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Encyclopedia > Electrical impedance

Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. The concept of electrical impedance generalizes Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number, however, like resistance, the unit of impedance is the ohm. Oliver Heaviside coined the term "impedance" in July of 1886. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... City lights viewed in a motion blurred exposure. ... Electric current is by definition the flow of electric charge. ... Ohms law states that, in an electrical circuit, the current passing through a conductor is directly proportional to the potential difference applied across them provided all physical conditions are kept constant. ... Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... An electrical network or electrical circuit is an interconnection of analog electrical elements such as resistors, inductors, capacitors, diodes, switches and transistors. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... The ohm (symbol: Î©) is the SI unit of electric resistance. ... Oliver Heaviside (May 18, 1850 â€“ February 3, 1925) was a self-taught English engineer, mathematician and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and magnetic...

Image File history File links A simple source and load circuit. ... Image File history File links A simple source and load circuit. ... Image File history File links A simple source and load circuit. ...

In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors and inductors (in short, all linearly behaving components) are solutions to a linear ordinary differential equation. It can be shown that if the voltage and/or current sources in the circuit are sinusoidal and of constant frequency, the solutions tend to a form referred to as AC steady state. Thus, all of the voltages and currents in the circuit are sinusoidal and have constant peak amplitude, frequency and phase. An ideal resistor is a component with an electrical resistance that remains constant regardless of the applied voltage or current flowing through the device. ... Various types of capacitors A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal but opposite electric charges have been placed. ... An inductor is a passive electrical device that stores energy in a magnetic field, typically by combining the effects of many loops of electric current. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

In AC steady state, v(t) is a sinusoidal function of time with constant amplitude Vp, constant frequency f, and constant phase $varphi$: Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... $v(t) = V_mathrm{p} cos left( 2 pi f t + varphi right) = Re left( V_mathrm{p} e^{j 2 pi f t} e^{j varphi} right)$

where

j represents the imaginary unit ( $sqrt{-1}$) $Re (z)$ means the real part of the complex number z.

The phasor representation of v(t) is the constant complex number V: In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... See wikibooks book on Phasors A phasor is a constant complex number representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. ... $V = V_mathrm{p} e^{j varphi} ,$

For a circuit in AC steady state, all of the voltages and currents in the circuit have phasor representations as long as all the sources are of the same frequency. That is, each voltage and current can be represented as a constant complex number. For DC circuit analysis, each voltage and current is represented by a constant real number. Thus, it is reasonable to suppose that the rules developed for DC circuit analysis can be used for AC circuit analysis by using complex numbers instead of real numbers. A resistive circuit is a circuit containing only resistors, ideal current sources, and ideal voltage sources. ... In mathematics, the real numbers may be described informally in several different ways. ...

## Definition of electrical impedance

The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element: $Z_mathrm{R} = frac{V_mathrm{r}}{I_mathrm{r}}$

It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.

For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor: $R = frac{V_mathrm{R}}{I_mathrm{R}}$

where

VR and IR above are DC (constant real) values.

Just as Ohm's law is generalized to AC circuits through the use of phasors, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem generalize to AC circuits. In electronics, a voltage divider or resistor divider is a design technique used to create a voltage (Vout) which is proportional to another voltage (Vin). ... Circuits Left: Series  | Right: Parallel Arrows indicate direction of current flow. ... Thevenins theorem for electrical networks states that any combination of voltage sources and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors. ... Nortons theorem for electrical networks states that any collection of voltage sources and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedances, not just...

The electric impedance is equal to: $z_e = sqrt{r_{e}^2 + x_{e}^2}$ , $varphi = arctan {frac{x_e}{r_e}}$

where $r_e = z_e cos varphi ,$ is the real part of the complex electric impedance, and $x_e = z_e sin varphi ,$ is the imaginary part of the complex electric impedance.

## Impedance of different devices

For a resistor: $Z_mathrm{resistor} = frac{V_mathrm{R}}{I_mathrm{R}} = R ,$

For a capacitor: $Z_mathrm{capacitor} = frac{V_mathrm{C}}{I_mathrm{C}} = frac{1}{j omega C} ,$

For an inductor: $Z_mathrm{inductor} = frac{V_mathrm{L}}{I_mathrm{L}} = j omega L ,$

For derivations, see Impedance of different devices (derivations). // For a resistor, we have the relation: That is, the ratio of the instantaneous voltage and current associated with a resistor is the value of the DC resistance denoted by R. Since R is constant and real, it follows that if v(t) is sinusoidal, i(t) is also sinusoidal...

## Reactance

Main article: Reactance

The term reactance refers to the imaginary part of the impedance. Some examples: It has been suggested that Electric reactance be merged into this article or section. ...

A resistor's impedance is R (its resistance) and its reactance is 0.

A capacitor's impedance is j (-1/ωC) and its reactance is -1/ωC.

An inductor's impedance is j ω L and its reactance is ω L.

It is important to note that the impedance of a capacitor or an inductor is a function of the frequency ω and is an imaginary quantity - however is certainly a real physical phenomenon relating the shift in phases between the voltage and current phasors due to the existence of the capacitor or inductor. Earlier it was shown that the impedance of a resistor is constant and real, in other words a resistor does not cause a phase shift between voltage and current as do capacitors and inductors.

When resistors, capacitors, and inductors are combined in an AC circuit, the impedances of the individual components can be combined in the same way that the resistances are combined in a DC circuit. The resulting equivalent impedance is in general, a complex quantity. That is, the equivalent impedance has a real part and an imaginary part. The real part is denoted with an R and the imaginary part is denoted with an X. Thus: $Z_mathrm{eq} = R_mathrm{eq} + jX_mathrm{eq} ,$

where

Req is termed the resistive part of the impedance
Xeq is termed the reactive part of the impedance.

It is therefore common to refer to a capacitor or an inductor as a reactance or equivalently, a reactive component (circuit element). Additionally, the impedance for a capacitance is negative imaginary while the impedance for an inductor is positive imaginary. Thus, a capacitive reactance refers to a negative reactance while an inductive reactance refers to a positive reactance.

A reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. That is, unlike a resistance, a reactance does not dissipate power.

It is instructive to determine the value of the capacitive reactance at the frequency extremes. As the frequency approaches zero, the capacitive reactance grows without bound so that a capacitor approaches an open circuit for very low frequency sinusoidal sources. As the frequency increases, the capacitive reactance approaches zero so that a capacitor approaches a short circuit for very high frequency sinusoidal sources.

Conversely, the inductive reactance approaches zero as the frequency approaches zero so that an inductor approaches a short circuit for very low frequency sinusoidal sources. As the frequency increases, the inductive reactance increases so that an inductor approaches an open circuit for very high frequency sinusoidal sources.

## Combining impedances

Combining impedances in series, parallel, or in delta-wye configurations, is the same as for resistors. The difference is that combining impedances involves manipulation of complex numbers. Left: Series / Right: Parallel Arrows indicate direction of current. ...

### In series

Combining impedances in series is simple: $Z_mathrm{eq} = Z_1 + Z_2 = (R_1 + R_2) + j(X_1 + X_2) ! .$

### In parallel

Combining impedances in parallel is much more difficult than combining simple properties like resistance or capacitance, due to a multiplication term. $Z_mathrm{eq} = Z_1 | Z_2 = left( {Z_mathrm{1}}^{-1} + {Z_mathrm{2}}^{-1}right) ^{-1} = frac{Z_mathrm{1}Z_mathrm{2}}{Z_mathrm{1}+Z_mathrm{2}} ! .$

In rationalized form the equivalent resistance is: $Z_mathrm{eq} = R_mathrm{eq} + j X_mathrm{eq} ! .$ $R_mathrm{eq} = { (X_1 R_2 + X_2 R_1) (X_1 + X_2) + (R_1 R_2 - X_1 X_2) (R_1 + R_2) over (R_1 + R_2)^2 + (X_1 + X_2)^2}$ $X_mathrm{eq} = {(X_1 R_2 + X_2 R_1) (R_1 + R_2) - (R_1 R_2 - X_1 X_2) (X_1 + X_2) over (R_1 + R_2)^2 + (X_1 + X_2)^2}$

