Impedance

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In electrical engineering Electrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. Its practitioners are called electrical engineers. Electrical engineering is a broad field that encompasses many subfields.

Subfields

Electronics

In the subfield of electronics, electrical engineers design and test electrical networks (more commonly known as circuits) that take advantage of electromagnetic properties of electrical components or elements (such as resistors, capacitors, inductors, transistors, diodes, semiconductors) to achieve the desired functionality.
..... Click the link for more information. , impedance is a measure for the manner and degree a component resists the flow of electrical current

In electricity, current is any flow of charge, usually through a metal wire or some other electrical conductor. Conventional current was defined early in the history of electrical science as a flow of positive charge, although we now know that, in the case of metallic conduction, current is caused by a flow of negatively charged electrons in the opposite direction. Despite this understanding, the original definition of conventional current still stands. The symbol typically used for the amount of current (the amount of charge flowing per unit of time) is I. Historically, the symbol for current,
..... Click the link for more information.  if a given voltage

In the physical sciences, potential difference is the difference in potential between two points in a conservative vector field. It can be described as the across variable, where flux is the through variable. The product of the flux and the potential difference is the power, which is the rate of change of the conserved quantity, e.g., energy.

..... Click the link for more information.  is applied. It is denoted by the symbol Z and is measured in ohms

The ohm is the SI unit of electrical resistance. Its symbol is the Greek capital letter omega (Ω). The ohm is named for Georg Ohm, a German physicist who discovered the relation between voltage and current, expressed in Ohm's Law.

By definition in Ohm's Law, 1 ohm equals 1 volt divided by 1 ampere. In other words, a device has a resistance of 1 ohm if a voltage of 1 volt will cause a current of 1 ampere to flow.
..... Click the link for more information. .

See also reactance

In the analysis of an alternating-current electrical circuit (for example a RLC series circuit), reactance is the imaginary part of impedance, and is caused by the presence of inductors or capacitors in the circuit. Reactance is denoted by the symbol X and is measured in ohms. If X > 0 the reactance is said to be inductive, and if X < 0 it is said to be capacitive. If X = 0, then the circuit is purely resistive, i.e. it has no reactance.
..... Click the link for more information. , inductance Inductance is a physical characteristic of an inductor, which produces a voltage proportional to the instantaneous change in current flowing through it.

The inductance of a solenoid (an idealization of a coil) is defined as:

   $L\; =$

μ is the permeability of the core, N is the number of turns, A is the cross sectional area of the coil, and l is the length.
..... Click the link for more information. . For a practical layman's introduction, see nominal impedance In electrical engineering, the nominal impendance of an input or ouput is the impedance of the output or input (respectively) that it is designed to accept.

This article is intended to be a layman's introduction, and focuses on audio frequencies at which cable impedance is not significant. See impedance for a more technical discussion. See also impedance matching, cable impedance.

Most equipment is designed to operate with the internal impedance of a signal source roughly equal to the impedance of the input to which it is connected. This provides the most efficient coupling, and is best in most but not all situations. The
..... Click the link for more information. .

If the applied voltage is constant, capacitors

A capacitor (historically known as a "condenser") is a device that stores energy in an electric field, by accumulating an internal imbalance of electric charge.

Physics of the capacitor

Overview

Typical designs consist of two electrodes or plates, each of which stores an opposite charge. These two plates are conductive and are separated by an insulator or dielectric. The charge is stored at the surface of the plates, at the boundary with the dielectric. Because each plate stores an equal but opposite charge, the total charge in the device is always zero.
..... Click the link for more information.  act like insulators and inductors 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.

Physics of the inductor

Construction

An inductor is usually constructed as a coil of conducting material, typically copper wire. A core of ferrous material is sometimes used, which increases the inductance. Inductors can also be built on integrated circuits using the same processes that are used to make computer chips. In these cases, aluminum is typically used as the conducting material. However, it is rare that actual inductors are built on ICs; practical constraints make it far more common to use a circuit called a "gyrator" which uses a capacitor to behave as if it were an inductor.
..... Click the link for more information.  act like conductors; the impedance is then due to resistors A resistor is an electrical component designed to have an electrical resistance that is independent of the current flowing through it. The common type of resistor is also designed to be independent of temperature and other factors. Resistors may be fixed or variable. Variable resistors are also called potentiometers or rheostats (see below).

Some resistors are long and thin, with the actual resisting material in the centre, and a conducting metal leg on each end. This is called an
..... Click the link for more information.  alone and is a real number In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number".

Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero.

Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247... (where the three dots express that there would still be more digits to come, no matter how many more might be added at the end).
..... Click the link for more information.  equal to the component's resistance

Resistance is thus a measure of the component's opposition to the flow of electric charge. The SI unit of electrical resistance is the ohm. Its reciprocal quantity is electrical conductance measured in siemens.
..... Click the link for more information. R.

If the applied voltage is changing over time (as in an AC An alternating current (AC) is an electrical current, where electrical charge oscillates (i.e., moves back and forth), rather than flowing continuously in one direction as is the case with direct current. The desired waveform of the oscillation is generally that of a perfect sine wave, as this results in the most efficient transmission of energy.

History

Alternating-current
..... Click the link for more information.  circuit), then the component may affect both the phase

..... Click the link for more information.  and the amplitude Amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation. In the following diagram,

the distance x is the amplitude of the wave. Sometimes that distance is called the "peak amplitude", distinguishing it from another concept of amplitude, used especially in electrical engineering: the root mean square amplitude, defined as the square root of the mean of the square of the maximum vertical distance of this graph from the horizontal axis.
..... Click the link for more information.  of the current, due to inductors and capacitors inside the component. In this case, the impedance is a complex number The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number i, the imaginary unit, with i²= −1, i.e., i is a square root of −1. Every complex number can be represented in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number respectively.
..... Click the link for more information.  (this is a mathematically convenient way of describing the amplitude Amplitude is a nonnegative scalar measure of a wave's magnitude of oscillation. In the following diagram,

the distance x is the amplitude of the wave. Sometimes that distance is called the "peak amplitude", distinguishing it from another concept of amplitude, used especially in electrical engineering: the root mean square amplitude, defined as the square root of the mean of the square of the maximum vertical distance of this graph from the horizontal axis.
..... Click the link for more information.  ratio and the phase difference The phase difference between two signals of the same frequency can be thought of as as delay or advance in the zero crossing of one signal with respect to another. Consider a graph of a sinusoidal waveform with amplitude on the y or vertical axis and time on the horizontal or x axis. If signals A and B begin at zero, build to a high positive value, fall through zero, build to a high negative value and return to zero at exactly the same time, the signals are of the same frequency and are said to be in phase, i.e. there is no phase difference between them.
..... Click the link for more information.  together in a single number). It is composed of the resistance R, the inductive reactance

In the analysis of an alternating-current electrical circuit (for example a RLC series circuit), reactance is the imaginary part of impedance, and is caused by the presence of inductors or capacitors in the circuit. Reactance is denoted by the symbol X and is measured in ohms. If X > 0 the reactance is said to be inductive, and if X < 0 it is said to be capacitive. If X = 0, then the circuit is purely resistive, i.e. it has no reactance.
..... Click the link for more information. X_L and the capacitive reactance X_C according to the formula

   $Z=R+j(X\_L-X\_C)$

where j is the imaginary unit In mathematics, the imaginary unit i allows the real number system R to be extended to the complex number system C. Its precise definition is dependent upon the particular method of extension.

The primary motivation for this extension is the fact that not every polynomial equation f(x) = 0 has a solution in the real numbers. In particular, the equation x² + 1 = 0 has no real solution. However, if we allow complex numbers as solutions, then this equation, and indeed every polynomial equation f(x) = 0 does have a solution. (See algebraic closure and fundamental theorem of algebra.)
..... Click the link for more information. , the square root of -1. Inductive reactance and capacitive reactance can be lumped together in a single quantity called reactance, X = X_L - X_C, so that we have

   $Z=R+jX$ .

Note that the reactance depends on the frequency Frequency is a measurement of the number of cycles repeated per event in a given time. To compute the frequency, one fixes a time interval, counts the number of occurrences of the event, and divides this count by the length of the time interval. The result is presented in units of hertz (Hz) after German physicist Heinrich Rudolf Hertz, where 1 Hz is an event that occurs once per second. Alternatively, one can measure the time between two occurrences of the event (the period) and then compute the frequency as the reciprocal of this time,    $f\; =\; \backslash \backslash frac$ , where T is the period.
..... Click the link for more information.  f of the applied voltage: the higher the frequency, the lower the capacitive reactance X_C and the higher the inductive reactance X_L.

