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The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is Template:Vanchor, 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 (Template:Math), while electrical conductance is measured in siemens (S) (formerly called the 'mho' and then represented by Template:Math).
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 conductance, while objects made of electrical conductors like metals tend to have very low resistance and high conductance. 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 Template:Mvar of an object is defined as the ratio of voltage Template:Mvar across it to current Template:Mvar through it, while the conductance Template:Mvar is the reciprocal: <math display=block>R = \frac{V}{I}, \qquad G = \frac{I}{V} = \frac{1}{R}.</math>
For a wide variety of materials and conditions, Template:Mvar and Template:Mvar are directly proportional to each other, and therefore Template:Mvar and Template:Mvar 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, incandescent light bulb or battery, Template:Mvar and Template:Mvar are not directly proportional. The ratio Template:Mvar is sometimes still useful, and is referred to as a chordal resistance or static resistance,<ref name="brown">Template:Cite book</ref><ref name=kaiser>Template:Cite book</ref> since it corresponds to the inverse slope of a chord between the origin and an [[current–voltage characteristic|Template:Mvar–Template:Mvar curve]]. In other situations, the derivative <math display="inline"> \frac{\mathrm{d}V}{\mathrm{d}I} </math> may be most useful; this is called the differential resistance.
IntroductionEdit
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 resistorsEdit
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 lawEdit
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For many materials, the current Template:Mvar through the material is proportional to the voltage Template:Mvar applied across it: <math display=block>I \propto V</math> 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 non-ohmic. Examples include diodes and fluorescent lamps.
Relation to resistivity and conductivityEdit
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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 Template:Math and conductance Template:Math of a conductor of uniform cross section, therefore, can be computed as
<math display=block>\begin{align}
R &= \rho \frac{\ell}{A}, \\[5pt]
G &= \sigma \frac{A}{\ell} \,.
\end{align}</math>
where <math>\ell</math> is the length of the conductor, measured in metres (m), Template:Mvar is the cross-sectional area of the conductor measured in square metres (m2), Template:Mvar (sigma) is the electrical conductivity measured in siemens per meter (S·m−1), and Template:Mvar (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: <math>\rho=1/\sigma</math>. 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,<ref>Template:Cite journal</ref> 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.
MeasurementEdit
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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 valuesEdit
| Component | Resistance (Ω) |
|---|---|
| 1 meter of copper wire with 1 mm diameter | 0.02Template:Efn |
| Template:Val overhead power line (typical) | 0.03<ref>Template:Cite book</ref> |
| AA battery (typical internal resistance) | 0.1Template:Efn |
| Incandescent light bulb filament (typical) | 200–1000<ref group=lower-alpha>A Template:Val light bulb (in the USA, with Template:Val mains electricity) draws RMS current Template:Sfrac = Template:Val, so its resistance is Template:Sfrac = Template:Val. The resistance of a Template:Val light bulb in Europe (Template:Val mains) is Template:Val. 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.</ref> |
| Human body | 1000–100,000Template:Efn |
Static and differential resistanceEdit
Template:See also Template:Multiple image
Many electrical elements, such as diodes and batteries do Template:Em satisfy Ohm's law. These are called non-ohmic or non-linear, and their current–voltage curves are Template:Em 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:<ref name=brown/><ref name=kaiser/>
Template:Glossary start Template:Term Template:Defn Template:Term Template:Defn Template:Glossary end
AC circuitsEdit
Impedance and admittanceEdit
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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:
<math display=block>\begin{array}{cl} u(t) &= \operatorname\mathcal{R_e} \left( U_0 \cdot e^{j\omega t}\right) \\ i(t) &= \operatorname\mathcal{R_e} \left( I_0 \cdot e^{j(\omega t + \varphi)}\right) \\ Z &= \frac{U}{\ I\ } \\ Y &= \frac{\ 1\ }{Z} = \frac{\ I\ }{U} \end{array}</math>
where:
- Template:Mvar is time;
- Template:Math and Template:Math are the voltage and current as a function of time, respectively;
- Template:Math and Template:Math indicate the amplitude of the voltage and current, respectively;
- <math>\omega</math> is the angular frequency of the AC current;
- <math>\varphi</math> is the displacement angle;
- Template:Mvar and Template:Mvar are the complex-valued voltage and current, respectively;
- Template:Mvar and Template:Mvar are the complex impedance and admittance, respectively;
- <math>\mathcal{R_e}</math> indicates the real part of a complex number; and
- <math>j \equiv \sqrt{-1\ }</math> is the imaginary unit.
