1/20/2005
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The Complex Propagation
Constant
γ
Recall that the current and voltage along a transmission line
have the form:
V ( z ) = V0+ e −γ z + V0− e + γ z
V0+ −γ z V0− + γ z
I( z ) =
e −
e
Z0
Z0
where Z0 and γ are complex constants that describe the
properties of a transmission line. Since γ is complex, we can
consider both its real and imaginary components.
γ = ( R + j ω L)( G + j ωC )
α + jβ
where α = Re {γ } and β = Im {γ } . Therefore, we can write:
e −γ z = e −( α + j β ) z = e −α z e − jBz
Since e − j β z =1, then e −α z alone determines the magnitude of
e −γ z .
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Complex Propagation Constant.doc
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I.E., e −γ z = e −α z .
e −α z
z
Therefore, α expresses the attenuation of the signal due to the
loss in the transmission line.
Since e −α z is a real function, it expresses the magnitude of
e −γ z only.
The relative phase φ (z ) of e −γ z is therefore
determined by e − j β z = e − j φ (z ) only (recall e − j β z = 1 ).
From Euler’s equation:
e j φ ( z ) = e j β z = cos( β z ) + j sin( β z )
Therefore, βz represents the relative phase φ (z ) of the
oscillating signal, as a function of transmission line position z.
Since phase φ (z ) is expressed in radians, and z is distance (in
meters), the value β must have units of :
β =
Jim Stiles
φ
z
radians
meter
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Complex Propagation Constant.doc
3/4
The wavelength λ of the signal is the distance ∆z 2π over which
the relative phase changes by 2π radians. So:
2π = φ (z + ∆z 2π )-φ (z ) =β ∆z 2π =β λ
or, rearranging:
β =
2π
λ
Since the signal is oscillating in time at rate ω rad sec , the
propagation velocity of the wave is:
vp =
ω
ωλ
=
= fλ
β
2π
rad m ⎞
⎛ m
=
⎜ sec
sec rad ⎟⎠
⎝
where f is frequency in cycles/sec.
Recall we originally considered the transmission line current and
voltage
as
a
function
of
time
and
position
(i.e., v (z ,t ) and i (z ,t ) ). We assumed the time function was
sinusoidal, oscillating with frequency ω :
v ( z ,t ) = Re {V ( z ) e j ωt }
i ( z ,t ) = Re {I ( z ) e j ωt }
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Complex Propagation Constant.doc
4/4
Now that we know V(z) and I(z), we can write the original
functions as:
v ( z ,t ) = Re {V0+e −α z e − j ( β z −ωt ) + V0−e α z e j ( β z +ωt ) }
⎧V0+ −α z − j ( β z −ωt ) V0+ α z j ( β z +ωt ) ⎫
i ( z ,t ) = Re ⎨ e e
−
e e
⎬
Z
Z
0
⎩ 0
⎭
The first term in each equation describes a wave propagating in
the +z direction, while the second describes a wave propagating
in the opposite (-z) direction.
Z0 ,γ
V0−e α z e j ( β z + ωt )
V0+e −α z e − j ( β z −ωt )
z
Each wave has wavelength:
λ =
2π
β
And velocity:
vp =
Jim Stiles
ω
β
The Univ. of Kansas
Dept. of EECS