Conductance
When students first see the parallel
resistance equation, the natural question to ask is, "Where
did that thing come from?" It is truly an odd piece
of arithmetic, and its origin deserves a good explanation.
Resistance, by definition, is the measure of
friction a component presents to the flow of
electrons through it. Resistance is symbolized by the
capital letter "R" and is measured in the unit of "ohm."
However, we can also think of this electrical property in
terms of its inverse: how easy it is for electrons to
flow through a component, rather than how difficult.
If resistance is the word we use to symbolize the
measure of how difficult it is for electrons to flow, then a
good word to express how easy it is for electrons to flow
would be conductance.
Mathematically, conductance is the
reciprocal, or inverse, of resistance:
The greater the resistance, the less the
conductance, and visa-versa. This should make intuitive
sense, resistance and conductance being opposite ways to
denote the same essential electrical property. If two
components' resistances are compared and it is found that
component "A" has one-half the resistance of component "B,"
then we could alternatively express this relationship by
saying that component "A" is twice as conductive as
component "B." If component "A" has but one-third the
resistance of component "B," then we could say it is
three times more conductive than component "B," and so
on.
Carrying this idea further, a symbol and
unit were created to represent conductance. The symbol is
the capital letter "G" and the unit is the mho, which
is "ohm" spelled backwards (and you didn't think electronics
engineers had any sense of humor!). Despite its
appropriateness, the unit of the mho was replaced in later
years by the unit of siemens (abbreviated by the
capital letter "S"). This decision to change unit names is
reminiscent of the change from the temperature unit of
degrees Centigrade to degrees Celsius, or the
change from the unit of frequency c.p.s. (cycles per
second) to Hertz. If you're looking for a pattern
here, Siemens, Celsius, and Hertz are all surnames of famous
scientists, the names of which, sadly, tell us less about
the nature of the units than the units' original
designations.
As a footnote, the unit of siemens is never
expressed without the last letter "s." In other words, there
is no such thing as a unit of "siemen" as there is in the
case of the "ohm" or the "mho." The reason for this is the
proper spelling of the respective scientists' surnames. The
unit for electrical resistance was named after someone named
"Ohm," whereas the unit for electrical conductance was named
after someone named "Siemens," therefore it would be
improper to "singularize" the latter unit as its final "s"
does not denote plurality.
Back to our parallel circuit example, we
should be able to see that multiple paths (branches) for
current reduces total resistance for the whole circuit, as
electrons are able to flow easier through the whole network
of multiple branches than through any one of those branch
resistances alone. In terms of resistance, additional
branches results in a lesser total (current meets with less
opposition). In terms of conductance, however,
additional branches results in a greater total (electrons
flow with greater conductance):
Total parallel resistance is less
than any one of the individual branch resistances because
parallel resistors resist less together than they would
separately:
Total parallel conductance is greater
than any of the individual branch conductances because
parallel resistors conduct better together than they would
separately:
To be more precise, the total conductance in
a parallel circuit is equal to the sum of the individual
conductances:
If we know that conductance is nothing more
than the mathematical reciprocal (1/x) of resistance, we can
translate each term of the above formula into resistance by
substituting the reciprocal of each respective conductance:
Solving the above equation for total
resistance (instead of the reciprocal of total resistance),
we can invert (reciprocate) both sides of the equation:
So, we arrive at our cryptic resistance
formula at last! Conductance (G) is seldom used as a
practical measurement, and so the above formula is a common
one to see in the analysis of parallel circuits.
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REVIEW:
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Conductance is the opposite of resistance:
the measure of how easy is it for electrons to flow
through something.
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Conductance is symbolized with the letter
"G" and is measured in units of mhos or Siemens.
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Mathematically, conductance equals the
reciprocal of resistance: G = 1/R
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