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Parallel conductance

Conductance is the reciprocal of resistance. The unit of conductance is the siemens (S). You can analyze parallel resistors by describing each resistor as a conductance. Written by Willy McAllister.
In a previous article we studied parallel resistors.
We derived this equation to combine parallel resistors into a single equivalent resistor,
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, 1, divided by, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, point, point, point, plus, start fraction, 1, divided by, start text, R, end text, start subscript, start text, N, end text, end subscript, end fraction, right parenthesis, end fraction
This is a fairly complex expression, with 1, slash, start text, R, end text terms embedded inside another reciprocal. There is an alternate way to approach this problem, using the concept of conductance.

Conductance

Ohm's Law, v, equals, i, start text, R, end text, defines resistance as the ratio of voltage over current,
start text, R, end text, equals, start fraction, v, divided by, i, end fraction
The term conductance is the inverse of this expression. It is the ratio of current over voltage,
start text, G, end text, equals, start fraction, i, divided by, v, end fraction
This gives us yet another way to write Ohm's Law,
i, equals, v, start text, G, end text
The unit of conductance is the siemens, abbreviated start text, S, end text. It is named after Werner von Siemens, founder of the German industrial electronics and telecommunications company that bears his name. There is an s at the end of siemens even if it is singular, 1, start text, s, i, e, m, e, n, s, end text. You may come across an older term, the mho, used as the unit of conductance. Mho is just "ohm" spelled backwards. That term isn't used anymore.
Using conductance instead of resistance for the same physical object simply emphasizes a different aspect of its behavior. Resistance reduces or impedes current flow, while conductance allows current to pass through. The terms are two aspects of the same idea.
A 100, \Omega resistor is the same as a conductance of start fraction, 1, divided by, 100, \Omega, end fraction equals, 0, point, 01, start text, S, end text.

Parallel conductance

In this section, we'll repeat the analysis of parallel resistors, but this time, instead of calling each component a resistor, we will call it a conductance. The result for parallel conductance will have a strong resemblance to series resistors.
Here is a circuit with conductances in parallel. We will analyze this circuit using the language of conductance, and the conductance form of Ohm's Law, i, equals, v, start text, G, end text, point
The value of current i is some given constant. We don't yet know v or how i splits up into three currents through the conductances.
Two things we do know are:
  • The three conductance currents add up to i.
  • Voltage v appears across all three conductances.
With just this little bit of knowledge, and the conductance form of Ohm's Law, we can write these expressions:
i, equals, i, start subscript, start text, G, 1, end text, end subscript, plus, i, start subscript, start text, G, 2, end text, end subscript, plus, i, start subscript, start text, G, 3, end text, end subscript
i, start subscript, start text, G, 1, end text, end subscript, equals, v, dot, start text, G, 1, end text, i, start subscript, start text, G, 2, end text, end subscript, equals, v, dot, start text, G, 2, end text, i, start subscript, start text, G, 3, end text, end subscript, equals, v, dot, start text, G, 3, end text
This is enough to get going. Combining equations:
i, equals, v, dot, start text, G, 1, end text, plus, v, dot, start text, G, 2, end text, plus, v, dot, start text, G, 3, end text
Factor out the voltage term and gather the conductance values in one place:
i, equals, v, left parenthesis, start text, G, 1, end text, plus, start text, G, 2, end text, plus, start text, G, 3, end text, right parenthesis
This looks just like Ohm's Law for a single conductance, with the parallel conductances appearing as a sum.
We conclude:
For conductances in parallel, the overall conductance is the sum of the individual conductances.
Notice how much this looks like the formula for resistors in series. Conductances in parallel are like resistances in series, they add.

Equivalent parallel conductances

We can imagine a new conductance equivalent to the sum of the parallel conductances. It is equivalent in the sense that the same voltage appears.
start text, G, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start text, G, 1, end text, plus, start text, G, 2, end text, plus, start text, G, 3, end text

Conductance example

Let's solve the same circuit we did for parallel resistors, but using the new representation.
This is the circuit with conductances, start text, G, end text, equals, start fraction, 1, divided by, start text, R, end text, end fraction
You can try to solve this yourself before looking at the answer. We want to find voltage v and the individual currents, i, start subscript, start text, G, 1, end text, end subscript, i, start subscript, start text, G, 2, end text, end subscript, and i, start subscript, start text, G, 3, end text, end subscript, using the conductance form of Ohm's Law, i, equals, v, start text, G, end text.
Find v and the current through the three conductances.
Show that the individual currents add up to i.

