Digital
Voltmeters
Digital Voltmeters (DVMs) are a special case of
A/Ds. DVMs are voltmeters  i.e. they measure voltage  and are
general purpose instruments commonly used to measure voltages in labs and in the
field. DVMs display the measured voltage using LCDs or LEDs to display the
result in a floating point format. They are an instrument of choice for
voltage measurements in all kinds of situations.
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.

Digital voltmeters usually have
scales that are 00.3v, 03v, 030v, 0300v, etc.
It is not clear why those ranges were chosen but they
are commonplace. Now, consider some of the implications of these facts.
Example
E1
Consider a voltmeter built around a 10 bit A/D converter. We will assume
the following.
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
03.072v. (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 (if the instrument also has a 030v range, for instance)
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 autoranging 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 030v 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 03v 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
03.072v  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.
A Note on Voltmeter Specifications
In the
example you saw a few typical voltmeter possibilities. For some reason
voltmeters have had scales like 03v, 030v, etc. for a long time. You
might have expected 01v and 010v, etc. to be more common. However,
that's not the way it is, and it probably won't change any time soon. That
situation has led to some interesting ways to specify voltmeters.
If
you had a voltmeter that had a 01v range, and it had ten bits, it would
probably be designed to have a range from 01.024v, and it would measure
voltages in steps of .001v. Then, the measurement results would be things
like 0.314v or 0.582v, things like that. Displayed values would all have
exactly three decimal places, and the instrument would be referred to as a
3 digit meter. If you use the same converter on a 010v scale (and
put the voltage through a 0.1x voltage divider!), then the results would be
things like 3.14v or 5.82v. You would get exactly the same number of
significant figures, and you would still refer to the meter as a 3 digit meter.
Let's think about this situation.

If you have a voltmeter with a
01v scale that can read increments of .001v the meter is a 3 digit meter.

If you have a voltmeter with a
01v scale that can read increments of .0001v the meter is a 4 digit meter.

If you have a voltmeter with a
010v scale that can read increments of .001v the meter is a 4 digit meter.

If you have a voltmeter with a
0100v scale that can read increments of .001v the meter is a 5 digit meter.
Now, what if you have a meter that has a 03v scale
that can read increments of .001v? How many digits is that meter?
The Number Of Digits In A DVM
You need to be able to answer the question in the last section. When you
buy a meter it may tell you the number of digits and you need to know what that
means, especially when the scales are 03v, etc. Here is the story.

A meter that reads in
increments of .001v and has a 01v range is a 3 digit meter.

A meter that reads in
increments of .001v and has a 010v range is a 4 digit meter.

A meter that reads in
increments of .001v and has a 0100v range is a 5 digit meter.
Notice the logarithmic nature
of the relationship, summarized in this table.
Range (v)

Digits
(for .001v)

01

3

010

4

0100

5

If the high limit of the
scale is 3, that's almost halfway between 1 and 10 on a logarithmic scale.
(The mid point is really at the square root of ten.) A meter that has a
range of 03v is said to be a 3 1/2 digit meter when it has intervals of .001v.
That's halfway between 3 and 4 digits.
There is another way to look at the question of digits. If you have a
meter that has a 010v scale that reads in increments of .01v that's a 3 bit
meter. That meter has 1000 steps, and 1000=10^{3}.
Let's repeat the table from above, but include the log_{10}
of the number of steps.
Range

Digits
(for .001v)

#Steps

log_{10}(#Steps)

01v

3

1000

3

010v

4

10,000

4

030v

4.5?

30,000

4.47

0100v

5

100,000

5

We included an extra row for a 030v meter. We also included the number of
steps and a suggestion for the number of digits we can claim for the meter.
It looks reasonable to call a 030v meter with 30,000 steps a 4.5 digit meter,
and that's the way they are sold.
That's it for digits in a voltmeter. That's the way that they are
specified, and that's what you pay for when you buy a DVM. The number of
digits is determined by the number of bits in the A/D, and we need to look at
that idea just a little bit more.
