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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.
In a previous article we studied parallel resistors.
We derived this equation to combine parallel resistors into a single equivalent resistor,
${\text{R}}_{\text{parallel}}=\frac{1}{\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\text{…}+\frac{1}{{\text{R}}_{\text{N}}}\right)}$
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.

## Conductance

Ohm's Law, $v=i\phantom{\rule{0.167em}{0ex}}\text{R}$, defines resistance as the ratio of voltage over current,
$\text{R}=\frac{v}{i}$
The term conductance is the inverse of this expression. It is the ratio of current over voltage,
$\text{G}=\frac{i}{v}$
This gives us yet another way to write Ohm's Law,
$i=v\phantom{\rule{0.167em}{0ex}}\text{G}$
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, $1\phantom{\rule{0.167em}{0ex}}\text{siemens}$. 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 $100\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }$ resistor is the same as a conductance of $\frac{1}{100\phantom{\rule{0.167em}{0ex}}\mathrm{\Omega }}$ $=0.01\phantom{\rule{0.167em}{0ex}}\text{S}$.

## 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, $i=v\phantom{\rule{0.167em}{0ex}}\text{G}.$
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:
• 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:
$i={i}_{\text{G1}}+{i}_{\text{G2}}+{i}_{\text{G3}}$
${i}_{\text{G1}}=v\cdot \text{G1}\phantom{\rule{2em}{0ex}}{i}_{\text{G2}}=v\cdot \text{G2}\phantom{\rule{2em}{0ex}}{i}_{\text{G3}}=v\cdot \text{G3}$
This is enough to get going. Combining equations:
$i=v\cdot \text{G1}\phantom{\rule{0.167em}{0ex}}+\phantom{\rule{0.167em}{0ex}}v\cdot \text{G2}\phantom{\rule{0.167em}{0ex}}+\phantom{\rule{0.167em}{0ex}}v\cdot \text{G3}$
Factor out the voltage term and gather the conductance values in one place:
$i=v\phantom{\rule{0.167em}{0ex}}\phantom{\rule{0.167em}{0ex}}\left(\text{G1}+\text{G2}+\text{G3}\right)$
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.
${\text{G}}_{\text{parallel}}=\text{G1}+\text{G2}+\text{G3}$

### Conductance example

Let's solve the same circuit we did for parallel resistors, but using the new representation.
This is the circuit with conductances, $\text{G}=\frac{1}{\text{R}}$
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, $i=v\phantom{\rule{0.167em}{0ex}}\text{G}$.
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:
${\text{G}}_{\text{parallel}}=\text{G1}+\text{G2}+\text{…}+{\text{G}}_{\text{N}}$
${\text{R}}_{\text{parallel}}=\frac{1}{\left(\frac{1}{\text{R1}}+\frac{1}{\text{R2}}+\text{…}+\frac{1}{{\text{R}}_{\text{N}}}\right)}$
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.

## Want to join the conversation?

• 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?
• 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.

Regards,

APD
• What about conductance in series?
(1 vote)
• Conductances in series have an equation very similar to resistors in parallel.
1/Gs = 1/G1 + 1/G2 + ... 1/Gn

See if you can derive this equation on your own, by combining what you learned in this article along with the theory in the Series Resistor article.
• 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)
• If you remove a resistor from a parallel arrangement, the total current through the remaining resistors goes down. By removing a resistor, the overall equivalent parallel resistance goes UP, so the current goes DOWN.
• In which situations is it more useful to use conductance intstead of resistance, when they essentially describe the same thing?
• Conductance might be helpful when a circuit is dominated by parallel connections. Adding conductance is simpler than an endless string of reciprocals of reciprocals for resistance.

I just wrote this article on Delta-Y transformation equations. One of the derivations uses conductance in a really elegant way. http://spinningnumbers.org/a/delta-wye-derivations.html. Look down at the last third of the article, the second derivation of Y-to-Delta.
• greetings
I would like to know more about conductance G which weren't given a full understandable explanation
(1 vote)
• "Conductance" is defined near the beginning of the article. Conductance is the reciprocal of Resistance. It is not a new type of component or a new idea, it is simply a different point of view of a resistor.
• So for conductance, is basically the inverse of resistance? you can calculate resistance and just inverse it?
(1 vote)
• 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".
• "...specified in units of mhos, which is just "ohms" spelled backwards." Wouldn't that be smho instead of mhos??
(1 vote)
• ohm --> mho
ohms --> mhos

Although I do like your spelling, too. It's fun to say. Too bad it's an old unit we don't use any more. That would be a pretty funny debate.
• What is the sum conductance
(1 vote)
• The sum of conductances is equivalent to the general formula for parallel resistors...

G_equiv = G1 + G2 + G3...

is the same as

1/R_equiv = 1/R1 + 1/R2 + 1/R3

If you have a circuit made of conductances in parallel you combine them by simply adding each conductance.
(1 vote)
• What would the applications of parallel conductance be?? Does it have the same properties like series resistance??
(1 vote)
• 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)
• Do we need to know all of the formulas to build circuits?
(1 vote)
• 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)