Electrical impedance

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OR:

In
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

, impedance is the opposition to
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
presented by the combined effect of resistance and in a . Quantitatively, the impedance of a two-terminal
circuit element Electrical elements are conceptual abstractions representing idealized electrical component An electronic component is any basic discrete device or physical entity in an Electronics, electronic system used to affect electrons or their associa ...
is the ratio of the representation of the
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...

voltage between its terminals, to the complex representation of the current flowing through it. In general, it depends upon the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...

of the sinusoidal voltage. Impedance extends the concept of resistance to alternating current (AC) circuits, and possesses both magnitude and phase, unlike resistance, which has only magnitude. Impedance can be represented as a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

, with the same units as resistance, for which the
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
is the
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm ( ...

(). Its symbol is usually , and it may be represented by writing its magnitude and phase in the form . However, Cartesian complex number representation is often more powerful for circuit analysis purposes. The notion of impedance is useful for performing AC analysis of electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law. In multiple
port A port is a maritime law, maritime facility comprising one or more Wharf, wharves or loading areas, where ships load and discharge Affreightment, cargo and passengers. Although usually situated on a sea coast or estuary, ports can a ...
networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still linearly related by the impedance matrix. The
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yield ...

of impedance is
admittance In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the multiplicative inverse, reciprocal of Electrical impedance, impedance, analogous to how Electrical Conductan ...
, whose unit is the
siemens Siemens AG ( ) is a German Multinational corporation, multinational Conglomerate (company), conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad. The principal ...
, formerly called mho. Instruments used to measure the electrical impedance are called
impedance analyzer An impedance analyzer is a type of electronic test equipment used to measure complex electrical impedance as a function of test frequency. Impedance is an important parameter used to characterize Electronic component, electronic components, Elect ...
s.

# History

Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the
Maxwell bridge A Maxwell bridge is a modification to a Wheatstone bridge used to measure an unknown inductance (usually of low Q value) in terms of calibrated Electrical resistance, resistance and inductance or resistance and capacitance. When the calibrated ...

. Wietlisbach avoided using differential equations by expressing AC currents and voltages as
exponential function The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...

s with imaginary exponents (see ). 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. The term ''impedance'' was coined by
Oliver Heaviside Oliver Heaviside Fellow of the Royal Society, FRS (; 18 May 1850 – 3 February 1925) was an English Autodidacticism, self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the La ...
in July 1886. Heaviside recognised that the "resistance operator" (impedance) in his
operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in Mathematical Analysis, analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial ...
was a complex number. In 1887 he showed that there was an AC equivalent to
Ohm's law Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...

.
Arthur Kennelly Arthur Edwin Kennelly (December 17, 1861 – June 18, 1939) was an American electrical engineer. Biography Kennelly was born December 17, 1861, in Colaba, in Bombay Presidency, British India, and was educated at University College School in Lond ...
published an influential paper on impedance in 1893. Kennelly arrived at a complex number representaion 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 Sir John Ambrose Fleming Fellow of the Royal Society, FRS (29 November 1849 – 18 April 1945) was an English electrical engineer and physicist who invented the first thermionic valve or vacuum tube, designed the radio transmitter with which ...
in 1889. Impedances could thus be added vectorially. Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers (
Argand diagram In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
). Problems in impedance calculation could thus be approached algebraically with a complex number representation. Later that same year, Kennelly's work was generalised to all AC circuits by
Charles Proteus Steinmetz Charles Proteus Steinmetz (born Karl August Rudolph Steinmetz, April 9, 1865 – October 26, 1923) was a German-born American mathematician and Electrical engineering, electrical engineer and professor at Union College. He fostered the deve ...

. 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. Steinmetz's work was highly influential in spreading the technique amongst engineers.

# 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 A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to t ...

s (
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
), and the electrostatic storage of charge induced by voltages between conductors (
capacitance Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
). The impedance caused by these two effects is collectively referred to as and forms the imaginary part of complex impedance whereas resistance forms the real part.

