Electrical impedance
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2008-07-16T09:49:32Z
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{{mergefrom|Impedance of different devices (derivations) |Talk:Electrical impedance#Merger proposal|date=June 2008}}
{{electromagnetism3|enet=true}}
'''Electrical impedance''', or simply '''impedance''', describes a measure of opposition to a sinusoidal [[alternating current]] (AC). Electrical impedance extends the concept of [[Electrical resistance|resistance]] to AC circuits, describing not only the relative [[amplitude]]s of the [[voltage]] and [[Electric current|current]], but also the relative [[Phase (waves)|phases]]. Impedance is a [[Complex number|complex]] quantity <math>\scriptstyle{\tilde{Z}}</math> and the term ''complex impedance'' may be used interchangeably; the [[Polar coordinates|polar form]] conveniently captures both magnitude and phase characteristics,
:<math>\tilde{Z} = Z e^{j\theta} \quad</math>
where the magnitude <math>\scriptstyle{Z}</math> gives the change in voltage amplitude for a given current amplitude, while the argument <math>\scriptstyle{\theta}</math> gives the phase difference between voltage and current. In [[Cartesian plane|Cartesian form]],
:<math>\tilde{Z} = R + j\Chi \quad</math>
where the [[real part]] of impedance is the resistance <math>\scriptstyle{R}</math> and the [[imaginary part]] is the [[reactance]] <math>\scriptstyle{\Chi}</math>. [[Dimensional analysis|Dimensionally]], impedance is the same as resistance; the [[SI unit]] is the [[ohm]]. The term ''impedance'' was coined by [[Oliver Heaviside]] in July 1886.
The dual of impedance is [[admittance]].
{|style="float:right"
|[[Image:Complex impedance plane.png|250px|thumb|A graphical representation of the [[Complex plane|complex impedance plane]]. Note that while reactance <math>\scriptstyle{\Chi}</math> can be either positive or negative, resistance <math>\scriptstyle{R}</math> is always positive.]]
|}
<!--[[Image:Complex impedance plane.png|thumb|right|250px|A graphical representation of the [[Complex plane|complex impedance plane]]. Note that while reactance <math>\scriptstyle{\Chi}</math> can be either positive or negative, resistance <math>\scriptstyle{R}</math> is always positive. [[Media:Complex impedance plane.png|Actual size]]]]-->
== Ohm's law ==
[[Image:General AC circuit.png|thumb|right|200px|An AC supply applying a voltage <math>\scriptstyle{V}</math>, across a [[load]] <math>\scriptstyle{Z}</math>, driving a current <math>\scriptstyle{I}</math>.]]
{{main|Ohm's law}}
We can understand this by substituting it into [[Ohm's law]].<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html AC Ohm's law], Hyperphysics</ref><ref name=HH1>{{cite book |last=Horowitz |first=Paul|coauthors=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |location= |isbn=0-521-37095-7 |pages=32-33 |chapter=1 }}</ref>
:<math>\tilde{V} = \tilde{I}\tilde{Z} = \tilde{I} Z e^{j\theta} \quad</math>
The magnitude of the impedance <math>\scriptstyle{Z}</math> acts just like resistance, giving the drop in voltage amplitude across an impedance <math>\scriptstyle{\tilde{Z}}</math> for a given current <math>\scriptstyle{\tilde{I}}</math>. The phase factor tells us that the current lags the voltage by a phase of <math>\theta</math> (i.e. in the time domain, the current signal is shifted <math>\frac{\theta T}{2 \pi}</math> to the right with respect to the voltage signal).<ref>[http://www.yokogawa.com/tm/tr/tm-tr0605_01.htm Capacitor/inductor phase relationships], Yokogawa</ref>
Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis such as [[Voltage divider|voltage division]], [[Current divider|current division]], [[Thevenin's theorem]], and [[Norton's theorem]], can also be extended to AC circuits by replacing resistance with impedance.
== Complex voltage and current ==
[[Image:Impedance symbol comparison.svg|thumb|right|200px|Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.]]
In order to simplify calculations, [[sinusoid]]al voltage and current waves are commonly represented as complex-valued functions of time denoted as <math>\scriptstyle{\tilde{V}}</math> and <math>\scriptstyle{\tilde{I}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html#c1 Complex impedance], Hyperphysics</ref><ref name=HH2>{{cite book |last=Horowitz |first=Paul|coauthors= Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |location= |isbn=0-521-37095-7 |pages=31-32 |chapter=1 }}</ref>
:<math>\ \tilde{V} = V_0e^{j(\omega t + \phi_V)}</math>
:<math>\ \tilde{I} = I_0e^{j(\omega t + \phi_I)}</math>
Impedance is defined as the ratio of these quantities.
