The lossless transmission line

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