Page 352 - Mechatronics with Experiments

JWST499-Cetinkunt
JWST499-c06
338 MECHATRONICS Printer: Yet to Come October 9, 2014 8:1 254mm×178mm
In general, ΔR∕R ≪ 1, and the above relation can be approximated,
o
ΔR∕R o
V ∕V = (6.36)
o
i
4
Another convenient way of expressing this relationship is
V i
V = ΔR (6.37)
o
4R
o
In Equation 6.20, if R = R = R = R is constant, but R = R +ΔR,thesame
1
0
3
4
0
2
output voltage and resistance variation relationship holds except the polarity of the output
voltage would be opposite (negative). Similarly, in Equation 6.20, if R = R = R = R is
4
0
2
1
constant, but R = R +ΔR, the same output voltage and resistance variation relationship
3 0
holds except the polarity of the output voltage would be opposite (negative). If R = R =
1 2
R = R is constant, but R = R +ΔR, the same output voltage and resistance variation
3 0 4 0
relationship holds except the polarity of the output voltage would be positive. This can be
easily shown by making these substitutions in Equation (6.33).
Notice that if the sensor signal conditioner has an ADC converter and an embedded
digital processor, the above approximation is not necessary. The more complicated rela-
tionship can be used in the software for a more accurate estimation of the measured variable
using Equation (6.33 or 6.35).
Now, let us relax the assumption on the measurement device resistance. Let us assume
that it is not infinite, but a large finite value. The result is same as the electrical loading
3
error condition. If the R is much larger than R , that is 10 times, the error introduced to
m
0
the measurement due to non-infinite R is negligible.
m
Example Consider that a RTD type temperature sensor is used to measure the temper-
ature of a location. The two terminals of the sensor are connected to the R position of a
1
Wheatstone bridge circuit. The sensor temperature–resistance relationship is as follows
R = R [1 + (T − T )] (6.38)
o
o
◦
◦ −1
where from the sensor calibration data it is known that = 0.004 C , T = 0 C reference
o
temperature, and R = 200 Ω at temperature T . Assume that V = 10 VDC, and R = R =
o
o
3
2
i
R = 200 Ω. What is the temperature when the V = 0.5 VDC?
4
o
Let us assume that the input resistance of the voltage measurement device is infinity,
ΔR∕R o
V = V ⋅ (6.39)
i
o
4
Find ΔR, then R = R +ΔR, calculate T from
o
R = R (1 + (T − T )) (6.40)
o
o
◦
The resulting numbers are ΔR = 40 Ω, T = 50 C.
Let us consider that the input resistance of the output voltage measuring device is
R = 1MΩ, instead of infinite. The resulting measurement would indicate the following
m
temperature,
ΔR∕R o
V = V ⋅ (6.41)
i
o
4(1 + R ∕R )
m
o
which gives
ΔR = 40 ⋅ 1.0002 Ω (6.42)