This page was generated using an imperfect translator. Text may be poorly positioned and mathematics will be barely readable. Illustrations will not show at all. If possible, view the PostScript or PDF versions of these notes.
10-1 Strain, Force, and Pressure *
* 10-1
Force is that which results in acceleration (when forces don't cancel).
Strain is the change in shape of an object : : :
: : :usually due to some force.
(Force is usually called stress in this context.)
Pressure is force per unit area.
10-1 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-1
10-2 Strain *
* 10-2
Consider an object in two situations: with and without a force applied.
Let a force be applied along a dimension.
Let L1 be the length of the object along the dimension when no force is applied.
Let L2 be the length when the force is applied.
L2 L1
Then the object's strain is defined to be __________.
L1
The symbol - is usually used to denote strain.
In most situations, strains of interest will be very small, j- j < 0:0001.
10-2 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-2
10-3 Strain Transducers *
* 10-3
Called strain gauges.
Symbol: no common symbol.
Construction
Flexible card with strip of some conductor arranged in special pattern.
Card is mounted (glued) onto the object being measured.
Conductor is usually a metal or semiconductor.
Pattern is chosen so that strain (to be measured) : : :
: : :occurs along direction of current flow.
Current is passed through conductor.
10-3 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-3
10-4 *
* 10-4
Principle of Operation For Both Types
Conductor maintains an almost constant volume with strain.
That is, conductor is not compressible.
Recall that the resistance of a conductor is R = ae L_A, where
L is its length,
A is its area, and
ae is its resistivity.
Suppose force causes length of the conductor to decrease.
Since volume does not change much, area must increase.
Thus, resistance decreases.
10-4 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-4
10-5 *
* 10-5
Model Function
Ht1(x) = R0(1 + Gf x),
where Gf is a constant : : :
: : :called the gauge factor.
For metal strain gauges, Gf = 2.
(An integer!)
For semiconductor strain gauges Gf is much higher.
10-5 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-5
10-6 *
* 10-6
Complementary Pairs
In some cases the strain : : :
: : :in two places on the object : : :
: : :will be of equal magnitude_but opposite sign.
For example, a cantilever beam:
The upper part of beam is stretched (positive strain) : : :
: : :and the lower part of beam is compressed (negative strain).
The two strain gauges therefore : : :
: : :form complementary pairs.
10-6 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-6
10-7 *
* 10-7
Derivation of Gauge Factor for Ideal Metal
Ideal metal's properties:
- Non-compressible. (Does not change volume.)
- Resistivity is constant.
Consider an Ideal Metal Block
Regardless of strain:
volume is S and resistivity is ae.
When unstrained:
call length L1, area A1, and resistance R1.
By standard resistivity formula: R1 = ae L1_A.
1
Since volume fixed: A1 = S=L1.
Resistance can be found in terms of length and area:
2
R1 = ae L1_S.
10-7 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-7
10-8 *
* 10-8
The Block Suffering Strain -
Call new length L2.
By definition of strain,
L2 = L1(1 + - ).
Resistance in terms of R1 and - :
R2 = ae L2_A
2
2
= ae L2_S
2
= ae (L1(1_+_-_))_S
2(1 + - )2
= ae L1_________S
= R1(1 + - )2
= R1(1 + 2- + - 2)
ssR1(1 + 2- )
When - is small simplified form close to the exact form.
10-8 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-8
10-9 *
* 10-9
Effect of Temperature
Physically, a strain gauge is not much different from an RTD : : :
: : :and so, alas, strain gauge affected by temperature : : :
: : :therefore temperature compensation needed.
Model function including temperature:
Ht1(x) = R0(T )(1 + Gf x),
where resistance R0(T ) is a function of temperature.
Conditioning circuit must "remove" R0(T ) term.
A bridge does this very well.
10-9 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 from*
* lsli10. 10-9
10-10 *
* 10-10
Typical Strain Gauge Conditioning Circuit
Consider model function including temperature effect:
Ht1(x) = R0(T )(1 Gf x),
(Note that the complementary pair is indicated here by a .)
Four strain gauges are placed in the bridge in the following way:
The temperature terms cancel, so:
vo = AvE Gf x.
10-10 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-10
10-11 Force *
* 10-11
Definition: that which causes a mass to accelerate, "F = m"a.
kg g
Units: Newton, 1 N j _______; dyne, 1 dy j ________; pound, etc.
m=s2 cm =s2
Types of Transducers
- Small displacement.
Force bends, compresses, or stretches a part of the transducer.
Change in shape usually measured using strain gauges.
- Large displacement.
Force moves a part of the transducer.
Movement measured using displacement sensors.
- Piezoelectric crystals.
Based on a material that emits charge when compressed.
10-11 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-11
10-12 Small-Displacement Force Transducers *
* 10-12
Also called load cells.
Typical construction:
Consists of a rigid framework,
for example a cantilever beam.
The force is applied at a predetermined point.
Strain gauges are placed : : :
: : :at locations chosen so that : : :
: : :their output is linearly related to force.
The choice of location for the strain gauges : : :
: : :and the derivation of the resulting load-cell model function : : :
: : :is beyond the scope of this course.
Load cells usually packaged : : :
: : :with strain gauges connected in bridge configuration.
10-12 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-12
10-13 *
* 10-13
Two other common load-cell configurations are illustrated below:
Load Cell Model Function
Load cells, as sold, usually include a bridge
circuit.
Therefore the bridge excitation voltage, vE ,
will appear in the model function.
Ht1(x) = x k vE , where k is a constant.
