Impedance
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.
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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,
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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.

In electrical engineering the potential difference is the
voltage.
 
In fluid systems the potential difference is the pressure.
 
in thermal systems the potential difference is the
temperature

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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.
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In the analysis of an alternatingcurrent 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.
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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.
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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
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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.
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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.
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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
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and is a real
number In mathematics, the real numbers
are intuitively defined as numbers that are in onetoone 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).
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to the component's resistance
Electrical resistance is the ratio of the potential difference
(i.e. voltage) across an electric component (such as a resistor) to the current
passing through it:
$R=V/I$
where R is the resistance, V the voltage and I
the current.
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
Alternatingcurrent
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then the component may affect both the phase

The phase of a waveform is the position of any peak or
trough compared to the same feature on a second waveform.
 
A phase of matter is a physically distinctive form of a
substance, such as the solid, liquid, and gaseous phases of ordinary matter.
Also sometimes included in this list are more exotic phases such as
superfluids.
 
Layers of immiscible liquids are called "phases."
 
A lunar phase is the appearance of the Moon as viewed from
the Earth. Similarly with planetary phases.
 

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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.
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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 nonconstant polynomials have roots.
The complex numbers contain a number i, the imaginary unit,
with i^{2}= −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.
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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.
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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.
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in a single number). It is composed of the resistance R, the inductive
reactance
In the analysis of an alternatingcurrent 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.
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and the capacitive reactance X_{C} according to the
formula
$Z=R+j(X\_LX\_C)$
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 x^{2} + 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.)
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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.
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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.
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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.
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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.
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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 R^{n} is
still an area of ongoing research,
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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.
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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:
$\backslash \backslash leftZ\backslash ight=\backslash \backslash sqrt\{R^2+X^2\}.$
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:
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(V_{RMS}) to RMS current (I_{RMS}):
$\backslash \backslash leftZ\backslash ight=V\_\{RMS\}/I\_\{RMS\}.$
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:

An abstract element of information, or more exactly usually
a flow of information (in either one or several dimensions). See Signal
(information theory)
 
In computing, an asynchronous event transmitted between one
process and another (in Linux, UNIX and other POSIXcompliant operating
systems, and also in several realtime operating system).
 
A means of controlling road vehicles, pedestrians or
trains. See Traffic signal, Pedestrian crossing or Railway signal.
 
In a partnership card game, a player's choice of card to
play at a particular time, which gives information to her partner. See
Signal (contract bridge).

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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,
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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 75ohm feeds
instead.
Further reading:
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 airspeed 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 twoterminal device which transports electrical charge between its
terminals at a timerate 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 twoterminal 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 realvalued 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 timedependent functions v(t) and i(t) obey the respective relationships,
v(t) = L di/dt, for the inductor and

(1)

i(t) = C dv/dt, for the capacitor,

(2)

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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