handbook of modern sensors

5.7 Bridge Circuits 193
determine the bridge sensitivity with respect to each impedance (calibration constant),
partial derivatives may be obtained from Eq. (5.41):
∂V out
Z 2
=
∂Z 1 (Z 1 + Z 2 ) 2 V ref ,
∂V out
Z 1
=−
∂Z 2 (Z 1 + Z 2 ) 2 V ref ,
∂V out
Z 4
=−
∂Z 3 (Z 3 + Z 4 ) 2 V ref ,
∂V out Z 3
=
∂Z 4 (Z 3 + Z 4 ) 2 V ref .
By summing these equations, we obtain the bridge sensitivity:
(5.43)
δV out
V ref
= Z 2δZ 1 − Z 1 δZ 2
(Z 1 + Z 2 ) 2 − Z 4δZ 3 − Z 3 δZ 4
(Z 3 + Z 4 ) 2 . (5.44)
A closer examination of Eq. (5.44) shows that only the adjacent pairs of impedances
(i.e., Z 1 and Z 2 , Z 3 and Z 4 ) have to be identical in order to achieve the ratiometric
compensation (such as the temperature stability, drift, etc.). It should be noted that
impedances in the balanced bridge do not have to be equal, as long as a balance of
the ratio (5.42) is satisfied. In many practical circuits, only one impedance is used as
a sensor; thus for Z 1 as a sensor, the bridge sensitivity becomes
δV out
V ref
= δZ 1
4Z 1
. (5.45)
The resistive bridge circuits are commonly used with strain gauges, piezoresistive
pressure transducers, thermistor thermometers, and other sensors when immunity
against environmental factors is required. Similar arrangements are used with the
capacitive and magnetic sensors for measuring force, displacement, moisture, and so
forth.
5.7.1 Disbalanced Bridge
A basic Wheatstone bridge circuit (Fig. 5.37A) generally operates with a disbalanced
bridge. This is called the deflection method of measurement. It is based on detecting the
voltage across the bridge diagonal. The bridge output voltage is a nonlinear function of
a disbalance ; however, for small changes ( < 0.05), which often is the case, it may
be considered quasilinear. The bridge maximum sensitivity is obtained when R 1 = R 2
and R 3 = R. When R 1 ≫ R 2 or R 2 ≫ R 1 , the bridge output voltage is decreased.
Assuming that k = R 1 /R 2 , the bridge sensitivity may be expressed as
α = V R
k
(k + 1) 2 . (5.46)