On Mar 23, 10:31 am, Cecil Moore wrote:
Keith Dysart wrote:
So yes, the phrase "conservation of power" is
appropriately descriptive and follows from
conservation of energy.

You have a contradiction built into your concepts.
You have argued that the instantaneous power dissipated
in the source resistor is not equal to the instantaneous
forward power component plus the instantaneous reflected
power because power must be conserved at each instant of
time.
That is simply not true.

Bzzzzzzzzzzztt.

I'm telling that energy must be conserved at each instant
of time but power does not have to be conserved at each
instant of time.

Bzzzzzzzzzzztt.

Energy can obviously be stored in a
battery or network reactance for dissipation later in
time.

Indeed. And forgetting to include such flows in the summation
would be a serious error.

For example, from the example,
Ps(t) = Prs(t) + Pg(t)
includes Pg(t) which accounts for the energy stored in the
line and later returned. There are no missing flows in
this equation. And Ps(t) also accounts for energy absorbed
in the voltage source. Only the source resistor has a
unidirectional energy flow.

Do you require that the power used to charge a battery
be instantaneously dissipated in the battery?

The energy flow into the battery is exactly and always
accounted for by the energy flow that heats the battery
and the energy flow consumed in the reversable chemical
reaction.

The instantaneous flows always sum appropriately to
satisfy conservation of energy. Of course, if one forgets
a flow, then the sum will not balance.

Of course
not! That's true for a dummy load but NOT for a battery.
There is no reason to require dissipation of power at
each instant of time to balance. Since energy can be
stored, there is no such thing as conservation of
instantaneous power,

Of course there is, but you must include the flows into
the elements that store energy as I have done.

only of instantaneous energy.

Where is "cos(theta)" in this?
And what "theta" is to be used?

How many times do I have to explain this? For instantaneous
values of voltage, if the sign of the two interfering voltages
are the same, theta is zero degrees and the cosine of theta is
+1.0. If the sign of the two interfering voltages are opposite,
theta is 180 degrees and the cosine of theta is -1.0.

A strange of way of looking at it. It seems easier just
to say that there is no theta. And add the voltages.

But no matter, I have figured out where your extra term
comes from.

Let us a consider a simple circuit with two voltage sources
(V.s1 and V.s2) in series, connected to a resistor R.

Using superposition we have
V.s1 = R * Ir.s1
and
V.s2 = R * Ir.s2

So
Vr.tot = V.s1 + V.s2
and
Ir.tot = I.s1 + I.s2
This is superposition, and all is well.

The power dissipated in the resistor is
Pr = (Vr.tot)**2 / R
but we could also derive Pr in terms of V.s1 and V.s2
Pr = (V.s1 + V.s2) (V.s1 + V.s2) / R
= ((V.s1)**2 + (V.s2)**2 + (2 * V.s1 * V.s2) ) / R
= (V.s1)**2 / R + (V.s2)**2 / R + (2 * V.s1 * V.s2) / R

Now some people attempt to compute a power for each of
the contributing voltages across the resistor and obtain
Pr.s1 = (V.s1)**2 / R
Pr.s2 = (V.s1)**2 / R
When these are added one obtains
Pr.false = (V.s1)**2 / R + (V.s2)**2 / R
which, by comparison with Pr above can easily be seen
not to be the power dissipated in the resistor. Pr.false
is missing the term ((2 * V.s1 * V.s2) / R) from Pr. It
is for this reason that it is said that one can not
superpose powers. Simply stated, when powers are dervived
from the constituent voltages that are superposed, it is
not valid to add the powers together to derive the total
power.

Of course for the most part, powers being added are not
powers derived from the constituent voltages of a total
voltage, so in most cases it is quite valid to add powers
and expect them to sum to the total power.

But what do you do if a circuit is superposing two
voltages and you are presented with information about
the circuit in terms of powers. Well then you can add
the powers and include a correction term.

Assume
Pr = Pr.false + Pr.correction
= Pr.s1 + Pr.s2 + Pr.correction
But can we find a Pr.correction? It has to correct for
the term missing from Pr.false but present in Pr,
i.e. ((2 * V.s1 * V.s2) / R).

Restated in terms of power, ((2 * V.s1 * V.s2) / R)
becomes
2 * sqrt(P.s1 * P.s2)
But sqrt has two solutions so we have to write
Pr = Pr.s1 + Pr.s2 +/- 2 * sqrt(P.s1 * P.s2)
which should look very familiar.

As you have correctly pointed out, the sign of
Pr.correction is negative when the signs of the
constituent voltages are different and positive when
they are the same. The reason for this can easily
be seen from the derivation of Pr.correction.

This Pr.correction term has nothing to do with
interference, it is the correction required when it is
desired to add two powers computed from the superposing
constituent voltages of an actual total voltage across
an element and derive the energy flow into the
element.

Note that there is no hint that Pr.correction needs to be
stored when it is negative nor come from somewhere when
it is positive. It is, after all, just a correction that
needs to be applied when one wants to compute the total
power given two powers derived from the constituent
voltages of superposition.

Since one needs to know the constituent voltages to
determine the sign of Pr.correction, why not just use
superposition to compute the total voltage and then
derive the power? It would be much simpler. With no
need for Pr.correction, interference, storage and
release of interference 'energy', ...
(Of course, over in optics land it is difficult to
measure the voltage, so suffering the pain of using
powers is probably appropriate).

This analysis also makes clear the nature of powers
computed from the constituent voltages of superposition.
These powers do not represent real energy flows. As
discussed far above, real energy flows can be summed
to test for conservation of energy. When energy flows
do not sum appropriately, then either an energy flow
is missing, or one is attempting to sum powers which
are not real, for example, having been computed from
the constituent voltages of superposition (e.g. Pfor
and Pref in a transmission line).

....Keith