"Dave" wrote in message
news:q7u8j.6941$xd.2942@trndny03...

"Roger" wrote in message
. ..
Richard Clark wrote:
On Thu, 13 Dec 2007 08:40:53 -0800, Roger wrote:


And just for completeness...
The fundamental equations also work when:
- the signal is not sinusoidal, e.g. pulse, step, square, ...
- rather than a load at one end, there is a source at each end
- the sources at each end produce different arbitrary functions
- the arbitrary functions at each end are DC sources
It is highly instructive to compute the forward and reverse
voltage and current (and then power) for a line with the same
DC voltage applied to each end.

...Keith

...Keith


Interesting! The important thing is to get answers that agree with
our experiments.

I have done some computations for DC voltage applied to transmission
lines. The real surprise for me came when I realized that
transmission
line impedance could be expressed as a function of capacitance and the
wave velocity. Z0 = 1/cC where c is the velocity of the wave and C
is
the capacitance of the transmission line per unit length.


Hi Roger,

This last round has piqued my interest when we dipped into DC. Those
"formulas" would lead us to a DC wave velocity?

73's
Richard Clark, KB7QHC

Hi Richard,

Here are two links to pages that cover the derivation of the formula Zo
= 1/cC and much more.

http://www.speedingedge.com/PDF-File..._Impedance.pdf
http://www.ece.uci.edu/docs/hspice/h...001_2-269.html

Here is the way I proposed to Kevin Schmidt nearly seven years ago after
seeing him use the formula on a web page:

*ASSUME*:
1) An electrical wave travels at the speed of light, c
2) A 'perfect' voltage source without impedance, V
3) A 'perfect' transmission line having no resistance but uniform
capacitance per unit length, C

*CONDITIONS AND SOLUTION*
The perfect voltage source has one terminal connected to the
transmission line prior to beginning the experiment. The experiment
begins by connecting the second terminal to the transmission line. The
voltage source drives an electrical wave down the transmission line at
the speed of light. Because of the limitation of speed, the wave
travels in the shape of a square wave containing all frequencies
required to create a square wave.

The square wave travels down the transmission line at the speed of light
(c). After time (T), the wave has traveled distance cT down the
transmission line, and has charged the distributed capacity CcT of the
line to voltage V over that distance. The total charge Q on the
distributed capacitor is VCcT.

Current (I) is expressed as charge Q per unit time. Therefore the
current into the transmission line can be expressed as

I = Q/T =
VCcT / T = VCc

Impedance (Zo) is the ratio of voltage (V) to current (I). Therefore
the impedance can be expressed as

Zo = V / I =
V / VCc = 1/Cc

We can generalize this by using the velocity of the electrical wave
rather than the speed of light, which allows the formula to be applied
to transmission line with velocities slower than the speed of light.

Of course, only the wave front and wave end of a DC wave can be
measured to have a velocity.

73, Roger, W7WKB

the OBVIOUS error is that the step when the second terminal is connected
DOES NOT travel down the line at c, it travels at some smaller percentage
of c given by the velocity factor of the line.

That IS what I said. Think of the velocity as a moving wall, with the
capacitor charged behind the wall, uncharged in front of the moving wall.

The second OBVIOUS error is the terminology 'DC wave'. you are measuring
the propagation velocity of a step function. this is a well defined
fields and waves 101 homework problem, not to be confused with the much
more common 'sinusoidal stead state' solution that most other arguments on
this group assume but don't understand.



Be real. This experiment can be performed, and the DC switched as
frequently as desired. How square the wave front will be depends upon real
world factors.

Go to a transmission line characteristics table and use the formula to
compare Zo, capacity per length, and line velocity. It will amaze you.

73, Roger, W7WKB