# Complex impedance

The impedance of a two-terminal circuit element is represented as a quantity $Z$. The
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

conveniently captures both magnitude and phase characteristics as :$\ Z = , Z, e^$ where the magnitude $, Z,$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument $\arg\left(Z\right)$ (commonly given the symbol $\theta$) gives the phase difference between voltage and current. $j$ is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimen ...
, and is used instead of $i$ in this context to avoid confusion with the symbol for
electric current An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving par ...

. In , impedance is defined as :$\ Z = R + jX$ where the
real part In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of impedance is the resistance and the
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
is the reactance . 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 conversion rules of complex numbers.

# Complex voltage and current

To simplify calculations,
sinusoid A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...

al voltage and current waves are commonly represented as complex-valued functions of time denoted as $V$ and $I$. :$\begin V &= , V, e^, \\ I &= , I, e^. \end$ The impedance of a bipolar circuit is defined as the ratio of these quantities: :$Z = \frac = \frace^.$ Hence, denoting $\theta = \phi_V - \phi_I$, we have :$\begin , V, &= , I, , Z, , \\ \phi_V &= \phi_I + \theta. \end$ 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 Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler ...

): :

## Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law. Assuming a two-terminal circuit element with impedance $Z$ is driven by a sinusoidal voltage or current as above, there holds :$\ V = I Z = I , Z, e^$ The magnitude of the impedance $, Z,$ acts just like resistance, giving the drop in voltage amplitude across an impedance $Z$ for a given current $I$. The
phase factor For any complex number written in Complex number#Polar complex plane, polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any mag ...
tells us that the current lags the voltage by a phase of $\theta = \arg\left(Z\right)$ (i.e., in the
time domain Time domain refers to the analysis of function (mathematics), mathematical functions, physical signal (information theory), signals or time series of economics, economic or environmental statistics, environmental data, with respect to time. In ...
, the current signal is shifted $\frac T$ 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 division Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a Electrostatics, static electric field, it corresponds to the Work (electrical), w ...
, current division,
Thévenin's theorem As originally stated in terms of direct-current resistive The electrical resistance of an object is a measure of its opposition to the flow of electric current An electric current is a stream of charged particles, such as electrons or io ...
and
Norton's theorem In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of ...
, can also be extended to AC circuits by replacing resistance with impedance.

## Phasors

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), 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
Ohm's law Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...
given above, recognising that the factors of $e^$ cancel.

# Device examples

## Resistor

The impedance of an ideal
resistor A resistor is a passivity (engineering), passive terminal (electronics), two-terminal electronic component, electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce c ...
is purely real and is called ''resistive impedance'': :$\ Z_R = R$ In this case, the voltage and current waveforms are proportional and in phase.

## Inductor and capacitor

Ideal
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges ...
s and
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passivity (engineering), passive electronic component with two termi ...
s have a purely imaginary ''reactive impedance'': the impedance of inductors increases as frequency increases; :$Z_L = j\omega L$ the impedance of capacitors decreases as frequency increases; :$Z_C = \frac$ 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''. Note the following identities for the imaginary unit and its reciprocal: :$\begin j &\equiv \cos + j\sin \equiv e^ \\ \frac \equiv -j &\equiv \cos + j\sin \equiv e^ \end$ Thus the inductor and capacitor impedance equations can be rewritten in polar form: :$\begin Z_L &= \omega Le^ \\ Z_C &= \frace^ \end$ 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 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 In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
, these derivations assume
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a curve, mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph of a function, graph. It is a type of continuous wave and also a Smoothness, smooth p ...
signals. In fact, this applies to any arbitrary periodic signals, because these can be approximated as a sum of sinusoids through
Fourier analysis In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
.

### Resistor

For a resistor, there is the relation :$v_\text \mathord\left\left( t \right\right) = i_\text \mathord\left\left( t \right\right) R$ which is
Ohm's law Ohm's law states that the electric current, current through a Electrical conductor, conductor between two points is directly Proportionality (mathematics), proportional to the voltage across the two points. Introducing the constant of proporti ...

