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Re: Vf,Zo,R,G,L,C.....



>My line is very lossy.  I add the DC resistance as the series R and
>adjusted the shunt conductance until the Q of the line matches what my coil
>measures.  I have seen a number of equations for the lossless cases but
>haven't seen any for the lossy case.  I assume they are not easy...


Hello Terry

I have not read any of the work that was published by the Corums, but I
have studied the physics of waves on transmission lines (I am currently
studying for a MSc in electrophysics). I hope that the following is of some
help for you:

The resistance in a transmission line can be accounted for by adding a
series resistance Ro ohms per unit length and a short circuiting or
shunting resistance between the wires, which is expressed as a shunt
conductance (inverse of resistance) written as Go, where Go has the
dimensions of siemens per metre.

Unfortunately the mathematics is a little to complicated to write in ASCII,
but the voltage-time dependance of the transmission line waves can be shown
to be:

V = A*exp(-a*x)*exp(i*(w*t-k*x)) + B*exp(a*x)*exp(i*(w*t+k*x))

The behaviour of V is a wave which travels with an amplitude decaying
exponentially with distance because of the term exp(-a*x) and a wave
travelling in the opposite direction with an amplitude decaying
exponentially with distance because of the term exp(a*x).

In the expression g=a+i*k, g is called the propogation constant, a is
called the attenuation or absoption coefficient and k is the wave number.

The behaviour of the current wave I is exactly sililar and since power is
the product VI, the power loss with distance varies as (exp(-a*x))^2, that
is, as exp(-2*a*x).

This behaviour is not unexpected...especially when compared to a damped
simple harmonic motion. When the transmission line properties are purely
inductive (inertial) and capacitive (elestic), a pure wave equation with a
sine or cosine solution will follow. The introduction of a resistive or
loss element produces an exponential decay with distance along the
transmission line in exactly the same way as an oscillator is damped with
time.

Such a loss mechanism, resistive, viscous, frictional or diffusive, will
always result in energy loss from the propogating wave. These are all
examples of random collision processes which operate in only one direction
in the sense that they are thermodynamically irreversible.

Safe coiling,

Gavin Hubbard