If the applied voltage is periodically changing with a fixed frequency f, according to a sine curve In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle, or, more generally still, as infinite series, or equally generally, as solutions of certain differential equations. All four approaches will be presented below.
..... Click the link for more information. , it is represented as the real part of a function of the form    $u(t)=ue^\{2\backslash \backslash pi\; jft\}$ where u is a complex number that encodes the phase and amplitude (see Euler's formula

Two unrelated results in mathematics are known as Euler's formula, after the mathematician Leonhard Euler.

Algebraic topology

In geometry and algebraic topology, there is a relationship called Euler's formula which relates the number of edges E, vertices V, and faces F of a simply connected polyhedron. Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: F - E + V = 2.
..... Click the link for more information. ). If the current is represented in an analogous manner as the real value of a function i(t), then the relation between current and voltage is given by    $Z=u(t)/i(t),$ an equation quite similar to Ohm's law Ohm's law (named after its discoverer Georg Ohm [1]) states that the voltage drop $V$ across a resistor is proportional to the current $I$ running through it:    $V\; =\; I\; \backslash \backslash cdot\; R$

where the proportionality constant $R$ is the electrical resistance of the device.

The law is strictly true only for resistors whose resistance does not depend on the applied voltage, which are called ohmic or ideal resistors. Fortunately, the conditions where Ohm's law holds are very common.
..... Click the link for more information. .

If the voltage is not a sine curve of fixed frequency, then one first has to perform Fourier analysis Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called "harmonics", hence the name "harmonic analysis."

The classical Fourier transform on Rⁿ is still an area of ongoing research,
..... Click the link for more information.  to find the signal components at the various frequencies. Each one is then represented as the real part of a complex function as above and divided by the impedance at the respective frequency. Adding the resulting current components yields a function i(t) whose real part is the current.

The notion of impedance can be useful even when the voltage/current is normally constant (as in many DC Direct current (DC) is the continuous flow of electricity through a conductor such as a wire from high to low potential. In direct current, the electric charges flow always in the same direction, which distinguishes it from alternating current (AC).

Direct current was used originally for electric power transmission after the discovery by Thomas Edison of the generation of electricity in the late nineteenth century. It has mostly been abandoned for this purpose in favor of alternating current (discovered and promoted by Nikola Tesla, see War of Currents), which is much more suited to transmission over long distances. DC power transmission is still used to link AC power networks with different frequencies.
..... Click the link for more information.  circuits), in order to study what happens at the instant when the constant voltage is switched on or off: generally, inductors cause the change in current to be gradual, while capacitors can cause large peaks in current.

If the internal structure of a component is known, its impedance can be computed using the same laws that are used for resistances: the total impedance of subcomponents connected in series is the sum of the subcomponents' impedances; the reciprocal of the total impedance of subcomponents connected in parallel is the sum of the reciprocals of the subcomponents' impedances. These simple rules are the main reason for using the formalism of complex numbers.

Often it is enough to know only the magnitude of the impedance:    $\backslash \backslash left|Z\backslash ight|=\backslash \backslash sqrt\{R^2+X^2\}.$ It is equal to the ratio of RMS

In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values.

The rms for a collection of N values is:
..... Click the link for more information.  voltage (V_RMS) to RMS current (I_RMS):    $\backslash \backslash left|Z\backslash ight|=V\_\{RMS\}/I\_\{RMS\}.$

The word "impedance" is often used for this magnitude; it is however important to realize that in order to compute this magnitude, one first computes the complex impedance as explained above and then takes the magnitude of the result. There are no simple rules that allow one to compute |Z| directly.

When fitting components together to carry electromagnetic

Electromagnetism is the physics of the electromagnetic field, including its effect on electrically charged particles.

While the electric and magnetic forces may sound fairly esoteric, almost all of the phenomena one encounters in daily life (with the exception of gravity) actually result from electromagnetism. The forces between atoms, including the attractive forces between atoms in
..... Click the link for more information.  signals

A signal may be:

..... Click the link for more information. , it is important to match impedance, which can be achieved with various matching devices. Failing to do so is known as impedance mismatch Impedance mismatch has two meanings.

It is a problem in electrical engineering that occurs when two transmission lines or circuits with different impedances are connected. This can cause signal reflection resulting in attenuation and noise. See also impedance matching.

In programming terminology it refers to the attempt to connect two systems that have very different conceptual bases,
..... Click the link for more information.  and results in signal loss.

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.

