Electrical impedance

Electromagnetism

Electricity · Magnetism
This box: view • talk • edit

Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal alternating current (AC). Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. An alternating current ( AC) is an Electric current whose direction reverses cyclically as opposed to Direct current, whose direction remains constant Electrical impedance extends the concept of resistance to AC circuits, describing not only the relative amplitudes of the voltage and current, but also the relative phases. Electrical resistance is a ratio of the degree to which an object opposes an Electric current through it measured in Ohms Its reciprocal quantity is Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In general impedance is a complex quantity $\scriptstyle{\tilde{Z}}$ and the term complex impedance may be used interchangeably; the polar form conveniently captures both magnitude and phase characteristics,

$\tilde{Z} = Z e^{j\theta} \quad$

where the magnitude $\scriptstyle{Z}$ gives the change in voltage amplitude for a given current amplitude, while the argument $\scriptstyle{\theta}$ gives the phase difference between voltage and current. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by In Cartesian form,

$\tilde{Z} = R + j\Chi \quad$

where the real part of impedance is the resistance $\scriptstyle{R}$ and the imaginary part is the reactance $\scriptstyle{\Chi}$ . In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z Dimensionally, impedance is the same as resistance; the SI unit is the ohm. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand The ohm (symbol Ω) is the SI unit of Electrical impedance or in the Direct current case Electrical resistance, The term impedance was coined by Oliver Heaviside in July 1886.

A graphical representation of the complex impedance plane. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Note that while reactance $\scriptstyle{\Chi}$ can be either positive or negative, resistance $\scriptstyle{R}$ is always positive.

1 Ohm's law
2 Complex voltage and current
- 2.1 Validity of complex representation
- 2.2 Phasors
3 Device examples
4 Resistance vs reactance
- 4.1 Resistance
- 4.2 Reactance
  - 4.2.1 Capacitive reactance
  - 4.2.2 Inductive reactance
5 Combining impedances
- 5.1 Series combination
- 5.2 Parallel combination
6 See also
7 External links
8 References

Ohm's law

An AC supply applying a voltage $\scriptstyle{V}$ , across a load $\scriptstyle{Z}$ , driving a current $\scriptstyle{I}$ .

Main article: Ohm's law

We can understand this by substituting it into Ohm's law. Ohm's law applies to Electrical circuits it states that the current through a conductor between two points is directly proportional to the Ohm's law applies to Electrical circuits it states that the current through a conductor between two points is directly proportional to the ^[1]^[2]

$\tilde{V} = \tilde{I}\tilde{Z} = \tilde{I} Z e^{j\theta} \quad$

The magnitude of the impedance $\scriptstyle{Z}$ acts just like resistance, giving the drop in voltage amplitude across an impedance $\scriptstyle{\tilde{Z}}$ for a given current $\scriptstyle{\tilde{I}}$ . The phase factor tells us that the current lags the voltage by a phase of $θ$ (i. e. in the time domain, the current signal is shifted $\frac{\theta T}{2 \pi}$ to the right with respect to the voltage signal). ^[3]

Just as impedance extends Ohm's law to cover AC circuits, other results from DC circuit analysis such as voltage division, current division, Thevenin's theorem, and Norton's theorem, can also be extended to AC circuits by replacing resistance with impedance. In Electronics, a voltage divider (also known as a potential divider) is a simple Linear circuit that produces an output Voltage ( In Electronics, a current divider is a simple Linear Circuit that produces an output Current ( I X that is a fraction In electrical circuit theory, Thévenin's theorem for linear Electrical networks states that any combination of Voltage sources Current sources Norton's theorem for Electrical networks states that any collection of Voltage sources Current sources and Resistors with two terminals is

Complex voltage and current

Generalized impedances in a circuit can be drawn with the same symbol as a resistor (US ANSI or DIN Euro) or with a labeled box.

In order to simplify calculations, sinusoidal voltage and current waves are commonly represented as complex-valued functions of time denoted as $\scriptstyle{\tilde{V}}$ and $\scriptstyle{\tilde{I}}$ . ^[4]^[5]

$\ \tilde{V} = V_0e^{j(\omega t + \phi_V)}$

$\ \tilde{I} = I_0e^{j(\omega t + \phi_I)}$

Impedance is defined as the ratio of these quantities.

