# Electrical impedance

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Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named volt. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for volt uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by V, but more often simply as V, for instance in the context of Ohm's or Kirchhoff's circuit laws.

## Contents

Quantitatively, the impedance of a two-terminal circuit element is the ratio of the complex representation of a sinusoidal voltage between its terminals to the complex representation of the current flowing through it. [1] In general, it depends upon the frequency of the sinusoidal voltage.

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six. Similarly, the ratio of lemons to oranges is 6:8 and the ratio of oranges to the total amount of fruit is 8:14.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and 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.

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

In physics and mathematics, the phase of a periodic function of some real variable is the relative value of that variable within the span of each full period.

Direct current (DC) is the unidirectional flow of an electric charge. A battery is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even through a vacuum as in electron or ion beams. The electric current flows in a constant direction, distinguishing it from alternating current (AC). A term formerly used for this type of current was galvanic current.

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 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. [2]

In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside circuit, that meets the port condition; the currents flowing into the two nodes must be equal and opposite.

Impedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal (linearized) response of non-linear networks. They are members of a family of similar parameters used in electronic engineering, other examples being: S-parameters, Y-parameters, H-parameters, T-parameters or ABCD-parameters.

Impedance is a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). Its symbol is usually Z, and it may be represented by writing its magnitude and phase in the form |Z|∠θ. However, cartesian complex number representation is often more powerful for circuit analysis purposes.

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often somewhat complex. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.

The ohm is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. Although several empirically derived standard units for expressing electrical resistance were developed in connection with early telegraphy practice, the British Association for the Advancement of Science proposed a unit derived from existing units of mass, length and time and of a convenient size for practical work as early as 1861. The definition of the ohm was revised several times. Today, the definition of the ohm is expressed from the quantum Hall effect.

The reciprocal of impedance is admittance, whose SI unit is the siemens, formerly called mho.

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse.

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 reciprocal of impedance. The SI unit of admittance is the siemens ; the older, synonymous unit is mho, and its symbol is ℧. Oliver Heaviside coined the term admittance in December 1887.

The siemens is the derived unit of electric conductance, electric susceptance, and electric admittance in the International System of Units (SI). Conductance, susceptance, and admittance are the reciprocals of resistance, reactance, and impedance respectively; hence one siemens is redundantly equal to the reciprocal of one ohm, and is also referred to as the mho. The 14th General Conference on Weights and Measures approved the addition of the siemens as a derived unit in 1971.

Instruments used to measure the electrical impedance are called impedance analyzers.

## Introduction

The term impedance was coined by Oliver Heaviside in July 1886. [3] [4] Arthur Kennelly was the first to represent impedance with complex numbers in 1893. [5]

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 fields (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 reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.

Impedance is defined as the frequency domain ratio of the voltage to the current. [6] In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency ω.

For a sinusoidal current or voltage input, the 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.

## Complex impedance

The impedance of a two-terminal circuit element is represented as a complex quantity ${\displaystyle \scriptstyle Z}$. The polar form conveniently captures both magnitude and phase characteristics as

${\displaystyle \ Z=|Z|e^{j\arg(Z)}}$

where the magnitude ${\displaystyle \scriptstyle |Z|}$ represents the ratio of the voltage difference amplitude to the current amplitude, while the argument ${\displaystyle \scriptstyle \arg(Z)}$ (commonly given the symbol ${\displaystyle \scriptstyle \theta }$) gives the phase difference between voltage and current. ${\displaystyle \scriptstyle j}$ is the imaginary unit, and is used instead of ${\displaystyle \scriptstyle i}$ in this context to avoid confusion with the symbol for electric current.

In Cartesian form, impedance is defined as

${\displaystyle \ Z=R+jX}$

where the real part of impedance is the resistance ${\displaystyle \scriptstyle R}$ and the imaginary part is the reactance ${\displaystyle \scriptstyle X}$.

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 conversion rules of complex numbers.

## Complex voltage and current

To simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as ${\displaystyle \scriptstyle V}$ and ${\displaystyle \scriptstyle I}$. [7] [8]

{\displaystyle {\begin{aligned}V&=|V|e^{j(\omega t+\phi _{V})},\\I&=|I|e^{j(\omega t+\phi _{I})}.\end{aligned}}}

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

${\displaystyle Z={\frac {V}{I}}={\frac {|V|}{|I|}}e^{j(\phi _{V}-\phi _{I})}.}$

Hence, denoting ${\displaystyle \theta =\phi _{V}-\phi _{I}}$, we have

{\displaystyle {\begin{aligned}|V|&=|I||Z|,\\\phi _{V}&=\phi _{I}+\theta .\end{aligned}}}

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):

${\displaystyle \ \cos(\omega t+\phi )={\frac {1}{2}}{\Big [}e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}{\Big ]}}$

The real-valued sinusoidal function representing either voltage or current may be broken into two complex-valued functions. By the principle of 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

${\displaystyle \ \cos(\omega t+\phi )=\Re {\Big \{}e^{j(\omega t+\phi )}{\Big \}}}$

### Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law. [9] [10] Assuming a two-terminal circuit element with impedance ${\displaystyle \scriptstyle Z}$ is driven by a sinusoidal voltage or current as above, there holds

${\displaystyle \ V=IZ=I|Z|e^{j\arg(Z)}}$

The magnitude of the impedance ${\displaystyle \scriptstyle |Z|}$ acts just like resistance, giving the drop in voltage amplitude across an impedance ${\displaystyle \scriptstyle Z}$ for a given current ${\displaystyle \scriptstyle I}$. The phase factor tells us that the current lags the voltage by a phase of ${\displaystyle \scriptstyle \theta \;=\;\arg(Z)}$ (i.e., in the time domain, the current signal is shifted ${\displaystyle \scriptstyle {\frac {\theta }{2\pi }}T}$ 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, current division, Thévenin's theorem and Norton's theorem, 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, 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 given above, recognising that the factors of ${\displaystyle \scriptstyle e^{j\omega t}}$ cancel.

## Device examples

### Resistor

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

${\displaystyle \ Z_{R}=R}$

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

### Inductor and capacitor

Ideal inductors and capacitors have a purely imaginary reactive impedance:

the impedance of inductors increases as frequency increases;

${\displaystyle \ Z_{L}=j\omega L}$

the impedance of capacitors decreases as frequency increases;

${\displaystyle \ Z_{C}={\frac {1}{j\omega C}}}$

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:

{\displaystyle {\begin{aligned}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(-{\frac {\pi }{2}})}\end{aligned}}}

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

{\displaystyle {\begin{aligned}Z_{L}&=\omega Le^{j{\frac {\pi }{2}}}\\Z_{C}&={\frac {1}{\omega C}}e^{j\left(-{\frac {\pi }{2}}\right)}\end{aligned}}}

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, 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

${\displaystyle v_{\text{R}}\left(t\right)={i_{\text{R}}\left(t\right)}R}$

which is Ohm's law.

Considering the voltage signal to be

${\displaystyle v_{\text{R}}(t)=V_{p}\sin(\omega t)}$

it follows that

${\displaystyle {\frac {v_{\text{R}}\left(t\right)}{i_{\text{R}}\left(t\right)}}={\frac {V_{p}\sin(\omega t)}{I_{p}\sin \left(\omega t\right)}}=R}$

This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is ${\displaystyle \scriptstyle R}$, and that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

${\displaystyle Z_{\text{resistor}}=R}$

#### Capacitor

For a capacitor, there is the relation:

${\displaystyle i_{\text{C}}(t)=C{\frac {\operatorname {d} v_{\text{C}}(t)}{\operatorname {d} t}}}$

Considering the voltage signal to be

${\displaystyle v_{\text{C}}(t)=V_{p}e^{j\omega t}\,}$

it follows that

${\displaystyle {\frac {\operatorname {d} v_{\text{C}}(t)}{\operatorname {d} t}}=j\omega V_{p}e^{j\omega t}}$

and thus, as previously,

${\displaystyle Z_{\text{capacitor}}={\frac {v_{\text{C}}\left(t\right)}{i_{\text{C}}\left(t\right)}}={1 \over j\omega C}.}$

Conversely, if the current through the circuit is assumed to be sinusoidal, its complex representation being

${\displaystyle i_{\text{C}}(t)=I_{p}e^{j\omega t}\,}$

then integrating the differential equation

${\displaystyle i_{\text{C}}(t)=C{\frac {\operatorname {d} v_{\text{C}}(t)}{\operatorname {d} t}}}$

${\displaystyle v_{C}(t)={1 \over j\omega C}I_{p}e^{j\omega t}+{\text{Const}}={1 \over j\omega C}i_{C}(t)+{\text{Const}}.}$

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

${\displaystyle Z_{\text{capacitor}}={1 \over j\omega C}.}$

#### Inductor

For the inductor, we have the relation (from Faraday's law):

${\displaystyle v_{\text{L}}(t)=L{\frac {\operatorname {d} i_{\text{L}}(t)}{\operatorname {d} t}}}$

This time, considering the current signal to be:

