Phase velocity

The solutions to the Maxwell’s equations for transmission media such as transmission lines and waveguides, there are two velocities: the phase velocity and the group velocity.  We are going to introduce the general definition of both concepts and, in this post, we will focus on the study  of the phase speed in a way you can easily understand the concept and apply it. Stay tuned to our website, as our next post will explain the group speed! 🙂

Difference between phase velocity and group velocity

The phase velocity is the speed which surfaces of equal phase are propagated with in a guided medium.

On the other hand, we know that in order to transport information we need to modulate the transmitted carrier. Due to this modulation, groups of frequencies are created, generally around the carrier, and they are propagated along the transmission medium. If the phase velocity is a function of the frequency, waves with different frequencies in the group will travel with slightly different velocities and they will be combined to form a modulation envelope that is propagated as a wave with group velocity.

In this post and the next, we will study both concepts highlighting their applications in real and practical examples.

Phase velocity

The phase velocity of a wave is the speed which its phase is propagated with. In other words, the velocity which the phase of any component on frequency is propagated with (it can be different for each frequency component).

If we take a particular phase of a wave (for instance, a maximum phase point), this will appear travelling at that velocity. The phase velocity is given in terms of the radial velocity, ω , and the wave vector, k, by the following relation:

phase-velocity-equation

In the following picture, we can observe the phase velocity in different points of time:

phase-velocity-concept

It’s important to notice that the phase velocity is not necessary equal to the group velocity of a wave, which is the propagation rate of the energy stored in the wave.

Therefore, for a wave that is propagated in a transmission line with no loses, it can be shown that the phase velocity equations as functions of the primary line constants are:

From the equations of a lossless transmission line:

eq1 eq2

We differentiate:

eq3 eq4

And solving the previous equations we obtain the following solution:

eq5

where

eq6

is the wave number, and

eq7

is the phase velocity.

Example 1: Generic Transmission Line

In a transmission line, the phase constant is 0.123 rad/m for a frequency of 2.5 MHz. The line length is 500 m. What is the wavelength and the phase velocity?

Easily, we can find the phase velocity by doing:

V=ω/β=2’298×10^8  m/s

and, as we know:

λ=2pi/β  with  β=ω/vf

Therefore:  λ = 51.08 m

 

Example 2: Coaxial Cable

The manufacturer of a coaxial cable provides the following characteristics:

eq8

Therefore, in order to obtain the phase velocity, we just need to operate:

eq9

We hope you found useful this post! Don’t hesitate in asking questions through the comments or sending us an email to contact@behindthesciences.com

See you in our next post, “Group Velocity”! 🙂

Add a Comment

Your email address will not be published. Required fields are marked *

Show Buttons
Hide Buttons