## Circuits with general sources

Impedance is defined by the ratio of two phasors where a phasor is the complex peak amplitude of a sinusoidal function of time. For more general periodic sources and even non-periodic sources, the concept of impedance can still be used. It can be shown that virtually all periodic functions of time can be represented by a Fourier series. Thus, a general periodic voltage source can be thought of as a (possibly infinite) series combination of sinusoidal voltage sources. Likewise, a general periodic current source can be thought of as a (possibly infinite) parallel combination of sinusoidal current sources. Periodicity is the quality of occurring at regular intervals (e. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

Using the technique of Superposition, each source is activated one at a time and an AC circuit solution is found using the impedances calculated for the frequency of that particular source. The final solutions for the voltages and currents in the circuit are computed as sums of the terms calculated for each individual source. However, it is important to note that the actual voltages and currents in the circuit do not have a phasor representation. Phasors can be added together only when each represents a time function of the same frequency. Thus, the phasor voltages and currents that are calculated for each particular source must be converted back to their time domain representation before the final summation takes place. The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page &#8212; a navigational aid which lists other pages that might otherwise share the same title. ...

This method can be generalized to non-periodic sources where the discrete sums are replaced by integrals. That is, a Fourier transform is used in place of the Fourier series. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

## Magnitude and phase of impedance

Complex numbers are commonly expressed in two distinct forms. The rectangular form is simply the sum of the real part with the product of j and the imaginary part: $Z = R + jX ,$

The polar form of a complex number the real magnitude of the number multiplied by the complex phase. This can be written with exponentials, or in phasor notation: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... Angle notation or phasor notation is a notation used in electronics using the âˆ  sign. ... $Z = left|Zright| e^ {j varphi} = left|Zright|angle varphi$

where $left|Zright| = sqrt{R^2+X^2} = sqrt{Z Z^*}$ is the magnitude of Z (Z* denotes the complex conjugate of Z), and $varphi = arctan bigg(frac{X}{R} bigg)$ is the angle.

In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

## Peak phasor versus rms phasor

A sinusoidal voltage or current has a peak amplitude value as well as an rms (root mean square) value. It can be shown that the rms value of a sinusoidal voltage or current is given by: RMS may refer to any of the following: Railway Mail Service, US mail transportation service until the mid-20th century Royal Mail Ship or Steamer, ship prefix for vessels that carry mail under contract to the British Royal Mail Royal Meteorological Society Royal Society of Miniature Painters Sculptors and Gravers... $V_mathrm{rms} = frac{V_mathrm{peak}}{sqrt{2}}$ $I_mathrm{rms} = frac{I_mathrm{peak}}{sqrt{2}}$

In many cases of AC analysis, the rms value of a sinusoid is more useful than the peak value. For example, to determine the amount of power dissipated by a resistor due to a sinusoidal current, the rms value of the current must be known. For this reason, phasor voltage and current sources are often specified as an rms phasor. That is, the magnitude of the phasor is the rms value of the associated sinusoid rather than the peak amplitude. Generally, rms phasors are used in electrical power engineering whereas peak phasors are often used in low-power circuit analysis.

In any event, the impedance is clearly the same whether peak phasors or rms phasors are used as the scaling factor cancels out when the ratio of the phasors is taken.

## Matched impedances

Main article: Impedance matching

When fitting components together to carry electromagnetic signals, it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch and results in signal loss and reflections. The existence of reflections allows the use of a time-domain reflectometer to locate mismatches in a transmission system. Impedance matching is the practice of attempting to make the output impedance of a source equal to the input impedance of the load to which it is ultimately connected, usually in order to maximise the power transfer and minimise reflections from the load. ... Electromagnetism is the force observed as static electricity, and causes the flow of electric charge (electric current) in electrical conductors. ... In telecommunication, signalling (or signaling) has the following meanings: The use of signals for controlling communications. ... Impedance mismatch has two meanings. ... Reflection in electricity is the result of impedance mismatch in electrical signals. ... In telecommunication, a time-domain reflectometer (TDR) is an electronic instrument used to characterize and locate faults in metallic cables ( twisted pair, coax). ...

