Electrical Engineering

Electrical resistivity (also known as resistivity, specific electrical resistance, or volume resistivity) is a property of a material; it quantifies how strongly the material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electric charge. The SI unit of electrical resistivity is the ohm⋅metre (Ω⋅m). It is commonly represented by the Greek letter ρ (rho).

Electrical conductivity or specific conductance is the reciprocal quantity, and measures a material's ability to conduct an electric current. It is commonly represented by the Greek letter σ (sigma), but κ (kappa) (especially in electrical engineering) or γ (gamma) are also occasionally used. Its SI unit is siemens per metre (S⋅m−1) and CGSE unit is reciprocal second (s−1).

Many resistors and conductors have a uniform cross section with a uniform flow of electric current and are made of one material. (See the diagram to the right.) In this case, the electrical resistivity ρ (Greek: rho) is defined as:

whereR is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω) is the length of the piece of material (measured in metres, m)A is the cross-sectional area of the specimen (measured in square metres, m²).

The reason resistivity is defined this way is that it makes resistivity a material property, unlike resistance. All copper wires, irrespective of their shape and size, have approximately the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity-for example rubber's resistivity is far larger than copper's.

In a hydraulic analogy, passing current through a high-resistivity material is like pushing water through a pipe full of sand, while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. But resistance is not solely determined by the presence or absence of sand; it also depends on how wide the pipe is (it is harder to push water through a skinny pipe than a wide one) and how long it is (it is harder to push water through a long pipe than a short one.)

The above equation can be transposed to get Pouillet's law:

The resistance of a given material will increase with the length, but decrease with increasing cross-sectional area. From the above equations, resistivity has SI units of ohm⋅metre. Other units like ohm⋅cm or ohm⋅inch are also sometimes used.

Conductivity is the inverse of resistivity:

Conductivity has SI units of Siemens per meter (S/m).

whereρ is the resistivity of the conductor material (measured in ohm⋅metres, Ω⋅m),E is the magnitude of the electric field (in volts per metre, V⋅m−1),J is the magnitude of the current density (in amperes per square metre, A⋅m−2),

in which E and J are inside the conductor.

Conductivity is the inverse:

For example, rubber is a material with large ρ and small σ, because even a very large electric field in rubber will cause almost no current to flow through it. On the other hand, copper is a material with small ρ and large σ, because even a small electric field pulls a lot of current through it.

Quantum mechanics states that electrons in an atom cannot take on any arbitrary energy value. Rather, there are fixed energy levels which the electrons can occupy, and values in between these levels are impossible[citation needed]. When a large number of such allowed energy levels are spaced close together (in energy-space) i.e. have similar (minutely differing energies) then we can talk about these energy levels together as an "energy band". There can be many such energy bands in a material, depending on the atomic number (number of electrons) and their distribution (besides external factors like environment modifying the energy bands). Two such bands important in the discussion of conductivity of materials are: the valence band and the conduction band (the latter is generally above the former)[citation needed]. Electrons in the conduction band may move freely throughout the material 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, even large voltages can yield relatively small currents.

Near room temperatures, metals have resistance. The primary cause of this resistance is the thermal motion of ions. This acts to scatter electrons (due to destructive interference of free electron waves on non-correlating potentials of ions)[citation needed]. Also contributing to resistance in metals with impurities are the resulting imperfections in the lattice. In pure metals this source is negligible[citation needed].

The larger the cross-sectional area of the conductor, the more electrons per unit length are available to carry the current. As a result, the resistance is lower in larger cross-section conductors. The number of scattering events encountered by an electron passing through a material is proportional to the length of the conductor. The longer the conductor, therefore, the higher the resistance. Different materials also affect the resistance.[1]

In metals, the Fermi level lies in the conduction band (see Band Theory, above) giving rise to free conduction electrons. However, in semiconductors the position of the Fermi level is within the band gap, approximately half-way between the conduction band minimum and valence band maximum for intrinsic (undoped) semiconductors. This means that at 0 kelvins, there are no free conduction electrons and the resistance is infinite. However, 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 concentration 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, hence highly doped semiconductors behave metallically. 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.

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 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.

The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In ordinary conductors, such as copper or silver, this decrease is limited by impurities and other defects. Even near absolute zero, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. An electric current flowing in a loop of superconducting wire can persist indefinitely with no power source.[2]

In 1986, it was discovered that some cuprate-perovskite ceramic materials have a critical temperature above 90 K (−183 °C). Such a high transition temperature is theoretically impossible for a conventional superconductor, leading the materials to be termed high-temperature superconductors. Liquid nitrogen boils at 77 K, facilitating many experiments and applications that are less practical at lower temperatures. In conventional superconductors, electrons are held together in pairs by an attraction mediated by lattice phonons. The best available model of high-temperature superconductivity is still somewhat crude. There is a hypothesis that electron pairing in high-temperature superconductors is mediated by short-range spin waves known as paramagnons.[3]

• A conductor such as a metal has high conductivity and a low resistivity.
• An insulator like glass has low conductivity and a high resistivity.
• The conductivity of a semiconductor is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of light, and, most important, with temperature and composition of the semiconductor material.

The degree of doping in semiconductors makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a solution of water is highly dependent on its concentration of dissolved salts, and other chemical species that ionize in the solution. Electrical conductivity of water samples is used as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower the conductivity (the higher the resistivity). Conductivity measurements in water are often reported as specific conductance, relative to the conductivity of pure water at 25 °C. An EC meter is normally used to measure conductivity in a solution. A rough summary is as follows:MaterialResistivity

ρ (Ω•m)Metals10−8SemiconductorsvariableElectrolytesvariableInsulators1016Superconductors0

This table shows the resistivity, conductivity and temperature coefficient of various materials at 20 °C (68 °F)Materialρ (Ω•m) at 20 °Cσ (S/m) at 20 °CTemperature

coefficient[note 1]

(K−1)ReferenceSilver1.59×10−86.30×1070.0038[4][5]Copper1.68×10−85.96×1070.0039[5]Annealed copper[note 2]1.72×10−85.80×107

[citation needed]Gold[note 3]2.44×10−84.10×1070.0034[4]Aluminium[note 4]2.82×10−83.5×1070.0039[4]Calcium3.36×10−82.98×1070.0041

Tungsten5.60×10−81.79×1070.0045[4]Zinc5.90×10−81.69×1070.0037[6]Nickel6.99×10−81.43×1070.006

Lithium9.28×10−81.08×1070.006

Iron1.0×10−71.00×1070.005[4]Platinum1.06×10−79.43×1060.00392[4]Tin1.09×10−79.17×1060.0045

Carbon steel (1010)1.43×10−76.99×106

[7]Lead2.2×10−74.55×1060.0039[4]Titanium4.20×10−72.38×106X

Grain oriented electrical steel4.60×10−72.17×106

[8]Manganin4.82×10−72.07×1060.000002[9]Constantan4.9×10−72.04×1060.000008[10]Stainless steel[note 5]6.9×10−71.45×106

[11]Mercury9.8×10−71.02×1060.0009[9]Nichrome[note 6]1.10×10−69.09×1050.0004[4]GaAs5×10−7 to 10×10−35×10−8 to 103

[12]Carbon (amorphous)5×10−4 to 8×10−41.25 to 2×103−0.0005[4][13]Carbon (graphite)[note 7]2.5e×10−6 to 5.0×10−6 //basal plane

3.0×10−3 ⊥basal plane2 to 3×105 //basal plane

3.3×102 ⊥basal plane

[14]Carbon (diamond)[note 8]1×1012~10−13

[15]Germanium[note 8]4.6×10−12.17−0.048[4][5]Sea water[note 9]2×10−14.8

[16]Drinking water[note 10]2×101 to 2×1035×10−4 to 5×10−2

[citation needed]Silicon[note 8]6.40×1021.56×10−3−0.075[4]Deionized water[note 11]1.8×1055.5×10−6

[17]Glass10×1010 to 10×101410−11 to 10−15?[4][5]Hard rubber1×101310−14?[4]Sulfur1×101510−16?[4]Air1.3×1016 to 3.3×10163×10−15 to 8×10−15

[18]Paraffin1×101710−18?

Fused quartz7.5×10171.3×10−18?[4]PET10×102010−21?

Teflon10×1022 to 10×102410−25 to 10−23?

The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C value is only an approximation when used at other temperatures. For example, the coefficient becomes lower at higher temperatures for copper, and the value 0.00427 is commonly specified at 0 °C.[19]

The extremely low resistivity (high conductivity) of silver is characteristic of metals. George Gamow tidily summed up the nature of the metals' dealings with electrons in his science-popularizing book, One, Two, Three...Infinity (1947): "The metallic substances differ from all other materials by the fact that the outer shells of their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current." More technically, the free electron model gives a basic description of electron flow in metals.

where is called the temperature coefficient of resistivity, is a fixed reference temperature (usually room temperature), and is the resistivity at temperature . The parameter is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, is different for different reference temperatures. For this reason it is usual to specify the temperature that was measured at with a suffix, such as , and the relationship only holds in a range of temperatures around the reference.[20] When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.

where 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. 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.

If more than one source of scattering is simultaneously present, Matthiessen's Rule (first formulated by Augustus Matthiessen in the 1860s) [21][22] says that the total resistance can be approximated by adding up several different terms, each with the appropriate value of n.

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 investigation of the low-temperature resistivity of metals was the motivation to Heike Kamerlingh Onnes's experiments that led in 1911 to discovery of superconductivity. For details see History of superconductivity.

In general, resistivity of intrinsic semiconductors decreases with increasing temperature. The electrons are bumped to the conduction energy band by thermal energy, where they flow freely and in doing so leave behind holes in the valence band which also flow freely. The electric resistance of a typical intrinsic (non doped) semiconductor decreases exponentially with the temperature:

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

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

This equation is used to calibrate thermistors.

Extrinsic (doped) semiconductors have a far more complicated temperature profile. As temperature increases 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.[23]

In non-crystalline semiconductors, conduction can occur by charges quantum tunnelling from one localised site to another. This is known as variable range hopping and has the characteristic form of,

where n = 2, 3, 4, depending on the dimensionality of the system.

Conversely, in such cases the conductivity must be expressed as a complex number (or even as a matrix of complex numbers, in the case of anisotropic materials) called the admittivity. Admittivity is the sum of a real component called the conductivity and an imaginary component called the susceptivity.

An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real permittivity. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more opaque the material is). For details, see Mathematical descriptions of opacity.

For an anisotropic material, it is not generally valid to use the scalar equations

For example, the current may not flow in exactly the same direction as the electric field. Instead, the equations are generalized to the 3D tensor form[25][26]

where the conductivity σ and resistivity ρ are rank-2 tensors (in other words, 3×3 matrices). The equations are compactly illustrated in component form (using index notation and the summation convention) [27]:

The σ and ρ tensors are inverses (in the sense of a matrix inverse). The individual components are not necessarily inverses; for example σxx may not be equal to 1/ρxx.

In cases like this, the formulas

need to be replaced with

where E and J are now vector fields. This equation, along with the continuity equation for J and the Poisson equation for E, form a set of partial differential equations. In special cases, an exact or approximate solution to these equations can be worked out by hand, but for very accurate answers in complex cases, computer methods like finite element analysis may be required.

product (nΩ•m•g/cm3)Sodium47.70.9746Lithium92.80.5349Calcium33.61.5552Potassium72.00.8964Beryllium35.61.8566Aluminium26.502.7072Magnesium43.901.7476.3Copper16.788.96150Silver15.8710.49166Gold22.1419.30427Iron96.17.874757

Silver, although it is the least resistive metal known, has a high density and does poorly by this measure. Calcium and the alkali metals have the best products, but are rarely used for conductors due to their high reactivity with water and oxygen. Aluminium is far more stable. Two other important attributes, price and toxicity, exclude the (otherwise) best choice: Beryllium. Thus, aluminium is usually the metal of choice when the weight of some required conduction (and/or the cost of conduction) is the driving consideration.

120 Watts in a 120 Volt system and 240 Watts in a 240 volt 1 phase system and 415 watts in a 240 Volt 3 phase system.

The SI unit for electric current, the ampere, is defined as 'that constant current which, if maintained in each of two, straight, parallel conductors of infinite length and negligible cross-sectional area, and placed one metre apart in a vacuum, would produce between them a force of 2 x 10-7 newtons per unit length'.