Properties
of A/D Converters
The
comparator simulation reveals a few important facts about A/D converters.

An A/D converter has a
range. The simulator has a range from 0v to 10v, or a total of ten
(10) volts. The total range of an A/D is the difference between the
highest and lowest voltages the A/D can convert.
An A/D has a resolution
that is determined by the largest count that the A/D's counter/register
can hold.
The largest count is determined
by the number of bits in the counter.
If there are N bits in
the counter, the largest count is 2^{N} -1.
In the comparator, there
is only one bit, and the largest count is 1, and there are only two different
outputs that are possible.
Clearly,
if you want a more accurate conversion the converter will need to have
a lot more than just one bit. Let's look at another simulation.
This simulation is a four-bit A/D converter.
Four
Bit A/D Converter Simulator
Sim2 Here
is a simulation of a four bit A/D converter. Note the following:

As in the comparator simulation,
we assume that the input voltage can range from zero (0) to ten (10) volts.
We have added some "fine
control" with two buttons that "nudge" the voltage up or down, but not
beyond the 0-10v range.
Try the simulator first,
and then we will examine what happens in a little more detail.
Now, consider the following observations about the converter above.
There are four bits in
the simulation converter.
The range is from 0v to
10v, or 10v total range.
With four bits, there
are 16 different count values (0 through 15).
Thus, 10v is divided into
16 different parts, each part being:
10/16 = 0.625 volts wide.
Now,
let's consider an example question.
Example
E1 How
many bits would you need to divide 10 v into .01 v intervals? To
get the answer to the question consider the following.

If you divide 10 v into
.01 v intervals you need 1000 intervals.
If you need 1000 intervals
you need to think about a power of 2 that is larger than 1000.
The smallest power of
2 that is larger than 1000 is 2^{10} which is equal to 1024.
That means that you need
10 bits in the converter, and the the count in the counter/register will
run from 0 to 1023.
And that leads us to observe
that real converters often go to 10.23v, not 10v because that gives perfect
.01v increments between resolvable voltages.
And another converter
might run from -5.12v to +5.11v for the same reason.
Almost all A/D converters use a scheme in which the analog voltage is first
converted into a binary integer (a "count", as in the simulator above)
when the conversion is done. In some cases, there may be some software
that gives you the actual count - the binary integer. In other cases
the conversion might be converted - using software - to a a numerical value
or a character representation of a numerical value.
Example
E2 A
GPIB (IEEE-488) voltmeter is used to measure a DC voltage. The instrument
uses an A/D converter that generates a binary number. Within the
instrument, that binary number is converted to a string of characters which
is,
in turn, transmitted to a computer connected to the instrument.

To illustrate the concept in the example, consider the revised simulator
below. In this simulator, the conversion algorithm - from the count
in the register to an analog voltage - is given by:
V_{meas}
= Count*0.625
Note: .625 = 10/16
= Range/#Counts
Here is the simulation.
Four Bit A/D Converter Simulator
(Revised)
Sim3 Here
is a revised simulation of a four bit A/D converter. Note the following:

V_{meas}
= Count*0.625
V_{meas}
is displayed on the simulation.

Of course, the simulation only raises a few more issues.
The largest issue is that
the computed voltage is always lower than the actual voltage.
The average error would
probably be less if we used a different way to compute the computed voltage.
The formula for the computed
voltage is:
Using
this expression for the calculated voltage, we can plot the calculated
voltage as a function of the count. That's shown in the figure below.

Even more interesting is the plot of the
calculated voltage against the voltage input that is being measured.
That's shown next. We've also included a plot (the blue line) that
shows what the output would be ideally (And that might take many, many
bits in the converter!). Notice that the calculated voltage is always
lower than the voltage input using the calculation method we assumed above.
Note also that the largest error occurs just before the converter switches
when the voltage is rising. For example, as the voltage rises from
0v to 3v, there is a time when the converter output switches from 0v to
0.625v. The largest error occurs just before that point, so a bound
on the error is:

Error < 0.625

We can reduce the error if we calculate the voltage differently.
We can examine what happens if we use a different expression for the computed
voltage. We will use:

V_{meas}
= Count*0.625 + 0.3125
The result is shown in the next simulator.

