5.5.4 Signal linearization
Several types of transducer used in
measuring instruments have an output that is a non-linear function of the
measured quantity input. In many cases, this non-linear signal can be converted
to a linear one by special operational amplifier configurations that have an
equal and opposite non-linear relationship between the amplifier input and
output terminals. For example, light intensity transducers typically have an
exponential relationship between the output signal and the input light
intensity, i.e.:
V0 = Ke-αQ (5.5)
where Q is the light intensity, V0
is the voltage level of the output signal, and K and α are constants. If a
diode is placed in the feedback path between the input and output terminals of
the amplifier as shown in Figure 5.13, the relationship between the amplifier
output voltage V0 and input voltage V1 is given by:
V0 = C loge (V1)
(5.6)
If the output of the light transducer
with characteristic given by equation (5.5) is conditioned by an amplifier of
characteristic given by equation (5.6), the voltage level
of the processed signal is given by:
V0 = C loge (K) – αCQ (5.7)
Expression (5.7) shows that the
output signal now varies linearly with light intensity Q but with an offset of
C loge (K). This offset would normally be removed by further signal
conditioning, as described below.
5.5.5 Bias (zero drift) removal
Sometimes, either because of the
nature of the measurement transducer itself, or as a result of other signal
conditioning operations, a bias (zero drift) exists in the output signal. This
can be expressed mathematically for a physical quantity x and measurement
signal y as:
y = Kx + C (5.8)
where C represents a bias in the
output signal that needs to be removed by signal processing. The bias removal
circuit shown in Figure 5.14 is a differential amplifier in which a
potentiometer is used to produce a variable voltage Vp equal to the bias on the
input voltage Vi. The differential amplification action thus removes
the bias. Referring to the circuit, for R1 = R2 and R3
= R4, the output V0 is given by:
V0 = (R3/R1) (Vp – Vi) (5.9)
where Vi is the unprocessed signal y
equal to (Kx + C) and Vp is the output voltage from a potentiometer supplied by
a known reference voltage Vref, that is set such that Vp = C. Now,
substituting these values for Vi and Vp into equation (5.9) and referring the
quantities back into equation (5.8) gives:
y = K’ x (5.10)
where the new constant K’ is related to
K according to K’ = - K (R3/R1). It is clear that a straight line relationship
now exists between the measurement signal y and the measured quantity x. Thus,
the unwanted bias has been removed.
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