:<math>\ \tilde{Z} = {\tilde{V} \over \tilde{I}}</math>
Substituting these into Ohm's law we have
:<math>
\begin{align}
V_0e^{j(\omega t + \phi_V)} &= I_0e^{j(\omega t + \phi_I)} Z e^{j\theta} \\
&= I_0 Z e^{j(\omega t + \phi_I + \theta)}
\end{align}
</math>
Noting that this must hold for all <math>t</math>, we may equate the magnitudes and phases to obtain
:<math>\ V_0 = I_0 Z \quad</math>
:<math>\ \phi_V = \phi_I + \theta \quad</math>
The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.
=== Validity of complex representation ===
This representation using complex exponentials may be justified by noting that (by [[Euler's formula]]):
:<math>\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]</math>
i.e. a real-valued sinusoidal function (which may represent our voltage or current waveform) may be broken into two complex-valued functions. By the principle of [[superposition principle|superposition]], we may analyse the behaviour of the sinusoid on the left-hand side by analysing the behaviour of the two complex terms on the right-hand side. Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that
:<math>\ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}</math>
In other words, we simply take the [[real part]] of the result.
=== Phasors ===
{{main|Phasor (electronics)}}
A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. Phasors are used by electrical engineers to simplify computations involving sinusoids, where they can often reduce a differential equation problem to an algebraic one.
The impedance of a circuit element can be defined as the ratio of the phasor voltage across the element to the phasor current through the element, as determined by the relative amplitudes and phases of the voltage and current. This is identical to the definition from [[Electrical impedance#Ohm's law|Ohm's law]] given above, recognising that the factors of <math>\scriptstyle{e^{j\omega t}}</math> cancel.
== Device examples ==
{{main|Impedance of different devices (derivations)}}
[[Image:VI phase.png|thumb|right|250px|The phase angles in the equations for the impedance of inductors and capacitors indicate that the voltage across a capacitor ''lags'' the current through it by a phase of <math>\scriptstyle{\pi/2}</math>, while the voltage across an inductor ''leads'' the current through it by <math>\scriptstyle{\pi/2}</math>. The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one.]]
The impedance of a [[resistor]] is purely real and is referred to as a ''resistive impedance''.
:<math>\tilde{Z}_R = R \quad</math>
[[Inductor]]s and [[capacitor]]s have a purely [[imaginary]] ''reactive impedance''.
:<math>\tilde{Z}_L = j\omega L \quad</math>
:<math>\tilde{Z}_C = {1 \over j\omega C}</math>
Note the following identities for the [[imaginary unit]] and its [[reciprocal]].
:<math>j = \cos{\left({\pi \over 2}\right)} + j\sin{\left({\pi \over 2}\right)} = e^{j{\pi \over 2}}</math>
:<math>{1 \over j} = -j = \cos{\left(-{\pi \over 2}\right)} + j\sin{\left(-{\pi \over 2}\right)} = e^{j(-{\pi \over 2})}</math>
Thus we can rewrite the inductor and capacitor impedance equations in polar form
:<math>\tilde{Z}_L = \omega Le^{j{\pi \over 2}}</math>
:<math>\tilde{Z}_C = {1 \over \omega C}e^{j(-{\pi \over 2})}.</math>
The magnitude tells us the change in voltage amplitude for a given current amplitude
through our impedance, while the exponential factors give the phase relationship.
== Resistance vs reactance ==
It is important to realize that resistance and reactance are not individually significant; '''together''' they determine the magnitude and phase of the impedance, through the following relations:
:<math>|\tilde{Z}| = \sqrt{\tilde{Z}\tilde{Z}^*} = \sqrt{R^2 + \Chi^2}</math>
:<math>\theta = \arctan{\left({\Chi \over R}\right)}</math>
In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.
=== Resistance ===
<!--[[Image:Resistors.jpg|thumb|right|200px|A pack of resistors. [[Media:Resistors.jpg|Actual size]]]]-->
{{main|Electrical resistance}}
Resistance <math>\scriptstyle{R}</math> is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.
:<math>R = Z \cos{\theta} \quad</math>
=== Reactance ===
{{main|Reactance}}
[[Reactance]] <math>\scriptstyle{\Chi}</math> is the imaginary part of the impedance; a component with a finite reactance induces a phase shift <math>\theta</math> between the voltage across it and the current through it.
:<math>\Chi = Z \sin{\theta} \quad</math>
A reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. A pure reactance will not dissipate any power.
==== Capacitive reactance ====
<!--[[Image:Photo-SMDcapacitors.jpg|thumb|right|200px|Capacitors: [[Surface-mount technology|SMD]] ceramic at top left; SMD tantalum at bottom left; [[through-hole]] tantalum at top right; through-hole electrolytic at bottom right. Major scale divisions are cm. [[Media:Resistors.jpg|Actual size]]]]-->
{{main|Capacitor}}
A [[capacitor]] has a purely reactive impedance which is [[Inversely proportional#Inverse proportionality|inversely proportional]] to the signal [[frequency]]. A capacitor consists of two [[Electrical conduction|conductor]]s separated by an [[Electrical insulation|insulator]], also known as a [[dielectric]].
At low frequencies a capacitor is [[open circuit]], as no charge flows in the dielectric. A DC voltage applied across a capacitor causes [[Electrical charge|charge]] to accumulate on one side; the [[electric field]] due to the accumulated charge is the source of the opposition to the current. When the [[potential]] associated with the charge exactly balances the applied voltage, the current goes to zero.
Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.
==== Inductive reactance ====
<!--[[Image:Inductors-photo.JPG|thumb|right|200px|Through-hole inductors. [[Media:Inductors-photo.JPG|Actual size]]]]-->
{{main|Inductor}}
An [[inductor]] has a purely reactive impedance which is [[proportional]] to the signal [[frequency]]. An inductor consists of a [[Coil#Electromagnetic coils|coiled conductor]]. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the back emf <math>\scriptstyle{\mathcal{E}}</math> (voltage opposing current) due to a rate-of-change of [[magnetic field]] <math>\scriptstyle{B}</math> through a current loop.
:<math>\mathcal{E} = -{{d\Phi_B} \over dt}.</math>
For an inductor consisting of a coil with <math>N</math> loops this gives.
:<math>\mathcal{E} = -N{d\Phi_B \over dt}.</math>
The back-emf is the source of the opposition to current flow. A constant [[direct current]] has a zero rate-of-change, and sees an inductor as a [[short-circuit]] (it is typically made from a material with a low [[resistivity]]). An alternating current has a time rate-of-change that is proportional to frequency and so the inductive reactance is proportional to frequency.
== Combining impedances ==
{{main|Series and parallel circuits}}
The total impedance of any network of components can be calculated using the rules for combining impedances in series and parallel. The rules are identical to those used for combining resistances, although they require some familiarity with [[complex number]]s.
=== Series combination ===
For components connected in series, the current through each circuit element is the same; the ratio of voltages across any two elements is the inverse ratio of their impedances.
[[Image:Impedances in series.svg]]
:<math>\tilde{Z}_{eq} = \tilde{Z}_1 + \tilde{Z}_2 = (R_1 + R_2) + j(\Chi_1 + \Chi_2) \quad</math>
=== Parallel combination ===
For components connected in parallel, the voltage across each circuit element is the same; the ratio of currents through any two elements is the inverse ratio of their impedances.
:[[Image:Impedances in parallel.svg]]
:<math>\tilde{Z}_{eq} = \tilde{Z}_1 \| \tilde{Z}_2 = \left(\tilde{Z}_1^{-1} + \tilde{Z}_2^{-1}\right)^{-1} = {\tilde{Z}_1 \tilde{Z}_2 \over \tilde{Z}_1 + \tilde{Z}_2} \quad</math>
The equivalent impedance <math>\scriptstyle{\tilde{Z}_{eq}}</math> can be calculated in terms of the equivalent resistance <math>\scriptstyle{R_{eq}}</math> and reactance <math>\scriptstyle{\Chi_{eq}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3 Parallel Impedance Expressions], Hyperphysics</ref>
:<math>\tilde{Z}_{eq} = R_{eq} + j \Chi_{eq} \quad</math>
:<math>R_{eq} = { (\Chi_1 R_2 + \Chi_2 R_1) (\Chi_1 + \Chi_2) + (R_1 R_2 - \Chi_1 \Chi_2) (R_1 + R_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}</math>
:<math>\Chi_{eq} = {(\Chi_1 R_2 + \Chi_2 R_1) (R_1 + R_2) - (R_1 R_2 - \Chi_1 \Chi_2) (\Chi_1 + \Chi_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}</math>
== See also ==
*[[Admittance]]
*[[Impedance matching]]
*[[Impedance bridging]]
*[[Characteristic impedance]]
*[[Electrical load]]
== External links ==
*[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html Explaining Impedance]
== References ==
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