$\ \tilde{Z} = {\tilde{V} \over \tilde{I}}$

Substituting these into Ohm's law we have

$\begin{align} V_0e^{j(\omega t + \phi_V)} &= I_0e^{j(\omega t + \phi_I)} Z e^{j\theta} \\ &= I_0 Z e^{j(\omega t + \phi_I + \theta)} \end{align}$

Noting that this must hold for all $t$ , we may equate the magnitudes and phases to obtain

$\ V_0 = I_0 Z \quad$

$\ \phi_V = \phi_I + \theta \quad$

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):

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

i. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic e. a real-valued sinusoidal function (which may represent our voltage or current waveform) 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. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems Given the symmetry, we only need to perform the analysis for one right-hand term; the results will be identical for the other. At the end of any calculation, we may return to real-valued sinusoids by further noting that

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

In other words, we simply take the real part of the result. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z

Phasors

Main article: Phasor (electronics)

A phasor is a constant complex number, usually expressed in exponential form, representing the complex amplitude (magnitude and phase) of a sinusoidal function of time. In Physics and Engineering, a phase vector ("phasor" is a representation of a Sine wave whose amplitude ( A) phase ( θ) 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 $\scriptstyle{e^{j\omega t}}$ cancel. Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal Alternating current (AC

Device examples

Main article: Impedance of different devices (derivations)

The phase angles in the equations for the impedance of inductors and capacitors indicate that the voltage across a capacitor lags the current through it by a phase of $\scriptstyle{\pi/2}$ , while the voltage across an inductor leads the current through it by $\scriptstyle{\pi/2}$ . The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one.

The impedance of a resistor is purely real and is referred to as a resistive impedance. |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| |

$\tilde{Z}_R = R \quad$

Inductors and capacitors have a purely imaginary reactive impedance. An inductor is a passive electrical component designed to provide Inductance in a circuit A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors

$\tilde{Z}_L = j\omega L \quad$

$\tilde{Z}_C = {1 \over j\omega C}$

Note the following identities for the imaginary unit and its reciprocal. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

$j = \cos{\left({\pi \over 2}\right)} + j\sin{\left({\pi \over 2}\right)} = e^{j{\pi \over 2}}$

${1 \over j} = -j = \cos{\left(-{\pi \over 2}\right)} + j\sin{\left(-{\pi \over 2}\right)} = e^{j(-{\pi \over 2})}$

Thus we can rewrite the inductor and capacitor impedance equations in polar form

$\tilde{Z}_L = \omega Le^{j{\pi \over 2}}$

$\tilde{Z}_C = {1 \over \omega C}e^{j(-{\pi \over 2})}.$

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

Resistance vs reactance

It is important to realize that resistance and reactance are not individually significant; together they determine the magnitude and phase of the impedance, through the following relations:

$|\tilde{Z}| = \sqrt{\tilde{Z}\tilde{Z}^*} = \sqrt{R^2 + \Chi^2}$

$\theta = \arctan{\left({\Chi \over R}\right)}$

In many applications the relative phase of the voltage and current is not critical so only the magnitude of the impedance is significant.

Resistance

Main article: Electrical resistance

Resistance $\scriptstyle{R}$ is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift between the voltage and current. Electrical resistance is a ratio of the degree to which an object opposes an Electric current through it measured in Ohms Its reciprocal quantity is

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

Reactance

Main article: Reactance

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

$\Chi = Z \sin{\theta} \quad$

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. A pure reactance will not dissipate any power.

Capacitive reactance

Main article: Capacitor

A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors This article is about proportionality the mathematical relation Frequency is a measure of the number of occurrences of a repeating event per unit Time. A capacitor consists of two conductors separated by an insulator, also known as a dielectric. Electrical conduction is the movement of electrically charged particles through a Transmission medium ( Electrical conductor) An insulator, also called a Dielectric, is a material that resists the flow of Electric current. A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday.