The impedance and admittance may be expressed as complex numbers that can be broken into real and imaginary parts: <math display=block>\begin{align} Z &= R + jX \\ Y &= G + jB ~. \end{align}</math>
where Template:Mvar is resistance, Template:Mvar is conductance, Template:Mvar is reactance, and Template:Mvar is susceptance. These lead to the complex number identities <math display=block>\begin{align} R &= \frac{G}{\ G^2 + B^2\ }\ , \qquad & X = \frac{-B~}{\ G^2 + B^2\ }\ , \\ G &= \frac{R}{\ R^2 + X^2\ }\ , \qquad & B = \frac{-X~}{\ R^2 + X^2\ }\ , \end{align}</math> which are true in all cases, whereas <math>\ R = 1/G\ </math> is only true in the special cases of either DC or reactance-free current.
The complex angle <math>\ \theta = \arg(Z) = -\arg(Y)\ </math> is the phase difference between the voltage and current passing through a component with impedance Template:Mvar. For capacitors and inductors, this angle is exactly -90° or +90°, respectively, and Template:Mvar and Template:Mvar are nonzero. Ideal resistors have an angle of 0°, since Template:Mvar is zero (and hence Template:Mvar also), and Template:Mvar and Template:Mvar reduce to Template:Mvar and Template:Mvar respectively. In general, AC systems are designed to keep the phase angle close to 0° as much as possible, since it reduces the reactive power, which does no useful work at a load. In a simple case with an inductive load (causing the phase to increase), a capacitor may be added for compensation at one frequency, since the capacitor's phase shift is negative, bringing the total impedance phase closer to 0° again.
Template:Mvar is the reciprocal of Template:Mvar (<math>\ Z = 1/Y\ </math>) for all circuits, just as <math>R = 1/G</math> for DC circuits containing only resistors, or AC circuits for which either the reactance or susceptance happens to be zero (Template:Mvar or Template:Math, respectively) (if one is zero, then for realistic systems both must be zero).
Frequency dependenceEdit
A key feature of AC circuits is that the resistance and conductance can be frequency-dependent, a phenomenon known as the universal dielectric response.<ref>Template:Cite journal</ref> 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 heatingEdit
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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: <math display=block>P=I^2R</math> where Template:Mvar is the power (energy per unit time) converted from electrical energy to thermal energy, Template:Mvar is the resistance, and Template:Mvar is the current through the resistor.
Dependence on other conditionsEdit
Temperature dependenceEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 Template:Mvar does not vary too much, a linear approximation is typically used: <math display=block>R(T) = R_0[1+\alpha (T - T_0)]</math> where <math>\alpha</math> is called the temperature coefficient of resistance, <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>R_0</math> is the resistance at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference.<ref>Template:Cite book</ref>
The temperature coefficient <math>\alpha</math> is typically Template:Val to Template:Val for metals near room temperature. It is usually negative for semiconductors and insulators, with highly variable magnitude.Template:Efn
Strain dependenceEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon strain.<ref>Template:Citation</ref> 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 dependenceEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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.
SuperconductivityEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Superconductors are materials that have exactly zero resistance and infinite conductance, because they can have Template:Math and Template:Math. 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 Template:Val with liquid helium for most metallic superconductors like niobium–tin alloys, or cooling to temperatures near Template:Val with liquid nitrogen for the expensive, brittle and delicate ceramic high temperature superconductors. Nevertheless, there are many technological applications of superconductivity, including superconducting magnets.
See alsoEdit
- Conductance quantum
- Von Klitzing constant (its reciprocal)
- Electrical measurements
- Contact resistance
- Electrical resistivity and conductivity for more information about the physical mechanisms for conduction in materials.
- Johnson–Nyquist noise
- Quantum Hall effect, a standard for high-accuracy resistance measurements.
- Resistor
- RKM code
- Series and parallel circuits
- Sheet resistance
- SI electromagnetism units
- Thermal resistance
- Voltage divider
- Voltage drop
FootnotesEdit
ReferencesEdit
External linksEdit
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