Page 94 - Mechanical Engineers' Handbook (Volume 2)

4 The Wheatstone Bridge 83

either ac or dc, but for now we assume it is dc so equations can be written in terms of
resistance R rather than a complex impedance. The condition for a balanced bridge with e o
equal to zero is
R 1 R 4
(5)
R 2 R 3
Next, an expression is presented for e due to small changes in R , R , R , and R :
o
1
2
3
4
e R dR 4 R dR 3 R dR 2 R dR 1 E
1
4
2
3

o
(R R ) 2 (R R ) 2 (R R ) 2 (R R ) 2 ex (6)
4
3
3
2
1
2
1
4
In many cases, the bridge circuit is made up of equal resistances. Substituting for individual
resistances, a strain gage resistance R, and using the definition of the gage factor from Eq.
(3), Eq. (6) becomes
FE
e ex ( ) (7)
o
2
3
4
1
4
The unbalance of the bridge is seen to be proportional to the sum of the strain (or
resistance changes) in opposite arms and to the difference of strain (or resistance changes)
in adjacent arms.
Equations (6) and (7) indicate one technique to compensate strain gage circuits to min-
imize the influence of temperature-induced strain. This was referred to in Section 3 as the
dummy gage method.
Assume that we have a bridge circuit with one active arm and arbitrarily let this arm
be number 4. Equation (7) becomes
FE
e ex ( ) (8)
o
4 4
Arm 4 responds to the total strain induced in it, which is comprised of both thermal (t) and
mechanical (m) strain
t (9)
m
4
A problem arises if it is desired to isolate the mechanical strain component. One solution
is to take another strain gage (the dummy gage) and mount it on a strain-isolated piece of
the same material as that on which gage 4 is mounted. If placed in the same thermal envi-
ronment as gage 4, the output from the dummy gage becomes simply . If the dummy gage
t
is wired in an adjacent bridge arm to 4 (1 or 3), Eq. (7) becomes
FE
e ex ( ) (10)
t
o
m
t
4
Equation (10) indicates that thermal strain effects are canceled. Similarly, in Fig. 2, four
gages were shown mounted on a transducer diaphragm. Equation (7) indicates that thermal
strain effects from this circuit should be canceled.
In reality, perfect temperature compensation is not achieved since no two strain gages
from a lot track one another identically. However, compensation adequate for many appli-
cations can be accomplished.
The biggest thermal problem with bridge transducers occurs in transient situations, such
as explosive or combustion environments. Here, due to individual physical locations, gages
in a bridge are not in the same time-varying temperature, and compensation cannot be