Propagation
constant
Updated November
16, 2007
Click
here to go to our main page on transmission lines
Click here
to go to our separate page on characteristic impedance
Click
here to go to our page on phase velocity
Click
here to go to our page on transmission line attenuation
Click
here to go to our page on wavenumber, which is often confused
with propagation phase constant!
Updated for December 2007:
We recently split the transmission
line model onto its own page. The propagation constant is an
important parameter associated with transmission lines. It is a
complex number denoted by Greek lower case
letter (gamma),
and is used to describe the behavior of an electromagnetic wave
along a transmission line.
Propagation,
attenuation and phase constants
The propagation constant is separated
into two components that have very different effects on signals:
The real part of the propagation
constant is the attenuation constant and is denoted by Greek
lowercase letter
(alpha). It causes a signal amplitude to decrease along a transmission
line. The natural units of the attenuation constant are Nepers/meter,
but we often convert to dB/meter in microwave engineering. To get
loss in dB/length, multiply Nepers/length by 8.686. Note that attenuation
constant is always a positive number, if it was negative you'd violate
the First Law of Thermodymamics (you never get something for nothing!)
The phase constant is
denoted by Greek lowercase letter
(beta) adds the imaginary component to the propagation constant.
It determines the sinusoidal amplitude/phase of the signal along
a transmission line, at a constant time. The phase constant's "natural"
units are radians/meter, but we often convert to degrees/meter.
A transmission line of length "l" will have an electrical
phase of l,
in radians or degrees. To convert from radians to degrees, multiply
by 180/.
The two parts of the propagation
constant have radically different effects on a wave. The amplitude
of a wave (frozen in time) goes as cosine(l).
In a lossless transmission line, the wave would propagate as a perfect
sine wave. In real life there is some loss to the transmission line,
and that is where the attenuation constant comes in. The amplitude
of the signal decays as Exp(l).
The composite behavior of the propagation constant is observed when
you multiply the effects of
and .
The graph below represents wave
propagation in a fairly lossy line, we made it lossy so you could
observe the familiar exponential curve of amplitude decay. In this
graph, =1
and =0.05.
In practice we usually want to minimize loss, but this example is
a very lossy line!
Phase constant versus wavenumber
We have a separate page on wavenumber.
Phase constant and wavenumber are often treated as the same thing.
Indeed, for TEM transmission lines (coax and stripline), the phase
constant and wavenumber are equal. Wavequide is one case where you
need to understand the difference between the two.
Wavenumber is denoted by lower
case "k", and is a measure of how many cycles a wave has
in a given length, for a traveling wave that is frozen in time.
Phase constant, phase velocity,
frequency and wavelength
Let's examine the relationships
between phase constant, frequency, phase velocity and wavelength,
Recall that there are 2
radians in a wavelength, therefore the relationship between phase
constant and wavelength is simply:
More to come!
Transmission line model
This discussion of how the propagation
constant can be derived from the tranmssion line model has been
moved to a separate page.
We have a separate page on transmission
line loss that deals further with the topic of splitting up the
attenuation constant, check it out!
