View source for Electrical impedance

{{Short description|Opposition of a circuit to a current when a voltage is applied}}
{{Electromagnetism|Network}}

In [[electrical engineering]], '''impedance''' is the opposition to [[alternating current]] presented by the combined effect of [[Electrical_resistance|resistance]] and [[Electrical_reactance|reactance]] in a [[electrical circuit|circuit]].<ref>Slurzberg; Osterheld (1950). [https://archive.org/details/essentialsofelec00slur/page/360/mode/2up ''Essentials of Electricity for Radio and Television'']. 2nd ed. McGraw-Hill. pp. 360 - 362</ref>

Quantitatively, the impedance of a two-terminal [[Electrical element|circuit element]] is the ratio of the [[phasor|complex]] representation of the [[Sine wave|sinusoidal]] voltage between its terminals, to the complex representation of the current flowing through it.<ref>Callegaro, L. (2012). Electrical Impedance: Principles, Measurement, and Applications. CRC Press, p. 5</ref> In general, it depends upon the [[frequency]] of the sinusoidal voltage. 

Impedance extends the concept of [[Electrical resistance|resistance]] to alternating current (AC) circuits, and possesses both magnitude and [[Phase (waves)|phase]], unlike resistance, which has only magnitude. 
 
Impedance can be represented as a [[complex number]], with the same units as resistance, for which the [[SI unit]] is the [[ohm]] ({{math|Ω}}).
Its symbol is usually {{math|''Z''}}, and it may be represented by writing its magnitude and phase in the [[Polar coordinate system|polar]] form {{math|{{!}}''Z''{{!}}''∠θ''}}. However, [[Complex plane|Cartesian complex number representation]] is often more powerful for circuit analysis purposes.

The notion of impedance is useful for performing AC analysis of [[Electrical network|electrical networks]], because it allows relating sinusoidal voltages and currents by a simple linear law.  
In multiple [[Port (circuit theory)|port]] networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still [[Linear relation|linearly related]] by the [[Impedance parameters|impedance matrix]].<ref>Callegaro, Sec. 1.6</ref>  

The [[Multiplicative inverse|reciprocal]] of impedance is [[admittance]], whose [[SI]] unit is the [[siemens (unit)|siemens]], formerly called [[mho]].

Instruments used to measure the electrical impedance are called [[impedance analyzer]]s.

==History==
Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the [[Maxwell bridge]].  Wietlisbach avoided using differential equations by expressing AC currents and voltages as [[exponential function]]s with [[imaginary number|imaginary]] exponents (see {{section link|#Validity of complex representation}}).  Wietlisbach found the required voltage was given by multiplying the current by a complex number (impedance), although he did not identify this as a general parameter in its own right.<ref>Kline, Ronald R., ''Steinmetz: Engineer and Socialist'', Johns Hopkins University Press, 1992 {{isbn|9780801842986}}, p. 78.</ref>

The term ''impedance'' was coined by [[Oliver Heaviside]] in July 1886.<ref>''Science'', p.&nbsp;18, 1888{{Full citation needed|date=March 2023}}{{failed verification|date = January 2024}}</ref><ref>Oliver Heaviside, ''The Electrician'', p.&nbsp;212, 23 July 1886, reprinted as ''Electrical Papers, Volume II'', p 64, AMS Bookstore, {{ISBN |0-8218-3465-7}}</ref>  Heaviside recognised that the "resistance operator" (impedance) in his [[operational calculus]] was a complex number.  In 1887 he showed that there was an AC equivalent to [[Ohm's law]].<ref>Kline, p. 79.</ref>