${\displaystyle i_{\text{L}}(t)=I_{p}\sin(\omega t)}$

it follows that:

${\displaystyle {\frac {\operatorname {d} i_{\text{L}}(t)}{\operatorname {d} t}}=\omega I_{p}\cos \left(\omega t\right)}$

This result is commonly expressed in polar form as

${\displaystyle \ Z_{\text{inductor}}=\omega Le^{j{\frac {\pi }{2}}}}$

or, using Euler's formula, as

${\displaystyle \ Z_{\text{inductor}}=j\omega L}$

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 this later case, integrating the differential equation above leads to a Const 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 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 . 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:

ElementImpedance expression
Resistor${\displaystyle R\,}$
Inductor${\displaystyle sL\,}$
Capacitor${\displaystyle {\frac {1}{sC}}\,}$

For a DC circuit, this simplifies to s = 0. For a steady-state sinusoidal AC signal s = .

## Resistance vs reactance

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

{\displaystyle {\begin{aligned}|Z|&={\sqrt {ZZ^{*}}}={\sqrt {R^{2}+X^{2}}}\\\theta &=\arctan {\left({\frac {X}{R}}\right)}\end{aligned}}}

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 ${\displaystyle R}$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current.

${\displaystyle \ R=|Z|\cos {\theta }\quad }$

### Reactance

Reactance ${\displaystyle X}$ is the imaginary part of the impedance; a component with a finite reactance induces a phase shift ${\displaystyle \theta }$ between the voltage across it and the current through it.

${\displaystyle \ X=|Z|\sin {\theta }\quad }$

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 to the signal frequency. A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

${\displaystyle X_{C}=-(\omega C)^{-1}=-(2\pi fC)^{-1}\quad }$

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 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

Inductive reactance ${\displaystyle X_{L}}$ is proportional to the signal frequency ${\displaystyle f}$ and the inductance ${\displaystyle L}$.

${\displaystyle X_{L}=\omega L=2\pi fL\quad }$

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf ${\displaystyle {\mathcal {E}}}$ (voltage opposing current) due to a rate-of-change of magnetic flux density ${\displaystyle B}$ through a current loop.

${\displaystyle {\mathcal {E}}=-{{d\Phi _{B}} \over dt}\quad }$

For an inductor consisting of a coil with ${\displaystyle N}$ loops this gives:

${\displaystyle {\mathcal {E}}=-N{d\Phi _{B} \over dt}\quad }$

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

${\displaystyle {X=X_{L}+X_{C}}}$ (note that ${\displaystyle X_{C}}$ is negative)

so that the total impedance is

${\displaystyle \ Z=R+jX}$

## 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 numbers. 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.

${\displaystyle \ Z_{\text{eq}}=Z_{1}+Z_{2}+\cdots +Z_{n}\quad }$

Or explicitly in real and imaginary terms:

${\displaystyle \ Z_{\text{eq}}=R+jX=(R_{1}+R_{2}+\cdots +R_{n})+j(X_{1}+X_{2}+\cdots +X_{n})\quad }$

### 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:

${\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}+\cdots +{\frac {1}{Z_{n}}}}$

or, when n = 2:

${\displaystyle {\frac {1}{Z_{\text{eq}}}}={\frac {1}{Z_{1}}}+{\frac {1}{Z_{2}}}={\frac {Z_{1}+Z_{2}}{Z_{1}Z_{2}}}}$
${\displaystyle \ Z_{\text{eq}}={\frac {Z_{1}Z_{2}}{Z_{1}+Z_{2}}}}$

The equivalent impedance ${\displaystyle \scriptstyle Z_{\text{eq}}}$ can be calculated in terms of the equivalent series resistance ${\displaystyle \scriptstyle R_{\text{eq}}}$ and reactance ${\displaystyle \scriptstyle X_{\text{eq}}}$. [11]

{\displaystyle {\begin{aligned}Z_{\text{eq}}&=R_{\text{eq}}+jX_{\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{aligned}}}

## 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" 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. [12]

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. [12]

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 tank circuit. The complex impedance of the circuit is

${\displaystyle Z(\omega )={j\omega L \over 1-\omega ^{2}LC}.}$

It is immediately seen that the value of ${\displaystyle {1 \over |Z|}}$ is minimal (actually equal to 0 in this case) whenever

${\displaystyle \omega ^{2}LC=1.}$

Therefore, the fundamental resonance angular frequency is

${\displaystyle \omega ={1 \over {\sqrt {LC}}}.}$

## Variable impedance

In general, neither impedance nor admittance can vary with time, since they are defined for complex exponentials in which -∞ < 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., varicaps 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.

