Page 117 - Electric Drives and Electromechanical Systems

110 Electric Drives and Electromechanical Systems

position measurement system with both high accuracy and repeatability to ensure that
the target point is measured. If the system has low resolution, Fig. 4.2, the uncertainty
regarding each measured point increases.
All measurement systems suffer from inherent inaccuracies; and estimation of the
uncertainty requires knowledge of the form that the error takes. In general, an error can be
classified either as a random or a systematic error. Random errors arise from chance or
random causes, and they must be considered using statistical methods. Systematic errors
are errors which shift all the readings in one direction; for example, a shift in the zero point
will cause all the readings to acquire a constant displacement from the true value. In
addition errors can be introduced by the measurement system itself, including any digi-
tisation process, the dynamic characteristics of the encoder or transducer, and the
interaction of the process with the measurement system.

4.1.1 Random errors
If a large set of data is taken from a transducer under identical conditions, and if the
errors generated by the measurement system are random, the distribution of values
about the mean will be Gaussian, Fig. 4.3. In this form of distribution, sixty-eight per cent
of the readings lie within 1 standard deviation of the mean and ninety five percent lie
within 2 standard deviations. In general, if a sample of n readings are taken with values
x 1 ,x 2 ..x n the mean x is given by,
n
1 X
x ¼ x i (4.1)
n
1¼1
and the standard deviation, s, by,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X n 2
ðx i xÞ
i¼1 (4.2)
s ¼
n 1
The mean value which is obtained is dependent on the number of samples taken and
on the spread; the true mean value can never be determined since this would require an

FIG. 4.2 Effect of resolution on the performance of a measurement system: the coarser the resolution (i.e. area of
the dot), the more uncertainty there is in the measurement. (A) Coarse resolution. (B) Fine resolution.