[[Arthur Kennelly]] published an influential paper on impedance in 1893. Kennelly arrived at a complex number representation in a rather more direct way than using imaginary exponential functions.  Kennelly followed the graphical representation of impedance (showing resistance, reactance, and impedance as the lengths of the sides of a right angle triangle) developed by [[John Ambrose Fleming]] in 1889.  Impedances could thus be added [[Vector (mathematics and physics)|vectorially]].  Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers ([[Argand diagram]]).  Problems in impedance calculation could thus be approached algebraically with a complex number representation.<ref>Kline, p. 81-2.</ref><ref>Kennelly, Arthur,[https://archive.org/details/transactionsame31engigoog/page/175/mode/1up "Impedance"], ''Transactions of the American Institute of Electrical Engineers'', vol. 10, pp. 175–232, 18 April 1893.</ref>  Later that same year, Kennelly's work was generalised to all AC circuits by [[Charles Proteus Steinmetz]].  Steinmetz not only represented impedances by complex numbers but also voltages and currents.  Unlike Kennelly, Steinmetz was thus able to express AC equivalents of DC laws such as Ohm's and Kirchhoff's laws.<ref>Kline, p. 85.</ref>  Steinmetz's work was highly influential in spreading the technique amongst engineers.<ref>Kline, p. 90-1.</ref>

==Introduction==
In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by the [[magnetic field]]s ([[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.

==Complex impedance==

[[File:Complex Impedance.svg|200px|thumb|right|A graphical representation of the [[Complex plane|complex impedance plane]]]]

The impedance of a two-terminal circuit element is represented as a [[Complex number|complex]] quantity <math>Z</math>. The [[Polar coordinates|polar form]] conveniently captures both magnitude and phase characteristics as

:<math>\ Z = |Z| e^{j \arg(Z) }</math>

where the magnitude <math>|Z|</math> represents the ratio of the voltage difference amplitude to the current amplitude, while the argument <math>\arg(Z)</math> (commonly given the symbol <math>\theta </math>) gives the phase difference between voltage and current. <math>j</math> is the [[imaginary unit]], and is used instead of <math>i</math> in this context to avoid confusion with the symbol for [[Ampere|electric current]].<ref name=":0">{{Cite book |last=Gross |first=Charles A. |url=https://www.worldcat.org/oclc/863646311 |title=Fundamentals of electrical engineering |date=2012 |publisher=CRC Press |others=Thaddeus Adam Roppel |isbn=978-1-4398-9807-9 |location=Boca Raton, FL |oclc=863646311}}</ref>{{Rp|page=21}}

In [[Cartesian plane|Cartesian form]], impedance is defined as

:<math>\ Z = R + jX</math>

where the [[real part]] of impedance is the resistance {{mvar|R}} and the [[imaginary part]] is the [[Reactance (electronics)|reactance]] {{mvar|X}}.

Where it is needed 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 [[Polar form|conversion rules of complex numbers]].

== Complex voltage and current ==
[[File:Impedance symbol comparison.svg|thumb|right|165px|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.]]
To simplify calculations, [[Sine wave|sinusoid]]al voltage and current waves are commonly represented as complex-valued functions of time denoted as <math>V</math> and <math>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 |author2=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-37095-0 |pages=[https://archive.org/details/artofelectronics00horo/page/31 31–32] |chapter=1 |chapter-url=https://archive.org/details/artofelectronics00horo |url=https://archive.org/details/artofelectronics00horo/page/31 }}</ref>

:<math>\begin{align}
  V &= |V|e^{j(\omega t + \phi_V)}, \\
  I &= |I|e^{j(\omega t + \phi_I)}.
\end{align}</math>

The impedance of a bipolar circuit is defined as the ratio of these quantities:

:<math> Z = \frac{V}{I} = \frac{|V|}{|I|}e^{j(\phi_V - \phi_I)}.</math>

Hence, denoting <math>\theta = \phi_V - \phi_I</math>, we have

:<math>\begin{align}
     |V| &= |I| |Z|, \\
  \phi_V &= \phi_I + \theta.
\end{align}</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>

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 are 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) = \operatorname\mathcal{R_e} \Big\{ e^{j(\omega t + \phi)} \Big\}</math>

=== Ohm's law ===
[[File:General AC circuit.svg|thumb|right|165px|An AC supply applying a voltage <math>V</math>, across a [[Electrical load|load]] <math>Z</math>, driving a current <math>I</math>]]
{{Main|Ohm's law}}

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 |author2=Hill, Winfield |title=The Art of Electronics |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-37095-0 |pages=[https://archive.org/details/artofelectronics00horo/page/32 32–33] |chapter=1 |chapter-url=https://archive.org/details/artofelectronics00horo |url=https://archive.org/details/artofelectronics00horo/page/32 }}</ref> Assuming a two-terminal circuit element with impedance <math>Z</math> is driven by a sinusoidal voltage or current as above, there holds  

:<math>\ V = I Z = I |Z| e^{j \arg (Z)}</math>

The magnitude of the impedance <math>|Z|</math> acts just like resistance, giving the drop in voltage amplitude across an impedance <math>Z</math> for a given current <math>I</math>. The [[phase factor]] tells us that the current lags the voltage by a phase of <math>\theta = \arg(Z)</math> (i.e., in the [[time domain]], the current signal is shifted <math display="inline">\frac{\theta}{2 \pi} T</math> later with respect to the voltage signal).

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.

=== Phasors ===
{{Main|Phasor}}
{{Further|Analytic representation}}

A phasor is represented by 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 (such as in AC circuits<ref name=":0" />{{Rp|page=53}}), 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>e^{j\omega t}</math> cancel.

== Device examples ==

===Resistor===
[[File:VI phase.svg|thumb|right|250px|The phase angles in the equations for the impedance of capacitors and inductors 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.]]

The impedance of an ideal [[resistor]] is purely real and is called ''resistive impedance'':

:<math>\ Z_R = R</math>

In this case, the voltage and current waveforms are proportional and in phase.

===Inductor and capacitor===
Ideal [[inductor]]s and [[capacitor]]s have a purely [[imaginary number|imaginary]] ''reactive impedance'':

the impedance of inductors increases as frequency increases;

:<math>Z_L = j\omega L</math>{{efn|<math>j</math> is the imaginary unit; i.e., <math>j = \sqrt{-1}</math> used in electrical engineering. The character <math>i</math> is not used as that is often used for current.}}

the impedance of capacitors decreases as frequency increases;

:<math>Z_C = \frac{1}{j\omega C}</math>

In both cases, for an applied sinusoidal voltage, the resulting current is also sinusoidal, but in [[quadrature phase|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''.

Note the following identities for the imaginary unit and its reciprocal:

:<math>\begin{align}
                      j &\equiv \cos{\left( \frac{\pi}{2}\right)} + j\sin{\left( \frac{\pi}{2}\right)} \equiv e^{j \frac{\pi}{2}} \\
  \frac{1}{j} \equiv -j &\equiv \cos{\left(-\frac{\pi}{2}\right)} + j\sin{\left(-\frac{\pi}{2}\right)} \equiv e^{j\left(-\frac{\pi}{2}\right)}
\end{align}</math>

Thus the inductor and capacitor impedance equations can be rewritten in polar form:

:<math>\begin{align}
  Z_L &= \omega Le^{j\frac{\pi}{2}} \\
  Z_C &= \frac{1}{\omega C}e^{j\left(-\frac{\pi}{2}\right)}
\end{align}</math>

The magnitude gives the change in voltage amplitude for a given current amplitude through the impedance, while the exponential factors give the phase relationship.

=== Deriving the device-specific impedances ===
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 assume [[sinusoidal]] signals. In fact, this applies to any arbitrary periodic signals, because these can be approximated as a sum of sinusoids through [[Fourier analysis]].

====Resistor====
For a resistor, there is the relation
:<math>v_\text{R} \mathord\left( t \right) = i_\text{R} \mathord\left( t \right) R</math>

which is [[Ohm's law]].

Considering the voltage signal to be
:<math>v_\text{R}(t) = V_p \sin(\omega t)</math>

it follows that
:<math>\frac{v_\text{R} \mathord\left( t \right)}{i_\text{R} \mathord\left( t \right)} = \frac{V_p \sin(\omega t)}{I_p \sin \mathord\left( \omega t \right)} = R</math>

This says that the ratio of AC voltage amplitude to [[alternating current]] (AC) amplitude across a resistor is <math>R</math>, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as
:<math>Z_\text{resistor} = R</math>

====Capacitor====
For a capacitor, there is the relation:
:<math>i_\text{C}(t) = C \frac{\mathrm{d}v_\text{C}(t)}{\mathrm{d}t}</math>

Considering the voltage signal to be
:<math>v_\text{C}(t) = V_p e^{j\omega t} </math>

it follows that
:<math>\frac{\mathrm{d}v_{\text{C}}(t)}{\mathrm{d}t} = j\omega V_p e^{j\omega t}</math>

and thus, as previously,
:<math>Z_\text{capacitor} = \frac{v_\text{C} \mathord\left( t \right)}{i_\text{C} \mathord\left( t \right)} = \frac{1}{j\omega C}.</math>

Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being 
:<math>i_\text{C}(t) = I_p e^{j\omega t} </math>
then integrating the differential equation 
:<math>i_\text{C}(t) = C \frac{\mathrm{d}v_\text{C}(t)}{\mathrm{d}t}</math>
leads to 
:<math>v_C(t) = \frac{1}{j\omega C}I_p e^{j\omega t} + \text{Const.} = \frac{1}{j\omega C} i_C(t) + \text{Const.}</math>
The ''Const'' term represents a fixed potential bias superimposed to the AC sinusoidal potential, that plays no role in AC analysis. For this purpose, this term can be assumed to be 0, hence again the impedance  
:<math>Z_\text{capacitor} = \frac{1}{j\omega C}.</math>

====Inductor====
For the inductor, we have the relation (from [[Faraday's law of induction|Faraday's law]]):
:<math>v_\text{L}(t) = L \frac{\mathrm{d}i_\text{L}(t)}{\mathrm{d}t}</math>

This time, considering the current signal to be:
:<math>i_\text{L}(t) = I_p \sin(\omega t)</math>

it follows that:
:<math>\frac{\mathrm{d}i_\text{L}(t)}{\mathrm{d}t} = \omega I_p \cos \mathord\left( \omega t \right)</math>

This result is commonly expressed in polar form as
:<math>Z_\text{inductor} = \omega L e^{j\frac{\pi}{2}}</math>

or, using Euler's formula, as
:<math>Z_\text{inductor} = j \omega L</math>

As in the case of capacitors, it is also possible to derive this formula directly from the complex representations of the voltages and currents, or by assuming a sinusoidal voltage between the two poles of the inductor. In the latter case, integrating the differential equation above leads to a constant term for the current, that represents a fixed DC bias flowing through the inductor. This is set to zero because AC analysis using frequency domain impedance considers one frequency at a time and DC represents a separate frequency of zero hertz in this context.

==Generalised s-plane impedance==
Impedance defined in terms of ''jω'' can strictly be applied only to circuits that 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:

{|class="wikitable" style="margin-left:3em;"
|-
! Element || Impedance expression
|-
| Resistor || <math>R \,</math>
|-
| Inductor || <math>sL \,</math>
|-
| Capacitor || <math>\frac{1}{sC} \,</math>
|}

For a DC circuit, this simplifies to {{nowrap|1=''s'' = 0}}. For a steady-state sinusoidal AC signal {{nowrap|1=''s'' = ''jω''}}.

=== Formal derivation ===
The impedance <math>Z</math> of an electrical component is defined as the ratio between the [[Laplace transform]]s of the voltage over it and the current through it, i.e.

:<math>Z(s) = \frac{\mathcal{L}\{v(t)\}}{\mathcal{L}\{i(t)\}} = \frac{V(s)}{I(s)} \qquad \text{(general impedance)}</math>

where <math>s = \sigma + j\omega</math> is the complex Laplace parameter. As an example, according to the I-V-law of a capacitor, <math>\mathcal{L}\{i(t)\} = \mathcal{L}\{C\,\mathrm{d}v(t)/\mathrm{d}t\} = sC\mathcal{L}\{v(t)\}</math>, from which it follows that <math>Z_C(s) = 1/sC</math>.

In the [[phasor]] regime (steady-state AC, meaning all signals are represented mathematically as simple [[complex exponential]]s <math>v(t) = \hat V\, e^{j\omega t}</math> and <math>i(t) = \hat I\, e^{j\omega t}</math> oscillating at a common frequency <math>\omega</math>), impedance can simply be calculated as the voltage-to-current ratio, in which the common time-dependent factor cancels out:

:<math>Z(\omega) = \frac{v(t)}{i(t)} = \frac{\hat V\, e^{j\omega t}}{\hat I\, e^{j\omega t}} = \frac{\hat V}{\hat I} \qquad \text{(phasor-regime impedance)}</math>

Again, for a capacitor, one gets that <math>i(t) = C\,\mathrm{d}v(t)/\mathrm{d}t = j\omega C\,v(t)</math>, and hence <math>Z_C(\omega) = 1/j\omega C</math>. The phasor domain is sometimes dubbed the frequency domain, although it lacks one of the dimensions of the Laplace parameter.<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 }}</ref> For steady-state AC, 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:
* The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude;
* The phase of the complex impedance is the [[phase shift]] by which the current lags the voltage.
These two relationships hold even after taking the real part of the complex exponentials (see [[phasor]]s), which is the part of the signal one actually measures in real-life circuits.

== Resistance vs reactance ==

Resistance and reactance together determine the magnitude and phase of the impedance through the following relations:

:<math>\begin{align}
     |Z| &= \sqrt{Z Z^*} = \sqrt{R^2 + X^2} \\
  \theta &= \arctan{\left(\frac{X}{R}\right)}
\end{align}</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 ===
<!--[[File:Resistors.jpg|thumb|right|200px|A pack of resistors. [[Media:Resistors.jpg|Actual size]]]]-->
{{Main|Electrical resistance}}

Resistance <math>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|Electrical reactance}}

Reactance <math> X</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>\ X = |Z| \sin{\theta} \quad</math>

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 does not dissipate any power.

==== Capacitive reactance ====
<!--[[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]]]]-->
{{Main|Capacitance}}

A capacitor has a purely reactive impedance that 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]].

:<math>X_\mathsf{C} = \frac{-1\ ~}{\ \omega\ C\ } = \frac{-1\ ~}{\ 2\pi\ f\ C\ } ~.</math>

The minus sign indicates that the imaginary part of the impedance is negative.

At low frequencies, a capacitor approaches an open circuit so no current flows through it.

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 accumulates only a limited charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge accumulates and the smaller the opposition to the current.

==== Inductive reactance ====
{{Main|Inductance}}

Inductive reactance <math>X_L</math> is [[Proportionality (mathematics)|proportional]] to the signal [[frequency]] <math>f</math> and the [[inductance]] <math>L</math>.

:<math>X_L = \omega L = 2\pi f L\quad</math>

An inductor consists of a coiled conductor. [[Faraday's law of induction|Faraday's law]] of electromagnetic induction gives the back [[Electromotive force|emf]] <math>\mathcal{E}</math> (voltage opposing current) due to a rate-of-change of [[magnetic flux density]] <math>B</math> through a current loop.

:<math>\mathcal{E} = -{{d\Phi_B} \over dt}\quad</math>

For an inductor consisting of a coil with <math>N</math> loops this gives:

:<math>\mathcal{E} = -N{d\Phi_B \over dt}\quad</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-averaged rate-of-change that is proportional to frequency, this causes the increase in inductive reactance with frequency.

==== Total reactance ====

The total reactance is given by

:<math>{X = X_L + X_C}</math> (note that <math>X_C</math> is negative)

so that the total impedance is

:<math>\ Z = R + jX</math>

== Combining impedances ==
{{Main|Series and parallel circuits}}

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 for combining resistances, except that the numbers in general are [[complex number]]s. The general case, however, requires [[equivalent impedance transforms]] in addition to series and parallel.

=== Series combination ===

For components connected in series, the current through each circuit element is the same; the total impedance is the sum of the component impedances.

[[File:Impedances in series.svg]]

:<math>\ Z_{\text{eq}} = Z_1 + Z_2 + \cdots + Z_n \quad</math>

Or explicitly in real and imaginary terms:
:<math>\ Z_{\text{eq}} = R + jX = (R_1 + R_2 + \cdots + R_n) + j(X_1 + X_2 + \cdots + X_n) \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.

