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Conductance definition – Electrical conductance theory

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conductance theory

Conductance (electrical conductivity) is the inverse of resistance.

Conductance is denoted by the letter G, and the unit of conductance is S (simens).

Conductance is a measure of the conductivity of an electric current in a material.

The conductivity is a measure of the susceptibility of a material to the flow of electric current. For the known geometric dimensions of a conductor and the conductivity of its material, its conductance is given by the conductance formula:

conductance formula
  • l – conductor length,
  • S – cross-sectional area of the element,
  • σ – conductivity of the material.

The above formula is specified only for macroscopic systems. For mesoscopic systems this quantity is expressed differently. For an ideal quantum wire it is expressed by the conductance formula:

conductance formula
  • e – elementary charge,
  • h – Planck’s constant,
  • N – number of open channels.

In this case, there is no direct dependence on the geometry of the system, only on the number of open conductivity channels. This quantity, in turn, depends stepwise on the transverse size of the conductor. The theory describing this phenomenon was given in 1957 by R. Landauer.

Conductance applies to DC circuits, and in AC circuits only to resistive elements (resistor). Generalization of the concept of conductance to AC circuits containing capacitive (capacitor) and inductive (coil) elements is called admittance.

Condutance vs resistance

Resistivity or specific resistance.

A unit of resistivity is Ω⋅m

  • The value of 1Ω⋅m tells you what resistance a conductor with a length of 1 meter and a cross-sectional area of 1m2,
  • Resistivity is a constant value that characterizes a material and has different values for different materials.

The symbol ϱ (ro) is used to define resistivity. It should be emphasized that resistivity and resistance are two different concepts, closely related but different. Do not confuse the terms resistivity (specific resistance) and resistance (electrical resistance).

The relationship between resistance and resistivity is defined by the formula:

ϱ – resistivity (specific resistance) [Ω⋅m],

R – resistance (electrical resistance) [Ω],

S – cross-sectional area of the solid under consideration [m2],

l – the length of the solid under consideration [m].

This formula applies to materials that satisfy Ohm’s law.

The inverse of resistivity is conductivity. The concept of conductivity is quite intuitive, the higher the conductivity, the better the material conducts current, the lower the conductivity, the worse the material conducts current. Exactly the opposite is the case with resistivity, the higher resistivity, the worse the body conducts current.

Equivalent resistance

When determining the equivalent resistance, the relationship between resistance and conductance is very useful.

G=1/R

R=1/G

In general, we distinguish between two elementary ways of combining resistances.

Series connection of resistors → the same current flows through the resistors, the equivalent resistance is equal to the algebraic sum of the component resistances.

R=R1+R2+…+Rn

Conductance in series:

G=1/G1 + 1/G2 + … + 1/Gn

Conductance in series

Parallel connection of resistors → the same voltage is applied to the resistor terminals, then the reciprocal of the equivalent resistance is equal to the algebraic sum of the reciprocals of the component resistances.

1/R=1/R1+1/R2+…+1/Rn

When determining the resistance of a parallel connection of resistors, it is convenient to use the relationship between resistance and conductance.

Conductance in parallel:

G=G1+G2+…+Gn

Conductance in parallel

Very often when analyzing electrical circuits, both direct current and alternating current, we treat resistance as a constant. In general, resistance depends on temperature R=R(T).

R(T)=R0-(1+α-(T-T0)

In the equation above, we represented the resistance of the conductor as a linear dependence on temperature. In reality, it does not have to be so colorful. Depending on the material and on the temperature range, the resistance can depend linearly on temperature, or it can be described by a polynomial equation. An example of satisfying the R(T) equation presented above is the characteristic of a parametric temperature sensor Pt100 for the temperature range 0[°C]-100[°C].

The resistance also depends on the frequency R=R(f) of the flowing current. In case of direct current with frequency f=0[Hz] this phenomenon does not occur. In the case of alternating current, resistance increases with increasing frequency f, this is related to the skin effect.

In the case of conductors, the resistance increases with increasing temperature. Conductors are not the only type of matter we encounter. Semiconductors are another example. In the case of semiconductors, their resistance can both increase and decrease with increasing temperature. In semiconductors, two mechanisms are responsible for the resistance. The first mechanism is related to the geometry of the semiconductor and the second mechanism is related to the charge carriers in the semiconductor. As the temperature increases in semiconductors, more charge carriers are released. Depending on which of the mechanisms in the semiconductor has the advantage it can in effect cause its resistance to increase or decrease with increasing temperature. A practical example is semiconductor temperature sensors:

  • PTC sensor (en. positive temperature coefficient) → resistance increases with increasing temperature
  • NTC sensor (en. negative temperature coefficient) → resistance decreases with increasing temperature

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