10.5.1 Analog-to-Digital Converters
These are the devices that convert a voltage level from a sensor to a digital word
usable by the computer. At the most basic level, all digital words are binary numbers
consisting of many “bits” that are set to either 1 or 0. Therefore, the task of the analog-
to-digital (A/D) converter at each sample time is to convert a voltage level to the correct
bit pattern and often to hold that pattern until the next sample time.
Of the many A/D conversion techniques that exist, the most common are based on
counting schemes or successive approximation schemes. In counting methods, the
input voltage may be converted to a train of pulses whose frequency is proportional to
the voltage level. The pulses are then counted over a fixed period by a binary counter,
which produces a binary representation of the voltage level. A variation on this
scheme is to start the count simultaneously with a linear (in time) voltage and to stop
the count when the voltage reaches the magnitude of the input voltage to be converted.
The successive approximation technique tends to be much faster than the counting
methods. It is based on successively comparing the input voltage to reference levels representing
the various bits in the digital word. The input voltage is first compared with a reference
which is half the maximum. If greater, the most significant bit is set and the signal is
then compared with a reference which is three-fourths of the maximum to determine the
next bit, and so on. These converters require one clock cycle to set each bit, thus they would
need 2n cycles for an n-bit converter. At the same clock rate, a counter-based converter
might require as many as two cycles, which would usually make the process much slower.
For either technique, it will take longer to perform the conversion for a higher
number of bits, i.e., a more accurate converter. Not surprisingly, the price of A/D converters
goes up with both speed and bit size.
If more than one channel of data need to be sampled and converted to digital words,
it is usually accomplished by use of a “Multiplexer” rather than multiple A/D converters.
This device sequentially connects the A/D converter into the channel being sampled.
10.5.2 Digital-to-Analog Converters
These devices are used to convert the digital words from the computer to a voltage
level for driving actuators or perhaps a recording device such as an oscilloscope or
strip-chart recorder. The basic idea is that the binary bits are used to cause switches
(electronic gates) to open or close, thus routing the electric current through an appropriate
network of resistors so that the correct voltage level is generated. Since no
counting or iteration is required for digital-to-analog (D/A) converters, they tend to be
much faster than A/D converters. In fact, D/A converters are a component in the A/D
converters based on successive approximation.
10.5.3 Analog Prefilters
This device is often placed between the sensor and the A/D converter. Its function is to
reduce the higher-frequency noise components in the analog signal so as to prevent the
frequency of the noise from being switched to a lower frequency by the sampling
process (called “aliasing” or “folding”).
An example of aliasing is shown in Fig. 10.15, where a 60-Hz oscillatory signal is
being sampled at 50 Hz. The figure shows that the result from the samples is a 10-Hz signal
and the mechanism by which the frequency of the signal was aliased from 60 to 10 Hz.
Aliasing will occur any time the sample rate is not at least twice as fast as any of the frequencies
in the signal being sampled. Therefore, to prevent aliasing of a 60-Hz signal,
the sample rate would have to be faster than 120 Hz rather than the 50 Hz in the figure.
This phenomenon is one of the consequences of the sampling theorem of Nyquist
and Shannon. The theorem basically states that a signal must have no frequency components
greater than half the sample rate (ωs/2) for the signal to be accurately reconstructed
from the samples. Another consequence is that the highest frequency that can
be represented by discrete samples is ωs/2, an idea that has already been discussed in
Sec. 10.2.3.
The consequence of aliasing on a digital control system could be substantial. Noise
components with a frequency much higher than the control system bandwidth normally
have a small effect because the system will not respond at the high frequency. However,
if the frequency of that noise was aliased down to the vicinity of the system bandwidth,
the system would respond, thus causing the noise to appear on the system’s output.
The solution is to place an analog prefilter before the sampler. In most cases a simple
first-order low-pass filter will do, i.e.,
where the break point a is selected lower than ωs/2 so that any noise present with frequencies
greater than ωs/2 would be attenuated by the prefilter. The lower the breakpoint
frequency selected, the more the noise above ωs/2 is attenuated. However, too
low a break point may reduce the control-system bandwidth. The prefilter does not
eliminate the aliasing, but by judicious choice of the prefilter break point and the sample
rate, the designer has the ability to reduce the magnitude of the aliased noise to some
acceptable level.
10.5.1 Analog-to-Digital Converters
These are the devices that convert a voltage level from a sensor to a digital word
usable by the computer. At the most basic level, all digital words are binary numbers
consisting of many “bits” that are set to either 1 or 0. Therefore, the task of the analog-
to-digital (A/D) converter at each sample time is to convert a voltage level to the correct
bit pattern and often to hold that pattern until the next sample time.
Of the many A/D conversion techniques that exist, the most common are based on
counting schemes or successive approximation schemes. In counting methods, the
input voltage may be converted to a train of pulses whose frequency is proportional to
the voltage level. The pulses are then counted over a fixed period by a binary counter,
which produces a binary representation of the voltage level. A variation on this
scheme is to start the count simultaneously with a linear (in time) voltage and to stop
the count when the voltage reaches the magnitude of the input voltage to be converted.
The successive approximation technique tends to be much faster than the counting
methods. It is based on successively comparing...
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