# Propagation Constant Expression in a Transmission Line

let's consider a generic transmission line:

Its wave vector Kz is equal to:

where

Then, I read also that it is true that:

where does this relation come from?

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.

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\$$.)

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