phase and group velocities
Updated March 30,
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If any of you experts see any
problems with the math, please get in touch with us, we want this
to be 100% correct.
By now you should have memorized
the speed of light in a free space ("c"), it's 2.9979E8
meters second, or about 186,000 miles per second, or as a rule
of thumb, about one foot per nanosecond. This is the speed limit
of the universe, no matter or information is allowed to travel faster.
Speaking of "c", we
recommend NOT to hire a microwave engineer candidate that doesn't
know the velocity of light, permeability and permittivity of free
space. Go to our table of physical constants
and commit them to memory. And don't fall for this trick question
on a quiz:
Question: what's the speed
Wrong answer: 300,000,000 meters/second
Correct answer: it depends
on the medium!
Note to Microwaves101 readers:
most textbooks use the term "phase velocity" denoted by
vp interchangeably to also mean "velocity of light
in a medium". This gets confusing, so we will avoid doing it
and denote "velocity of light in a medium" by vlight.
The velocity of light is only
a function of the permeability and permittivity of the medium:
In free space, the values for
are used, and the speed is denoted by the constant "c":
Velocity of light from lumped
transmission line model
Before we reveal this knowledge,
let us point out that the lumped transmission
line model now has its own page!
Velocity of light in a transmission
line can be derived from the inductance and capacitance per unit
length. Going back to our lumped element transmission line model...
Under the normal conditions of:
The lumped model reduces to:
The velocity of
light in the transmission line is simply:
Note the similarity to this equation:
In a media other than free space,
light is slowed down, never sped up. We can define a velocity factor
"VF" to account for this. The velocity factor will always
be less than unity, remember that the speed of light in vacuum is
the speed limit of the universe.
Most media that we are interested
in have relative permeability=1, therefore the velocity factor is
Typically, coax vendors will
quote a velocity factor for their wares. If the "stuffing"
in a coax cable is PTFE, with a dielectric constant (relative permittivity)
of 2.07, the velocity factor is 69.5%.
Velocity of light for quasi-TEM
transmission lines such as microstrip and coplanar waveguide is
equal to the speed of light divided by the square root of the effective
For a description of what is
meant by effective relative permittivity, go
here. Sometimes it is denoted as keff, sometimes
Frequency doesn't change when
a signal encounters material and transmission line effects, but
wavelength does. The change in wavelength compared to free space
is inversely proportional to velocity factor.
The phase velocity is a the speed
at which a point of fixed phase propagates, which is not always
the speed that electromagnetic information travels. In TEM waves,
the phase velocity is the same as the velocity of light in the chosen
For waveguide, the phase velocity
is more complicated:
After some rearranging we can
We like the more complicated
form of the equation on the left, because it has been reduced to
a function of velocity factor VF and cutoff wavenumber which related
to dimensions and material properties. Here you can see that the
velocity factor has two effects, it directly reduces the phase velocity
in the numerator, while it's placement in the denominator acts to
increase phase velocity. The form of the equation on the right,
the velocity factor term in the denominator is built in to the cutoff
frequency term fc. From this you can see that when the cutoff frequency
is reached, the phase velocity goes to infinity.
For air-filled waveguide, the
phase velocity reduces further to:
Group velocity is the speed at
which electromagnetic information travels. Group delay is the length
of a circuit or transmission line, divided by its group velocity.
Although we aren't smart enough to bore you with the details of
how this is derived (order a copy of Pozar from our book
page), group velocity is slowed by the inverse of the factor
that phase velocity is increased. In the general case,
Another way to express this is:
It can be seen that this implies
that group velocity is less than the speed of light, while phase
velocity is greater than the speed of light.
Going back to the case of air-filled
waveguide, the group velocity is:
The inequality of group velocity,
speed of light in vacuum, and phase velocity is then: