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let's consider a generic transmission line: enter image description here

Its wave vector Kz is equal to:

enter image description here

where

enter image description here

Then, I read also that it is true that:

enter image description here

where does this relation come from?

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2 Answers 2

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where does [the] relation [\$k_z=\frac{2\pi}{\lambda}\$] ​come from?

The variable \$k\$ is used to refer to different things by different authors.

In many texts, \$k\$ (usually with no subscript) is defined as the wavenumber of a wave, i.e. the number of waves per unit length. With that definition,

$$k=\frac{1}{\lambda}$$ where \$\lambda\$ is the wavelength.

In the text where you saw, $$k=\frac{2\pi}{\lambda}$$ \$k\$ is just modified by a convenience factor of \$2\pi\$ just like we use $$\omega = 2\pi f$$. In that case, \$k\$ is the phase constant for a particular sinusoidal wave. It represents the change in phase angle per unit length.

In the equation $$k_z=\sqrt{(R+j\omega L)(G + j\omega C)}$$ \$k_z\$ is the propagation constant for a sinusoidal wave along the transmission line.

The real part of $$k_z = \alpha + j\beta$$ (i.e. \$\alpha\$) is the attenuation factor. It informs us how much a sinusoidal signal is attenuated per unit length.

The \$\beta\$ in the above equation is the phase constant for the wave under consideration. It is identical to the \$k\$ in the equation

$$k=\frac{2\pi}{\lambda}$$

Thus, when you see in different places,

$$k_z = \alpha + j\beta = \sqrt{(R+j\omega L)(G + j\omega C)}$$

and

$$k=\frac{2\pi}{\lambda}$$

The \$k\$s mean different (but related) things.

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The wavenumber \$k\$ is the result of solving the differential equations of the (general) electromagnetic wave for \$E\$ and \$H\$ in e.g. free space. The result for \$k\$ consists of the material characteristics \$\mu,\epsilon,\sigma\$.

When solving the telegrapher equations for the coaxial cable, you also get \$k\$, but this time \$k\$ consists of an R,L,G & C expression.

So the wavenumber \$k\$ comes either from the definition of a general wave or from the definition of a generic transmission line like e.g. a coaxial cable.

(The phase constant \$\beta\$ is just the imaginary component of \$k\$.)

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  • \$\begingroup\$ But where does the link k= √(YZ) = k = 2π/lambda? Why are they necessarily equal in case of TEM mode? \$\endgroup\$
    – Kinka-Byo
    Nov 16, 2019 at 11:39
  • \$\begingroup\$ @Kinka-Byo I refined my anwer to give you a bit more insight \$\endgroup\$ Nov 18, 2019 at 5:56

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