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Encyclopedia > Impedance

In electrical engineering, Impedance is a measure of opposition to a sinusoidal electric current. The concept of impedance generalizes Ohm's law in AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit is a complex number. Oliver Heaviside coined the term impedance in July of 1886. Electrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... Ohms law, named after its discoverer Georg Ohm , states that the potential difference or Voltage drop V between the ends of a conductor and the current, I flowing through the conductor are proportional at a given temperature: where V is the voltage and I is the current; the... 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, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... 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...

In general, the solutions for the voltages and currents in a circuit containing resistors, capacitors and inductors (in short, all linear 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. In this case, all of the voltages and currents in the circuit are sinusoidal and constant in peak amplitude, frequency and phase.

Let v(t) be a sinusoidal function of time with constant peak amplitude Vp, constant frequency f, and constant phase φ.

To simplify notation, usually one calculates with angular speed (in radians per second) instead of frequency: Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... See Radian (band) for the Austrian trio. ... The second (symbol s) is a unit for time, and one of seven SI base units. ...

ω = 2πf

The function v(t) can then be written as:

where j represents the imaginary unit () and means the real part (i.e. resistance) of Z. In mathematics, the imaginary unit i (sometimes also represented by j, but in this article i will be used exclusively) 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. ...

Now, let the complex number V be given by:

V is called the phasor representation of v(t). V is a constant complex number. 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. In physics a Phasor describes the phase of a particle in a simple harmonic motion or a wave motion. ... A resistive circuit is a circuit containing only resistors, ideal current sources, and ideal voltage sources. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

Definition of 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:

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:

where the 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. The theorem can also be applied to general impedances, not just resistors. ...

Impedance of a resistor

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 with the same frequency and phase. Thus, we have that the impedance of a resistor is equal to R:

Impedance of a capacitor

For a capacitor, we have the relation:

Now, Let

It follows that

Using phasor notation and the result above, write our first equation as:

It follows that the impedance of a capacitor is:

Impedance of an inductor

For the inductor, we have:

By the same reasoning used in the capacitor example above, it follows that the impedance on an inductor is:

Reactance

It is important to note that the impedance of a capacitor or an inductor is a function of the frequency f 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 existance 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:

Req is termed the resistive part of the impedance while 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.

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. ... The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. ...

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 — 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. The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i. ...

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:

The polar form of a complex number is the product of a real number called the magnitude and another complex number called the phase:

Where the magnitude is given by:

and the angle is given by:

Alternately, the magnitude is given by:

Where Z* denotes the complex conjugate of Z: .

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 mean: root mean square, a concept in statistics and electronics Richard M. Stallman, a computer programmer and founder of the GNU project. ...

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

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. Quadrupole (four-pole) magnet, focus particle beams in a particle accelerator. ... Signaling, or signal, may mean: Look up signal in Wiktionary, the free dictionary. ... Impedance mismatch has two meanings. ... In telecommunication, a time-domain reflectometer (TDR) is an electronic instrument used to characterize and locate faults in metallic cables ( twisted pair, coax). ...

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. Conductance can refer to: Electrical conductance, the reciprocal of electrical resistance. ... In electrical engineering, the admittance (Y) is the inverse or reciprocal of the impedance (Z). ... In electrical engineering, the susceptance (B) is the imaginary part of the admittance. ...

Acoustic impedance & data-transfer 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. ...

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. ... Results from FactBites:

 Impedance (189 words) For a pure resistor, Z = R. Because the phase affects the impedance and because the contributions of capacitors and inductors differ in phase from resistive components by 90 degrees, a process like vector addition (phasors) is used to develop expressions for impedance. Combining impedances has similarities to the combining of resistors, but the phase relationships make it practically necessary to use the complex impedance method for carrying out the operations. Impedances may be combined using the complex impedance method.
 Impedance matching - Wikipedia, the free encyclopedia (2322 words) 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. In general, impedance is a complex number, which means that loads generally have a resistance to the source that is in phase with the source signal and a reactance to the source that is in quadrature to the phase of the source. Impedance bridging is unsuitable for RF connections because it causes power to be reflected back to the source from the boundary between the high impedance and the low impedance.
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