Now, here is a plot of the calculated voltage
against the voltage input that is measured. Notice that the error
limit is half of what it was using the first calculation method.

The
conclusion is that the way the voltage is computed can affect the average
error when an A/D is used to measure voltage values.

Whenever you buy an A/D converter, or a voltmeter, or a data acquisition
unit, you need to be cognizant of how the data presented to the user is
actually computed. That's not usually a problem in instruments, but
there are A/D computer cards that use the first method above to calculate
voltage, and you should be aware of that when it happens. It is less
important when the number of bits in the converter is higher, but when
you have high requirements for accuracty you should be thinking of what
might be taking place.

The
Effect of Number of Bits
The number of bits used in the counter also affects the accuracy of the
conversion. Again, we will use a simulation to show the effect of
the number of bits.

Six Bit
A/D Converter Simulator
Sim4
Here is a simulation of a six bit A/D converter.

Run the simulator,
and compare results with the four bit simulators above.

We can
think a little bit about error in an A/D. First, note that the error
limit depends upon the range and the number of bits.
If there are N bits, the
number of divisions is 2^{N} -1.
If the total range of
voltage is V_{Range} , then the size of a division is:
SmallestDivision = V_{Range} /2^{N}
The error is related to
the Division size.
If the calculated voltage
is SmallestDivision*Count, the error is the same size as the smallest division.
If the calculated voltage
is SmallestDivision*Count + SmallestDivision/2, the error is SmallestDivision/2,
i.e. half the size of the smallest division.
See the simulations above
for a better appreciation/understanding of the error.
We're
not going to give you any more simulators with more bits. Actually,
we've almost reached the resolution limit for the screen. The simulators
with the "nudge buttons" just nudge the controls by a single pixel, and
that is too much for a twelve bit converter, for example.
Example
E3
A popular A/D board gives a count and requires you to perform the computation
shown in the line of C code shown below. The variable binary is the
count value that is returned.

MeasuredVolts = (((binary-2048)*20)/4096);

The A/D converts voltages
from -10v to +10v. We can conclude the following.

The converter has 12 bits.
You get that conclusion by noting that the count can vary from 0 to 4095.
When the count is 0, MeasuredVolts = 0.
Let's sum up a few points.
More is better when it
comes to bits in an A/D. The indicated voltage will be closer to
the actual value of the voltage being measured when there are more bits.
More is pricier when it
comes to bits in an A/D. It takes more parts, and they have to be
made more accurately if there are more bits.
How
Does An A/D Work?
While we have discussed A/D converters above, we haven't yet given you
any insight into how you could build an A/D. Here we will discuss
a simple way to build an A/D. This might not be the fastest A/D possible,
but it will start to give you some insight into what happens inside an
A/D.

The circuit is shown in the simulator below.

Simulator - Showing How You Might
Construct An A/D
Sim 5
This simulator is a four bit A/D, and it consists of the following components.

A pulse generator that
produces a sequence of 0's and 1's.
A counter which counts
the pulses and produces an increasing count.
The counter has a "Stop"
input. When that input goes high, the counter ceases to count.
A D/A which produces an
analog signal proportional to the count output from the counter.
An adjustable input voltage
source.
A comparator which produces
a "1" when the D/A output is larger than the input voltage.
The output of the comparator
is used to stop the counter.
Here is the circuit.