Desirable Characteristic
Accurate.
Undesirable Characteristic
Expensive.
10-13 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-13
10-14 *
* 10-14
Large-Displacement Force Transducers
Typical construction and principle of operation:
Consists of a flexible structure.
The force is applied at a predetermined point.
Force causes structure to flex, this measured
by a displacement transducer.
Structure chosen so that displacement is linearly related to force.
The design of this part is within the scope of the course.
One candidate for the flexible structure is, of course, a spring.
Undesirable Characteristics
Requires a large displacement. Not practical for all systems. (E.g., a bathroom scale.)
Higher calibration and repeatability error than small-displacement transducers.
Desirable Characteristic
Inexpensive.
10-14 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-14
10-15 *
* 10-15
Ideal Springs
One end of spring mounted to a fixed point.
Force applied to other end.
Ideal springs obey Hooke's Law: F = kx,
where x is the displacement,
F is the force, and
k is the force constant for the spring.
In this class, all springs will be ideal. (When used as directed.)
10-15 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-15
10-16 Large-Displacement Force Transducer Sample Problem *
* 10-16
Design a system to convert process variable x 2 [0 N; 33 N] to a floating point number,
1
H(x) = x ___, to be written into variable force. The precision must be at least 0:5 N.
N
Solution:
Use a large-displacement force transducer consisting of a spring with k = 1 _N___mm.
Use a coded displacement sensor to measure spring displacement.
10-16 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-16
10-17 *
* 10-17
Number of distinct forces to measure at least 33=:5 + 1 = 67. Therefore, use a 7-bit binary
coded-displacement transducer.
Spring displacement range: [0; 33 mm ]. Choose CDT which matches this range.
j x k
Combined response of spring and CDT: Htc(x) = _____331N27.
Interface-routine code:
raw = readInterface();
force = raw * 0.2598;
10-17 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-17
10-18 *
* 10-18
Piezoelectric Crystals
Symbol:
Construction and Operation
Consists of a crystal of a material with piezoelectric properties.
Material can be quartz, or special ceramics.
Contacts are placed along two faces of crystal.
Pressure is applied to the crystal.
Charge appears on the surface of the crystal, proportional to the force.
Model Function
Ht1(x) = kx, where k is a constant with dimensions charge per pressure.
That is, the output of the transducer is charge.
10-18 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-18
10-19 *
* 10-19
Typical Conditioning Circuit
The output of the crystal converted to voltage
using a capacitor.
Recall, V = Q_C.
Capacitor chosen to get desired voltage range.
Very high input impedance amplifier needed.
Because of leakage : : :
: : :through capacitor, amplifier, etc. : : :
: : :even when an unchanging force is applied : : :
: : :voltage will decrease over time.
Therefore, piezoelectric crystals best used : : :
: : :for measuring changes in force : : :
: : :exempli gratia, vibration.
10-19 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-19
10-20 *
* 10-20
Pressure
Definition: force per unit area.
Here, only pressure exerted by fluids (liquids, gases, and solids under certain condi-
tions) will be considered.
Common units,
N
Pascal 1 Pa = 1 ____ ,
m2
kilopascal ( kPa ),
mm Hg,
bar,
pound per square inch.
Types of Pressure Transducers
- Large-Displacement Transducers.
These consists of a variety of flexible containers that change size with pressure.
- Small-Displacement Transducers.
These usually consist of a diaphragm and a strain gauge.
10-20 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-20
10-21 Types of Pressure *
* 10-21
Pressure usually specified
as a difference between
the process variable mea-
sured and some reference
pressure.
This can be visualized as the pressure on a diaphragm, with the pressure being mea-
sured on one side and the reference pressure on the other.
That reference pressure determines (defines) the type of pressure measured:
- Absolute. Reference pressure is zero.
- Gauge. Reference pressure is the environment air pressure.
Automobile tire pressure is gauge pressure. (A zero on your pressure gauge mea-
sured from a flat tire does not mean there is a vacuum inside the tire. Instead it
means the pressure is the same inside and outside the tire.)
- Differential. The reference pressure is a second process variable being measured.
10-21 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-21
10-22 Large-Displacement Pressure Transducers *
* 10-22
Construction and Operation:
Key part is a flexible container.
Inside of flexible container is exposed to the
pressure.
Outside of flexible container is exposed to ref-
erence pressure.
Size of container changes with pressure.
Size is measured by displacement transducer.
Desirable Characteristic
Inexpensive.
Undesirable Characteristic
High calibration and repeatability errors.
10-22 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-22
10-23 *
* 10-23
Small-Displacement Pressure Transducers
Construction and Operation:
Consists of a diaphragm, one side exposed to pressure being measured.
Other side exposed to reference pressure.
Either:
Displacement of diaphragm measured by capacitive or inductive displacement transducer.
Or strain of diaphragm measured using strain gauges.
Placement of Strain Gauges
Center of diaphragm convex, part near edge is
convex.
Strain gauges can be placed so that there are
complementary pairs.
10-23 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-23
10-24 *
* 10-24
Integrated Pressure Sensors
Uses a small-displacement pressure transducer.
Entire sensor is fabricated on one silicon chip.
Diaphragm is etched into silicon.
Strain gauges are fabricated on silicon.
Conditioning circuit on same chip.
10-24 EE 4770 Lecture Transparency. Formatted 8:35, 19 February 1999 fr*
*om lsli10. 10-24
| David M. Koppelman - koppel@ee.lsu.edu | Modified 19 Feb 1999 8:37 (14:37 UTC) |