# Electrical resistance and conductance

Last updated

The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is electrical conductance, measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with mechanical friction. The SI unit of electrical resistance is the ohm (), while electrical conductance is measured in siemens (S) (formerly called "mho"s and then represented by ).

## Contents

The resistance of an object depends in large part on the material it is made of. Objects made of electrical insulators like rubber tend to have very high resistance and low conductivity, while objects made of electrical conductors like metals tend to have very low resistance and high conductivity. This relationship is quantified by resistivity or conductivity. The nature of a material is not the only factor in resistance and conductance, however; it also depends on the size and shape of an object because these properties are extensive rather than intensive. For example, a wire's resistance is higher if it is long and thin, and lower if it is short and thick. All objects resist electrical current, except for superconductors, which have a resistance of zero.

The resistance R of an object is defined as the ratio of voltage V across it to current I through it, while the conductance G is the reciprocal:

${\displaystyle R={\frac {V}{I}},\qquad G={\frac {I}{V}}={\frac {1}{R}}}$

For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constants (although they will depend on the size and shape of the object, the material it is made of, and other factors like temperature or strain). This proportionality is called Ohm's law, and materials that satisfy it are called ohmic materials.

In other cases, such as a transformer, diode or battery, V and I are not directly proportional. The ratio V/I is sometimes still useful, and is referred to as a chordal resistance or static resistance, [1] [2] since it corresponds to the inverse slope of a chord between the origin and an I–V curve. In other situations, the derivative ${\displaystyle {\frac {\mathrm {d} V}{\mathrm {d} I}}}$ may be most useful; this is called the differential resistance.

## Introduction

In the hydraulic analogy, current flowing through a wire (or resistor) is like water flowing through a pipe, and the voltage drop across the wire is like the pressure drop that pushes water through the pipe. Conductance is proportional to how much flow occurs for a given pressure, and resistance is proportional to how much pressure is required to achieve a given flow.

The voltage drop (i.e., difference between voltages on one side of the resistor and the other), not the voltage itself, provides the driving force pushing current through a resistor. In hydraulics, it is similar: The pressure difference between two sides of a pipe, not the pressure itself, determines the flow through it. For example, there may be a large water pressure above the pipe, which tries to push water down through the pipe. But there may be an equally large water pressure below the pipe, which tries to push water back up through the pipe. If these pressures are equal, no water flows. (In the image at right, the water pressure below the pipe is zero.)

The resistance and conductance of a wire, resistor, or other element is mostly determined by two properties:

• geometry (shape), and
• material

Geometry is important because it is more difficult to push water through a long, narrow pipe than a wide, short pipe. In the same way, a long, thin copper wire has higher resistance (lower conductance) than a short, thick copper wire.

Materials are important as well. A pipe filled with hair restricts the flow of water more than a clean pipe of the same shape and size. Similarly, electrons can flow freely and easily through a copper wire, but cannot flow as easily through a steel wire of the same shape and size, and they essentially cannot flow at all through an insulator like rubber, regardless of its shape. The difference between copper, steel, and rubber is related to their microscopic structure and electron configuration, and is quantified by a property called resistivity.

In addition to geometry and material, there are various other factors that influence resistance and conductance, such as temperature; see below.

## Conductors and resistors

Substances in which electricity can flow are called conductors. A piece of conducting material of a particular resistance meant for use in a circuit is called a resistor. Conductors are made of high-conductivity materials such as metals, in particular copper and aluminium. Resistors, on the other hand, are made of a wide variety of materials depending on factors such as the desired resistance, amount of energy that it needs to dissipate, precision, and costs.

## Ohm's law

For many materials, the current I through the material is proportional to the voltage V applied across it:

${\displaystyle I\propto V}$

over a wide range of voltages and currents. Therefore, the resistance and conductance of objects or electronic components made of these materials is constant. This relationship is called Ohm's law, and materials which obey it are called ohmic materials. Examples of ohmic components are wires and resistors. The current–voltage graph of an ohmic device consists of a straight line through the origin with positive slope.

Other components and materials used in electronics do not obey Ohm's law; the current is not proportional to the voltage, so the resistance varies with the voltage and current through them. These are called nonlinear or nonohmic. Examples include diodes and fluorescent lamps. The current-voltage curve of a nonohmic device is a curved line.

## Relation to resistivity and conductivity

The resistance of a given object depends primarily on two factors: What material it is made of, and its shape. For a given material, the resistance is inversely proportional to the cross-sectional area; for example, a thick copper wire has lower resistance than an otherwise-identical thin copper wire. Also, for a given material, the resistance is proportional to the length; for example, a long copper wire has higher resistance than an otherwise-identical short copper wire. The resistance R and conductance G of a conductor of uniform cross section, therefore, can be computed as

{\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}},\\[5pt]G&=\sigma {\frac {A}{\ell }}.\end{aligned}}}

where ${\displaystyle \ell }$ is the length of the conductor, measured in metres (m), A is the cross-sectional area of the conductor measured in square metres (m2), σ (sigma) is the electrical conductivity measured in siemens per meter (S·m−1), and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m). The resistivity and conductivity are proportionality constants, and therefore depend only on the material the wire is made of, not the geometry of the wire. Resistivity and conductivity are reciprocals: ${\displaystyle \rho =1/\sigma }$. Resistivity is a measure of the material's ability to oppose electric current.

This formula is not exact, as it assumes the current density is totally uniform in the conductor, which is not always true in practical situations. However, this formula still provides a good approximation for long thin conductors such as wires.

Another situation for which this formula is not exact is with alternating current (AC), because the skin effect inhibits current flow near the center of the conductor. For this reason, the geometrical cross-section is different from the effective cross-section in which current actually flows, so resistance is higher than expected. Similarly, if two conductors near each other carry AC current, their resistances increase due to the proximity effect. At commercial power frequency, these effects are significant for large conductors carrying large currents, such as busbars in an electrical substation, [3] or large power cables carrying more than a few hundred amperes.

The resistivity of different materials varies by an enormous amount: For example, the conductivity of teflon is about 1030 times lower than the conductivity of copper. Loosely speaking, this is because metals have large numbers of "delocalized" electrons that are not stuck in any one place, so they are free to move across large distances. In an insulator, such as Teflon, each electron is tightly bound to a single molecule so a great force is required to pull it away. Semiconductors lie between these two extremes. More details can be found in the article: Electrical resistivity and conductivity. For the case of electrolyte solutions, see the article: Conductivity (electrolytic).

Resistivity varies with temperature. In semiconductors, resistivity also changes when exposed to light. See below.

## Measurement

An instrument for measuring resistance is called an ohmmeter. Simple ohmmeters cannot measure low resistances accurately because the resistance of their measuring leads causes a voltage drop that interferes with the measurement, so more accurate devices use four-terminal sensing.

## Typical values

ComponentResistance (Ω)
1 meter of copper wire with 1 mm diameter0.02 [lower-alpha 1]
1 km overhead power line (typical)0.03 [5]
AA battery (typical internal resistance )0.1 [lower-alpha 2]
Incandescent light bulb filament (typical)200–1000 [lower-alpha 3]
Human body1000–100,000 [lower-alpha 4]

## Static and differential resistance

The current–voltage curve of a non-ohmic device (purple). The static resistance at point A is the inverse slope of line B through the origin. The differential resistance at A is the inverse slope of tangent line C.
The current–voltage curve of a component with negative differential resistance, an unusual phenomenon where the current–voltage curve is non-monotonic.

Many electrical elements, such as diodes and batteries do not satisfy Ohm's law. These are called non-ohmic or non-linear, and their current–voltage curves are not straight lines through the origin.

Resistance and conductance can still be defined for non-ohmic elements. However, unlike ohmic resistance, non-linear resistance is not constant but varies with the voltage or current through the device; i.e., its operating point. There are two types of resistance: [1] [2]

Static resistance (also called chordal or DC resistance)
This corresponds to the usual definition of resistance; the voltage divided by the current
${\displaystyle R_{\mathrm {static} }={\frac {~U~}{I}}\,}$.

