity vt  and voltage vt 
are equivalent variables, usually called analogous variables, and the systems
are analogous systems.
The concept of analogous systems is a very useful and powerful technique for
system modelling. Analogous systems with similar solutions exist for
electrical, mechanical, thermal and fluid systems. The existence of analogous
systems and solutions allows us to extend the solution of one system to all
analogous systems with the same describing differential equation.

Index
25 - System Modelling

Differential Equations of Physical Systems

PMcL
2015

25.5
25.2 Linear Approximations of Physical Systems
Many physical systems are linear within some range of variables. However, all
systems ultimately become nonlinear as the variables are increased without
limit. For example, the spring-mass-damper system is linear so long as the
mass is subjected to small deflections y t  . However, if y t  were continually
increased, eventually the spring would be overextended and break. Therefore,
the question of linearity and the range of applicability must be considered for
each system.
A necessary condition for a system to be linear can be determined in terms of a
forcing function xt  and a response y t  . A system is linear if and only if:

ax1 t   bx2 t   ay1 t   by2 t 

(25.1)

That is, linear systems obey the principle of superposition, [excitation by

x1 t   x2 t  results in y1 t   y 2 t  ] and they also satisfy the homogeneity
property [excitation by ax1 t  results in ay1 t  ].
It may come as a surprise that a system obeying the relation y  mx  b is not
linear, since it does not satisfy the homogeneity property. However, the system
may be considered linear about an operating point x0 , y0  for small changes
x and y . When x  x0  x and y  y0  y , we have:

y  mx  b
y0  y  mx0  mx  b

(25.2)

and, since y0  mx0  b , then y  mx , which is linear.
In general, we can often linearize nonlinear elements by assuming small-signal
conditions. This approach is the normal approach used to obtain linear
equivalent circuits for electronic circuits and transistors.

PMcL
2015

Linear Approximations of Physical Systems

Index
25 - System Modelling