2.1 Strain Gauge Sensors 31
After easy rearrangement, we get
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++++
−−+
=
4
4
2
2
3
3
1
1
3241
ΔΔ
Δ
Δ
24
Δ
T
T
T
T
T
T
T
T
RRRR
in
out
R
R
R
R
R
R
R
R
V
V
εεεε
(2.6)
The increment of resistance in elastic limits of the gauge material and of the object
under test may change only a fraction of a percent. Because of it, we can assume that
4
ΔΔ
Δ
Δ
2
4
4
2
2
3
3
1
1
<<
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++
T
T
T
T
T
T
T
T
R
R
R
R
R
R
R
R
(2.7)
and finally Eq. (2.6) can be simplified to the form
)(
4
1
Δ
3241 RRRR
in
out
V
V
εεεε
−−+=
(2.8)
or
)(
4
Δ
3241
εεεε
−−+=
k
V
V
in
out
(2.9)
where strain gauge constant .2≈k
2.1.1 Temperature Compensation
Strain gauges should be glued in to an object under test, and connected in a bridge
circuit, in a special way indicated by Eq. (2.9). If the strains
1
ε
and ,
4
ε
related to
the gauges
R
T1
and R
T4
of the bridge shown in Fig. 2.2a, are positive, then the
strains
2
ε
and
3
ε
of the gauges
2T
R and
3T
R should be negative. This way the
strains add together and the output voltage has a maximum value. At the same
time, such a connection makes possible the compensation of thermal effect. The
temperature effect causes a change of strain in each of the strain gauges involved.
The change denoted
T
ε
+ is due to thermal expansion of the object under test.
Including this effect into Eq. (2.9), we can write
)(
4
)]()()()[(
4
Δ
3241
3241
εεεε
εεεεεεεε
−−+=
+−+−+++=
k
k
V
V
TTTT
in
out
(2.10)
Examining the Eq. (2.10), it is easy to notice that the influence of temperature
in such a circuit is compensated. The same reasoning can be applied to the half
a bridge circuit and the connection of strain gauges into it. They should be
connected e.g. in the branch
1T
R for
1
ε
+ and the branch
2T
R for .
2
ε
− It renders
certain the temperature compensation because