It is the slope of the line (chord) from the origin through the point on the curve. Static resistance determines the power dissipation in an electrical component. Points on the current–voltage curve located in the 2nd or 4th quadrants, for which the slope of the chordal line is negative, have negative static resistance. Passive devices, which have no source of energy, cannot have negative static resistance. However active devices such as transistors or op-amps can synthesize negative static resistance with feedback, and it is used in some circuits such as gyrators.

Differential resistance (also called dynamic, incremental or small signal resistance)
Differential resistance is the derivative of the voltage with respect to the current; the slope of the current–voltage curve at a point
${\displaystyle R_{\mathrm {diff} }={\frac {~{\mathrm {d} }\,U~}{{\mathrm {d} }\,I}}\,}$.

If the current–voltage curve is nonmonotonic (with peaks and troughs), the curve has a negative slope in some regions—so in these regions the device has negative differential resistance . Devices with negative differential resistance can amplify a signal applied to them, and are used to make amplifiers and oscillators. These include tunnel diodes, Gunn diodes, IMPATT diodes, magnetron tubes, and unijunction transistors.

## AC circuits

When an alternating current flows through a circuit, the relation between current and voltage across a circuit element is characterized not only by the ratio of their magnitudes, but also the difference in their phases. For example, in an ideal resistor, the moment when the voltage reaches its maximum, the current also reaches its maximum (current and voltage are oscillating in phase). But for a capacitor or inductor, the maximum current flow occurs as the voltage passes through zero and vice versa (current and voltage are oscillating 90° out of phase, see image below). Complex numbers are used to keep track of both the phase and magnitude of current and voltage:

${\displaystyle u(t)={\mathfrak {Re}}\left(U_{0}\cdot e^{j\omega t}\right),\quad i(t)={\mathfrak {Re}}\left(I_{0}\cdot e^{j(\omega t+\varphi )}\right),\quad {\underline {Z}}={\frac {\underline {U}}{\underline {I}}},\quad {\underline {Y}}={\frac {\underline {I}}{\underline {U}}}}$

where:

• t is time,
• u(t) and i(t) are, respectively, voltage and current as a function of time,
• U0 and I0 indicate the amplitude of voltage respective current,
• ${\displaystyle \omega }$ is the angular frequency of the AC current,
• ${\displaystyle \varphi }$ is the displacement angle,
• U, I, Z, and Y are complex numbers,
• Z is called impedance,
• Re indicates real part,
• ${\displaystyle j={\sqrt {-1}}}$ is the imaginary unit.

The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts:

${\displaystyle {\underline {Z}}=R+jX,\quad {\underline {Y}}=G+jB}$

where R and G are resistance and conductance respectively, X is reactance, and B is susceptance. For ideal resistors, Z and Y reduce to R and G respectively, but for AC networks containing capacitors and inductors, X and B are nonzero.

${\displaystyle {\underline {Z}}=1/{\underline {Y}}}$ for AC circuits, just as ${\displaystyle R=1/G}$ for DC circuits.

### Frequency dependence

A key feature of AC circuits is that the resistance and conductance can be frequency-dependent, a phenomenon known as the universal dielectric response. [8] One reason, mentioned above is the skin effect (and the related proximity effect). Another reason is that the resistivity itself may depend on frequency (see Drude model, deep-level traps, resonant frequency, Kramers–Kronig relations, etc.)

## Energy dissipation and Joule heating

Resistors (and other elements with resistance) oppose the flow of electric current; therefore, electrical energy is required to push current through the resistance. This electrical energy is dissipated, heating the resistor in the process. This is called Joule heating (after James Prescott Joule), also called ohmic heating or resistive heating.

The dissipation of electrical energy is often undesired, particularly in the case of transmission losses in power lines. High voltage transmission helps reduce the losses by reducing the current for a given power.

