Waveguide mathematics
Updated July 31,
2006
Our other info on waveguide can
be found on these pages:
Waveguide
primer (main waveguide page)
Waveguide
construction
Waveguide
dimensions and letter bands
This page contains some of important
equations for rectangular waveguide. Here is an index to the subject
of waveguide mathematics:
Cutoff frequencies
Guide wavelength
Phase velocity
and group velocity
Group delay
in waveguide
Waveguide
loss (new for July 2006, separate page)
Note that in the equations on
this page we have kept the units simple and consistent, and you
might want to do the same.... for time, use nanoseconds. For distance,
use cm. For velocity, use cm/nanosecond. For frequency, use GHz.
Cutoff
frequencies
Waveguide can support many modes
of transmission. All microwave textbooks will tell you about this,
but we don't really care. The usual mode of transmission in rectangular
waveguide is called TE01. The lower cutoff wavelength (and frequency)
for this mode is very simply:
The upper cutoff frequency is
exactly one octave above the lower. We'll let you do the math on
this (multiply lower cutoff frequency by two...) now it's time for
another Microwaves101 rule of thumb:
Waveguide operating band
The accepted limits of operation
for rectangular waveguide are (approximately) between 125% and 189%
of the lower cutoff frequency. Thus for WR-90, the cut-off is 6.557
GHz, and the accepted band of operation is 8.2 to 12.4 GHz. Remember,
at the lower cutoff the guide simply stops working. See our page
on waveguide loss for more information.
Guide wavelength
Guide wavelength is defined as
the distance between two equal phase planes along the waveguide.
The guide wavelength is a function of operating wavelength (or frequency)
and the lower cutoff wavelength, and is always longer than the wavelength
would be in free-space. Here's the equation for guide wavelength:
Guide wavelength
is used when you design distributed structures in waveguide. For
example, if you are making a PIN diode switch with two shunt diodes
spaces 3/4 wavelength apart, use the 3/4 of a guide wavelength in
your design.
The guide wavelength
in waveguide is longer than wavelength in free space. This isn't
intuitive, it seems like the dielectric constant in waveguide must
be less than unity for this to happen... don't think about this
too hard you will get a headache.
Here is a way to
imagine why this is... picture yourself at Zuma Beach in Malibu.
The waves are coming in at an angle to the beach.... check out the
intersection of the wavefront with the beach, it is zipping along
faster than you can run... yes, it's apparently faster than the
waves are moving if you look straight at them. Maybe it's time for
us all to go to the beach and check this out... send us an good
mpg video of this and we'll send you $100!
Phase
velocity and group velocity
Phase velocity is an almost
useless piece of information you'll find in waveguide mathematics;
here you multiply frequency times guide wavelength, and come up
with a number that exceeds the speed of light!
Be assured that the energy in
your wave is not exceeding the speed of light, because it travels
at what is called the group velocity of the waveguide:
The group velocity
is always less than the speed of light, we like to think of that
this is because the EM wave is ping-ponging back and forth as it
travels down the guide. Note that group velocity x phase velocity
= c2.
Group velocity in a waveguide
is speed at which EM energy travels in the guide. Plotted below
as a percentage of the speed of light (c), we see how group velocity
varies across the band for WR-90 (X-band) waveguide. Note that the
recommended operating band of WR-90 is from 8.2 to 12.4 GHz. At
8.2 GHz the signal is slowed to 60% of the free-space speed of light.
At the lower cutoff (6.56 GHz), the wave is slowed to zero, and
you can outrun it without breathing hard.
Group
delay in waveguide
Click
here to check out our page on group delay!
Now that we know
the group velocity, we can calculate the group delay of any piece
of waveguide, noting that time is distance divided by velocity:
The group delay of rectangular
waveguide components is a function of the frequency you are applying.
Near the lower cutoff, the group delay gets longer and longer, as
the EM wave ping-pongs down the guide, and can easily be 10X the
free-space group delay. But at the upper end of a waveguide's band,
the group delay approaches the free-space group delay, which follows
the rule-of-thumb, approximately
one foot per nanosecond, independent of frequency.
To compare with the one nanosecond/foot
rule of thumb, below is a plot of the group delay of one foot of
WR-90 waveguide. At the upper end of the band you will see that
very nearly the free-space group delay is achieved.
The problem of electromagnetic
energy traveling at different speeds over frequency is commonly
called dispersion. Soon we will have a page on this topic as well.
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