Electrical resistance and conductance: Difference between revisions - Wikipedia
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{{use dmy dates|date=July 2020}}
{{Electromagnetism|Network}}
{{Infobox physical quantity
| name = Electric resistance
| unit = ohm (Ω)
| symbols = {{mvar|R}}
| baseunits = kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−2</sup>
| dimension = <math>\mathsf{M} \mathsf{L}^2 \mathsf{T}^{-3} \mathsf{I}^{-2}</math>
}}
{{Infobox physical quantity
| name = Electric conductance
| unit = siemens (S)
| symbols = {{mvar|G}}
| dimension = <math>\mathsf{M}^{-1} \mathsf{L}^{-2} \mathsf{T}^3 \mathsf{I}^2</math>
|baseunits=kg<sup>−1</sup>⋅m<sup>−2</sup>⋅s<sup>3</sup>⋅A<sup>2</sup>|derivations=<math>G = \frac{1}{R}</math>}}
The '''electrical resistance''' of an object is a measure of its opposition to the flow of [[electric current]]. Its [[Multiplicative inverse|reciprocal]] quantity is '''{{vanchor|electrical conductance|CONDUCTANCE}}''', measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels with mechanical [[friction]]. The [[International System of Units|SI]] unit of electrical resistance is the [[ohm]] ({{math|[[Omega|Ω]]}}), while electrical conductance is measured in [[siemens (unit)|siemens]] (S) (formerly called the 'mho' and then represented by {{math|℧}}).
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The resistance {{mvar|R}} of an object is defined as the ratio of [[voltage]] {{mvar|V}} across it to [[electric current|current]] {{mvar|I}} through it, while the conductance {{mvar|G}} 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, {{mvar|V}} and {{mvar|I}} are directly proportional to each other, and therefore {{mvar|R}} and {{mvar|G}} are [[Constant (mathematics)|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 [[Deformation (mechanics)|strain]]). This proportionality is called [[Ohm's law]], and materials that satisfy it are called ''ohmic'' materials.
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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 [[resistor]]s. The [[current–voltage characteristic|current–voltage graph]] of an ohmic device consists of a straight line through the origin with positive [[slope (mathematics)|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 [[diode]]s and [[fluorescent lamp]]s. The current-voltage curve of a nonohmic device is a curved line.
== Relation to resistivity and conductivity ==
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|-
| [[AA battery]] (''typical [[internal resistance]]'')
| 0.1{{efn|For a fresh Energizer E91 AA alkaline battery, the internal resistance varies from {{val|0.9|u=Ω}} at {{val|−40|u=°C}}, to {{val|0.1|u=Ω}} at {{val|+40|u=°C}}.<ref>{{cite report |url=http://data.energizer.com/PDFs/BatteryIR.pdf |title=Battery internal resistance |publisher=Energizer Corp. |access-date=13 December 2011 |archive-date=11 January 2012 |archive-url=https://web.archive.org/web/20120111122430/http://data.energizer.com/PDFs/BatteryIR.pdf |url-status=dead }}</ref>}}
|-
| [[Incandescent light bulb]] filament (''typical'')
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{{glossary start}}
{{term|Static resistance}}
{{defn|1=
{{ghat|Also called '''chordal''' or '''DC resistance'''}}
{{defn|This corresponds to the usual definition of resistance; the voltage divided by the current
<math display=block>R_\mathrm{static} = {V\frac{U}{over I} \,.</math>
It is the slope of the line ([[chord (geometry)|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''. [[Passivity (engineering)|Passive]] devices, which have no source of energy, cannot have negative static resistance. However active devices such as transistors or [[op-amp]]s can synthesize negative static resistance with feedback, and it is used in some circuits such as [[gyrator]]s.
}}
{{term|Differential resistance}}
{{defn|1=
{{ghat|Also called '''dynamic''', '''incremental''', or '''small-signal resistance'''}}
{{defn|[[Electrical resistance#Static and differential resistance|Differential resistance]]It is the derivative of the voltage with respect to the current; the [[slope]] of the current–voltage curve at a point
<math display=block>R_\mathrm{diff} = \frac { {\mathrm ddV}U}{ \over{\mathrm ddI}I} \,.</math>
If the current–voltage curve is non-[[monotonic]] (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 diode]]s, [[Gunn diode]]s, [[IMPATT diode]]s, [[magnetron]] tubes, and [[unijunction transistor]]s.
}}
{{glossary end}}
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[[File:VI phase.svg|thumb|right|300px|The voltage (red) and current (blue) versus time (horizontal axis) for a [[capacitor]] (top) and [[inductor]] (bottom). Since the [[amplitude]] of the current and voltage [[Sine wave|sinusoids]] are the same, the [[absolute value]] of [[electrical impedance|impedance]] is 1 for both the capacitor and the inductor (in whatever units the graph is using). On the other hand, the [[phase (waves)|phase difference]] between current and voltage is −90° for the capacitor; therefore, the [[argument (complex analysis)|complex phase]] of the [[electrical impedance|impedance]] of the capacitor is −90°. Similarly, the [[phase (waves)|phase difference]] between current and voltage is +90° for the inductor; therefore, the complex phase of the impedance of the inductor is +90°.]]
