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Today I would like to present some definitions (besides being and engineers I teach engineering) that many times either are not well remembered or we never learned them in details.
Systems and devices in general must conform to a set of
definitions and standards to describe their behavior and characteristics. This
is necessary in order for professionals to effectively interchange ideas. In
this offer I will present some important definitions and standards that may be useful to all engineers.
Error
This is one of the most important quantities when
determining the accuracy of a measurement. In this context it is defined as the
difference between the actual value of a variable (quantity) and its measured
value. In control systems it is defined as the difference between the actual
value and the desired value of a variable. Figure 1 shows a calibration curve
of a sensor where the input variable is X and output variable is Y. The figure
shows the actual value of a measurement and the measured value. Note that the
error represents a range of values around the actual value and is normally
represented as a percentage.
Figure 1: Error Determination
Transfer Function
A general system can be represented by blocks that
describe different interconnected parts. Sensors are followed by transducers,
for instance. Each one of these blocks is a graphical representation of the
mathematical function of the relationship between the output and the input of
the block. In technical terms such a block is called a Transfer
Function.
Figure 2: General Transfer Function
Figure 2 shows a typical transfer function of a general
device. The transfer function T is a function of the input, the output, and --
in the case of a dynamic transfer function -- it is also a function of time, t. T describes the
relationship between the output and the input of the block. In general, for
linear systems, this relationship is simply the ratio of the output over the
input as it is indicated in the figure.
Transfer functions are composed of two parts (in some
systems one of these component does not exist): the dynamic part and the static
part. The dynamic transfer function is the relationship between the output and
the input when the input changes with time. The static transfer function is the
relationship between output and input when the input is not a function of time.
Static transfer function can always be represented by simple fraction of output
over input. Dynamic transfer functions are normally represented by differential
equation.
Example 1: Static Transfer Function
Figure 3 is an example of a static transfer function. This
transfer function represents the relationship between the distance traveled by
a car and the amount of gasoline in the tank. As it can be seen there is a
linear relationship between the amount of gasoline in the tank and the distance
traveled. Therefore the transfer function is static.
Figure 3: Example 1
Accuracy
The accuracy of a device is the combined maximum overall
error (non--linearity, Hysteresis, repeatability, etc.) that is expected when
measuring a variable. This term is normally expressed as the inaccuracy or
uncertainty of the measurement. It can be expressed in several ways. The most
common are as follows:
·
Measured variable:The
inaccuracy is a percentage of any value measured.
·
Full-scale percentage (FS): The
inaccuracy or uncertainty of any measure is expressed as a percentage of the
full scale reading of the instrument.
·
Span percentage: The
uncertainty is expressed as percentage of the range of measuring capability of
the instrument.
·
Percentage of actual reading:
In this form the inaccuracy is a percentage of the value of the actual reading.
An example follows.
Example 2:
A voltmeter with a reading span of 5 -- 20 volts is used
to make a measurement. The reading results in a value of 8 V. Determine the
error and the possible range of voltages for the reading if the accuracy is (a) ±2% of any measurement,
(b) ±0.5% FS, (c) ± 0.6% of span, and (d) ± 0.8 of reading.
Solution:
1) Error
= ± (0.02)(8 V) = ± 0.16 V. The possible
range of voltages for this reading is: 7.84 - 8.16 V.
2) Error
= ± 0.005)(20 V) = ± 0.1 V. The possible
range of voltages for the reading is: 7.9 - 8.1 V.
3) Error
= ± 0.006)(20 - 5) V
= ± 0.09 V. The possible
range of voltages for the reading is: 7.91 - 8.09 V.
4) Error
= ± 0.008)(8 V) = ± 0.064 V. The possible
range of voltages for the reading is: 7.936 - 8.064 V.
Accuracy for Digital Signals
For digital systems the most important source of error is
in the inaccuracy of the digital representation of analog signals. Based on
this the accuracy is defined as the percentage deviation of
the analog signal per bit of the digital signal. For example, suppose an
analog-to-digital converter (ADC) has a resolution of 0.525 volts per bit, and
an accuracy of ± 2%. This implies that
in order to set an output bit an input (analog) voltage change of 0.525 ± (0.525)(0.01) =
0.525 ± 0.005 V (0.52 -
0.53 V) should be applied.
Sensitivity
Sensitivity is defined as the amount of change in the
output of a device for a given change in the input. A device is highly
sensitive if it exhibits a big change in the output for a small change in the
input signal. Normally, sensitivity is equivalent to the value of the transfer
function of an instrument or sensor. For example, if an LVDT outputs 5 V for every
2 mm of motion, then its sensitivity and its transfer function are both equal a
to 2.5 V/mm.
Hysteresis
Some instruments (mechanical sensors, for instance)
exhibit the peculiarity that a different reading results for a specific input,
depending on weather the input value is approached from higher or lower values.
This phenomenon is called hysteresis (derived from
the Greek word that means deficiency), and it is
related to the history of the instrument. Figure 4 shows a typical graph of
hysteresis for an instrument. Notice that for the same input we get two different
outputs depending on weather we approach the input from curve A (increasing)
values or curve B (decreasing) values.
Figure 4: Hysteresis
Resolution
The resolution of a measurement is defined as the minimum
measurable input value. For instance, if the slider of a potentiometer moves in
such a way that for every turn of the winding the resistance changes by 2 Ω , then the
potentiometer cannot provide a resistance smaller than 2 Ω .
Linearity
The output of sensors and other devices is a function of
the input values. It is very important that for every different output value
there is a unique input value that is related to this output. Therefore, a
simple linear relationship between the output and the input is desirable to
ensure that for each input value there is only one output value associated with
it.
A linear relationship between two variables can be
expressed algebraically by the equation of a straight line. Figure 5 represents
a general sensor whose input is X and its output is Y.
Figure 5: General sensor transfer function
The mathematical relationship between these two variables
is given by
Y = mX + b
(1)
where m is the slope of the
line and b is its intercept with
the vertical axis. Figure 6 shows the graph of Eq. (1).
Figure 6: Linear plot (a calibration curve)
The following example illustrates the general procedure to
determine the linear relationship between the output and the input.
Example 3:
A thermistor (temperature sensor) changes linearly
from 200 Ω to 500 Ω as the temperature
changes from 20° to 120°. Determine the linear
equation relating resistance (output) to temperature (input).
Solution:
Using equation (1) we set up two equations as follows:
200 Ω = (20°)m + b
500 Ω = (120°)m + b
Solving these two equations for the two unknowns - m and b - we get
m = 3 Ω / °
b = 140 Ω
Then the resulting equation is
R = 3m + b
(I will treat dynamic transfer functions in a coming blog.)
Bibliography:
"Process Control Instrumentations Technology", Curtis D. Johnson. Prentice Hall. ISBN:0-13-938200-3
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