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| {{electromagnetism|cTopic=[[Electrical network|Electrical Network]]}}
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| [[File:Complex Impedance.svg|200px|thumb|right|A graphical representation of the [[Complex plane|complex impedance plane]]]]
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| '''Electrical impedance''' is the measure of the opposition that a [[electrical circuit|circuit]] presents to a [[electrical current|current]] when a [[voltage]] is applied.
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| In quantitative terms, it is the [[complex number|complex]] [[ratio]] of the voltage to the current in an [[alternating current]] (AC) circuit. Impedance extends the concept of [[Electrical resistance|resistance]] to AC circuits, and possesses both magnitude and [[Phase (waves)|phase]], unlike resistance, which has only magnitude. When a circuit is driven with [[direct current]] (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero [[phase angle]].
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| It is necessary to introduce the concept of impedance in AC circuits because there are two additional impeding mechanisms to be taken into account besides the normal resistance of DC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents ([[inductance]]), and the electrostatic storage of charge induced by voltages between conductors ([[capacitance]]). The impedance caused by these two effects is collectively referred to as [[electrical reactance|reactance]] and forms the [[imaginary number|imaginary]] part of complex impedance whereas resistance forms the [[real number|real]] part.
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| The symbol for impedance is usually {{math|''Z''}} and it may be represented by writing its magnitude and phase in the form {{math|{{!}}''Z''{{!}}''∠θ''}}. However, complex number representation is often more powerful for circuit analysis purposes. The term ''impedance'' was coined by [[Oliver Heaviside]] in July 1886.<ref>''Science'', p. 18, 1888</ref><ref>Oliver Heaviside, ''The Electrician'', p. 212, 23 July 1886, reprinted as ''Electrical Papers'', p 64, AMS Bookstore, ISBN 0-8218-3465-7</ref> [[Arthur Kennelly]] was the first to represent impedance with complex numbers in 1893.<ref>[http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4768008 Kennelly, Arthur. ''Impedance'' (AIEE, 1893)]</ref>
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| Impedance is defined as the [[frequency domain]] ratio of the voltage to the current.<ref>{{Cite book | last = Alexander | first = Charles | last2 = Sadiku | first2 = Matthew | title = Fundamentals of Electric Circuits | publisher = McGraw-Hill | year = 2006 | edition = 3, revised | pages =387–389 | isbn = 978-0-07-330115-0 | postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> In other words, it is the voltage–current ratio for a single [[complex exponential]] at a particular frequency {{math|''ω''}}. In general, impedance will be a complex number, with the same [[dimensional analysis|units]] as resistance, for which the [[SI unit]] is the [[ohm]] ({{math|Ω}}). For a sinusoidal current or voltage input, the [[Complex number#Notation_of_the_polar_form|polar form]] of the complex impedance relates the amplitude and phase of the voltage and current. In particular,
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| * The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
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| * The phase of the complex impedance is the [[phase shift]] by which the current lags the voltage.
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| The [[Multiplicative inverse|reciprocal]] of impedance is [[admittance]] (i.e., admittance is the current-to-voltage ratio, and it conventionally carries units of [[siemens (unit)|siemens]], formerly called [[mho]]s).
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| ==Complex impedance==
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| Impedance is represented as a [[Complex number|complex]] quantity <math>\scriptstyle Z</math> and the term ''complex impedance'' may be used interchangeably; the [[Polar coordinates|polar form]] conveniently captures both magnitude and phase characteristics,
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| :<math>\ Z = |Z| e^{j\arg (Z)}</math>
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| where the magnitude <math>\scriptstyle |Z|</math> represents the ratio of the voltage difference amplitude to the current amplitude, while the argument <math>\scriptstyle \arg (Z)</math> (commonly given the symbol <math>\scriptstyle \theta </math>) gives the phase difference between voltage and current. <math>\scriptstyle j</math> is the [[imaginary unit]], and is used instead of <math>\scriptstyle i</math> in this context to avoid confusion with the symbol for [[Ampere|electric current]]. In [[Cartesian plane|Cartesian form]],
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| :<math>\ Z = R + jX</math>
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| where the [[real part]] of impedance is the resistance <math>\scriptstyle R</math> and the [[imaginary part]] is the [[Reactance (electronics)|reactance]] <math>\scriptstyle X</math>.
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| Where it is required to add or subtract impedances the cartesian form is more convenient, but when quantities are multiplied or divided the calculation becomes simpler if the polar form is used. A circuit calculation, such as finding the total impedance of two impedances in parallel, may require conversion between forms several times during the calculation. Conversion between the forms follows the normal [[Complex number#Polar form|conversion rules of complex numbers]].
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| == Ohm's law ==
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| [[File:General AC circuit.svg|thumb|right|169px|An AC supply applying a voltage <math>\scriptstyle V</math>, across a [[Electrical load|load]] <math>\scriptstyle Z</math>, driving a current <math>\scriptstyle I</math>.]]
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| {{Main|Ohm's law}}
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| The meaning of electrical impedance can be understood 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>
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| :<math>\ V = I Z = I |Z| e^{j \arg (Z)}</math>
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| The magnitude of the impedance <math>\scriptstyle |Z|</math> acts just like resistance, giving the drop in voltage amplitude across an impedance <math>\scriptstyle Z</math> for a given current <math>\scriptstyle I</math>. The phase factor tells us that the current lags the voltage by a phase of <math>\scriptstyle \theta \;=\; \arg (Z)</math> (i.e., in the [[time domain]], the current signal is shifted <math>\scriptstyle \frac{\theta}{2 \pi} T</math> later with respect to the voltage signal).
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| 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]], [[Thévenin's theorem]], and [[Norton's theorem]] can also be extended to AC circuits by replacing resistance with impedance.
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| == Complex voltage and current ==
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| [[File: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.]]
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| In order to simplify calculations, [[Sine wave|sinusoid]]al voltage and current waves are commonly represented as complex-valued functions of time denoted as <math>\scriptstyle V</math> and <math>\scriptstyle 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>
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| :<math>\begin{align}
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| V &= |V|e^{j(\omega t + \phi_V)} \\
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| I &= |I|e^{j(\omega t + \phi_I)}
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| \end{align}</math>
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| Impedance is defined as the ratio of these quantities.
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| :<math>\ Z = \frac{V}{I}</math>
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| Substituting these into Ohm's law we have
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| :<math>
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| \begin{align}
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| |V| e^{j(\omega t + \phi_V)} &= |I| e^{j(\omega t + \phi_I)} |Z| e^{j\theta} \\
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| &= |I| |Z| e^{j(\omega t + \phi_I + \theta)}
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| \end{align}
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| </math>
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| Noting that this must hold for all <math>t</math>, we may equate the magnitudes and phases to obtain
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| :<math>\begin{align}
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| |V| &= |I| |Z| \\
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| \phi_V &= \phi_I + \theta
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| \end{align}</math>
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| The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.
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| === Validity of complex representation ===
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| This representation using complex exponentials may be justified by noting that (by [[Euler's formula]]):
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| :<math>\ \cos(\omega t + \phi) = \frac{1}{2} \Big[ e^{j(\omega t + \phi)} + e^{-j(\omega t + \phi)}\Big]</math>
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| The real-valued sinusoidal function representing either voltage or current 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
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| :<math>\ \cos(\omega t + \phi) = \Re \Big\{ e^{j(\omega t + \phi)} \Big\}</math>
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| === Phasors ===
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| {{Main|Phasor (electronics)}}
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| 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.
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| 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.
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| == Device examples ==
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| [[File: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>\pi/2</math>, while the voltage across an inductor ''leads'' the current through it by <math>\pi/2</math>. The identical voltage and current amplitudes indicate that the magnitude of the impedance is equal to one.]]
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| The impedance of an ideal [[resistor]] is purely real and is referred to as a ''resistive impedance'':
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| :<math>\ Z_R = R</math>
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| In this case, the voltage and current waveforms are proportional and in phase.
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| Ideal [[inductor]]s and [[capacitor]]s have a purely [[Imaginary number|imaginary]] ''reactive impedance'':
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| the impedance of inductors increases as frequency increases;
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| :<math>\ Z_L = j\omega L</math>
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| the impedance of capacitors decreases as frequency increases;
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| :<math>\ Z_C = \frac{1}{j\omega C}</math>
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| In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in quadrature, 90 degrees out of phase with the voltage. However, the phases have opposite signs: in an inductor, the current is ''lagging''; in a capacitor the current is ''leading''.
