1/20/2005
The Lossless Transmission Line.doc
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The Lossless
Transmission Line
Say a transmission line is lossless (i.e., R=G=0); the transmission
line equations are then significantly simplified!
Characteristic Impedance
Z0 =
R + j ωL
G + j ωC
=
j ωL
j ωC
=
L
C
Note the characteristic impedance of a lossless transmission
line is purely real (i.e., Im{Z0} =0)!
Propagation Constant
γ = ( R + j ω L)( G + j ωC )
= ( j ω L)( j ωC )
= −ω 2LC
= j ω LC
The wave propagation constant is purely imaginary!
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Lossless Transmission Line.doc
2/4
In other words, for a lossless transmission line:
α = 0 and β = ω LC
Voltage and Current
The complex functions describing the magnitude and phase of
the voltage/current at every location z along a transmission line
are for a lossless line are:
V ( z ) = V0+ e − j β z + V0− e + j β z
V0+ − j β z V0− + j β z
−
I( z ) =
e
e
Z0
Z0
Line Impedance
The complex function describing the impedance at every point
along a lossless transmission line is:
V0+ e − j β z + V0− e + j β z
V(z)
Z(z) =
= Z0 + − j βz
I( z )
V0 e
− V0− e + j β z
Reflection Coefficient
The complex function describing the reflection at every point
along a lossless transmission line is:
V0− e + j β z V0− + j 2 β z
Γ (z ) = + − j β z = + e
V0 e
V0
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Lossless Transmission Line.doc
3/4
Wavelength and Phase Velocity
We can now explicitly write the wavelength and propagation
velocity of the two transmission line waves in terms of
transmission line parameters L and C:
λ=
2π
β
vp =
=
1
f LC
ω
1
=
β
LC
Q: Oh please, continue wasting my
valuable time. We both know that a
perfectly lossless transmission line
is a physical impossibility.
A: True! However, a low-loss line is
possible—in fact, it is typical! If
R ω L and G ωC , we find that the
lossless transmission line equations are
excellent approximations!
Unless otherwise indicated, we will use the lossless equations to
approximate the behavior of a low-loss transmission line.
Jim Stiles
The Univ. of Kansas
Dept. of EECS
1/20/2005
The Lossless Transmission Line.doc
4/4
The lone exception is when determining the attenuation of a
long transmission line. For that case we will use the
approximation:
⎞
1⎛ R
α≈ ⎜
+ GZ 0 ⎟
2 ⎝ Z0
⎠
where Z 0 = L C .
A summary of lossless transmission line equations
Z0 =
V ( z ) = V0 e
+
− j βz
L
C
+ V0 e
−
γ = j ω LC
+ j βz
V0+ − j β z V0− + j β z
I( z ) =
e
−
e
Z0
Z0
V0+ e − j β z + V0− e + j β z
Z ( z ) = Z0 + −j βz
− V0− e + j β z
V0 e
V + ( z ) = V0+ e − j β z
V − ( z ) = V0− e + j β z
V0− + j 2 β z
Γ (z ) = + e
V0
β = ω LC
Jim Stiles
λ=
1
f LC
The Univ. of Kansas
vp =
1
LC
Dept. of EECS