Cecil Moore wrote:

Here's the equation that Ramo and Whinnery says is an approximation
for low-loss lines.

Z0 = SQRT(L/C)[1 + j(G/2wC - R/2wL)]

If G = C * R / L then Z0 = SQRT(L/C)

So why did Ramo and Whinnery say it is an approximation for low-loss
lines?

My response was to your earlier post:

+Reg Edwards wrote:

+ It was obvious I introduced G = C * R / L simply to show that a
+line's Zo
+ can be purely resistive even when it is NOT lossless. It can have
+any loss
+ you like.

To which you responded:

+Ramo and Whinnery are the authors of my 50's college textbook on fields
+and waves. Of course it could be a misprint, but they say your above
+formula is an approximation that is good for low-loss lines.

+Apparently, something additional happens for high-loss lines. Chipman
+seems to agree with Ramo and Whinnery when he introduces some +additional
+interference terms (discussed some time ago on this newsgroup). At the
+time, I didn't realize the additional terms were interference terms but
+the impedance of the load apparently somehow interacts with the
+characteristic impedance of the high-loss transmission line to upset
+the ideal relationships in your equation above.

As Reg points out, if G = C * R / L then Zo is purely resistive for any
frequency for which the relationship holds. This is the well known
'distortionless' condition.

The formula just introduced to the thread

Z0 = SQRT(L/C)[1 + j(G/2wC - R/2wL)]

is an approximation of the well known exact formula:

Z0 = sqrt( (R + jwL) / (G + jwC)) which can be found in any treatise on
transmission lines.

The equation just introduced from Ramo & Whinnery consists of the first
two terms of a particular series expansion of the exact formula for Zo,
and is therefore an approximation to Zo.

Reg's exact formula for the distortionless condition can be easily
derived by noting Zo is purely real when arctan(wL/R) = arctan(wC/G) and
working through the algebra.

Setting the imaginary part of Zo to zero in the approximate formula for
Zo will in general result in an approximation of the conditions required
for the distortionless case (Zo purely real). In this particular case
it turns out that the conditions derived from the approximation to Zo
are in fact exact. A fact which can be easily verified by direct
algebraic manipulation of the exact formula for Zo. Use of an
approximation developed for a specific purpose to draw conclusions
outside the validity of the approximation will likely lead to incorrect
conclusions and provide very little insight into the system being
analyzed.

bart
wb6hqk

On Mon, 29 Nov 2004 09:26:50 -0600, Cecil Moore
wrote:
Is it that difficult to find their exact solution for any lines?
Maybe the math is easier for the approximation?
Can only guess? That's OK, Bart did it for you. You should at least
thank him for filling this gap.

Bart Rowlett wrote:
Z0 = SQRT(L/C)[1 + j(G/2wC - R/2wL)]

is an approximation of the well known exact formula:

Z0 = sqrt( (R + jwL) / (G + jwC)) which can be found in any treatise on
transmission lines.

The equation just introduced from Ramo & Whinnery consists of the first
two terms of a particular series expansion of the exact formula for Zo,
and is therefore an approximation to Zo.

Thanks, Bart, that answers my question. I jumped to the conclusion that
since [(G/2wC - R/2wL) = 0] satisfied Reg's equation, it followed that it
must be part of the approximation mentioned by Ramo and Whinnery. I just
didn't read far enough: "For many important problems, losses are finite
but relatively small. If R/wL 1 and G/wC 1, the following approximations
are obtained by retaining up to second-order terms in the binomial expansions ...
it is often sufficient to retain only first-order correction terms ..."
which is what the above equation does.
--
73, Cecil http://www.qsl.net/w5dxp

Richard Clark wrote:

Cecil Moore wrote:
Maybe the math is easier for the approximation?

Can only guess?

Nope, rhetorical question.

Use of an
approximation developed for a specific purpose to draw conclusions
outside the validity of the approximation will likely lead to incorrect
conclusions and provide very little insight into the system being
analyzed.

bart
wb6hqk

====================================

.. . . . . and there are still people about who think the Smith Chart is
the best thing since sliced bread.
----
Reg.