Wave impedance
Encyclopedia
The wave impedance of an electromagnetic wave is the ratio
of the transverse components of the electric
and magnetic field
s (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic (TEM
) plane wave
traveling through a homogeneous medium
, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms
. The symbol η (eta
) may be used instead of Z for wave impedance to avoid confusion with electrical impedance
.
The wave impedance is given by
where
is the electric field and 
is the magnetic field, in phasor representation.
In terms of the parameters of an electromagnetic wave and the medium it travels through, the wave impedance is given by
where μ is the magnetic permeability, ε is the electric permittivity
and σ is the electrical conductivity of the material the wave is travelling through. In the equation, j is the imaginary unit
, and ω is the angular frequency
of the wave. In the case of a dielectric
(where the conductivity is zero), the equation reduces to
As usual for any electrical impedance, the ratio is defined only for the frequency domain
and never in the time domain
.
H/m and 
F/m. So, the value of wave impedance in free space is
,
H/m and 
F/m. So, the value of wave impedance in a perfect dielectric is
In a perfect dielectric
, the wave impedance can be found by dividing Z0 by the square root of the dielectric constant
. In anything else, the formula becomes larger and a complex number is the result.
in the form of a hollow metal tube, (such as rectangular guide, circular guide, or double-ridge guide), the wave impedance of a travelling wave is dependent on the frequency
, but is the same throughout the guide. For transverse electric (TE
) modes of propagation the wave impedance is
where fc is the cut-off frequency of the mode, and for (TM
) modes
Above the cut-off (f > fc), the impedance is real (resistive) and the wave carries energy. Below cut-off the impedance is imaginary (reactive) and the wave is evanescent. These expressions neglect the effect of resistive loss in the walls of the waveguide. For a waveguide entirely filled with a homogeneous dielectric medium, similar expressions apply, but with the wave impedance of the medium replacing Z0. The presence of the dielectric also modifies the cut-off frequency fc.
For a waveguide or transmission line containing more than one type of dielectric medium (such as microstrip
), the wave impedance will in general vary over the cross-section of the line.
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
of the transverse components of the electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic (TEM
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
) plane wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
traveling through a homogeneous medium
Medium (optics)
An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it...
, the wave impedance is everywhere equal to the intrinsic impedance of the medium. In particular, for a plane wave travelling through empty space, the wave impedance is equal to the impedance of free space. The symbol Z is used to represent it and it is expressed in units of ohms
Ohms
OHMS may refer to:* The plural of ohm, a unit of resistance, named after Georg Ohm* Ohm's Law of electric currents, first proposed by Georg Ohm* O.H.M.S., On His/Her Majesty's Service...
. The symbol η (eta
ETA
ETA , an acronym for Euskadi Ta Askatasuna is an armed Basque nationalist and separatist organization. The group was founded in 1959 and has since evolved from a group promoting traditional Basque culture to a paramilitary group with the goal of gaining independence for the Greater Basque Country...
) may be used instead of Z for wave impedance to avoid confusion with electrical impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical 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 circuit...
.
The wave impedance is given by
where
is the electric field and is the magnetic field, in phasor representation.In terms of the parameters of an electromagnetic wave and the medium it travels through, the wave impedance is given by
where μ is the magnetic permeability, ε is the electric permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
and σ is the electrical conductivity of the material the wave is travelling through. In the equation, j is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, and ω is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the wave. In the case of a dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
(where the conductivity is zero), the equation reduces to
As usual for any electrical impedance, the ratio is defined only for the frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....
and never in the time domain
Time domain
Time domain is a term used to describe the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the case of continuous time, or at various...
.
Wave impedance of free space
In free space, H/m and F/m. So, the value of wave impedance in free space is-
.
Wave impedance in an unbounded dielectric
In a perfect dielectricDielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
,
H/m and F/m. So, the value of wave impedance in a perfect dielectric is-
.
In a perfect dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
, the wave impedance can be found by dividing Z0 by the square root of the dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
. In anything else, the formula becomes larger and a complex number is the result.
Wave impedance in a waveguide
For any waveguideWaveguide (electromagnetism)
In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves...
in the form of a hollow metal tube, (such as rectangular guide, circular guide, or double-ridge guide), the wave impedance of a travelling wave is dependent on the frequency
, but is the same throughout the guide. For transverse electric (TETransverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
) modes of propagation the wave impedance is
where fc is the cut-off frequency of the mode, and for (TM
Transverse mode
A transverse mode of a beam of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam...
) modes
Above the cut-off (f > fc), the impedance is real (resistive) and the wave carries energy. Below cut-off the impedance is imaginary (reactive) and the wave is evanescent. These expressions neglect the effect of resistive loss in the walls of the waveguide. For a waveguide entirely filled with a homogeneous dielectric medium, similar expressions apply, but with the wave impedance of the medium replacing Z0. The presence of the dielectric also modifies the cut-off frequency fc.
For a waveguide or transmission line containing more than one type of dielectric medium (such as microstrip
Microstrip
Microstrip is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals. It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. Microwave components...
), the wave impedance will in general vary over the cross-section of the line.