. Considering the voltage signal to be :$v_\text\left(t\right) = V_p \sin\left(\omega t\right)$ it follows that :$\frac = \frac = R$ This says that the ratio of AC voltage amplitude to
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
(AC) amplitude across a resistor is $R$, and that the AC voltage leads the current across a resistor by 0 degrees. This result is commonly expressed as :$Z_\text = R$

### Capacitor

For a capacitor, there is the relation: :$i_\text\left(t\right) = C \frac$ Considering the voltage signal to be :$v_\text\left(t\right) = V_p e^$ it follows that :$\frac = j\omega V_p e^$ and thus, as previously, :$Z_\text = \frac = \frac.$ Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being :$i_\text\left(t\right) = I_p e^$ then integrating the differential equation :$i_\text\left(t\right) = C \frac$ leads to :$v_C\left(t\right) = \fracI_p e^ + \text = \frac i_C\left(t\right) + \text$ 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 :$Z_\text = \frac.$

### Inductor

For the inductor, we have the relation (from Faraday's law): :$v_\text\left(t\right) = L \frac$ This time, considering the current signal to be: :$i_\text\left(t\right) = I_p \sin\left(\omega t\right)$ it follows that: :$\frac = \omega I_p \cos \mathord\left\left( \omega t \right\right)$ This result is commonly expressed in polar form as :$Z_\text = \omega L e^$ or, using Euler's formula, as :$Z_\text = j \omega L$ 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 In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time ...
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 In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time ...
of the
time domain Time domain refers to the analysis of function (mathematics), mathematical functions, physical signal (information theory), signals or time series of economics, economic or environmental statistics, environmental data, with respect to time. In ...
expression of the signal. The impedance of the basic circuit elements in this more general notation is as follows: For a DC circuit, this simplifies to . For a steady-state sinusoidal AC signal .

## Formal derivation

The impedance $Z$ of an electrical component is defined as the ratio between the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time ...
s of the voltage over it and the current through it, i.e. :$Z\left(s\right) = \frac = \frac \qquad \text$ where $s = \sigma + j\omega$ is the complex Laplace parameter. As an example, according to the I-V-law of a capacitor, $\mathcal\ = \mathcal\ = sC\mathcal\$, from which it follows that $Z_C\left(s\right) = 1/sC$. In the
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sine wave, sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and Phase (waves), initial phase (''θ'') are time-inva ...
regime (steady-state AC, meaning all signals are represented mathematically as simple
complex exponential The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
s $v\left(t\right) = \hat V\, e^$ and $i\left(t\right) = \hat I\, e^$ oscillating at a common frequency $\omega$), impedance can simply be calculated as the voltage-to-current ratio, in which the common time-dependent factor cancels out: :$Z\left(\omega\right) = \frac = \frac = \frac \qquad \text$ Again, for a capacitor, one gets that $i\left(t\right) = C\,\mathrmv\left(t\right)/\mathrmt = j\omega C\,v\left(t\right)$, and hence $Z_C\left(\omega\right) = 1/j\omega C$. The phasor domain is sometimes dubbed the frequency domain, although it lacks one of the dimensions of the Laplace parameter. For steady-state AC, the
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
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 In physics and mathematics, the phase of a periodic function F of some real number, real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a sca ...
by which the current lags the voltage. These two relationships hold even after taking the real part of the complex exponentials (see
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sine wave, sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and Phase (waves), initial phase (''θ'') are time-inva ...
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: :$\begin , Z, &= \sqrt = \sqrt \\ \theta &= \arctan \end$ In many applications, the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

## Resistance

Resistance $R$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current. :$\ R = , Z, \cos \quad$

## Reactance

Reactance $X$ is the imaginary part of the impedance; a component with a finite reactance induces a phase shift $\theta$ between the voltage across it and the current through it. :$\ X = , Z, \sin \quad$ 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

A capacitor has a purely reactive impedance that is
inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a Constant (mathematics), constant ratio, which is called the coefficient of proportionality or p ...
to the signal
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...

. A capacitor consists of two conductors separated by an insulator, also known as a
dielectric In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electr ...
. :$X_\mathsf = \frac = \frac ~.$ 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 charge to accumulate on one side; the
electric field An electric field (sometimes E-field) is the field (physics), physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the ...
due to the accumulated charge is the source of the opposition to the current. When the
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
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

Inductive reactance $X_L$ is proportional to the signal
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in Hertz (unit), hertz (H ...

$f$ and the
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
$L$. :$X_L = \omega L = 2\pi f L\quad$ An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf $\mathcal$ (voltage opposing current) due to a rate-of-change of
magnetic flux density A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to t ...
$B$ through a current loop. :$\mathcal = -\quad$ For an inductor consisting of a coil with $N$ loops this gives: :$\mathcal = -N\quad$ The back-emf is the source of the opposition to current flow. A constant
direct current Direct current (DC) is one-directional electric current, flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor (material), conductor such as a wire, but can also flow throug ...
has a zero rate-of-change, and sees an inductor as a
short-circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit An electrical network is an interconnection of electronic component, electrical components (e.g., battery (electricity), batteries, resistors, inductors, capa ...
(it is typically made from a material with a low
resistivity Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
). An
alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which ...
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 : (note that $X_C$ is negative) so that the total impedance is :$\ Z = R + jX$

# Combining impedances

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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...

s. The general case, however, requires
equivalent impedance transforms An equivalent impedance is an equivalent circuit of an electrical network of electrical impedance, impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes transformat ...
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. :$\ Z_ = Z_1 + Z_2 + \cdots + Z_n \quad$ Or explicitly in real and imaginary terms: :$\ Z_ = R + jX = \left(R_1 + R_2 + \cdots + R_n\right) + j\left(X_1 + X_2 + \cdots + X_n\right) \quad$

## 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. : Hence the inverse total impedance is the sum of the inverses of the component impedances: :$\frac = \frac + \frac + \cdots + \frac$ or, when n = 2: :$\frac = \frac + \frac = \frac$ :$\ Z_ = \frac$ The equivalent impedance $Z_$ can be calculated in terms of the equivalent series resistance $R_$ and reactance $X_$.Parallel Impedance Expressions
Hyperphysics
:$\begin Z_ &= R_ + j X_ \\ R_ &= \frac \\ X_ &= \frac \end$

# Measurement

The measurement of the impedance of devices and transmission lines is a practical problem in
radio Radio is the technology of signaling and telecommunication, communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device ...
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 In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver (radio), receiver. In Transmission (telecommunicati ...
, the
standing wave ratio In radio engineering and telecommunication Telecommunication is the transmission of information by various types of technologies over wire, radio, Optical system, optical, or other Electromagnetism, electromagnetic systems. It has its origin ...
or
reflection coefficient In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wa ...
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" methods, similar to the direct-current
Wheatstone bridge A Wheatstone bridge is an electrical circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component. The primary benefit of the circuit is its ability to provide ...
; 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. The use of an impulse response may be used in combination with the
fast Fourier transform A fast Fourier transform (FFT) is an algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perf ...
(FFT) to rapidly measure the electrical impedance of various electrical devices. The
LCR meter An LCR meter is a type of electronic test equipment used to measure the inductance (L), capacitance (C), and electrical resistance, resistance (R) of an electrical component, electronic component. In the simpler versions of this instrument the ...
(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
tank A tank is an armoured fighting vehicle intended as a primary offensive weapon in front-line ground combat. Tank designs are a balance of heavy firepower, strong vehicle armour, armour, and good battlefield mobility (military), mobility prov ...
circuit. The complex impedance of the circuit is :$Z\left(\omega\right) = \frac.$ It is immediately seen that the value of  is minimal (actually equal to 0 in this case) whenever :$\omega^2 LC = 1.$ Therefore, the fundamental resonance angular frequency is :$\omega = .$

# Variable impedance

In general, neither impedance nor admittance can vary with time, since they are defined for complex exponentials in which . 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 In electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using Electronic component, electronic devices. Electronics uses Passivity (engineering ...
s that are used in radio tuners) may exhibit non-linear or time-varying voltage to current ratios that seem to be 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.

* * * * * * * * * * * * * * Transmission line impedance *

# References

*Kline, Ronald R., ''Steinmetz: Engineer and Socialist'', Plunkett Lake Press, 2019 (ebook reprint of Johns Hopkins University Press, 1992 ).