PowerPedia:Electrical resistance

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A 750-kohm , as identified by its . A could be used to verify this value.
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A 750-kohm resistor, as identified by its electronic color code. A multimeter could be used to verify this value.

Electrical resistance is a measure of the degree to which an object opposes the passage of an electric current. It is often futile. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.

The quantity of resistance in an electric circuit determines the amount of current flowing in the circuit for any given voltage applied to the circuit.

R = \frac{V}{I}

where

R is the resistance of the object, usually measured in ohms, equivalent to J·s/C2
V is the potential difference across the object, usually measured in volts
I is the current passing through the object, usually measured in amperes

For a wide variety of materials and conditions, the electrical resistance does not depend on the amount of current flowing or the amount of applied voltage. V can either be measured directly across the object or calculated from a subtraction of voltages relative to a reference point. The former method is simpler for a single object and is likely to be more accurate. There may also be problems with the latter method if the voltage supply is AC and the two measurements from the reference point are not in phase with each other.

Table of contents
1 Resistive loss
2 Resistance of a conductor

2.1 DC resistance
2.2 AC resistance

3 Causes of resistance

3.1 In metals
3.2 In semiconductors and insulators
3.3 In ionic liquids/electrolytes
3.4 Resistance of various materials
3.5 Band theory

4 Differential resistance
5 Temperature-dependence
6 Resistivity

6.1 Definitions
6.2 Table of resistivities
6.3 Temperature dependence
6.4 Complex resistivity

7 Related
8 External articles and references
9 See also
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Resistive loss

When a current, I, flows through an object with resistance, R, electrical energy is converted to heat at a rate (power) equal to

P = {I^{2} \cdot R} \,

where

P is the power measured in watts
I is the current measured in amperes
R is the resistance measured in ohms

This effect is useful in some applications such as incandescent lighting and electric heating, but is undesirable in power transmission. Common ways to combat resistive loss include using thicker wire and higher voltages. Superconducting wire is used in special applications.

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Resistance of a conductor

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DC resistance

As long as the current density is totally uniform in the conductor, the DC resistance R of a conductor of regular cross section can be computed as

R = {L \cdot \rho \over A} \,

where

L is the length of the conductor, measured in meters
A is the cross-sectional area, measured in square meters
Ï? (Greek: rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm · meter. Resistivity is a measure of the material's ability to oppose the flow of electric current.

For practical reasons, almost any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.

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AC resistance

If a wire conducts high-frequency alternating current then the effective cross sectional area of the wire is reduced. This is because of the skin effect.

This formula applies to isolated conductors. In a conductor close to others, the actual resistance is higher because of the proximity effect.

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Causes of resistance

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In metals

A metal consists of a lattice of atoms, each with a shell of electrons. This can also be known as positive ionic lattice. The outer electrons are free to dissociate from their parent atoms and travel through the lattice, creating a 'sea' of electrons, making the metal a conductor. When an electrical potential difference (a voltage) is applied across the metal, the electrons drift from one end of the conductor to the other under the influence of the electric field.

In a metal the thermal motion of ions is the primary source of scattering of electrons (due to destructive interference of free electron wave on non-correlating potentials of ions) - thus the prime cause of metal resistance. Imperfections of lattice also contribute into resistance, although their contribution in pure metals is negligible.

The larger the cross-sectional area of the conductor, the more electrons are available to carry the current, so the lower the resistance. The longer the conductor, the more scattering events occur in each electron's path through the material, so the higher the resistance. [1] (http://www.ias.ac.in/resonance/Sept2003/pdf/Sept2003p41-48.pdf)

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In semiconductors and insulators

In metals the fermi level lies in the conduction band giving rise to free conduction electrons. However in semiconductors the position of the fermi level is within the band gap, exactly half way between the conduction band minimum and valence band maximum for intrinsic(undoped) semiconductors. This means that at 0 Kelvin, there are no free conduction electrons and the resistance is infinite. However, as the temperature is increased, charge carriers are thermally excited to the conduction band giving rise to a non-zero resistance. The resistance will continue to decrease as the charge carrier density in the conduction band increases. In extrinsic (doped) semiconductors, dopant atoms increase the majority charge carrier by donating electrons to the conduction band or accepting holes in the valence band. For both types of donor or acceptor atoms, increasing the dopant density leads to a reduction in the resistance. Highly doped semiconductors hence behave metallic. At very high temperatures, the contribution of thermally generated carriers will dominate over the contribution from dopant atoms and the resistance will decrease exponentially with temperature.

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In ionic liquids/electrolytes

In electrolytes, electrical conduction happens not by band electrons or holes, but by full atomic species (ions) traveling, each carrying an electrical charge. The resistivity of ionic liquids varies tremendously by the salt concentration - while distilled water is almost an insulator, salt water is a very efficient electrical conductor. In biological membranes, currents are carried by ionic salts. Small holes in the membranes, called ion channels, are selective to specific ions and determine the membrane resistance.

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Resistance of various materials

Material Resistivity, ρ
ohm-meter
Metals 10 − 8
Semiconductors variable
Electrolytes variable
Insulators 1016
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Band theory

Electron energy levels in an insulator.
Electron energy levels in an insulator.

Quantum mechanics states that the energy of an electron in an atom cannot be any arbitrary value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible. The energy levels are grouped into two bands: the valence band and the conduction band (the latter is generally above the former). Electrons in the conduction band may move freely throughout the substance in the presence of an electrical field.

In insulators and semiconductors, the atoms in the substance influence each other so that between the valence band and the conduction band there exists a forbidden band of energy levels, which the electrons cannot occupy. In order for a current to flow, a relatively large amount of energy must be furnished to an electron for it to leap across this forbidden gap and into the conduction band. Thus, large voltages yield relatively small currents.

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Differential resistance

When resistance may depend on voltage and current, differential resistance, incremental resistance or slope resistance is defined as the slope of the V-I graph at a particular point, thus:

R = \frac {{d}V} {{d}I} \,

This quantity is sometimes called simply resistance, although the two definitions are equivalent only for an ohmic component such as an ideal resistor. If the V-I graph is not monotonic (i.e. it has a peak or a trough), the differential resistance will be negative for some values of voltage and current. This property is often known as negative resistance, although it is more correctly called negative differential resistance, since the absolute resistance V/I is still positive.

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Temperature-dependence

Near room temperature, the electric resistance of a typical metal conductor increases linearly with the temperature:

R = R_0(1 + aT) \,,

where a is the thermal resistance coefficient.

The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:

R= R_0 e^{a/T}\,

Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increased starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures it will behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.

The electric resistance of electrolytes and insulators is highly nonlinear, and case by case dependent, therefore no generalized equations are given.

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Resistivity

Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electrical charge. The SI unit of electrical resistivity is the ohm meter.

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Definitions

The electrical resistivity Ï? (rho) of a material is usually defined by the following:

{R={\rho l\over A}}

where

Ï? is the static resistivity (measured in ohm meters - Ωm)
R is the electrical resistance of a uniform specimen of the material (measured in ohms - Ω)
l is the length of the specimen (measured in meters - m)
A is the cross-sectional area of the specimen (measured in square meters - m²)

Electrical resistivity can also be defined as:

\rho={E \over J}

where

E is the magnitude of the electric field (measured in volts per meter - V/m)
J is the magnitude of the current density (measured in amperes per square meter A/m²)

Finally, electrical resistivity is also defined as the inverse of the conductivity σ (sigma), of the material, or:

\rho = {1 \over \sigma}.
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Table of resistivities

This table shows the resistivity and temperature coefficient of various materials. The values are correct at 20 degrees Celsius.

MaterialResistivity (Ωm)Temperature coefficient per kelvin *
Silver1.59 × 10-8.0038
Copper1.72 × 10-8.0039
Gold2.44 × 10-8.0034
Aluminium2.82 × 10-8.0039
Tungsten5.6 × 10-8.0045
Iron1.0 × 10-7.005
Brass0.8 × 10-7.0015
Platinum1.1 × 10-7.00392
Lead2.2 × 10-7.0039
Manganin4.4 × 10-7.000002
Constantan<4.9 × 10-7.00001
Mercury9.8 × 10-7.0009
Nichrome1.10 × 10-6.0004
Carbon3.5 × 10-5-.0005
Germanium4.6 × 10-1-.048
Silicon6.40 × 102-.075
Glass1010 to 1014nil
Hard rubberapproximately 1013nil
Sulfur1015nil
Quartz (fused)7.5 × 1017nil
PETapproximately 1 × 1020nil
Teflonapproximately 1 × 1022 to 1 × 1024nil
  • The numbers in this column increase or decrease the significand portion of the resistivity. For example, at 21°C (294.15K), the resistivity of silver is 1.5938 x 10-8
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Temperature dependence

In general, electrical resistivity of metals increases with temperature, while the resistivity of semiconductors decreases with increasing temperature. In both cases, electron-phonon interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity Ï? of a metal is given by the Bloch-Gruneissen formula :

\rho(T)=\rho(0)+A(\frac{T}{\Theta_R})^n\int_0^{\frac{\Theta_R}{T}}\frac{x^n}{(e^x-1)(1-e^{-x})}dx

where ρ(0) is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the fermi surface, the Debye radius and the number density of electrons in the metal. ΘR is the Debye temperature as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction:

  1. n=5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)
  2. n=3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)
  3. n=2 implies that the resistance is due to electron-electron interaction.

As the temperature of the metal is sufficiently reduced (so as to 'freeze' all the phonons), the resistivity usually reaches a constant value, known as the residual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as superconductivity.

An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the Steinhart-Hart equation:

1/T = A + B \ln(\rho) + C (\ln(\rho))^3 \,

where A, B and C are the so-called Steinhart-Hart coefficients.

This equation is used to calibrate thermistors.

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Complex resistivity

When analysing the response of materials to alternating electric fields, as is done in certain types of tomography, it is necessary to replace resistivity with a complex quantity called impeditivity, in analogy to electrical impedance. Impeditivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity (reactance) [2] (http://www.otto-schmitt.org/OttoPagesFinalForm/Sounds/Speeches/MutualImpedivity.htm).

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Related

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External articles and references

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See also

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