#import "../../core/core.typ"

=== Ohmmeter design

Though mechanical ohmmeter (resistance meter) designs are rarely used
today, having largely been superseded by digital instruments, their
operation is nonetheless intriguing and worthy of study.

The purpose of an ohmmeter, of course, is to measure the resistance
placed between its leads. This resistance reading is indicated through a
mechanical meter movement which operates on electric current. The
ohmmeter must then have an internal source of voltage to create the
necessary current to operate the movement, and also have appropriate
ranging resistors to allow just the right amount of current through the
movement at any given resistance.

Starting with a simple movement and battery circuit, let's see how it
would function as an ohmmeter:

#image("static/00174.png")

When there is infinite resistance (no continuity between test leads),
there is zero current through the meter movement, and the needle points
toward the far left of the scale. In this regard, the ohmmeter
indication is "backwards" because maximum indication (infinity) is on
the left of the scale, while voltage and current meters have zero at the
left of their scales.

If the test leads of this ohmmeter are directly shorted together
(measuring zero $Omega$), the meter movement will have a maximum amount of
current through it, limited only by the battery voltage and the
movement's internal resistance:

#image("static/00175.png")

With 9 volts of battery potential and only 500 $Omega$ of movement resistance,
our circuit current will be 18 mA, which is far beyond the full-scale
rating of the movement. Such an excess of current will likely damage the
meter.

Not only that, but having such a condition limits the usefulness of the
device. If full left-of-scale on the meter face represents an infinite
amount of resistance, then full right-of-scale should represent zero.
Currently, our design "pegs" the meter movement hard to the right when
zero resistance is attached between the leads. We need a way to make it
so that the movement just registers full-scale when the test leads are
shorted together. This is accomplished by adding a series resistance to
the meter's circuit:

#image("static/00176.png")

To determine the proper value for R, we calculate the total circuit
resistance needed to limit current to 1 mA (full-scale deflection on
the movement) with 9 volts of potential from the battery, then subtract
the movement's internal resistance from that figure:

$
  R_"total" &= E / I = (9 V) / (1 m A) \
  R_"total" &= 9 k Omega \ 
  R &= R_"total" - 500 Omega = 8.5 k Omega \
$

Now that the right value for R has been calculated, we're still left
with a problem of meter range. On the left side of the scale we have
"infinity" and on the right side we have zero. Besides being
"backwards" from the scales of voltmeters and ammeters, this scale is
strange because it goes from nothing to everything, rather than from
nothing to a finite value (such as 10 volts, 1 amp, etc.). One might
pause to wonder, "what does middle-of-scale represent? What figure lies
exactly between zero and infinity?" Infinity is more than just a
_very big_ amount: it is an incalculable quantity, larger than any
definite number ever could be. If half-scale indication on any other
type of meter represents 1/2 of the full-scale range value, then what is
half of infinity on an ohmmeter scale?

The answer to this paradox is a _logarithmic scale_. Simply put,
the scale of an ohmmeter does not smoothly progress from zero to
infinity as the needle sweeps from right to left. Rather, the scale
starts out "expanded" at the right-hand side, with the successive
resistance values growing closer and closer to each other toward the
left side of the scale:

#image("static/00177.png")

Infinity cannot be approached in a linear (even) fashion, because the
scale would _never_ get there! With a logarithmic scale, the amount
of resistance spanned for any given distance on the scale increases as
the scale progresses toward infinity, making infinity an attainable
goal.

We still have a question of range for our ohmmeter, though. What value
of resistance between the test leads will cause exactly 1/2 scale
deflection of the needle? If we know that the movement has a full-scale
rating of 1 mA, then 0.5 mA (500 µA) must be the value needed for
half-scale deflection. Following our design with the 9 volt battery as a
source we get:

$
  R_"total" = E / I = (9 V) / (500 mu A) \
  R_"total" = 18 k Omega
$

With an internal movement resistance of 500 $Omega$ and a series range
resistor of 8.5 k$Omega$, this leaves 9 k$Omega$ for an external (lead-to-lead)
test resistance at 1/2 scale. In other words, the test resistance giving
1/2 scale deflection in an ohmmeter is equal in value to the (internal)
series total resistance of the meter circuit.

Using Ohm's Law a few more times, we can determine the test resistance
value for 1/4 and 3/4 scale deflection as well:


1/4 scale deflection (0.25 mA of meter current):

$
  R_"total" &= E / I = (9 V) / (250 mu A) \
  R_"total" &= 36 k Omega
$

$
  R_"test" &= R_"total" - R_"internal" \
  R_"test" &= 36 k Omega - 9 k Omega \
  R_"test" &= 27 k Omega \
$


3/4 scale deflection (0.75 mA of meter current):

$
  R_"total" &= E / I = (9 V) / (750 mu A) \
  R_"total" &= 12 k Omega \
$

$ 
  R_"test" &= R_"total" - R_"internal" \
  R_"test" &= 12k Omega - 9k Omega \
  R_"test" &= 3 k Omega \
$


So, the scale for this ohmmeter looks something like this:

#image("static/00178.png")


One major problem with this design is its reliance upon a stable battery
voltage for accurate resistance reading. If the battery voltage
decreases (as all chemical batteries do with age and use), the ohmmeter
scale will lose accuracy. With the series range resistor at a constant
value of 8.5 k$Omega$ and the battery voltage decreasing, the meter will no
longer deflect full-scale to the right when the test leads are shorted
together (0 $Omega$). Likewise, a test resistance of 9 k$Omega$ will fail to
deflect the needle to exactly 1/2 scale with a lesser battery voltage.

There are design techniques used to compensate for varying battery
voltage, but they do not completely take care of the problem and are to
be considered approximations at best. For this reason, and for the fact
of the logarithmic scale, this type of ohmmeter is never considered to
be a precision instrument.

One final caveat needs to be mentioned with regard to ohmmeters: they
only function correctly when measuring resistance that is not being
powered by a voltage or current source. In other words, you cannot
measure resistance with an ohmmeter on a "live" circuit! The reason
for this is simple: the ohmmeter's accurate indication depends on the
only source of voltage being its internal battery. The presence of any
voltage across the component to be measured will interfere with the
ohmmeter's operation. If the voltage is large enough, it may even
damage the ohmmeter.

#core.review[
- Ohmmeters contain internal sources of voltage to supply power in
  taking resistance measurements.
- An analog ohmmeter scale is "backwards" from that of a voltmeter or
  ammeter, the movement needle reading zero resistance at full-scale and
  infinite resistance at rest.
- Analog ohmmeters also have logarithmic scales, "expanded" at the low
  end of the scale and "compressed" at the high end to be able to span
  from zero to infinite resistance.
- Analog ohmmeters are not precision instruments.
- Ohmmeters should _never_ be connected to an energized circuit
  (that is, a circuit with its own source of voltage). Any voltage
  applied to the test leads of an ohmmeter will invalidate its reading.
]
