In
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, impedance is the opposition to
alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
presented by the combined effect of
resistance and
reactance in a
circuit.
Quantitatively, the impedance of a two-terminal
circuit element
Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electri ...
is the ratio of the
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
representation of the
sinusoidal
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often i ...
voltage between its terminals, to the complex representation of the current flowing through it. In general, it depends upon the
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
of the sinusoidal voltage.
Impedance extends the concept of
resistance to alternating current (AC) circuits, and possesses both magnitude and
phase, unlike resistance, which has only magnitude.
Impedance can be represented as a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
, with the same units as resistance, for which the
SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
is the
ohm
Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm.
Ohm or OHM may also refer to:
People
* Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm''
* Germán Ohm (born 1936), Mexican boxer
* Jörg Ohm (bor ...
().
Its symbol is usually , and it may be represented by writing its magnitude and phase in the
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
*Polar climate, the cli ...
form . However,
Cartesian complex number representation is often more powerful for circuit analysis purposes.
The notion of impedance is useful for performing AC analysis of
electrical networks, because it allows relating sinusoidal voltages and currents by a simple linear law.
In multiple
port
A port is a maritime facility comprising one or more wharves or loading areas, where ships load and discharge cargo and passengers. Although usually situated on a sea coast or estuary, ports can also be found far inland, such as H ...
networks, the two-terminal definition of impedance is inadequate, but the complex voltages at the ports and the currents flowing through them are still
linearly related by the
impedance matrix Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electri ...
.
The
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of impedance is
admittance
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittanc ...
, whose
SI unit is the
siemens
Siemens AG ( ) is a German multinational conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad.
The principal divisions of the corporation are ''Industry'', ''E ...
, formerly called
mho
The siemens (symbol: S) is the unit of electric conductance, electric susceptance, and electric admittance in the International System of Units (SI). Conductance, susceptance, and admittance are the reciprocals of resistance, reactance, and ...
.
Instruments used to measure the electrical impedance are called
impedance analyzers.
History
Perhaps the earliest use of complex numbers in circuit analysis was by Johann Victor Wietlisbach in 1879 in analysing the
Maxwell bridge
A Maxwell bridge is a modification to a Wheatstone bridge used to measure an unknown inductance (usually of low Q value) in terms of calibrated resistance and inductance or resistance and capacitance. When the calibrated components are a paral ...
. Wietlisbach avoided using differential equations by expressing AC currents and voltages as
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
s with
imaginary exponents (see ). Wietlisbach found the required voltage was given by multiplying the current by a complex number (impedance), although he did not identify this as a general parameter in its own right.
The term ''impedance'' was coined by
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently develope ...
in July 1886. Heaviside recognised that the "resistance operator" (impedance) in his
operational calculus was a complex number. In 1887 he showed that there was an AC equivalent to
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...
.
Arthur Kennelly
Arthur Edwin Kennelly (December 17, 1861 – June 18, 1939) was an American electrical engineer.
Biography
Kennelly was born December 17, 1861, in Colaba, in Bombay Presidency, British India, and was educated at University College School in Lond ...
published an influential paper on impedance in 1893. Kennelly arrived at a complex number representaion in a rather more direct way than using imaginary exponential functions. Kennelly followed the graphical representation of impedance (showing resistance, reactance, and impedance as the lengths of the sides of a right angle triangle) developed by
John Ambrose Fleming
Sir John Ambrose Fleming FRS (29 November 1849 – 18 April 1945) was an English electrical engineer and physicist who invented the first thermionic valve or vacuum tube, designed the radio transmitter with which the first transatlantic r ...
in 1889. Impedances could thus be added
vectorially. Kennelly realised that this graphical representation of impedance was directly analogous to graphical representation of complex numbers (
Argand diagram). Problems in impedance calculation could thus be approached algebraically with a complex number representation. Later that same year, Kennelly's work was generalised to all AC circuits by
Charles Proteus Steinmetz
Charles Proteus Steinmetz (born Karl August Rudolph Steinmetz, April 9, 1865 – October 26, 1923) was a German-born American mathematician and electrical engineer and professor at Union College. He fostered the development of alternati ...
. Steinmetz not only represented impedances by complex numbers but also voltages and currents. Unlike Kennelly, Steinmetz was thus able to express AC equivalents of DC laws such as Ohm's and Kirchhoff's laws. Steinmetz's work was highly influential in spreading the technique amongst engineers.
Introduction
In addition to resistance as seen in DC circuits, impedance in AC circuits includes the effects of the induction of voltages in conductors by the
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s (
inductance
Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of th ...
), and the electrostatic storage of charge induced by voltages between conductors (
capacitance
Capacitance is the capability of a material object or device to store electric charge. It is measured by the change in charge in response to a difference in electric potential, expressed as the ratio of those quantities. Commonly recognized are ...
). 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
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
part.
Complex impedance
![Complex Impedance](https://upload.wikimedia.org/wikipedia/commons/c/c4/Complex_Impedance.svg)
The impedance of a two-terminal circuit element is represented as a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
quantity
. The
polar form conveniently captures both magnitude and phase characteristics as
:
where the magnitude
represents the ratio of the voltage difference amplitude to the current amplitude, while the argument
(commonly given the symbol
) gives the phase difference between voltage and current.
is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, and is used instead of
in this context to avoid confusion with the symbol for
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movi ...
.
In
Cartesian form, impedance is defined as
:
where the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
of impedance is the resistance and the
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
is the
reactance .
Where it is needed 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 as finding 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.
Complex voltage and current
![Impedance symbol comparison](https://upload.wikimedia.org/wikipedia/commons/a/ae/Impedance_symbol_comparison.svg)
To simplify calculations,
sinusoid
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
al voltage and current waves are commonly represented as complex-valued functions of time denoted as
and
.
:
The impedance of a bipolar circuit is defined as the ratio of these quantities:
:
Hence, denoting
, we have
:
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
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that ...
):
: