TEM mode solution of Maxwell's equations

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Maxwell's equations in a space among conductors of transmission line are:

The transverse electric and magnetic (TEM) mode solutions are:

where the suffix T indicates the transverse components:

The scalar potential satisfies the 2-D Laplace equation and the boundary condition:

We will derive the TEM mode solution in a transmission line.

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In a space among conductors of transmission line, electric charges or currents do not exist. Then Maxwell's equations are:

Decomposing the fields into frequency components by Fourier transformation:

we obtain Maxwell's equations in the frequency domain:

Note that the last two equations automatically hold because of the first two equations.

For convenience, decomposing the fields into the transverse (suffix T) and longitudinal (suffix z) components:

, we obtain the four equations:

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Assuming the transverse electric and magnetic (TEM) mode :

, the equations are simplified:

Eliminating BT between (2) and (4):

, we finally obtain the wave equation for ET. Noting eq (1), ET can be expressed as a gradient of a scalar potential (x,y):

Substituting this into the wave equation:

, we obtain the TEM mode solution for ET.

Noting that the divergence of E is zero, the scalar potential (x,y) must satisfy the 2D-Laplace's equation:

Note that u is the propagation speed:

Substituting the solution ET into (2), we obtain BT :

The directions of the electromagnetic fields and propagation direction z are depicted as follows:


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