On Jan 3, 2:44*pm, "Frank" wrote:
Frank
Please keep in mind the following
NEC is based totally on the extremely thin wire where various
assumption can be made
such as equations being equal to zero in the limit., These same
assumptions can not be held to
when dealing with thick radiators despite the closeness of the
approximations.
Best regards
Art

The reference athttp://www.nec2.org/other/nec2prt1.pdf*p 21 deals
with the accuracey of NEC 2 in respect to the "Thin wire approximation".
From the NEC-4, theory manual, p 21, para 4: ".... the NEC-4 wire model
employes the extended boundary condition in the thin wire approximation,
so that the current is treated as a tubular distribution on the wire
surface......."

Calculus is based on homogenous materials or planes where you can
refer dy/dx to
some thing aproaching zero. In the case of using this aproach where
the antenna diameter aproaches zero
this is an invalid aproach for accuracy but O.K. for aproximations. So
much for the foibles of theoretical mathematics.

Your comments about calculus are confusing. *A derivative
is always non-zero -- unless you are differentiating a constant.
The homogeneity, or otherwise, of a material is irrelevant
to the process of differentiation.
That is exactly my point. The skin is not hogenoius even if you
consider the resistive action to be constant in depth thus you cannot
put a limit on the thicknes
or diameter of the radiator! If you do put a limit anyway on skin
depth then you cannot apply the reasoning to a hollow tube.
We can talk back and forwards for ever on the analogy provided with
vanishingly thin radiators but until we break apart the mathematics
such that there is a reflection at the end of a radiator the posters
question cannot be answered.
If one is to model the situation as Cecil suggests we must first
determine how and where the reflection is created and the
applied math provided to support it. I can see no reference via
mathematics that shows the reversal or reflection of current flow
prior to the end of a cycle.If there were such an instance then there
must be a determination of the resistance radiation or otherwise
so that any assumption made is factual.




The vanishing thin radiator cannot be applied directly to a non
homogenous material because at the limits of the *the diameter
is unable to support the presence of eddy currents(skin depth) . In
other words the assumption of limi tess ness cannot be held if the
presence of skin effect is true.

Most conductors are homogeneous. *In fact I cannot think of
a non-homogeneous conductor. *Even in plated conductors
the current flows in the plating.

No that is not true as homogenous implies equilibrium and for skin
depth the value (e) comes into beingor what so0me would refer to as
decay

Of course if skin effect is not present then you
have a DC current where only copper losses are present.
As always with mathematics assumptions and preconditions are alway
subject to examination. This in no way takes away from the advantages
oif the NEC programs.
Art

Copper loss still exists for high frequency currents.

Very true Frank but the radiation resistance plus the resistance
encoutered by surface flow is not related/
proportional to the pure copper losses where skin resistance is not
present where in the absence of
skin depth leaves one with DC pulses.

I am ofcourse still interested what the NEC programs show for
reflection and consequental resistance
which I believe was in Cecil's thoughts to determine the truth.
Hopefully the dialogue between you and I will not drop to the level of
David's where he contendes that Gaussian law of STATICS
is one of the basic laws that Maxwell applied/used without the
required proof..On top of which he denies the applicability of statics
with electro magnetics thus any mathematical aproach cannot be
applicable which is absolutely crazy
The thrust of this thread is solely on the difference of radiation
with respect to hollow radiators and solid radiators and it should be
kept at that to provide a reasonable answer as required in any formal
debate.

73, *Frank