Analog-to-Digital converters - a.k.a. A/D converters - are widely used by many
engineers and scientists of all types, often without their realizing it.
Whenever they make a measurement of a voltage, and that measurement is taken
into a computer, an A/D is used.

If
you're going to take measurements - and just about every engineer will do a lot
of that - then you will be better off if you understand some of the basic ideas
behind A/D converters. There are two simple goals for this lesson.

Given an A/D converter with a given range and number of bits,

To be able to calculate the resolution of the converter.

Given an A/D converter in the laboratory,

To be able to determine the resolution of the converter and the number of
bits used in the converter.

What Are A/D Converters?

A/D
converters are electrical circuits that have the following characteristics.

The input to the A/D converter
is a voltage.

A/D converters may be designed
for voltages from 0 to 10v, from -5 to +5v, etc., but they almost always take a
voltage input. (Some rare exceptions occur with current inputs!) In
any event, the input is an analog voltage signal for most cases.

The output of the A/D converter
is a binary signal, and that binary signal encodes the analog input voltage.
So, the output is some sort of digital number.

A
comparator can be used as a simple one-bit A/D converter. Although a
converter with just one bit isn't particularly useful, you can begin to see how
an A/D converter works by puttering with it for a moment. If you read the
lesson on comparators you encountered a simluation of a comparator. That
simulator is reproduced below.

Comparator
Simulator

Sim1
Here is the comparator simulator. You can think of a comparator as a
one-bit A/D converter. The input is an analog signal, and the output is a
one bit digital representation of the analog signal. In the simulator, you
can control a simulated voltage source that is the input the the comparator, and
the digital output bit is indicated with a simulated LED. Notice the
following.

The input can range from zero
(0) to ten (10) volts.

When the input voltage goes
above five (5) volts, the output is a binary one (1) and the LED lights.
When the input voltage is less than five volts, the output is a binary zero (0)
and the LED does not light.

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:

V_{meas}
= Count*0.625

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.