On the other hand, Joule heating is sometimes useful, for example in electric stoves and other electric heaters (also called resistive heaters). As another example, incandescent lamps rely on Joule heating: the filament is heated to such a high temperature that it glows "white hot" with thermal radiation (also called incandescence).

The formula for Joule heating is:

${\displaystyle P=I^{2}R}$

where P is the power (energy per unit time) converted from electrical energy to thermal energy, R is the resistance, and I is the current through the resistor.

## Dependence on other conditions

### Temperature dependence

Near room temperature, the resistivity of metals typically increases as temperature is increased, while the resistivity of semiconductors typically decreases as temperature is increased. The resistivity of insulators and electrolytes may increase or decrease depending on the system. For the detailed behavior and explanation, see Electrical resistivity and conductivity.

As a consequence, the resistance of wires, resistors, and other components often change with temperature. This effect may be undesired, causing an electronic circuit to malfunction at extreme temperatures. In some cases, however, the effect is put to good use. When temperature-dependent resistance of a component is used purposefully, the component is called a resistance thermometer or thermistor. (A resistance thermometer is made of metal, usually platinum, while a thermistor is made of ceramic or polymer.)

Resistance thermometers and thermistors are generally used in two ways. First, they can be used as thermometers: By measuring the resistance, the temperature of the environment can be inferred. Second, they can be used in conjunction with Joule heating (also called self-heating): If a large current is running through the resistor, the resistor's temperature rises and therefore its resistance changes. Therefore, these components can be used in a circuit-protection role similar to fuses, or for feedback in circuits, or for many other purposes. In general, self-heating can turn a resistor into a nonlinear and hysteretic circuit element. For more details see Thermistor#Self-heating effects.

If the temperature T does not vary too much, a linear approximation is typically used:

${\displaystyle R(T)=R_{0}[1+\alpha (T-T_{0})]}$

where ${\displaystyle \alpha }$ is called the temperature coefficient of resistance, ${\displaystyle T_{0}}$ is a fixed reference temperature (usually room temperature), and ${\displaystyle R_{0}}$ is the resistance at temperature ${\displaystyle T_{0}}$. The parameter ${\displaystyle \alpha }$ is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, ${\displaystyle \alpha }$ is different for different reference temperatures. For this reason it is usual to specify the temperature that ${\displaystyle \alpha }$ was measured at with a suffix, such as ${\displaystyle \alpha _{15}}$, and the relationship only holds in a range of temperatures around the reference. [9]

The temperature coefficient ${\displaystyle \alpha }$ is typically +3×10−3 K−1 to +6×10−3 K−1 for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude. [lower-alpha 5]

### Strain dependence

Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain. By placing a conductor under tension (a form of stress that leads to strain in the form of stretching of the conductor), the length of the section of conductor under tension increases and its cross-sectional area decreases. Both these effects contribute to increasing the resistance of the strained section of conductor. Under compression (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on strain gauges for details about devices constructed to take advantage of this effect.

### Light illumination dependence

Some resistors, particularly those made from semiconductors, exhibit photoconductivity , meaning that their resistance changes when light is shining on them. Therefore, they are called photoresistors (or light dependent resistors). These are a common type of light detector.

## Superconductivity

Superconductors are materials that have exactly zero resistance and infinite conductance, because they can have V = 0 and I ≠ 0. This also means there is no joule heating, or in other words no dissipation of electrical energy. Therefore, if superconductive wire is made into a closed loop, current flows around the loop forever. Superconductors require cooling to temperatures near 4 K with liquid helium for most metallic superconductors like niobium–tin alloys, or cooling to temperatures near 77 K with liquid nitrogen for the expensive, brittle and delicate ceramic high temperature superconductors. Nevertheless, there are many technological applications of superconductivity, including superconducting magnets.