:[[File:Impedances in parallel.svg]]

Hence the inverse total impedance is the sum of the inverses of the component impedances:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}</math>

or, when n = 2:
:<math>\frac{1}{Z_{\text{eq}}} = \frac{1}{Z_1} + \frac{1}{Z_2} = \frac{Z_1 + Z_2}{Z_1 Z_2}</math>
:<math>\ Z_{\text{eq}} = \frac{Z_1 Z_2}{Z_1 + Z_2}</math>

The equivalent impedance <math>Z_{\text{eq}}</math> can be calculated in terms of the equivalent series resistance <math>R_{\text{eq}}</math> and reactance <math>X_{\text{eq}}</math>.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3 Parallel Impedance Expressions], Hyperphysics</ref>

:<math>\begin{align}
  Z_{\text{eq}} &= R_{\text{eq}} + j X_{\text{eq}} \\
  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} \\
  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}
\end{align}</math>

== Measurement ==
The measurement of the impedance of devices and transmission lines is a practical problem in [[radio]] technology and other fields. 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. The measurement of impedance requires the 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.

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 |author=George Lewis Jr. |author2=George K. Lewis Sr. |author3=William Olbricht |name-list-style=amp |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|page=105102 |doi=10.1088/0957-0233/19/10/105102 |pmid=19081773 |pmc=2600501 |bibcode = 2008MeScT..19j5102L }}</ref>

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"/>

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.

===Example===
Consider an LC [[LC circuit|tank]] circuit. 
The complex impedance of the circuit is
:<math>Z(\omega) = \frac{j\omega L}{1 - \omega^2 LC}.</math>
It is immediately seen that the value of <math display="inline">{1 \over |Z|}</math> is minimal (actually equal to 0 in this case) whenever 
:<math>\omega^2 LC = 1.</math>
Therefore, the fundamental resonance angular frequency is  
:<math>\omega = {1 \over \sqrt{LC}}.</math>

== Variable impedance ==
{{see also|Network analysis (electrical circuits)#Time-varying components}}

In general, neither impedance nor admittance can vary with time, since they are defined for complex exponentials in which {{math|1= −∞ < ''t'' < +∞}}. If the complex exponential voltage to current ratio changes over time or amplitude, the circuit element cannot be described using the frequency domain. However, many components and systems (e.g., [[varicap]]s that are used in [[Tuner (radio)|radio tuners]]) may exhibit non-linear or time-varying voltage to current ratios that seem to be [[LTI system theory|linear time-invariant (LTI)]] for small signals and over small observation windows, so they can be roughly described as-if they had a time-varying impedance. This description is an approximation: Over large signal swings or wide observation windows, the voltage to current relationship will not be LTI and cannot be described by impedance.

== See also ==
*{{annotated link|Bioelectrical impedance analysis}} 
*{{annotated link|Characteristic impedance}} 
*{{annotated link|Electrical characteristics of dynamic loudspeakers}} 
*{{annotated link|High impedance}} 
*{{annotated link|Immittance}} 
*{{annotated link|Impedance analyzer}} 
*{{annotated link|Impedance bridging}} 
*{{annotated link|Impedance cardiography}} 
*{{annotated link|Impedance control}} 
*{{annotated link|Impedance matching}} 
*{{annotated link|Impedance microbiology}} 
*{{annotated link|Negative impedance converter}} 
*{{annotated link|Resistance distance}} 
*[[Reflections of signals on conducting lines#trasm line imp equn|Transmission line impedance]] 
*{{annotated link|Universal dielectric response}}

== Notes ==
{{notelist}}

== References ==
{{reflist|25em}}
*Kline, Ronald R., ''Steinmetz: Engineer and Socialist'', Plunkett Lake Press, 2019 (ebook reprint of Johns Hopkins University Press, 1992 {{isbn|9780801842986}}).

== External links ==
*[http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf ECE 209: Review of Circuits as LTI Systems]&nbsp;&ndash; Brief explanation of Laplace-domain circuit analysis; includes a definition of impedance.

{{Authority control}}

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[[Category:Electrical resistance and conductance]]
[[Category:Physical quantities]]
[[Category:Antennas (radio)]]