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| Note the following identities for the imaginary unit and its reciprocal:
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| :<math>\begin{align}
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| j &\equiv \cos{\left( \frac{\pi}{2}\right)} + j\sin{\left( \frac{\pi}{2}\right)} \equiv e^{j \frac{\pi}{2}} \\
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| \frac{1}{j} \equiv -j &\equiv \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} \equiv e^{j(-\frac{\pi}{2})}
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| \end{align}</math>
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| Thus the inductor and capacitor impedance equations can be rewritten in polar form:
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| :<math>\begin{align}
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| Z_L &= \omega Le^{j\frac{\pi}{2}} \\
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| Z_C &= \frac{1}{\omega C}e^{j\left(-\frac{\pi}{2}\right)}
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| \end{align}</math>
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| The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.
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| === Deriving the device-specific impedances ===
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| What follows below is a derivation of impedance for each of the three basic [[Electrical network|circuit]] elements: the resistor, the capacitor, and the inductor. Although the idea can be extended to define the relationship between the voltage and current of any arbitrary [[Signal (electrical engineering)|signal]], these derivations will assume [[sinusoidal]] signals, since any arbitrary signal can be approximated as a sum of sinusoids through [[Fourier analysis]].
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| ====Resistor====
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| For a resistor, there is the relation:
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| :<math>v_{\text{R}} \left( t \right) = {i_{\text{R}} \left( t \right)}R</math>
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| This is [[Ohm's law]].
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| Considering the voltage signal to be
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| :<math>v_{\text{R}}(t) = V_p \sin(\omega t)</math>
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| it follows that
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| :<math>\frac{v_{\text{R}} \left( t \right)}{i_{\text{R}} \left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \left( \omega t \right)} = R</math>
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| This says that the ratio of AC voltage amplitude to [[alternating current]] (AC) amplitude across a resistor is <math>\scriptstyle R</math>, and that the AC voltage leads the current across a resistor by 0 degrees.
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| This result is commonly expressed as
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| :<math>Z_{\text{resistor}} = R</math>
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| ====Capacitor====
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| For a capacitor, there is the relation:
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| :<math>i_{\text{C}}(t) = C \frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t}</math>
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| Considering the voltage signal to be
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| :<math>v_{\text{C}}(t) = V_p \sin(\omega t) \,</math>
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| it follows that
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| :<math>\frac{\operatorname{d}v_{\text{C}}(t)}{\operatorname{d}t} = \omega V_p \cos \left( \omega t \right)</math>
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| And thus
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| :<math>\frac{v_{\text{C}} \left( t \right)}{i_{\text{C}} \left( t \right)} = \frac{V_p \sin(\omega t)}{\omega V_p C \cos \left( \omega t \right)}= \frac{\sin(\omega t)}{\omega C \sin \left( \omega t + \frac{\pi}{2}\right)}</math>
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| This says that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is <math>\scriptstyle \frac{1}{\omega C}</math>, and that the AC voltage lags the AC current across a capacitor by 90 degrees (or the AC current leads the AC voltage across a capacitor by 90 degrees).
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| This result is commonly expressed in [[polar form]], as
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| :<math>\ Z_{\text{capacitor}} = \frac{1}{\omega C} e^{-j \frac{\pi}{2}}</math>
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| or, by applying Euler's formula, as
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| :<math>\ Z_{\text{capacitor}} = -j\frac{1}{\omega C} = \frac{1}{j \omega C}</math>
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| ====Inductor====
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| For the inductor, we have the relation:
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| :<math>v_{\text{L}}(t) = L \frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t}</math>
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| This time, considering the current signal to be
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| :<math>i_{\text{L}}(t) = I_p \sin(\omega t)</math>
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| it follows that
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| :<math>\frac{\operatorname{d}i_{\text{L}}(t)}{\operatorname{d}t} = \omega I_p \cos \left( \omega t \right)</math>
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| And thus
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| :<math>\frac{v_{\text{L}} \left( t \right)}{i_{\text{L}} \left( t \right)} = \frac{\omega I_p L \cos(\omega t)}{I_p \sin \left( \omega t \right)} = \frac{\omega L \sin \left( \omega t + \frac{\pi}{2}\right)}{\sin(\omega t)}</math>
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| This says that the ratio of AC voltage amplitude to AC current amplitude across an inductor is <math>\scriptstyle \omega L</math>, and that the AC voltage leads the AC current across an inductor by 90 degrees.