## Footnotes

1. The resistivity of copper is about 1.7×10−8 Ωm. [4]
2. For a fresh Energizer E91 AA alkaline battery, the internal resistance varies from 0.9 Ω at −40 °C, to 0.1 Ω at +40 °C. [6]
3. A 60 W light bulb (in the USA, with 120-V mains electricity) draws RMS current 60 W/120 V = 500 mA, so its resistance is 120 V/500 mA = 240 Ω. The resistance of a 60 W light bulb in Europe (230 V mains) is 900 Ω. The resistance of a filament is temperature-dependent; these values are for when the filament is already heated up and the light is already glowing.
4. 100,000 Ω for dry skin contact, 1000 Ω for wet or broken skin contact. High voltage breaks down the skin, lowering resistance to 500 Ω. Other factors and conditions are relevant as well. For more details, see the electric shock article, and NIOSH 98-131. [7]
5. See Electrical resistivity and conductivity for a table. The temperature coefficient of resistivity is similar but not identical to the temperature coefficient of resistance. The small difference is due to thermal expansion changing the dimensions of the resistor.

## Related Research Articles

An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving particles are called charge carriers, which may be one of several types of particles, depending on the conductor. In electric circuits the charge carriers are often electrons moving through a wire. In semiconductors they can be electrons or holes. In a electrolyte the charge carriers are ions, while in plasma, an ionized gas, they are ions and electrons.

An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil.

A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active elements, and terminate transmission lines, among other uses. High-power resistors that can dissipate many watts of electrical power as heat, may be used as part of motor controls, in power distribution systems, or as test loads for generators. Fixed resistors have resistances that only change slightly with temperature, time or operating voltage. Variable resistors can be used to adjust circuit elements, or as sensing devices for heat, light, humidity, force, or chemical activity.

A thermistor is a type of resistor whose resistance is strongly dependent on temperature, more so than in standard resistors. The word is a combination of thermal and resistor. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting overcurrent protectors, and self-regulating heating elements. An operational temperature range of a thermistor is dependent on the probe type and is typically in between −100 °C (173 K) and 300 °C (573 K).

In electrical engineering, electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied.

Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

Electrical resistivity is a fundamental property of a material that measures how strongly it resists electric current. Its inverse, called electrical conductivity, quantifies how well a material conducts electricity. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the Greek letter ρ (rho). The SI unit of electrical resistivity is the ohm-meter (Ω⋅m). For example, if a 1 m solid cube of material has sheet contacts on two opposite faces, and the resistance between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m.

Thermal conduction is the transfer of internal energy by microscopic collisions of particles and movement of electrons within a body. The colliding particles, which include molecules, atoms and electrons, transfer disorganized microscopic kinetic and potential energy, jointly known as internal energy. Conduction takes place in all phases: solid, liquid, and gas.

In electric and electronic systems, reactance is the opposition of a circuit element to the flow of current due to that element's inductance or capacitance. Greater reactance leads to smaller currents for the same voltage applied. Reactance is similar to electric resistance in this respect, but differs in that reactance does not lead to dissipation of electrical energy as heat. Instead, energy is stored in the reactance, and a quarter-cycle later returned to the circuit, whereas a resistance continuously loses energy.

In physics and electrical engineering, a conductor is an object or type of material that allows the flow of charge in one or more directions. Materials made of metal are common electrical conductors. Electrical current is generated by the flow of negatively charged electrons, positively charged holes, and positive or negative ions in some cases.

Skin effect is the tendency of an alternating electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor and decreases exponentially with greater depths in the conductor. The electric current flows mainly at the "skin" of the conductor, between the outer surface and a level called the skin depth. Skin depth depends on the frequency of the alternating current; as frequency increases, current flow moves to the surface, resulting in less skin depth. Skin effect reduces the effective cross-section of the conductor and thus increases its effective resistance. Skin effect is caused by opposing eddy currents induced by the changing magnetic field resulting from the alternating current. At 60 Hz in copper, the skin depth is about 8.5 mm. At high frequencies the skin depth becomes much smaller.

The lumped-element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions. It is useful in electrical systems, mechanical multibody systems, heat transfer, acoustics, etc.

