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.
- REVIEW:
- Conductance is the opposite of resistance: the measure of how
easy is it for electrons to flow through something.
- Conductance is symbolized with the letter "G" and is measured
in units of mhos or Siemens.
- Mathematically, conductance equals the reciprocal of
resistance: G = 1/R
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