Summary

Conductances in parallel combine with a simple sum. The two ways to combine parallel resistors are:
start text, G, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start text, G, 1, end text, plus, start text, G, 2, end text, plus, point, point, point, plus, start text, G, end text, start subscript, start text, N, end text, end subscript
start text, R, end text, start subscript, start text, p, a, r, a, l, l, e, l, end text, end subscript, equals, start fraction, 1, divided by, left parenthesis, start fraction, 1, divided by, start text, R, 1, end text, end fraction, plus, start fraction, 1, divided by, start text, R, 2, end text, end fraction, plus, point, point, point, plus, start fraction, 1, divided by, start text, R, end text, start subscript, start text, N, end text, end subscript, end fraction, right parenthesis, end fraction
The sum of conductances is simpler than the "reciprocal of reciprocals" we came up with for parallel resistors, and there are no special-case formulas to remember. This is the main reason to introduce the concept of conductance. The reciprocals did not go away, we just did them at the beginning when we derived start text, G, end text values from the given start text, R, end text's. Using conductance represents a rearrangement of the same computation.
How you choose to analyze parallel circuits, start text, G, end text or start text, R, end text, is a matter of convenience and simplicity.

Want to join the conversation?

  • blobby green style avatar for user alexis.trousil
    In the example problem, I'm confused about the use of the variable G vs the variable S. What are the meanings of these two variables, and are they interchangeable?
    (4 votes)
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    • purple pi purple style avatar for user APDahlen
      Hello Alexis,

      Electrical conductance (G) is measured in units of Siemens (S). This is the inverse of resistance (R) which is measured in units of Ohms (Ω).

      Personally, I prefer the old unit unit called the mho abbreviated ℧. Note that mho is ohm spelled backwards. This older definition always made the inverse relationship between resistance and conductance easier to understand.

      Please leave a comment below if you have any questions.

      Regards,

      APD
      (9 votes)
  • piceratops sapling style avatar for user BNgetich69
    What about conductance in series?
    (1 vote)
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  • blobby green style avatar for user jaxwarner
    hi, if one branch of parallel circuit were to open circuit what would happen to total current ? i'm assuming because it's parallel ( other branches are still connected to nodes) and current are usually divided in parallel series, the current would increase and as a result power would increase (power = current times volt) Can you confirm?
    (1 vote)
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  • blobby green style avatar for user Riles
    So for conductance, is basically the inverse of resistance? you can calculate resistance and just inverse it?
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      Yes. Conductance is the inverse of resistance. The main reason we talk about conductance is the equation for parallel conductance. That equation (sum of conductances) looks just like the formula for computing series resistance (sum of resistances). That is a nice observation to make and understand. In electrical engineering this kind of relationship between two equations is called a "dual".
      (2 votes)
  • blobby green style avatar for user Majed Emad
    greetings
    I would like to know more about conductance G which weren't given a full understandable explanation
    (1 vote)
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  • duskpin ultimate style avatar for user Paul Breeden
    "...specified in units of mhos, which is just "ohms" spelled backwards." Wouldn't that be smho instead of mhos??
    (1 vote)
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  • leafers seed style avatar for user roschrocksds
    What would the applications of parallel conductance be?? Does it have the same properties like series resistance??
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      Parallel conductance is simply a different viewpoint of parallel resistors. It's the same circuit configuration, but uses a different equation. The "conductance" viewpoint looks at a parallel resistor and instead of seeing a 1/R term in the parallel resistor formula, it sees a G = 1/R term.

      When you use the conductance viewpoint, the formula (G1 + G2 + G3... = Gequiv) strongly resembles the formula for series resistors. That doesn't mean its the same, its just a strong resemblance.

      Relationships like the one between parallel conductance and series resistance are common in electrical engineering. They are called "duals" of each other.

      I personally have never used the conductance approach to resistor circuits. It is good to know about just for the improvement in mental flexibility.
      (1 vote)
  • orange juice squid orange style avatar for user Robert Rondahl
    In which situations is it more useful to use conductance intstead of resistance, when they essentially describe the same thing?
    (1 vote)
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  • winston default style avatar for user fluffyplatypus
    Do we need to know all of the formulas to build circuits?
    (1 vote)
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    • purple pi purple style avatar for user APDahlen
      Hello Dragoncgn,

      To build - no!

      To design working circuits - sometimes.

      Often we start electronics by building existing circuits and then experimenting with the circuit by substituting parts and seeing how the circuit changes. This is an excellent way to start. It gives you an intuitive sense for the circuit operation and later provides a foundation upon which to add the math.

      Have fun for the design!

      Regards,

      APD
      (1 vote)
  • blobby green style avatar for user Anwesha Donna
    Since conductance is in Siemens, and we are looking to find the voltage in Volts, aren't we supposed to convert the current (which is in mA) into Amperes?
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      The units in Ohm's Law v=IR are volts, amperes, and ohms. So yes, when you have a current in mA you have to be sure to treat the units correctly. In the example near the end the current is given as 100mA. You have to be sure to treat that as 0.1 A in the calculations.
      (1 vote)