# Transverse electromagnetic wave

The transverse electromagnetic wave or TEM wave ( transverse electromagnetic mode ) is a special case of an electromagnetic wave in which both the electrical and the magnetic field disappear in the direction of propagation . Instead, the magnetic and electric fields are exclusively in planes perpendicular ( transversal ) to the direction of propagation. This type of electromagnetic wave forms as a guided wave e.g. B. between the outer and inner conductors of a lossless coaxial cable . The plane wave is also an example of TEM waves.

In general, two separate, ideal conductors must exist for a TEM line , the arrangement of which is uniform in the direction of propagation and is located in a homogeneous space. The electric field in the line cross-section then corresponds to the electrostatic field for the given geometry, which however oscillates and spreads along the line.

In Cartesian coordinates , the vector components of a TEM wave with the direction of propagation  z can be expressed as:

{\ displaystyle {\ begin {alignedat} {2} {\ vec {E}} & = (E_ {x}, E_ {y}, 0) \ qquad {\ vec {D}} & = (D_ {x} , D_ {y}, 0) \\ {\ vec {H}} & = (H_ {x}, H_ {y}, 0) \ qquad {\ vec {B}} & = (B_ {x}, B_ {y}, 0) \\ {\ vec {S}} & = (0,0, S_ {z}) \ end {alignedat}}}

The field vectors are here

• ${\ displaystyle {\ vec {E}}}$for the electric field strength , for the electric flux density${\ displaystyle {\ vec {D}}}$
• ${\ displaystyle {\ vec {H}}}$for the magnetic field strength , for the magnetic flux density .${\ displaystyle {\ vec {B}}}$
• The Poynting vector expresses the energy flux density , which in this case takes place in the z direction without exception .${\ displaystyle {\ vec {S}} = {\ vec {E}} \ times {\ vec {H}}}$

The wave impedance of a TEM wave is: ${\ displaystyle Z _ {\ mathrm {w}}}$

${\ displaystyle Z _ {\ mathrm {w}} = {\ sqrt {\ frac {\ mu} {\ varepsilon}}}}$

With

If and are real , one speaks of field wave resistance . ${\ displaystyle \ mu}$${\ displaystyle \ varepsilon}$

The propagation constant in TEM waves is:

{\ displaystyle {\ begin {aligned} \ gamma & = \ pm j \, k \\ & = \ pm j \, \ omega \, {\ sqrt {\ mu \ varepsilon}} \ end {aligned}}}

With

• the wave number ${\ displaystyle k}$
• the imaginary unit ${\ displaystyle j}$
• the angular frequency .${\ displaystyle \ omega}$

## Relation to line sizes

With TEM waves, there is a simple relationship between the field sizes and the line sizes .

In general, for time-varying electric fields, the non-zero rotation (i.e. the electric field is not eddy-free ) depends on the integration path . In this case, no line size, such as B. specify a voltage clearly.

However, since TEM waves have no field component in the direction of propagation, and consequently also no component of the rotation in the direction of propagation (see induction law ), the orbital integral of the electric field strength disappears in the transverse plane. This allows a potential field to be defined across the direction of propagation , regardless of the integration path. The field size of the electrical field strength can therefore be specified as a line size in the form of an electrical voltage between the conductors along the direction of propagation, for example along a coaxial cable.

The line size of the electric current in the conductor can be defined in a similar manner , since with a TEM wave no displacement current flows through the transverse plane (law of flux ) and the tangential component of the magnetic field strength disappears on the conductor surface. This means that the two currents, for example in the inner and outer conductors of a coaxial cable, must be constant and opposite in the direction of propagation.

To implement the ability to field sizes of the electric and magnetic field strength in line sizes as an electric voltage and current, plays in the transmission line theory a significant role, since so leave the complex in general field conditions reduced to simple manageable line sizes. The equivalent description of the conditions at an electric gate , either by wave sizes or, equivalent to this, by line sizes, as in the case of the scattering parameters , is based on this fact.

## Transverse electric and transverse magnetic waves

Besides the TEM waves there are:

• transverse electric waves (TE waves); here only the electrical component disappears in the direction of propagation, while the magnetic component can assume values ​​other than 0.
• transverse magnetic waves (TM waves); here only the magnetic component disappears in the direction of propagation, while the electrical component can assume values ​​other than 0.

Such waves can be found z. B. in waveguides and on the single-wire waveguide .

TE waves also describe the wave propagation in lasers or laser beams and optical waveguides and are also referred to as H waves , particularly in waveguides . Similarly, TM waves are also referred to as E waves .

TEM waves can always be broken down into a TE and a TM part when specifying a reference area. With the normal vector of the reference surface, the following applies to the PD component: ${\ displaystyle {\ vec {n}}}$

${\ displaystyle {\ vec {E}} \ cdot {\ vec {n}} = 0}$.

That is synonymous with

${\ displaystyle {\ vec {E_ {TE}}} = ({\ vec {n}} \ times {\ vec {E}}) \ times {\ vec {n}}}$.

The following applies to the TM part:

${\ displaystyle {\ vec {H}} \ cdot {\ vec {n}} = 0}$

or.

${\ displaystyle {\ vec {H_ {TM}}} = ({\ vec {n}} \ times {\ vec {H}}) \ times {\ vec {n}}}$.

The transversal parts do not contain any components in the direction of the surface normals of the reference surface; instead, the transversal parts are perpendicular to this normal or parallel to the reference surface. ${\ displaystyle {\ vec {n}}}$

## literature

• Károly Simonyi: Theoretical electrical engineering . 10th edition. Barth Verlagsgesellschaft, 1993, ISBN 3-335-00375-6 .
• Karl Küpfmüller, Wolfgang Mathis, Albrecht Reibiger: Theoretical electrical engineering . 18th edition. Springer, 2008, ISBN 978-3-540-78589-7 .