Electrical impedance - Wikipedia, the free encyclopedia.pdf

7/30/2019 Electrical impedance - Wikipedia, the free encyclopedia.pdf

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A graphical representation of the

complex impedance plane

Electrical impedanceFrom Wikipedia, the free encyclopedia

Electrical impedance is the measure of the opposition that a circuit

presents to the passage of a current when a voltage is applied. In

quantitative terms, it is the complex ratio of the voltage to the current in

an alternating current (AC) circuit. 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.

It is necessary to introduce the concept of impedance in AC circuits

because there are other mechanisms impeding the flow of current besides

the normal resistance of DC circuits. There are an additional two

impeding mechanisms to be taken into account in AC circuits: theinduction of voltages in conductors self-induced by the magnetic fields of

currents (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.

The symbol for impedance is usually and it may be represented by writing its magnitude and phase in the form

. However, complex number representation is often more powerful for circuit analysis purposes. The term

impedance was coined by Oliver Heaviside in July 1886.[1][2] Arthur Kennelly was the first to represent

impedance with complex numbers in 1893.[3]

Impedance is defined as the frequency domain ratio of the voltage to the current.[4] In other words, it is the

voltagecurrent ratio for a single complex exponential at a particular frequency . In general, impedance will be

a complex number, with the same units as resistance, for which the SI unit is the ohm (). 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 is ahead ofthe voltage.

The reciprocal of impedance is admittance (i.e., admittance is the current-to-voltage ratio, and it conventionallycarries units of siemens, formerly called mhos).

Contents

1 Complex impedance

2 Ohm's law

3 Complex voltage and current

3.1 Validity of complex representation3.2 Phasors

4 Device examples

4.1 Deriving the device-specific impedances
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2/6/12 Electrical impedance - Wik ipedia, the f ree encyc lopedia

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4.1.1 Resistor

4.1.2 Capacitor

4.1.3 Inductor

5 Generalised s-plane impedance

6 Resistance vs reactance

6.1 Resistance

6.2 Reactance

6.2.1 Capacitive reactance6.2.2 Inductive reactance

7 Combining impedances

7.1 Series combination

7.2 Parallel combination

8 Measurement

9 Variable impedance

10 See also

11 References

12 External links

Complex impedance

Impedance is represented as a complex quantity and the term complex impedance may be used

interchangeably; the polar form conveniently captures both magnitude and phase characteristics,

where the magnitude represents the ratio of the voltage difference amplitude to the current amplitude, while

the argument gives the phase difference between voltage and current. is the imaginary unit, and is used

instead of in this context to avoid confusion with the symbol for electric current. In Cartesian form,

where the real part of impedance is the resistance and the imaginary part is the reactance .

Where it is required 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 asfinding 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.

Ohm's law

Main article: Ohm's law

The meaning of electrical impedance can be understood by substituting it into Ohm's law.[5][6]

The magnitude of the impedance acts just like resistance, giving the drop in voltage amplitude across an

impedance for a given current . The phase factor tells us that the current lags the voltage by a phase of
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An AC supply applying a

voltage , across a load ,

driving a current .

Generalized impedances in a

circuit can be drawn with thesame symbol as a resistor (US

ANSI or DIN Euro) or with a

labeled box.

(i.e., in the time domain, the current signal is shifted 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, Thevenin's theorem, and Norton's theorem can also be extended to AC

circuits by replacing resistance with impedance.

Complex voltage and current

In order to simplify calculations, sinusoidal voltage and current waves are

commonly represented as complex-valued functions of time denoted as and

.[7][8]

Impedance is defined as the ratio of these quantities.

Substituting these into Ohm's law we have

Noting that this must hold for all , we may equate the magnitudes andphases to obtain

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):

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 will be identical for the other.
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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

, while the voltage across an inductor

leads the current through it by . The

identical voltage and current amplitudes

indicate that the magnitude of the

impedance is equal to one.

At the end of any calculation, we may return to real-valued sinusoids by further noting that

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. 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 cancel.

Device examples

The impedance of an ideal resistor is purely real and is referred

to as a resistive impedance:

In this case, the voltage and current waveforms are proportional

and in phase.

Ideal inductors and capacitors have a purely imaginary reactive

impedance:

the impedance of inductors increases as frequency increases;

the impedance of capacitors decreases as frequency increases;

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:
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en.wikipedia.org/wiki/Electrical_impedance

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

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 will assume sinusoidal signals, since any arbitrary signal can

be approximated as a sum of sinusoids through Fourier analysis.

Resistor

For a resistor, there is the relation:

This is Ohm's law.

Considering the voltage signal to be

it follows that

This says that the ratio of AC voltage amplitude to alternating current (AC) amplitude across a resistor is , and

that the AC voltage leads the current across a resistor by 0 degrees.

This result is commonly expressed as

Capacitor

For a capacitor, there is the relation:

Considering the voltage signal to be
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it follows that

And thus

This says that the ratio of AC voltage amplitude to AC current amplitude across a capacitor is , and that the

AC voltage lags the AC current across a capacitor by 90 degrees (or the AC current leads the AC voltage

across a capacitor by 90 degrees).

This result is commonly expressed in polar form, as

or, by applying Euler's formula, as

Inductor

For the inductor, we have the relation:

This time, considering the current signal to be

it follows that

And thus

This says that the ratio of AC voltage amplitude to AC current amplitude across an inductor is , and that the

AC voltage leads the AC current across an inductor by 90 degrees.

This result is commonly expressed in polar form, as
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or, using Euler's formula, as

Generalised s-plane impedance

Impedance defined in terms ofj can strictly only be applied to circuits which are energised 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 ofj. Complex frequency is given the symbols 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:

Element Impedance expression

Resistor

Inductor

Capacitor

For a DC circuit this simplifies tos = 0. For a steady-state sinusoidal AC signals =j.

Resistance vs reactance

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

relations:

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 is the real part of impedance; a device with a purely resistive impedance exhibits no phase shift

between the voltage and current.

Reactance

Main article: Electrical reactance
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Reactance is the imaginary part of the impedance; a component with a finite reactance induces a phase shift

between the voltage across it and the current through it.

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 will not dissipate any power.

Capacitive reactance

Main article: Capacitance

A capacitor has a purely reactive impedance which is inversely proportional to the signal frequency. A capacitor

consists of two conductors separated by an insulator, also known as a dielectric.

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. When the potential associated with the charge exactly balances the

applied voltage, the current goes to zero.

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: Inductance

Inductive reactance is proportional to the signal frequency and the inductance .

An inductor consists of a coiled conductor. Faraday's law of electromagnetic induction gives the back emf

(voltage opposing current) due to a rate-of-change of magnetic flux density through a current loop.

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

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). Analternating current has a time-averaged rate-of-change that is proportional to frequency, this causes the increase

in inductive reactance with frequency.
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Combining impedances

Main article: Series and parallel circuits

The total impedance of many simple networks of components can be calculated using the rules for combining

impedances in series and parallel. The rules are identical to those used for combining resistances, except that the

numbers in general will be complex numbers. In the general case however, equivalent impedance transforms in

addition to series and parallel will be required.

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.

Or explicitly in real and imaginary terms:

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:

or, when n = 2:
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The equivalent impedance can be calculated in terms of the equivalent series resistance and reactance

.[9]

Measurement

The measurement of the impedance of devices and transmission lines is a practical problem in radio technology

and others. 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. Measurement ofimpedance requires 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.[10]

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.[10]

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.

Variable impedanceIn general, neither impedance nor admittance can be time varying as they are defined for complex exponentials

for < t< +. If the complex exponential voltagecurrent ratio changes over time or amplitude, the circuit

element cannot be described using the frequency domain. However, many systems (e.g., varicaps that are used

in radio tuners) may exhibit non-linear or time-varying voltagecurrent ratios that appear to be linear time-

invariant (LTI) for small signals over small observation windows; hence, they can be roughly described as having

a time-varying impedance. That is, this description is an approximation; over large signal swings or observation

windows, the voltagecurrent relationship is non-LTI and cannot be described by impedance.

