handbook of modern sensors

192 5 Interface Electronic Circuits
the numerator is also zero). In practice, the denominator is a signal from a reference
sensor, which usually is of a constant value.
Division has long been the most difficult of the four arithmetic functions to implement
with analog circuits. This difficulty stems primarily from the nature of division:
the magnitude of a ratio becomes quite large, approaching infinity, for a denominator
that is approaching zero (and a nonzero numerator). Thus, an ideal divider must have
a potentially infinite gain and infinite dynamic range. For a real divider, both of these
factors are limited by the magnification of drift and noise at low values of V D ; that
is, the gain of a divider for a numerator is inversely dependent on the value of the
denominator (Fig. 5.35B). Thus, the overall error is the net effect of several factors,
such as gain dependence of denominator, numerator and denominator input errors,
like offsets, noise, and drift (which must be much smaller than the smallest values of
the input signals). In addition, the output of the divider must be constant for constant
ratios of numerator and denominator, independent of their magnitudes; for example,
10/10 = 0.01/0.01 = 1 and 1/10 = 0.001/0.01 = 0.1.
5.7 Bridge Circuits
The Wheatstone bridge circuits are popular and very effective implementations of
the ratiometric technique or a division technique on a sensor level. A basic circuit is
shown in Fig. 5.36. Impedances Z may be either active or reactive; that is, they may
be either simple resistances, as in piezoresistive gauges, or capacitors, or inductors.
For the resistor, the impedance is R; for the ideal capacitor, the magnitude of its
impedance is equal to 1/2πf C; and for the inductor, it is 2πf L, where f is the
frequency of the current passing through the element. The bridge output voltage is
represented by
(
Z1
V out = − Z )
3
V ref , (5.41)
Z 1 + Z 2 Z 3 + Z 4
The bridge is considered to be in a balanced state when the following condition
is met:
Z 1
= Z 3
. (5.42)
Z 2 Z 4
Under the balanced condition, the output voltage is zero. When at least one impedance
changes, the bridge becomes unbalanced and the output voltage goes either in a
positive or negative direction, depending on the direction of the impedance change. To
Fig. 5.36. General circuit of a Wheatstone bridge.