1/20/2005 The Lossless Transmission Line.doc 1/4

Jim Stiles The Univ. of Kansas Dept. of EECS

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

0R j LZG j Cj Lj CLC

ωω

ωω

+=

+

=

=

Note the characteristic impedance of a lossless transmission line is purely real (i.e., Im{Z0} =0)! Propagation Constant

2

( R j L)( G j C )

j

( j L)( j C )LC

LC

+ +

=

=

=

= −

ω ωγ

ω ω

ω

ω

The wave propagation constant is purely imaginary!

1/20/2005 The Lossless Transmission Line.doc 2/4

Jim Stiles The Univ. of Kansas Dept. of EECS

In other words, for a lossless transmission line:

0 and LCα β ω= =

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:

0 0

0 0

0 0

j z j z

j z j z

V ( z ) V e V e

V VI ( z ) e eZ Z

β β

β β

+ − − +

+ −− +

= +

= −

Line Impedance The complex function describing the impedance at every point along a lossless transmission line is:

0 00

0 0

j z j z

j z j zV e V eV ( z )Z ( z ) Z

I ( z ) V e V e

β β

β β

+ − − +

+ − − +

+= =

−

Reflection Coefficient The complex function describing the reflection at every point along a lossless transmission line is:

( ) 20 0

0 0

j zj z

j zV e Vz eV e V

ββ

β

+− −+

−+ +Γ = =

1/20/2005 The Lossless Transmission Line.doc 3/4

Jim Stiles The Univ. of Kansas Dept. of EECS

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:

2 1f LC

πλβ

= =

1

pvLC

ωβ

= =

Unless otherwise indicated, we will use the lossless equations to approximate the behavior of a low-loss transmission line.

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 Lω and G Cω , we find that the lossless transmission line equations are excellent approximations!

1/20/2005 The Lossless Transmission Line.doc 4/4

Jim Stiles The Univ. of Kansas Dept. of EECS

The lone exception is when determining the attenuation of a long transmission line. For that case we will use the approximation:

00

12

R GZZ

α⎛ ⎞

≈ +⎜ ⎟⎝ ⎠

where 0Z L C= .

A summary of lossless transmission line equations

0LZC

= j LC=γ ω

0 0

0 00 0

j z j z j z j zV VV ( z ) V e V e I ( z ) e eZ Z

β β β β+ −

− + − ++ −= + = −

0 0

00 0

j z j z

j z j zV e V eZ ( z ) ZV e V e

β β

β β

− ++ −

− ++ −

+=

−

( ) ( )0 0j z j zV z V e V z V eβ β− ++ + − −= =

( ) 20

0

j zVz eV

β−

++Γ =

LCβ ω= 1f LC

λ = 1pv

LC=