On Jan 2, 4:57*pm, Jim Kelley wrote:
Keith Dysart wrote:
On Dec 29, 2:31 pm, Cecil Moore wrote:

Roger wrote:

Are there reflections at point "+"? *Traveling waves going in opposite
directions must pass here, therefore they must either pass through one
another, or reflect off one another.

In the absence of a real physical impedance discontinuity,
they cannot "reflect off one another". In a constant Z0 transmission
line, reflections can only occur at the ends of the line and only
then at an impedance discontinuity.

Roger: an astute observation. And Cecil thinks he has
the ONLY answer. Allow me to provide an alternative.

Many years ago, when I first encountered this news group
and started really learning about transmission lines, I
found it useful to consider not only sinusoidallly
excited transmission lines, but also pulse excitation.
It sometimes helps remove some of the confusion and
clarify the thinking. So for this example, I will use
pulses.

Consider a 50 ohm transmission line that is 4 seconds
long with a pulse generator at one end and a 50 ohm
resistor at the other.

The pulse generator generates a single 1 second pulse
of 50 volts into the line. Before and after the pulse
its output voltage is 0. While generating the pulse,
1 amp (1 coulomb/s) is being put into the line, so
the generator is providing 50 watts to the line.

After one second the pulse is completely in the line.
The pulse is one second long, contains 1 coulomb of
charge and 50 joules of energy. It is 50 volts with
1 amp: 50 watts.

Let's examine the midpoint (2 second) on the line.
At two seconds the leading edge of the pulse arrives
at the midpoint. The voltage rises to 50 volts and
the current becomes 1 amp. One second later, the
voltage drops back to 0, as does the current. The
charge and the energy have completely passed the
midpoint.

When the pulse reaches the end of the line, 50
joules are dissipated in the terminating resistor.

Notice a key point about this description. It is
completely in terms of charge. There is not a single
mention of EM waves, travelling or otherwise.

Now we expand the experiment by placing a pulse
generator at each end of the line and triggering
them to each generate a 50V one second pulse at
the same time. So after one second a pulse has
completely entered each end of the line and these
pulse are racing towards each other at the speed
of light (in the line). In another second these
pulses will collide at the middle of the line.

What will happen? Recall one of the basics about
charge: like charge repel. So it is no surprise
that these two pulses of charge bounce off each
and head back from where they came. At the center
of the line, for one second the voltage is 100 V
(50 V from each pulse), while the current is
always zero. No charge crossed the mid-point. No
energy crossed the mid-point (how could it if
the current is always zero (i.e. no charge
moves) at the mid-point.

It is a minor extension to have this model deal
with sinusoidal excitation.

What happens when these pulses arrive back at the
generator? This depends on generator output
impedance. If it is 50 ohms (i.e. equal to Z0),
then there is no reflection and 1 joule is
dissipated in each generator. Other values
of impedance result in more complicated
behaviour.

So do the travelling waves "reflect" off each
other? Save the term "reflect" for those cases
where there is an impedance discontinuity and
use "bounce" for those cases where no energy
is crossing a point and even Cecil may be
happy. But bounce it does.

...Keith

It's fairly safe to make this argument when both pulses are identical.
* I challenge you to obtain this result when they are not. *:-)

The example was carefully chosen to illustrate the
point, of course. But that is the value of particular
examples.

When the pulses are not identical, the energy that crosses
the point is exactly sufficient to turn one pulse
into the other. The remainder of the energy must bounce
because it does not cross the mid-point.

...Keith