Parallel conductance

In a previous article we studied parallel resistors.

We derived this equation to combine parallel resistors into a single equivalent resistor,

This is a fairly complex expression, with $1/\text{R}$‍ terms embedded inside another reciprocal. There is an alternate way to approach this problem, using the concept of conductance.

Ohm's Law, $v=i\text{R}$‍, defines resistance as the ratio of voltage over current,

The term conductance is the inverse of this expression. It is the ratio of current over voltage,

This gives us yet another way to write Ohm's Law,

The unit of conductance is the siemens, abbreviated $\text{S}$‍. 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, $1\text{siemens}$‍. 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 $100\mathrm{\Omega }$‍ resistor is the same as a conductance of ${\displaystyle \frac{1}{100\mathrm{\Omega }}}$‍ $=0.01\text{S}$‍.

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, $i=v\text{G}.$‍

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:

With just this little bit of knowledge, and the conductance form of Ohm's Law, we can write these expressions:

This is enough to get going. Combining equations:

Factor out the voltage term and gather the conductance values in one place:

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.

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.

Let's solve the same circuit we did for parallel resistors, but using the new representation.

This is the circuit with conductances, $\text{G}={\displaystyle \frac{1}{\text{R}}}$‍

You can try to solve this yourself before looking at the answer. We want to find voltage $v$‍ and the individual currents, $i_{\text{G1}}$‍, $i_{\text{G2}}$‍, and $i_{\text{G3}}$‍, using the conductance form of Ohm's Law, $i=v\text{G}$‍.

Conductances in parallel combine with a simple sum. The two ways to combine parallel resistors are:

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 $\text{G}$‍ values from the given $\text{R}$‍'s. Using conductance represents a rearrangement of the same computation.

How you choose to analyze parallel circuits, $\text{G}$‍ or $\text{R}$‍, is a matter of convenience and simplicity.