On 1/8/2011 9:09 AM, Jack Mandelman wrote:
> Forget about the formulas! None of the discussed formulas is
> sufficiently accurate of the wide ranges of S/d discussed. All are
> approximations that are reasonably valid only over their limited
> domains.
The log formula gives industry accepted precision for impedances above
about 250 or 275 ohms. Spacing about 4 times diameter out to arbitrarily
large spacing. The inverse hyperbolic cosh formula matches the impedance
for all that range yet also matches the near zero impedance with the
conductor spacing is infinitesimal at the low Z domain boundary.
>
> A much more practical engineering approach would be to apply a
> finite-element analysis of LaPlace's equation using the boundary
> conditions appropriate to the geometry of interest. For these types
> of problems a 2-D quasi-static analysis provides much better accuracy
> than any formulas presented.
Are you sure that the finite element stepped analysis is better than
Harold Wheeler's 2d analysis of 1938? That gave a new formula based on
the inv hyp cos for capacitance and the effects of inductance have been
neglected taking Z0 = sqrt (L/C). Are you sure that he didn't do exactly
a 2 D static analysis to come up with that formula? I think he did
without the discreteness of finite element approximations. Harold
Wheeler was a transmission line guru of the highest class with many
papers to his credit, but when interviewed later in life he was most
proud of the inverse hyperbolic cosh formula for capacitance of closely
spaced parallel wires. His career ran another 30 or 40 years after that
paper and he had plenty of time to revisit it but didn't change the
results while investigating many other transmission line shapes and
publishing curves and formulas for their parameters.
Its conceptually easy to check the available formulae, simply construct
samples of transmission lines and measure their characteristic
impedance, probably as quarter wave transformers with a known load. Or
by varying the line or the load to present the same impedance when a
quarter and a half wave long. Then you might have to work out the
complications of the balun you measure the balanced line through or
create a balanced bridge and the effects of spacers and conductor
mountings at the ends and any intermediate points. 1/2" copper tubing
would be stiff enough to allow measuring at 6 meters without needing
intermediate supports. Perhaps the most precise measuring instrument
would use the line with a traveling differential voltage probe loosely
coupled like a slotted line without shielding. Perhaps that voltage
probe could be made arbitrarily small to include a microwave transmitter
to send a measure of the detected voltage to the user without having to
use wires to mess up the balance of the line under test.
73, Jerry, K0CQ
>
> Jack K1VT
>
>> Correct!
>
>> I believe it was Jerry who pointed me to the accurate formula a
>> couple of years ago on this very List. I was considering how to
>> build low impedance balanced line with a Zo in the range 50 Ohms to
>> 70 Ohms for a Hexbeam application. One look at the published curves
>> told me it couldn't be done; but Jerry put me right!
>
>> Steve G3TXQ
>
>
>> On 08/01/2011 02:05, Dr. Gerald N. Johnson wrote:
>>> / Like my curves its center to center. The upper trace is using
>>> the 276/ / log formula and the lower one is the 120 inv
>>> hyperbolic cosine function./ / Mine has the spacings on a log
>>> scale his is a liner scale. The upper/ / trace formula is
>>> increasingly inaccurate below 300 ohms impedance with/ / 87,000
>>> percent error at a spacing just a hair wider than the
>>> conductors/ / touching. The wrong formula says you can't get a
>>> Z0 less than 87 ohms/ / (as published in QST last summer again),
>>> but the correct formula gets/ / down to practically zero Z0 just
>>> before the conductors make contact./ /
>>> http://www.geraldj.networkiowa.com/papers/CSVHF2010/lztl1.JPG// /
>>> http://www.geraldj.networkiowa.com/papers/CSVHF2010/lztl2.JPG//
>>>
>>> / 73, Jerry, K0CQ/
>>>
>>> / On 1/7/2011 7:46 PM, Jack Mandelman wrote:/ /> Steve,/ />
>>> In your plot how are you defining S, the spacing between
>>> conductors? Is it/ /> center to center? Is it consistent for
>>> both the approximated and actual/ /> curves? It appears that
>>> only the actual curve uses the center to center/ /> definition;
>>> in the limit as S-->d the inner edge to inner edge spacing
>>> goes/ /> to zero, and Zo also goes to zero./ />/ /> Jack
>>> K1VT/
>
>
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