Wave impedance

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The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (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. In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six. Similarly, the ratio of lemons to oranges is 6∶8 and the ratio of oranges to the total amount of fruit is 8∶14. An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature. A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

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The wave impedance is given by

$Z={E_{0}^{-}(x) \over H_{0}^{-}(x)}$ where $E_{0}^{-}(x)$ is the electric field and $H_{0}^{-}(x)$ is the magnetic field, in phasor representation. The impedance is, in general, a complex number. In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude (A), angular frequency (ω), and initial phase (θ) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, and sinor or even complexor. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

In terms of the parameters of an electromagnetic wave and the medium it travels through, the wave impedance is given by

$Z={\sqrt {j\omega \mu \over \sigma +j\omega \varepsilon }}$ where μ is the magnetic permeability, ε is the (real) electric permittivity and σ is the electrical conductivity of the material the wave is travelling through (corresponding to the imaginary component of the permittivity multiplied by omega). In the equation, j is the imaginary unit, and ω is the angular frequency of the wave. Just as for electrical impedance, the impedance is a function of frequency. In the case of an ideal dielectric (where the conductivity is zero), the equation reduces to the real number In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a given medium. A charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium. The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex number is 2 + 3i. In physics, angular frequencyω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function. Angular frequency is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity.

$Z={\sqrt {\mu \over \varepsilon }}.$ Wave impedance in free space

In free space the wave impedance of plane waves is:

$Z_{0}={\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}$ (where ε0 is the permittivity constant in free space and μ0 is the permeability constant in free space) and:

$c_{0}={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}=299,792,458{\text{ m/s}}$ (by the SI definition of the metre)

hence, because the values of $c_{0}$ and $\mu _{0}$ are exact, the value of $Z_{0}$ in ohms is exactly:

$Z_{0}=\mu _{0}c_{0}=4\pi \times 10^{-7}{\text{ H/m}}\times 299,792,458{\text{ m/s}}=376.730313\ldots ~\Omega \approx 120\pi ~\Omega$ Wave impedance in an unbounded dielectric

In an isotropic, homogeneous dielectric with negligible magnetic properties, i.e. $\mu =\mu _{0}=4\pi \times 10^{-7}$ H/m and $\varepsilon =\varepsilon _{r}\times 8.854\times 10^{-12}$ F/m. So, the value of wave impedance in a perfect dielectric is 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 they do in an electrical conductor but only slightly shift from their average equilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displaced in the direction of the field and negative charges shift in the direction opposite to the field. This creates an internal electric field that reduces the overall field within the dielectric itself. If a dielectric is composed of weakly bonded molecules, those molecules not only become polarized, but also reorient so that their symmetry axes align to the field.

$Z={\sqrt {\mu \over \varepsilon }}={\sqrt {\mu _{0} \over \varepsilon _{0}\varepsilon _{r}}}={Z_{0} \over {\sqrt {\varepsilon }}_{r}}\approx {377 \over {\sqrt {\varepsilon _{r}}}}\,\Omega$ ,

where $\varepsilon _{r}$ is the relative dielectric constant.

Wave impedance in a waveguide

For any waveguide 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 $f$ , but is the same throughout the guide. For transverse electric (TE) modes of propagation the wave impedance is: [ citation needed ]

$Z={\frac {Z_{0}}{\sqrt {1-\left({\frac {f_{c}}{f}}\right)^{2}}}}\qquad {\mbox{(TE modes)}},$ where fc is the cut-off frequency of the mode, and for transverse magnetic (TM) modes of propagation the wave impedance is:[ citation needed ]

$Z=Z_{0}{\sqrt {1-\left({\frac {f_{c}}{f}}\right)^{2}}}\qquad {\mbox{(TM modes)}}$ 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.

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