Joule heating, also known as resistive, resistance, or Ohmic heating, is the process by which the passage of an electric current through a conductor produces heat.

Resistance thermometers, also called resistance temperature detectors (RTDs), are sensors used to measure temperature. Many RTD elements consist of a length of fine wire wrapped around a ceramic or glass core but other constructions are also used. The RTD wire is a pure material, typically platinum, nickel, or copper. The material has an accurate resistance/temperature relationship which is used to provide an indication of temperature. As RTD elements are fragile, they are often housed in protective probes.

Voltage drop is the decrease of electrical potential along the path of a current flowing in an electrical circuit. Voltage drops in the internal resistance of the source, across conductors, across contacts, and across connectors are undesirable because some of the energy supplied is dissipated. The voltage drop across the electrical load is proportional to the power available to be converted in that load to some other useful form of energy.

Copper loss is the term often given to heat produced by electrical currents in the conductors of transformer windings, or other electrical devices. Copper losses are an undesirable transfer of energy, as are core losses, which result from induced currents in adjacent components. The term is applied regardless of whether the windings are made of copper or another conductor, such as aluminium. Hence the term winding loss is often preferred. The term load loss is closely related but not identical, since an unloaded transformer will have some winding loss.

The telegrapher's equations are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who developed the transmission line model starting with an August 1876 paper, On the Extra Current. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can form along the line.

There are a number of possible ways to measure thermal conductivity, each of them suitable for a limited range of materials, depending on the thermal properties and the medium temperature. Three classes of methods exist to measure the thermal conductivity of a sample: steady-state, time-domain, and frequency-domain methods.

Soil resistivity is a measure of how much the soil resists or conducts electric current. It is a critical factor in design of systems that rely on passing current through the Earth's surface. An understanding of the soil resistivity and how it varies with depth in the soil is necessary to design the grounding system in an electrical substation, or for lightning conductors. It is needed for design of grounding (earthing) electrodes for substations and High-voltage direct current transmission systems. It was formerly important in earth-return telegraphy. It can also be a useful measure in agriculture as a proxy measurement for moisture content.

Performance modelling is the abstraction of a real system into a simplified representation to enable the prediction of performance. The creation of a model can provide insight into how a proposed or actual system will or does work. This can, however, point towards different things to people belonging to different fields of work.

## References

1. Brown, Forbes T. (2006). Engineering System Dynamics: A Unified Graph-Centered Approach (2nd ed.). Boca Raton, Florida: CRC Press. p. 43. ISBN   978-0-8493-9648-9.
2. Kaiser, Kenneth L. (2004). Electromagnetic Compatibility Handbook. Boca Raton, Florida: CRC Press. pp. 13–52. ISBN   978-0-8493-2087-3.
3. Fink & Beaty (1923). "Standard Handbook for Electrical Engineers". Nature (11th ed.). 111 (2788): 17–19. Bibcode:1923Natur.111..458R. doi:10.1038/111458a0. hdl:. S2CID   26358546.
4. Cutnell, John D.; Johnson, Kenneth W. (1992). (2nd ed.). New York: Wiley. p. 559. ISBN   978-0-471-52919-4.
5. McDonald, John D. (2016). Electric Power Substations Engineering (2nd ed.). Boca Raton, Florida: CRC Press. pp. 363ff. ISBN   978-1-4200-0731-2.
6. Battery internal resistance (PDF) (Report). Energizer Corp.
7. "Worker Deaths by Electrocution" (PDF). National Institute for Occupational Safety and Health. Publication No. 98-131. Retrieved 2 November 2014.
8. Zhai, Chongpu; Gan, Yixiang; Hanaor, Dorian; Proust, Gwénaëlle (2018). "Stress-dependent electrical transport and its universal scaling in granular materials". Extreme Mechanics Letters. 22: 83–88. arXiv:. doi:10.1016/j.eml.2018.05.005. S2CID   51912472.
9. Ward, M.R. (1971). Electrical Engineering Science. McGraw-Hill. pp. 36–40.