Here is how the A/D Simulator
works.
First, set a DC voltage
from the adjustable source.
Second, click the "Convert"
button.
The counter begins counting,
and the count is displayed on the LEDs and the numerical value of the count
is displayed.
The digital count signals
(four bits of signals in this case) form the input to a D/A converter which
produces an output voltage proportional to the count.
The output voltage from
the D/A is compared to the input voltage.
When the output voltage
from the D/A becomes larger than the input voltage the comparator develops
a "Stop" signal that is used to stop the counter.
The count that remains
in the counter is a digital representation of the input voltage.
Try the simulator, using
different voltages within the allowable range, i.e. 0-5v.
The simulator lets you see the inner workings of one type of A/D converter.
Note the following about any A/D converter.
The conversion takes time.
In the simulator, there is a variable amount of time until the counter
reaches the correct count. We have slowed down the simulator so you
can see what happens, but even in the best A/Ds there is some time that
must elapse between the start of a conversion and the end of a conversion.
That time limits how many conversions can be done per second. People
have worked on that problem and there are other kinds of converters that
convert more quickly.
The conversion is not
exact, and the accuracy depends upon the number of bits in the counter.
The result of the conversion
is an integer, and that integer still must be translated into an equivalent
voltage (although the conversion isn't hard to do).
Practical
A/Ds
Now, consider what happens in a typical application of an A/D. We'll
look at a voltmeter. In a voltmeter, this is what happens.

A voltage is applied to
the voltmeter, and an A/D converts the voltage to a count.
The count is converted
to a floating point number.
The floating point number
is displayed on an LCD or LED display.
If the voltmeter is connected
to a computer (say through an IEEE-488 bus) the following also takes place.
The floating point data
is converted to a character representation to transmit over an IEEE-488
or internet connection.
You
can see that there are a few conversions that have to take place in this
process. There are a number of other considerations here as well.
The number of bits in
the A/D converter will affect the accuracy of the voltmeter. There
is a lot of legend and lore in that area. Click
here to examine that.
You need to be aware of
how to make conversions. Actually, you need to become knowledgeable
about the different representations as well as how to program those conversions
in various programming environments like C, C++, Visual Basic and LabView.

A/D converters are found in many places - including places where you might
not think that you would find them. Here are a few.
Voltmeters are digital
today. To display a digital result for a voltage measurement the
voltage is first converted to digital form in an A/D, and then it is changed
to a decimal format to be displayed.
A/Ds are used in digital
thermometers, so if you've spent some time in the hospital, you've used
an A/D.
And, you get the idea.
However, there is an important situation where you need to dig deeper into
how A/Ds operate, and how they interface with the software and hardware
that is often used in measurement and control situations. Let's think
about a few typical situations.

Assume that you are recording
temperature at four points in a heat-treating oven. You need to do
the following.
You need to get the data
into a computer and you need to compute and record the average temperature
(the four temperature data points).
You have three tanks used
for chemical processing. You need to control the level of the liquid
in the three tanks. Each tank has a small computer with an A/D board
that measures and controls the liquid level in each tank and also measures
the temperature. Each computer has a network card that connects it
to an Ethernet LAN. There is a central computer on the LAN and you
need to send information from each of the control computers to the central
computer.
You need to take the data
at each computer and you need to transmit it over the network to the central
computer so that it can be recorded there and be available for any computations
that might be necessary.
In
both of these situations you need to consider how to manipulate the data
that you measure and want to store. Here are some of the considerations.
If you take data with
an instrument connected to a computer (like a data acquisition unit or
a voltmeter) the instrument may be connected to the computer with an IEEE-488
(GPIB) connection. (Click
here to learn more about IEEE-488.) IEEE-488 connections have
some considerations.
Most IEEE-488 instruments
convert data into strings of characters and those strings are transmitted
to the computer.
If you send data over
a network, you must send strings of characters.
The
conclusion that you need to reach is that you need to be aware of how data
has to be converted in these kinds of situations. Let's look at the
sequence of operations you might face.
The A/D converter gets
some raw data or information and it is in the form of an integer
count . Often it is simply a binary number
represented with some fixed number of bits.
The count could be a number
like 10001000 (or 129 in decimal numbers).
If the A/D converter has
eight (8) bits or less, the data will take a single byte.
If the A/D converter has
more than eight bits (which is typical) the data will take more than one
byte. Usually, for most converters, the data will take two bytes,
since most converters are 10, 12 or more bits.
The count is converted
to a number that represents the voltage (or possibly some other physical
variable). That takes some computer code (And there will probably
be some sort of computing chip in the instrument.) and the process results
in a number that is probably a floating point
number .
The floating point number
might be something like 3.1416v.
If you transmit that floating
point number you need to generate a string of characters.
The string of characters
might be something like a "3", a period ("."), a "1", a "4", etc.
In many cases you may
need to add one or more characters to signal the end of a string.
That could be a carriage return and a line feed, for example.
Representing
Instrument Data
Taking data with an instrument immediately leads you to consider several
things about data. Here is a sample.