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 [[Phase (waves)|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 number]]s are used to keep track of both the phase and magnitude of current and voltage:
<math display=block>\begin{array}{cl}
u(t) &= \mathfrakoperatorname\mathcal{ReR_e} \left( U_0 \cdot e^{j\omega t}\right) \\
i(t) &= \mathfrakoperatorname\mathcal{ReR_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>
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* {{mvar|t}} is time;
* {{math|''u''(''t'')}} and {{math|''i''(''t'')}} are the voltage and current as a function of time, respectively;
* {{math|''U''<sub>0</sub>}} and {{math|''I''<sub>0</sub>}} indicate the amplitude of the voltage and current, respectively;electronic powerful circuit
* <math>\omega</math> is the [[angular frequency]] of the AC current;
* <math>\varphi</math> is the displacement angle;
* {{mvar|U}} and {{mvar|I}} are the complex-valued voltage and current, respectively;
* {{mvar|Z}} and {{mvar|Y}} are the complex [[electrical impedance|impedance]] and [[admittance]], respectively;
* <math>\mathfrakmathcal{ReR_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 {{mvar|R}} is resistance, {{mvar|G}} is conductance, {{mvar|X}} is [[electrical reactance|reactance]], and {{mvar|B}} is [[susceptance]]. ForThese ideal resistors, {{mvar|Z}} and {{mvar|Y}} reducelead to {{mvar|R}} and {{mvar|G}} respectively. But for AC networks containingthe [[capacitor]]scomplex and [[inductornumber]]s, {{mvar|X}} and {{mvar|B}} are nonzero.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 [[argument (complex analysis)|complex angle]] <math>\ \theta = \arg(Z) = -\arg(Y)\ </math> is the phase difference between the voltage and current passing through a component with impedance {{mvar|Z}}. For [[capacitor]]s and [[inductor]]s, this angle is exactly -90° or +90°, respectively, and {{mvar|X}} and {{mvar|B}} are nonzero. Ideal resistors have an angle of 0°, since {{mvar|X}} is zero (and hence {{mvar|B}} also), and {{mvar|Z}} and {{mvar|Y}} reduce to {{mvar|R}} and {{mvar|G}} respectively. In general, AC systems are designed to keep the phase angle close to 0° as much as possible, since it reduces the [[AC_power#Reactive power|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.
▲where {{mvar|R}} is resistance, {{mvar|G}} is conductance, {{mvar|X}} is [[electrical reactance|reactance]], and {{mvar|B}} is [[susceptance]]. For ideal resistors, {{mvar|Z}} and {{mvar|Y}} reduce to {{mvar|R}} and {{mvar|G}} respectively. But for AC networks containing [[capacitor]]s and [[inductor]]s, {{mvar|X}} and {{mvar|B}} are nonzero.
{{mvar|Y}} is the reciprocal of {{mvar|Z}} (<math>\ Z = 1/Y\ </math>) for ACall 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 ({{mvar|X}} or {{math|''B'' {{=}} 0}}, respectively) (if one is zero, then for realistic systems both must be zero).
===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]].<ref>{{cite journal |title=Stress-dependent electrical transport and its universal scaling in granular materials |journal=Extreme Mechanics Letters |volume=22 |pages=83–88 |doi=10.1016/j.eml.2018.05.005 |year=2018 |last1=Zhai |first1=Chongpu |last2=Gan |first2=Yixiang |last3=Hanaor |first3=Dorian |last4=Proust |first4=Gwénaëlle |arxiv=1712.05938 |bibcode=2018ExML...22...83Z |s2cid=51912472 }}</ref> One reason, mentioned above is the [[skin effect]] (and the related [[proximity effect (electromagnetism)|proximity effect]]). Another reason is that the resistivity itself may depend on frequency (see [[Drude model]], [[deep-level trap]]s, [[resonant frequency]], [[Kramers–Kronig relations]], etc.)
==Energy dissipation and Joule heating==
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=== Strain dependence ===
{{main|Strain gauge}}
Just as the resistance of a conductor depends upon temperature, the resistance of a conductor depends upon [[strain (materials science)|strain]].<ref>{{Citation|last=Meyer|first=Sebastian|display-authors = etal|date=2022|pages=114712|language=en|doi=10.1016/j.scriptamat.2022.114712|title=Volume 215|series=Scripta Materialia|chapter=Characterization of the deformation state of magnesium by electrical resistance|volume=215 |s2cid=247959452 |doi-access=free}}</ref> By placing a conductor under [[tension (mechanics)|tension]] (a form of [[stress (physics)|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 (physics)|compression]] (strain in the opposite direction), the resistance of the strained section of conductor decreases. See the discussion on [[strain gauge]]s for details about devices constructed to take advantage of this effect.
=== Light illumination dependence ===
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[[Category:Electrical resistance and conductance| ]]
[[Category:Electricity]]
[[Category:PhysicalElectromagnetic quantities]]
[[Category:Electromagnetism]]