Page 797 - Mechatronics with Experiments

LABORATORY EXPERIMENTS 783
where parameter is the resistivity of the material. Taking the logarithm of Equation 11.114
and differentiating, we obtain:

ln R =ln +ln L −ln A (11.115)
dR d dL dA
= + − (11.116)
R L A
ΔR Δ ΔL ΔA
= + − (11.117)
R L A
Assuming that the section of the conductor wire is circular, A = D 2 , it follows
4
ΔA 2 ΔD ΔL
= =− 2 (11.118)
A D L
where it can be shown that
dA d
= (ln A) (11.119)
A dA
d 2
= (ln( D ∕4)) (11.120)
dD
( ∕4)2D ⋅ dD
= (11.121)
( ∕4)D 2
2dD
= (11.122)
D
Hence,
ΔA 2ΔD
= (11.123)
A D
In Equation 11.118 the coefficient of Poisson, , of the material has been used in order to
relate longitudinal and transverse deformations of the wire.
Substituting Equation 11.118 into 11.117 the following is obtained:
ΔR ΔL Δ
= (1 + 2 ) + (11.124)
R L
It is common to define the gauge factor G as:
ΔR∕R Δ ∕
G = = (1 + 2 ) + (11.125)
ΔL∕L ΔL∕L
ΔR∕R
= (11.126)

Then,
ΔR
= G ⋅ (11.127)
R
Equation 11.125 shows that the sensor’s gauge factor G depends on a geometric
Δ ∕
term 1 + 2 and on a microstructural term that relates the variation of resistivity to
ΔL∕L
deformation. This term characterizes the piezoresistive behavior of the material.
For some materials (like constantan), the piezoresistive component is much smaller
than the geometric one. The strain gauges obtained from these materials are called “metal-
lic.” They have a relatively small gauge factor (G = 2). They are stable under temperature
variations and linear in the operating range.
Other strain gauges, made from different materials (i.e., semiconductors) exhibit
gauge factors in the range of G = 70–200. Despite the increased sensitivity, their use is
more difficult since they are nonlinear and may need temperature compensation.