If you take a lot of data,
you need to know how the data is represented.
If you take a lot of data,
you need to know how the data is transmitted from the instrument to a computer.
If you take a lot of data,
you need to know how the data you get will be stored if you store it in
a computer file.
Digital
Voltmeters
Digital
Voltmeters are a special case of A/Ds. Obviously, if voltage
measurements are taken and the results are displayed digitally with LED
or LCD displays, the instrument has to contain an A/D converter.
Digital voltmeters have some characteristics that you might need to understand.
You can click here for more material on digits
in digital voltmeters.

Digital voltmeters usually
have scales that are 0-0.3v, 0-3v, 0-30v, 0-300v, etc.
You need to learn a little terminology and the
reasons for the terminology. Let's take a look at a sample voltmeter.
Example
E4
Consider a voltmeter built around a 10 bit A/D converter. We will
assume the following.

The range of the voltmeter
is from 0-3v, and it does DC voltage measurements. It does not measure
negative voltages.
Then, with 10 bits we
can draw these inferences.
Ten bits will produce
2^{10} intervals. That's 1024 intervals.
If there are 1024 intervals
over a range of 3v, each interval will be 3/1024 = .00293v.
It is easier to compute
the displayed voltage if the interval is adjusted to .003v.
That would make the range
0-3.072v. (That's .003 x 1024.)
If you are measuring a
voltage that varies around 3v, that would allow you to keep the range the
same, but still change the range when the voltage got large enough.
Manufacturers like to build in a little "hysteresis" to prevent constant
range changes in situations like that and it might be especially hard on
auto-ranging meters.
If you wanted to measure
negative voltages and have the range be from -3v to +3v, you would have
intervals of .006v, and the meter would measure from -3.072v to +3.072v.
If you wanted to measure
voltages on a 0-30v scale, you would probably use a voltage divider or
some other way to reduce the voltage by a factor of (exactly) 10 (i.e.,
multiply it by exactly 0.1) and then use the same converter as on the 0-3v
scale.
If we could use a 12 bit
A/D, then some conclusions would change.
Twelve bits will produce
2^{12} intervals. That's 4096 intervals.
If there are 4096 intervals
over a range of 3v, each interval will be 3/4096 = .000732v.
It is easier to compute
the displayed voltage if the interval is adjusted to .0075v.
That would make the range
0-3.072v - just as it was in the case of the 10 bit converter,
That produces the same
advantages as you had with the 10 bit converter.
If you wanted to measure
negative voltages and have the range be from -3v to +3v, you would have
intervals of .0015v, and the meter would measure from -3.072v to +3.072v.
At
this point, you should have a good idea about how the number of bits in
an A/D converter determines the accuracy of the converter - i.e. the resolution
of the converter.
You also need to learn how to determine the resolution of an instrument
in the laboratory, and relate the resolution of lab instruments to what
you know about the number of bits in a converter. You can click
here for an laboratory exercise which leads you through getting the
number of bits in the A/D converter inside a voltmeter.

Problems
Links
to Related Laboratory Work
Links
to Related Lessons
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