Roger Sparks wrote:
Roy Lewallen wrote:
. . .
The more general case where the line is some length L, rather than the
integral number of wavelengths in the example,

vf1(t, x) = sin(wt - x)
vr1(t, x) = Gl * sin(wt + x - 2L)

vf2(t, x) = Gs * Gl * sin(wt - x - 2L)
vr2(t, x) = Gs * Gl^2 * sin(wt + x - 4L)

Should we add an L to vf1(t, x, L) to keep the notation consistant in that we are considering 3 phase components. Rewriting,

Yes, that's fine. It can be viewed as either a variable or a constant,
but it won't change during the course of a single analysis.

vf1(t, x, L) = sin(wt - x + 0*L) (input point is congruent with refection point)

It's not necessary to explicitly add the 0*L term, even if you consider
vf to be a function of a variable L. And it adds unnecessary clutter and
potential confusion without any effect on the result.

vr1(t, x, L) = Gl * sin(wt + x - 2L)(2L evaluates the entire line travel time)

The equation is correct and what I wrote. I don't understand the
parenthetical comment.

In general,

(1) vfn(t, x) = (Gs * Gl)^n * sin(wt - x - 2nL)
(2) vrn(t, x) = Gl * (Gs * Gl)^n * sin(wt + x - (2n + 1)L)

where L is expressed in the same units as x and wt (degrees or radians).

These equations are correct with x being the distance from the input end
of the line.

I get your drift here. You are writting the general equation as if the first event is event zero, computer programming style.

Sorry, I don't know what event you're talking about. And it's not
"computer programming style", but standard notation as you'll find in
any text. I am making the assumption that the line is initially
discharged, if that's what you mean.

Shouldn't these be written like this, changing the term "(2n + 1)L"?

(1) vfn(t, x) = (Gs * Gl)^n * sin(wt - x - 2nL)
(2) vrn(t, x) = Gl * (Gs * Gl)^n * sin(wt + x - 2(n+1)L)


You're right. I made an error -- thanks for spotting it. The reflected
wave has an additional 2L delay relative to the corresponding forward
wave, and I didn't write it correctly. I apologize for the error.

You could, as I mentioned, use a different reference, for example x' = L
- x, where L is the line length in radians or degrees (same units as x
and wt). Then you have, simply by substituting L - x' for x:

vf1(t, x') = sin(wt - L + x')
vr1(t, x') = sin(wt + L - x' - 2L) = sin(wt - L - x')

and so forth, and for the general case,

(3) vfn(t, x') = (Gs * Gl)^n * sin(wt + x' - (2n + 1) * L)
(4) vrn(t, x') = (Gs * Gl)^n * sin(wt - x' - 2nL)

Equation 3 has the same error.

or, you can use x for the forward wave and x' for the reverse wave or
vice-versa in order to reference to the point the wave was reflected
from or where it will be reflected from. Any combination of the
equations is equally valid and will give correct results. You can't,
however, simply redefine the reference point without a corresponding
change in the equation. In general, equation 1 and equation 3 will give
different results if you put in the same value for x and x'; likewise
equations 2 and 4. There are some special cases, as you showed, where
you can change the reference without modifying the equations and not
have any impact on the sum of the waves. However, you can see from the
equations that this won't usually work.

Yes, the discussion becomes confusing quickly. If we were to have a rigorus discussion, we would need diagrams locating the points and directions. Lacking that, we are adrift.

It depends on your ability to visualize equations, which usually
improves as you work with them. But sketches of the waves are definitely
very helpful in keeping track of what's going on. I hope my little
program will also prove helpful for this.

This has been very productive for me Roy. I am gaining a much better appreciation for the whole subject, especially the use of phasors. Your use of the phase angle (posted with the corrections) was particularly helpful. I am planning to find Chipman's book.

Thanks for your efforts.

You're welcome. I'm glad to help when I can. Chipman's book might be
hard to find, but it's well worth the search. I have more than a dozen
texts dealing with transmission lines, but Chipman's has material, like
the concise development of steady state from startup, that you'll find
in few others.

I haven't used phasors in these postings at all, but would have to in
order to sum the waves for the general case. In phasor notation,

vfn(t, x) = (Gs * Gl)^n * exp(j(-x - 2nL))
vrn(t, x) = Gl * (Gs * Gl)^n * exp(j(x - 2(n + 1)L))

where Gs and Gl are complex. So the ratio of successive terms vfn(t, x)
/ vfn-1(t, x) = vrn(t, x) / vrn-1(t, x) is a multiplier term Gs * Gl *
exp(-j2nL). So we can use the formula for summing an infinite series to get

vf(ss)(t, x) = vf1 / (1 - Gs * Gl * exp(-j2nL))
vr(ss)(t, x) = vr1 / (1 - Gs * Gl * exp(-j2nL))

which can be evaluated as complex numbers and converted back to time
functions for evaluation. While the summation of the infinite series
could almost certainly be done by clever application of trig identities,
it's trivial with phasors.

Roy Lewallen, W7EL