Further reading:

SOURCE

What is an impedance?
What is a mono directional impedance, and why is it so much easier to simulate?
How can one simulate a mono directional impedance?
Some sample simulators

Impedance is a very useful concept in the subject of power delivery. In general it provides information about the load being driven by the power source. For the output torque of an automobile transmission, the impedance is the output torque divided by the angular velocity that such torque will sustain. For a jet engine, the impedance is the thrust (force) divided by the air-speed that such thrust will sustain, and for a fluid pump, the impedance is the pressure it delivers divided by the volume flow rate that such pressure sustains. In general, an impedance is the ratio of a force or other physical imposition capable of power delivery, to the reaction that such imposition can sustain, where the reaction is defined such that the product of the imposition and sustained reaction has the units of energy per unit time, or power.

For most mechanical systems, a device's impedance varies with the conditions of the situation (such as what slope the automobile is climbing, or the viscosity of the fluid being pumped by the pump), but an electrical impedance will either be a constant value or it will depend on the frequency component of the driving signal. As illustrated in Figure 1, below, an electrical impedance Z is a two-terminal device which transports electrical charge between its terminals at a time-rate I, measured in Coulombs per second (Amperes), such that I is proportional to the voltage V (electrical pressure) applied across the two terminals. Each circle represents a two-terminal charge pump known as a voltage source, which can sustain the electrical pressure difference given by its indicated voltage V, E1 or E2 as indicated.
The value of the impedance is Z, and as shown above it represents the constant of proportionality in the relationship between the voltage V and the current I. This relationship is known as Ohm's Law, which states:

V = ZI,

where V is the difference in the electrical pressures applied across the two terminals, and Z is measured in Ohms (Volts per Ampere). In Fig 1(a), the pressure difference V is applied directly across the terminals of the impedance device Z, but at (b), each pressure E1 and E2 is generated with respect to an ambient (ground) pressure. Thus, E1 and E2 are referred to as the electric "potentials" of the terminals connected to the impedance. This is the more typical means of signal measurement used in electronic circuits. Thus, the electrical pressure difference V applied across the two terminals is usually measured as the potential difference E1 - E2. For any given potential difference (voltage) across the two terminals, as the impedance Z increases, the current I decreases proportionately. Likewise, for any given impedance Z, if the voltage is increased, the current must increase proportionately.

In general, the values of E, V and I are expressed as complex, phasor values, having a common sinusoidal frequency throughout the equation. As such, any real-valued voltage applied across the impedance can be accurately represented as a superposition of sinusoidal components, as implied by the Fourier Integral Theorem.

The use of impedance theory (aka classical network theory) has concentrated its interests in three natural and theoretically fundamental types of impedance. The simplest of these forms is the resistance, R, whose current at any given time is proportional to the applied voltage at that time. The other two impedances are known as the capacitance, C, and the inductance, L. For these, the time-dependent functions v(t) and i(t) obey the respective relationships,

where the conventional dx/dt notation denotes the time rate of change in the arbitrary variable x. Because the inductor cannot change its current rapidly in the absence of a large voltage, and because the capacitor cannot change its voltage rapidly in the absence of a large current, these devices have some very useful capabilities in frequency discrimination circuits. Fundamental theory of Laplace Transforms readily shows that the capacitor's impedance has the magnitude of 1/wC, where w is the angular frequency component under consideration; and the inductor's impedance has the magnitude wL. It also follows that for any given signal frequency component, the inductor's current lags its voltage by 90 degrees in phase, whereas in contrast the capacitor's current leads it's own voltage by 90 degrees in phase. As such, for a series wiring of a capacitor and an inductor, where the current i(t) is the same in both, their voltage components for any given frequency are of opposite sign, so they tend to cancel each other out as seen by the external circuitry. So their series impedance is consequently smaller than either of their individual impedances. Because their voltages subtract in accordance with Kirchhoff's Loop Law, the impedance of the series combination is wL - 1/wC. Likewise, when the two are wired in parallel, so that they have the same voltage, their currents are of opposite sign and thereby partially cancel each other out. As such, the impedance of their parallel combination as seen externally is larger than the impedance of either one component. These neutralization characteristics are, for both these wirings, most profound at the angular frequency given by the reciprocal of the square root of the product LC, for this is the frequency at which their impedances are equal and opposite.

We have long been able to manufacture capacitors that approximate the equation (2), for the most part with suitable precision; but the manufacture of inductors that closely approximate the description in equation (1) has been plagued by numerous barriers, especially at low frequencies. As such it has been found advantageous for many applications to simulate inductors electronically. Simulation of impedances is also helpful for such purposes as realizing capacitors and/or inductors whose values are continuously variable.

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