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| This result is commonly expressed in polar form, as
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| :<math>\ Z_{\text{inductor}} = \omega L e^{j \frac{\pi}{2}}</math>
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| or, using Euler's formula, as
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| :<math>\ Z_{\text{inductor}} = j \omega L</math>
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| ==Generalised s-plane impedance==
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| Impedance defined in terms of ''jω'' can strictly only be applied to circuits which are driven with a steady-state AC signal. The concept of impedance can be extended to a circuit energised with any arbitrary signal by using [[complex frequency]] instead of ''jω''. Complex frequency is given the symbol ''s'' and is, in general, a complex number. Signals are expressed in terms of complex frequency by taking the [[Laplace transform]] of the [[time domain]] expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows:
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| {|class="wikitable" style="margin-left:3em;"
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| |-
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| !Element||Impedance expression
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| |-
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| |Resistor||<math>R \,</math>
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| |-
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| |Inductor||<math>sL \,</math>
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| |-
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| |Capacitor||<math>\frac{1}{sC} \,</math>
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| |}
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| For a DC circuit this simplifies to {{nowrap|1=''s'' = 0}}. For a steady-state sinusoidal AC signal {{nowrap|1=''s'' = ''jω''}}.
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| == Resistance vs reactance ==
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| Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:
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| :<math>|Z| = \sqrt{Z Z^*} = \sqrt{R^2 + X^2}</math>
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| :<math>\theta = \arctan{\left(\frac{X}{R}\right)}</math>
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| In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.
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| === Resistance ===
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| <!--[[File:Resistors.jpg|thumb|right|200px|A pack of resistors. [[Media:Resistors.jpg|Actual size]]]]-->
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| {{Main|Electrical resistance}}
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| 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.
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| :<math>\ R = |Z| \cos{\theta} \quad</math>
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| === Reactance ===
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| {{Main|Electrical reactance}}
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| Reactance <math>\scriptstyle X</math> is the imaginary part of the impedance; a component with a finite reactance induces a phase shift <math>\scriptstyle \theta</math> between the voltage across it and the current through it.
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| :<math>\ X = |Z| \sin{\theta} \quad</math>
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| A purely reactive component is distinguished by the sinusoidal voltage across the component being 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.
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| ==== Capacitive reactance ====
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| <!--[[File: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]]]]-->
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| {{Main|Capacitance}}
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| 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]].
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| :<math>X_C = (\omega C)^{-1} = (2\pi f C)^{-1}\quad</math> | |
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| 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.
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| 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.
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| ==== Inductive reactance ====
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| {{Main|Inductance}}
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| Inductive reactance <math>\scriptstyle{X_L}</math> is [[Proportionality (mathematics)|proportional]] to the signal [[frequency]] <math>\scriptstyle{f}</math> and the [[inductance]] <math>\scriptstyle{L}</math>.
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| :<math>X_L = \omega L = 2\pi f L\quad</math>
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| An inductor consists of a [[Coil#Electromagnetic coils|coiled conductor]]. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the back [[Electromotive force|emf]] <math>\scriptstyle{\mathcal{E}}</math> (voltage opposing current) due to a rate-of-change of [[magnetic flux density]] <math>\scriptstyle{B}</math> through a current loop.
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| :<math>\mathcal{E} = -{{d\Phi_B} \over dt}\quad</math>
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| For an inductor consisting of a coil with <math>N</math> loops this gives.
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| :<math>\mathcal{E} = -N{d\Phi_B \over dt}\quad</math>
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| 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-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.
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| ==== Total reactance ====
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| The total reactance is given by
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| :<math>{X = X_L - X_C}</math>
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| so that the total impedance is
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| :<math>\ Z = R + jX</math>
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| == Combining impedances ==
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| {{Main|Series and parallel circuits}}
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| The total impedance of many simple networks 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, except that the numbers in general will be complex numbers. In the general case however, [[equivalent impedance transforms]] in addition to series and parallel will be required.
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| === Series combination ===
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| For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.
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| [[File:Impedances in series.svg]]
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| :<math>\ Z_{\text{eq}} = Z_1 + Z_2 + \cdots + Z_n \quad</math>
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| Or explicitly in real and imaginary terms:
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| :<math>\ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \quad</math>
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| === Parallel combination ===
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| 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.
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| :[[File:Impedances in parallel.svg]]
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| Hence the inverse total impedance is the sum of the inverses of the component impedances:
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| :<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}</math>
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| or, when n = 2:
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| :<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}</math>
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| :<math>\ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}</math>
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| The equivalent impedance <math>\scriptstyle Z_{\text{eq}}</math> can be calculated in terms of the equivalent series resistance <math>\scriptstyle R_{\text{eq}}</math> and reactance <math>\scriptstyle X_{\text{eq}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3 Parallel Impedance Expressions], Hyperphysics</ref>
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| :<math>\begin{align}
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| Z_{\text{eq}} &= R_{\text{eq}} + j X_{\text{eq}} \\
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| R_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2} \\
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| X_{\text{eq}} &= \frac{(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2)}{(R_1 + R_2)^2 + (X_1 + X_2)^2}
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| \end{align}</math>
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| == Measurement ==
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| The measurement of the impedance of devices and transmission lines is a practical problem in [[radio]] technology and others. Measurements of impedance may be carried out at one frequency, or the variation of device impedance over a range of frequencies may be of interest. The impedance may be measured or displayed directly in ohms, or other values related to impedance may be displayed; for example in a [[radio antenna]] the [[standing wave ratio]] or [[reflection coefficient]] may be more useful than the impedance alone. Measurement of impedance requires measurement of the magnitude of voltage and current, and the phase difference between them. Impedance is often measured by [[Bridge circuit |"bridge" methods]], similar to the direct-current [[Wheatstone bridge]]; a calibrated reference impedance is adjusted to balance off the effect of the impedance of the device under test. Impedance measurement in power electronic devices may require simultaneous measurement and provision of power to the operating device.
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| The impedance of a device can be calculated by complex division of the voltage and current. The impedance of the device can be calculated by applying a sinusoidal voltage to the device in series with a resistor, and measuring the voltage across the resistor and across the device. Performing this measurement by sweeping the frequencies of the applied signal provides the impedance phase and magnitude.<ref name="LewisJr">{{cite journal |last=Lewis Jr. |first=George |authorlink= |coauthors=George K. Lewis Sr. and William Olbricht |date=August 2008 |title=Cost-effective broad-band electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound transducers |journal=Measurement Science and Technology |volume=19 |issue= 10|pages=105102 |id= |url=http://www.iop.org/EJ/abstract/0957-0233/19/10/105102/ |accessdate=2008-09-15 |quote= |doi=10.1088/0957-0233/19/10/105102 |pmid=19081773 |pmc=2600501 |bibcode = 2008MeScT..19j5102L }}</ref>
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| The use of an impulse response may be used in combination with the [[fast Fourier transform]] (FFT) to rapidly measure the electrical impedance of various electrical devices.<ref name="LewisJr"/>
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| The [[LCR meter]] (Inductance (L), Capacitance (C), and Resistance (R)) is a device commonly used to measure the inductance, resistance and capacitance of a component; from these values the impedance at any frequency can be calculated.
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| == Variable impedance ==
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| In general, neither impedance nor admittance can be time varying as they are defined for complex exponentials for –∞ < ''t'' < +∞. If the complex exponential voltage–current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many systems (e.g., [[varicap]]s that are used in [[Tuner (radio)|radio tuners]]) may exhibit non-linear or time-varying voltage–current ratios that appear to be [[LTI system theory|linear time-invariant (LTI)]] for small signals over small observation windows; hence, they can be roughly described as having a time-varying impedance. That is, this description is an approximation; over large signal swings or observation windows, the voltage–current relationship is non-LTI and cannot be described by impedance.
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| == See also ==
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| *[[Characteristic impedance]]
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| *[[Electrical characteristics of dynamic loudspeakers]]
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| *[[Immittance]]
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| *[[Impedance bridging]]
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| *[[Impedance cardiography]]
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| *[[Impedance matching]]
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| *[[Negative impedance converter]]
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| *[[Resistance distance]]
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| == References ==
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| {{Reflist|1}}
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| == External links ==
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| *[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html Explaining Impedance]
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| *[http://www.antenna-theory.com/basics/impedance.php Antenna Impedance]
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| *[http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems] – Brief explanation of Laplace-domain circuit analysis; includes a definition of impedance.
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| {{DEFAULTSORT:Electrical Impedance}}
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| [[Category:Electronics]]
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| [[Category:Physical quantities]]
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| [[Category:Antennas (radio)]]
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