At low frequencies a capacitor is open circuit, as no charge flows in the dielectric. 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. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can When the potential associated with the charge exactly balances the applied voltage, the current goes to zero. The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical

Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes sign and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

Inductive reactance

Main article: Inductor

An inductor has a purely reactive impedance which is proportional to the signal frequency. An inductor is a passive electrical component designed to provide Inductance in a circuit An inductor is a passive electrical component designed to provide Inductance in a circuit Frequency is a measure of the number of occurrences of a repeating event per unit Time. An inductor consists of a coiled conductor. A coil is a series of loops A coiled coil is a structure where the coil itself is in turn also looping Faraday's law of electromagnetic induction gives the back emf $\scriptstyle{\mathcal{E}}$ (voltage opposing current) due to a rate-of-change of magnetic field $\scriptstyle{B}$ through a current loop. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges

$\mathcal{E} = -{{d\Phi_B} \over dt}.$

For an inductor consisting of a coil with $N$ loops this gives.

$\mathcal{E} = -N{d\Phi_B \over dt}.$

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). Direct current ( DC) is the unidirectional flow of Electric charge. Short Circuit is a 1986 comedy Science fiction film starring Ally Sheedy and Steve Guttenberg and directed by Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of Electric current. An alternating current has a time rate-of-change that is proportional to frequency and so the inductive reactance is proportional to frequency.

Combining impedances

Main article: Series and parallel circuits

The total impedance of any network of components can be calculated using the rules for combining impedances in series and parallel. If two or more circuit components are connected end to end like a daisy chain it is said they are connected in series. The rules are identical to those used for combining resistances, although they require some familiarity with complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Series combination

For components connected in series, the current through each circuit element is the same; the ratio of voltages across any two elements is the inverse ratio of their impedances.

$\tilde{Z}_{eq} = \tilde{Z}_1 + \tilde{Z}_2 = (R_1 + R_2) + j(\Chi_1 + \Chi_2) \quad$

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.

$\tilde{Z}_{eq} = \tilde{Z}_1 \| \tilde{Z}_2 = \left(\tilde{Z}_1^{-1} + \tilde{Z}_2^{-1}\right)^{-1} = {\tilde{Z}_1 \tilde{Z}_2 \over \tilde{Z}_1 + \tilde{Z}_2} \quad$

The equivalent impedance $\scriptstyle{\tilde{Z}_{eq}}$ can be calculated in terms of the equivalent resistance $\scriptstyle{R_{eq}}$ and reactance $\scriptstyle{\Chi_{eq}}$ . ^[6]

$\tilde{Z}_{eq} = R_{eq} + j \Chi_{eq} \quad$

$R_{eq} = { (\Chi_1 R_2 + \Chi_2 R_1) (\Chi_1 + \Chi_2) + (R_1 R_2 - \Chi_1 \Chi_2) (R_1 + R_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$

$\Chi_{eq} = {(\Chi_1 R_2 + \Chi_2 R_1) (R_1 + R_2) - (R_1 R_2 - \Chi_1 \Chi_2) (\Chi_1 + \Chi_2) \over (R_1 + R_2)^2 + (\Chi_1 + \Chi_2)^2}$

External links

Explaining Impedance

References

^ AC Ohm's law, Hyperphysics
^ Horowitz, Paul; Hill, Winfield (1989). In Electrical engineering, the admittance ( Y) is the inverse of the impedance ( Z) Impedance matching is the electronics design practice of setting the Output impedance ( Z S of a signal source equal to the Input impedance ( In electronics especially audio and Sound recording, a high impedance bridging, voltage bridging, or simply bridging Connection is The characteristic impedance or surge impedance of a uniform Transmission line, usually written Z_0 is the ratio of the amplitudes of a single If an electric circuit has a well-defined output terminal the circuit connected to this terminal (or its Input impedance) is the load. "1", The Art of Electronics. Cambridge University Press, 32-33. ISBN 0-521-37095-7.
^ Capacitor/inductor phase relationships, Yokogawa
^ Complex impedance, Hyperphysics
^ Horowitz, Paul; Hill, Winfield (1989). "1", The Art of Electronics. Cambridge University Press, 31-32. ISBN 0-521-37095-7.
^ Parallel Impedance Expressions, Hyperphysics

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents