| |
 |
|
|
Science Forum Index » Physics - Electromagnetic Forum » Propagation constant (beta) and wave numbr (k)
Page 1 of 1
|
| Author |
Message |
| Guest |
 Posted: Thu Feb 22, 2007 11:12 pm |
|
|
|
|
Hi,
I'm currently taking course in microwave engineering (and maybe I
should already know this) but, what exactly is the difference between
the wavenumber k and the propagation constant beta. Is it that
k=w*sqrt(permeability*permittivity) and beta is the real part of k
given that k is complex?
And following that questions, why do they refer to k as the "wave
number"? Would you refer to waves as wave #1, wave #2???
Thanks,
Tiff |
|
|
| Back to top |
|
| Timo Nieminen |
| Posted: Fri Feb 23, 2007 12:36 am |
|
|
|
Guest
|
On Fri, 22 Feb 2007 tlarosa11@gmail.com wrote:
Quote: Hi,
I'm currently taking course in microwave engineering (and maybe I
should already know this) but, what exactly is the difference between
the wavenumber k and the propagation constant beta. Is it that
k=w*sqrt(permeability*permittivity) and beta is the real part of k
given that k is complex?
That's one usage of the term.
It's also used for waves in waveguides, and is the propagation constant
along the waveguide. This is probably what is meant by it in your course.
You can think of the "local" wavevector pointing at an angle to the
direction along the waveguide, and the propagation constant is the
component of the wavevector in this direction.
Quote: And following that questions, why do they refer to k as the "wave
number"? Would you refer to waves as wave #1, wave #2???
Because the wavenumber is (originally) the number of cycles of the wave in
a unit length, k = 1/lambda. Basically, the spatial equivalent of
frequency (cf f = 1/T). k=1/lambda is still used in spectroscopy, but in
electromagnetics and optics, it's usually k=2*pi/lambda. Note that these
are also fields that use angular frequency instead of frequency.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html |
|
|
| Back to top |
|
| Wimpie |
| Posted: Fri Feb 23, 2007 7:55 pm |
|
|
|
Guest
|
On 23 feb, 04:12, tlaros...@gmail.com wrote:
Quote: Hi,
I'm currently taking course in microwave engineering (and maybe I
should already know this) but, what exactly is the difference between
the wavenumber k and the propagation constant beta. Is it that
k=w*sqrt(permeability*permittivity) and beta is the real part of k
given that k is complex?
And following that questions, why do they refer to k as the "wave
number"? Would you refer to waves as wave #1, wave #2???
Thanks,
Tiff
Hello Tiff,
About wave propagation. When you are studying microwaves from an
engineering background (rather then a physical background), a Gamma
(complex propagation constant) is defined.
Gamma = Alpha + j*Beta
Where Alpha is the attenuation constant and Beta is sometimes called
wave number (Beta = 2*pi/lambda). Gamma is the complex propagation
constant that holds both attenuation and phase shift due to retarded
time.
It fits in the formula (E is complex amplitude of E field, d =
distance from reference point zero, d is chosen in the direction of
propagation):
E(d)/E(0) = exp(-Gamma*d) = exp(-(Alpha + j*Beta)d)
E(d)/E(0) = exp(-Alpha*d) * Exp(-j*beta*d).
The "exp(-alpha*d)" part is the gain (1/attenuation). When alpha*d=1,
the wave is attenuated with factor e (2.7182...). The bigger Alpha, the
higher the attenuation in a certain media, waveguide or cable.
The "exp(-j*beta*d)" is the phase shift part (radians) of the E field
at distance d (meters) from the reference point. When, for example
Beta*d = pi, the phase at d meter from the reference point is 180
degrees behind. To have this formula working, Beta must be defined as
2*pi/lambda.
The propagation constant (Gamma) is related to material properties
via:
Gamma = j*w*sqrt(epsilon*permeability). w = Omega (2*pi*f).
With the above formula, you can calculate the plane wave propagation
into any media (also metal). Conductivity in a medium is converted to
a lossy part in epsilon (so you get an e' and e'', like is done for
magnetics [u' , u'']). With the propagation formula, you can also
derive, for example, the skin depth and surface impedance formulas for
good and bad conductors.
The complex part in e (e'') is -j*1/(rho*w).
In physics books, you may see a complex k (wave number). The complex k
is linked to the propagation constant via:
Gamma = j*k.
I also saw k (wave number) defined as 2*pi/lambda. So there is some
confusion.
When both complex epsilon and permeability are real (no losses), Gamma
becomes imaginary (no real part). When you fill this in the formula
with exponents, alpha will be zero and:
E(d)/E(0) = Exp(-Gamma*d) = exp(-
j*w*sqrt(epsilon*permeability)*d ).
Hence there will be no attenuation and the phase delay is:
Phase delay = w*sqrt(epsilon*permeability)*d [radians].
As v = d/t (v=speed, d=distance, t = time), and Phase delay = w*t, you
can derive that v = 1/(epsilon*permeability) [m/s].
I hope this will help you a bit and that a did not make errors.
Best Regards,
Wim
PA3DJS |
|
|
| Back to top |
|
| |
|
Page 1 of 1
All times are GMT - 5 Hours
The time now is Sun Oct 12, 2008 4:31 pm
|
|