# Electric conductance: Wikis

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Electrical conductance is a measure of how easily electricity flows along a certain path through an electrical element. The SI derived unit of conductance is the siemens. Conductance may also be expressed as mho, because it is the reciprocal of electrical resistance (measured in ohms). Oliver Heaviside coined the term conductivity in September 1885.[1]

Electrical conductance is related to but should not be confused with conduction, which is the mechanism by which charge flows, or with conductivity, which is a property of a material.

## Relation to other quantities

For purely resistive circuits conductance is related to resistance by:

$G = \frac{1}{R}$

where R is the electrical resistance (Note: this is not true where the impedance is non-real)

Furthermore, conductance is related to susceptance and admittance by the equation:

$Y = G + j B \,$

or

$G = Re(Y) \,$

where:

j is the imaginary unit,
B is the susceptance.

## Combining conductances

The conductance G of an object of cross-sectional area A and length $\ell$ can be determined from the material's conductivity σ by the formula,

$G=\frac{\sigma \, A}{\ell}.$

From Kirchhoff's circuit laws we can deduce the rules for combining conductances. For two conductances G1 and G2 in parallel the voltage across them is the same and from Kirchoff's Current Law the total current is

$I_{Eq} = I_1 + I_2.\ \,$

Substituting Ohm's law for conductances gives

$G_{Eq} V = G_1 V + G_2 V\ \,$

and the equivalent conductance will be,

$G_{Eq} = G_1 + G_2.\ \,$

For two conductances G1 and G2 in series the current through them will be the same and Kirchhoff's Voltage Law tells us that the voltage across them is the sum of the voltages across each conductance, that is,

$V_{Eq} = V_1 + V_2.\ \,$

Substituting Ohm's law for conductance then gives,

$\frac {I}{G_{Eq}} = \frac {I}{G_1} + \frac {I}{G_2}$

which in turn gives the formula for the equivalent conductance,

$\frac {1}{G_{Eq}} = \frac {1}{G_1} + \frac {1}{G_2}.$

This equation can be rearranged slightly, though this is a special case that will only rearrange like this for two components.

$G_{Eq} = \frac{G_1 G_2}{G_1+G_2}.$

## Small-signal device conductances

The term conductance is applied to electronic devices such as transistors and diodes, where it usually refers to a small-signal model that is a linearization of the underlying device equations about a selected DC operating point or Q-point. This conductance is the reciprocal of the small-signal device resistance. See Early effect and channel length modulation.

## References

• Halliday, David (1960). Physics Part II. John Wiley and Sons.
1. ^ Nahin, Paul J. (2002). "7. Heaviside's Electrodynamics" (online). Oliver Heaviside: the life, work, and times of an electrical genius of the Victorial age. Johns Hopkins. p. 118. ISBN 0-8018-6909-9. "general formula for energy current ...[was] proved by me for conductors in the summer of 1884, and in January 1885 extended to all media non-homogeneous as regards capacity, conductivity and permeability"

Electrical conductance is a measurement of how easily electricity flows along a certain path through an electrical element. The SI derived unit of conductance is the siemens. This unit was historically referred to as the mho, because it is the reciprocal of electrical resistance (measured in ohms). Oliver Heaviside coined the term conductivity in September 1885.[1]

Electrical conductance is related to but should not be confused with conduction, which is the mechanism by which charge flows, or with conductivity, which is a property of a material.

## Relation to other quantities

For purely resistive circuits conductance is related to resistance by:

$G = \frac\left\{1\right\}\left\{R\right\}$

where $R$ is the electrical resistance (Note: this is not true where the impedance is non-real)

Furthermore, conductance is related to susceptance and admittance by the equation:

$Y = G + j B \,$

or

$G = Re\left(Y\right) \,$

where:

$j$ is the imaginary unit,
B is the susceptance.

## Combining conductances

The conductance $G$ of an object of cross-sectional area $A$ and length $\ell$ can be determined from the material's conductivity $\sigma$ by the formula,

$G=\frac\left\{\sigma \, A\right\}\left\{\ell\right\}.$

From Kirchhoff's circuit laws we can deduce the rules for combining conductances. For two conductances $G_1$ and $G_2$ in parallel the voltage across them is the same and from Kirchoff's Current Law the total current is

$I_\left\{Eq\right\} = I_1 + I_2.\ \,$

Substituting Ohm's law for conductances gives

$G_\left\{Eq\right\} V = G_1 V + G_2 V\ \,$

and the equivalent conductance will be,

$G_\left\{Eq\right\} = G_1 + G_2.\ \,$

For two conductances $G_1$ and $G_2$ in series the current through them will be the same and Kirchhoff's Voltage Law tells us that the voltage across them is the sum of the voltages across each conductance, that is,

$V_\left\{Eq\right\} = V_1 + V_2.\ \,$

Substituting Ohm's law for conductance then gives,

$\frac \left\{I\right\}\left\{G_\left\{Eq\right\}\right\} = \frac \left\{I\right\}\left\{G_1\right\} + \frac \left\{I\right\}\left\{G_2\right\}$

which in turn gives the formula for the equivalent conductance,

$\frac \left\{1\right\}\left\{G_\left\{Eq\right\}\right\} = \frac \left\{1\right\}\left\{G_1\right\} + \frac \left\{1\right\}\left\{G_2\right\}.$

This equation can be rearranged slightly, though this is a special case that will only rearrange like this for two components.

$G_\left\{Eq\right\} = \frac\left\{G_1 G_2\right\}\left\{G_1+G_2\right\}.$

## Small-signal device conductances

The term conductance is applied to electronic devices such as transistors and diodes, where it usually refers to a small-signal model that is a linearization of the underlying device equations about a selected DC operating point or Q-point. This conductance is the reciprocal of the small-signal device resistance. See Early effect and channel length modulation.

## References

• Halliday, David (1960). Physics Part II. John Wiley and Sons.
1. ^ Nahin, Paul J. (2002). "7. Heaviside's Electrodynamics" (online). Oliver Heaviside: the life, work, and times of an electrical genius of the Victorial age. Johns Hopkins. p. 118. ISBN 0-8018-6909-9. "general formula for energy current ...[was] proved by me for conductors in the summer of 1884, and in January 1885 extended to all media non-homogeneous as regards capacity, conductivity and permeability"