See also

Impedance matching

Impedance cardiography
http://en.wikipedia.org/wiki/Impedance_cardiographyhttp://en.wikipedia.org/wiki/Impedance_matchinghttp://en.wikipedia.org/wiki/LTI_system_theoryhttp://en.wikipedia.org/wiki/Tuner_(radio)http://en.wikipedia.org/wiki/Varicaphttp://en.wikipedia.org/wiki/LCR_meterhttp://en.wikipedia.org/wiki/Electrical_impedance#cite_note-LewisJr-10http://en.wikipedia.org/wiki/Fast_Fourier_transformhttp://en.wikipedia.org/wiki/Electrical_impedance#cite_note-LewisJr-10http://en.wikipedia.org/wiki/Wheatstone_bridgehttp://en.wikipedia.org/wiki/Bridge_circuithttp://en.wikipedia.org/wiki/Reflection_coefficienthttp://en.wikipedia.org/wiki/Standing_wave_ratiohttp://en.wikipedia.org/wiki/Radio_antennahttp://en.wikipedia.org/wiki/Radiohttp://en.wikipedia.org/wiki/Electrical_impedance#cite_note-9


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Impedance bridging

Characteristic impedance

Negative impedance converter

Immittance

Resistance distance

References

1. ^Science, p. 18, 1888

2. ^ Oliver Heaviside, The Electrician, p. 212, 23 July 1886, reprinted as Electrical Papers, p 64, AMS

Bookstore, ISBN 0-8218-3465-7

3. ^ Kennelly, Arthur.Impedance (AIEE, 1893) (http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4768008)

4. ^ Alexander, Charles; Sadiku, Matthew (2006). Fundamentals of Electric Circuits (3, revised ed.). McGraw-

Hill. pp. 387389. ISBN 978-0-07-330115-0

5. ^ AC Ohm's law (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html) , Hyperphysics

6. ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 3233.

ISBN 0-521-37095-7.7. ^ Complex impedance (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html#c1) , Hyperphysics

8. ^ Horowitz, Paul; Hill, Winfield (1989). "1". The Art of Electronics. Cambridge University Press. pp. 3132.

ISBN 0-521-37095-7.

9. ^ Parallel Impedance Expressions (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html#c3) ,

Hyperphysics

10. ^ ab Lewis Jr., George; George K. Lewis Sr. and William Olbricht (August 2008). "Cost-effective broad-band

electrical impedance spectroscopy measurement circuit and signal analysis for piezo-materials and ultrasound

transducers" (http://www.iop.org/EJ/abstract/0957-0233/19/10/105102/) . Measurement Science and

Technology19 (10): 105102. Bibcode 2008MeScT..19j5102L

(http://adsabs.harvard.edu/abs/2008MeScT..19j5102L) . doi:10.1088/0957-0233/19/10/105102

(http://dx.doi.org/10.1088%2F0957-0233%2F19%2F10%2F105102) . PMC 2600501(//www.ncbi.nlm.nih.gov/pmc/articles/PMC2600501) . PMID 19081773

(//www.ncbi.nlm.nih.gov/pubmed/19081773) . http://www.iop.org/EJ/abstract/0957-0233/19/10/105102/.

Retrieved 2008-09-15.

External links

Explaining Impedance (http://hyperphysics.phy-astr.gsu.edu/hbase/electric/imped.html)

Antenna Impedance (http://www.antenna-theory.com/basics/impedance.php)

ECE 209: Review of Circuits as LTI Systems(http://www.tedpavlic.com/teaching/osu/ece209/support/circuits_sys_review.pdf) Brief explanation of

Laplace-domain circuit analysis; includes a definition of impedance.

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