For example, a conventional radio frequency antenna for carrying broadcast television in North America was standardized to 300 ohms, using balanced, unshielded, flat wiring. However cable television systems introduced the use of 75 ohm unbalanced, shielded, circular wiring, which could not be plugged into most TV sets of the era. To use the newer wiring on an older TV, small devices known as baluns were widely available. Today most TVs simply standardize on 75 ohm feeds instead. Rough plot of Earths atmospheric transmittance (or opacity) to various wavelengths of electromagnetic radiation, including radio waves. ... A yagi antenna Most simply, an antenna is an electronic component designed to send or receive radio waves. ... Coaxial cable is often used to transmit cable television into the house. ... A balun is a device designed to convert between balanced and unbalanced electrical signals, such as between coaxial cable and ladder line. ...

## Inverse quantities

The reciprocal of a non-reactive resistance is called conductance. Similarly, the reciprocal of an impedance is called admittance. The conductance is the real part of the admittance, and the imaginary part is called the susceptance. Conductance and susceptance are not the reciprocals of resistance and reactance in general, but only for impedances that are purely resistive or purely reactive; in the latter case a change of sign is required. Conductance can refer to: Electrical conductance, the reciprocal of electrical resistance. ... In electrical engineering, the admittance (Y) is the inverse of the impedance (Z). ... In electrical engineering, the susceptance (B) is the imaginary part of the admittance. ...

## Origin of impedances

The origin of j was found by calculating an electrical circuit by the direct method, without using impedances or phasors. The circuit is formed by a resistance an inductance and a capacitor in series The circuit is connected to a sinusoidal voltage source and we have waited long enough so that all the transitory phenomena have faded away. It is now in steady sinusoidal state. As the system is linear, the steady state current will be also sinusoidal and of the same frequency of the voltage source. The only two quantities that we ignore are the amplitude of the current and its phase relative to the voltage source. If the voltage source is $scriptstyle{V=V_circcos(omega t)}$ the current will be of the form $scriptstyle{I=I_circcos(omega t+varphi)}$, where $scriptstyle{varphi}$ is the relative phase of the current, which is unknown. The equation of the circuit is: Electrical resistance is a measure of the degree to which an electrical component opposes the passage of current. ... Inductance (or electric inductance) is a measure of the amount of magnetic flux produced for a given electric current. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... $V_circcos(omega t)= V_R+V_L+V_C$

where $scriptstyle{V_R}$, $scriptstyle{V_L}$ and $scriptstyle{V_C}$ are the voltages across the resistance, the inductance and the capacitor. $V_R,$ is equal to $RI_circcos(omega t+varphi)$

The definition of inductance says: $V_L=Ltextstyle{{dIover dt}}= Ltextstyle{{dleft(I_circcos(omega t+varphi)right)over dt}}= -omega L I_circsin(omega t+varphi)$.

The definition of capacitance says that $scriptstyle{I=C{dV_Cover dt}}$. It is easy to verify (taking the expression derivative) that: $V_C=textstyle{{1over omega C}} I_circsin(omega t+varphi)$.

Then the equation to solve is: $V_circcos(omega t)= RI_circcos(omega t+varphi) -omega L I_circsin(omega t+varphi)+ textstyle{{1over omega C}} I_circsin(omega t+varphi)$

That is, we have to find the two values $scriptstyle{I_circ}$ and $scriptstyle{varphi}$ that makes this equation true for all values of time $scriptstyle{t}$.

To do this, another circuit must be considered, identical to the former and fed by a voltage source whose only difference with the former is that it started with a lag of a quarter of a period. The voltage of this source is $scriptstyle{V=V_circcos(omega t - {pi over 2} ) = V_circsin(omega t) }$. The current in this circuit will be the same as in the former one but for a lag of a quarter of period: $I=I_circcos(omega t + varphi - {pi over 2})= I_circsin(omega t + varphi) ,$.

The equation for this retarded circuit is: $V_circsin(omega t)= RI_circsin(omega t+varphi) +omega L I_circcos(omega t+varphi)- textstyle{{1over omega C}} I_circcos(omega t+varphi)$

Some of the signs have changed because a retarded cosine becomes a sine, but a retarded sine becomes −cosine.

The first equation is added with the second one multiplied by j, to try to replace expressions with the form $scriptstyle{cos x+jsin x}$ by $scriptstyle{e^{jx} }$, using the les Euler's formula. This gives: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... $V_circ e^{jomega t} =RI_circ e^{jleft(omega t+varphiright)}+jomega LI_circ e^{jleft(omega t+varphiright)} +textstyle{{1over jomega C}}I_circ e^{jleft(omega t+varphiright)}$

As $scriptstyle{e^{jomega t} }$ is not zero we can divide all the equation by this factor: $V_circ =RI_circ e^{jvarphi}+jomega LI_circ e^{jvarphi} +textstyle{{1over jomega C}}I_circ e^{jvarphi}$

This gives: $I_circ e^{jvarphi}= textstyle{V_circ over R + jomega L + scriptstyle{{1 over jomega C}}}$

The left side of the equation contains the two values we are trying to deduce: the modulus and the phase of the current. The amplitude is the modulus of the complex number at the right and its phase is the argument of the complex number at the right. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

The formula at right is the habitual formula which is written when doing circuit equations using phasors and impedances. The denominator of the equation is the impedances of the resistance, inductor and capacitor.

Even though the formula $I= textstyle{V_circ over R + jomega L + scriptstyle{{1 over jomega C}}}$

contains imaginary parts, at least some of the imaginary numbers will become real in the circuit (j*j = -1), which means that the previously stated formula can not be simplified to just $I= textstyle{V_circ over R}$

## Analogous impedances

### Electromagnetic impedance

In problems of electromagnetic wave propagation in a homogeneous medium, the intrinsic impedance of the medium is defined as: Electromagnetic radiation is a propagating wave in space with electric and magnetic components. ... $eta = sqrt{frac{mu}{varepsilon}}$

where

μ and ε are the permeability and permittivity of the medium, respectively.

In electromagnetism, permeability is the degree of magnetisation of a material that responds linearly to an applied magnetic field. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

### Acoustic impedance

Main article: Acoustic impedance

In complete analogy to the electrical impedance discussed here, one also defines acoustic impedance, a complex number which describes how a medium absorbs sound by relating the amplitude and phase of an applied sound pressure to the amplitude and phase of the resulting sound flux. The acoustic impedance Z (or sound impedance) is the ratio of sound pressure p to particle velocity v in a medium or acoustic component. ... The acoustic impedance Z (or sound impedance) is the ratio of sound pressure p to particle velocity v in a medium or acoustic component. ...

### Data-transfer impedance

Another analogous coinage is the use of impedance by computer programmers to describe how easy or difficult it is to pass data and flow of control between parts of a system, commonly ones written in different languages. The common usage is to describe two programs or languages/environments as having a low or high impedance mismatch. In computing, a programmer is someone who does computer programming and develops computer software. ... Impedance mismatch has two meanings. ...

## Application to physical devices

Note that the equations above only apply to theoretical devices. Real resistors, capacitors, and inductors are more complex and each one may be modeled as a network of theoretical resistors, capacitors, and inductors. Rated impedances of real devices are actually nominal impedances, and are only accurate for a narrow frequency range, and are typically less accurate for higher frequencies. Even within its rated range, an inductor's resistance may be non-zero. Above the rated frequencies, resistors become inductive (power resistors more so), capacitors and inductors may become more resistive. The relationship between frequency and impedance may not even be linear outside of the device's rated range. In electrical engineering or audio, the nominal impedance of an input or output is the equivalent impedance of all of the output or input circuitry of a device lumped into one (imaginary) component. ... Results from FactBites:

 Medical Policy MED.00044 | Electrical Impedance Scanning of the Breast (1620 words) Electrical impedance scanning of the breast involves the transmission of continuous electricity into the body using either an electrical patch attached to the arm or a hand held cylinder. The majority of patients studied with electrical impedance of the breast were involved in the study presented to the FDA during the approval process for the T-Scan 200 device. Electrical impedance scanning of the breast involves the transmission of a continuous low level electrical current through the body, using either an electrically charged skin patch attached to the arm or a hand-held cylinder.
 Electrical impedance - Wikipedia, the free encyclopedia (1958 words) Electrical impedance or simply impedance is a measure of opposition to a 'sinusoidal' electric current. A resistor's impedance is R (its resistance) and its reactance is 0. Impedance is defined by the ratio of two phasors where a phasor is the complex peak amplitude of a sinusoidal function of time.
More results at FactBites »

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