## Related Research Articles

The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

In electric and electronic systems, reactance is the opposition of a circuit element to a change in current or voltage, due to that element's inductance or capacitance. The notion of reactance is similar to electric resistance, but it differs in several respects.

A resistor–capacitor circuit, or RC filter or RC network, is an electric circuit composed of resistors and capacitors driven by a voltage or current source. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

In electronics, a voltage divider is a passive linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

The Smith chart, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.

In the power systems analysis field of electrical engineering, a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit do not change when they are referred from one side of a transformer to the other. This can be a pronounced advantage in power system analysis where large numbers of transformers may be encountered. Moreover, similar types of apparatus will have the impedances lying within a narrow numerical range when expressed as a per-unit fraction of the equipment rating, even if the unit size varies widely. Conversion of per-unit quantities to volts, ohms, or amperes requires a knowledge of the base that the per-unit quantities were referenced to. The per-unit system is used in power flow, short circuit evaluation, motor starting studies etc.

A Colpitts oscillator, invented in 1918 by American engineer Edwin H. Colpitts, is one of a number of designs for LC oscillators, electronic oscillators that use a combination of inductors (L) and capacitors (C) to produce an oscillation at a certain frequency. The distinguishing feature of the Colpitts oscillator is that the feedback for the active device is taken from a voltage divider made of two capacitors in series across the inductor.

In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and sinor or even complexor.

Power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductors and capacitors may result in periodic reversals of the direction of energy flow.

In microwave and radio-frequency engineering, a stub or resonant stub is a length of transmission line or waveguide that is connected at one end only. The free end of the stub is either left open-circuit or short-circuited. Neglecting transmission line losses, the input impedance of the stub is purely reactive; either capacitive or inductive, depending on the electrical length of the stub, and on whether it is open or short circuit. Stubs may thus function as capacitors, inductors and resonant circuits at radio frequencies.

Ripple in electronics is the residual periodic variation of the DC voltage within a power supply which has been derived from an alternating current (AC) source. This ripple is due to incomplete suppression of the alternating waveform after rectification. Ripple voltage originates as the output of a rectifier or from generation and commutation of DC power.

Passive integrator circuit is a simple four-terminal network consisting of two passive elements. It is also the simplest (first-order) low-pass filter.

A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals.

Zobel networks are a type of filter section based on the image-impedance design principle. They are named after Otto Zobel of Bell Labs, who published a much-referenced paper on image filters in 1923. The distinguishing feature of Zobel networks is that the input impedance is fixed in the design independently of the transfer function. This characteristic is achieved at the expense of a much higher component count compared to other types of filter sections. The impedance would normally be specified to be constant and purely resistive. For this reason, Zobel networks are also known as constant resistance networks. However, any impedance achievable with discrete components is possible.

The gyrator–capacitor model is a lumped-element model for magnetic fields, similar to magnetic circuits, but based on using elements analogous to capacitors rather than elements analogous to resistors to represent the magnetic flux path. Windings are represented as gyrators, interfacing between the electrical circuit and the magnetic model.

The primary line constants are parameters that describe the characteristics of conductive transmission lines, such as pairs of copper wires, in terms of the physical electrical properties of the line. The primary line constants are only relevant to transmission lines and are to be contrasted with the secondary line constants, which can be derived from them, and are more generally applicable. The secondary line constants can be used, for instance, to compare the characteristics of a waveguide to a copper line, whereas the primary constants have no meaning for a waveguide.

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

## References

1. Callegaro, p. 2
2. Callegaro, Sec. 1.6
3. Science, p. 18, 1888
4. Oliver Heaviside, The Electrician, p. 212, 23 July 1886, reprinted as Electrical Papers, Volume II, p 64, AMS Bookstore, ISBN   0-8218-3465-7
5. Alexander, Charles; Sadiku, Matthew (2006). Fundamentals of Electric Circuits (3, revised ed.). McGraw-Hill. pp. 387–389. ISBN   978-0-07-330115-0.
6. Complex impedance, Hyperphysics
7. Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 31–32. ISBN   978-0-521-37095-0.
8. AC Ohm's law, Hyperphysics
9. Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 32–33. ISBN   978-0-521-37095-0.
10. Parallel Impedance Expressions, Hyperphysics
11. George Lewis Jr.; George K. Lewis Sr. & William Olbricht (August 2008). "Cost-effective broad-band electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound transducers". Measurement Science and Technology. 19 (10): 105102. Bibcode:2008MeScT..19j5102L. doi:10.1088/0957-0233/19/10/105102. PMC  . PMID   19081773 . Retrieved 2008-09-15.