On Jun 28, 5:27*pm, Keith Dysart wrote:
And yet, you have not located any errors in the math or the models.

Superposition of power *IS* an error! You add and subtract powers
willy-nilly as if that mathematical step were valid which it is not.
Two coherent 50w waves do not add up to a 100w wave, even using
average powers, except for the special case of zero interference where
the waves are 90 degrees out of phase with each other.

In order to use power as an energy tracking tool, we must be very
careful to ensure that there is a one-to-one correspondence between
energy and power, i.e. every joule passing a point in one second must
result in one watt of power (no VARs allowed). If that one-to-one
correspondence doesn't exist, no valid conclusion can be drawn from
tracking the power and any valid conclusion must be based on tracking
the energy which is no small task. The key is that there is no such
thing as imaginary energy. All energy is real. Some "power" is not
real.

A one-to-one correspondence does not exist in a standing wave.
Therefore, tracking power as if it were equivalent to energy in
standing waves is invalid. You have made that error for years.

One-to-one correspondence also does not exist over a fraction of a
wave. Therefore, instantaneous power is irrevelent in tracking the
energy. That's your latest error which is the same conceptual error as
before. In general, average power in the traveling waves over at least
one complete cycle (or over many cycles) has a one-to-one
correspondence to the average energy in the traveling waves. But that
one-to-one correspondence is more often than not violated within a
fraction of each cycle.

Here's a quote from "Optics", by Hecht, concerning power density
(irradiance).

"If however, the 'T' is now divided out, a highly practical quantity
results, one that corresponds to the average energy per unit area per
unit time, namely 'I'." - where 'I' is the irradiance (*AVERAGE* power
density).

If I calculate the Z0 of a 1/4WL transformer, I get two roots when I
take the square root of R1*R2. One of the roots is negative. If I ask
you to prove something is in error with the math that yielded a
negative Z0, could you find the math error? If not, does it follow
that you can find the transmission line with the negative
characteristic impedance existing in reality?

That's your argument in a nutshell. There may (or may not) be an error
in your math but it doesn't matter either way. The conclusions that
you reach from your math do not match reality so your math is a moot
point, i.e. there is no one-to-one correspondence between your math
and the real world.

If you have forgotten the importance of the one-to-one correspondence
concept in mathematics, now would be a good time to review that
concept. Without a one-to-one correspondence to reality, math is just
fantasy existing only in your mind.
--
73, Cecil, w5dxp.com

On Jun 29, 9:46*am, Cecil Moore wrote:
On Jun 28, 5:27*pm, Keith Dysart wrote:

And yet, you have not located any errors in the math or the models.

Superposition of power *IS* an error!

You have really latched on to that haven't you. I remember the good
ol'
days when you were superposing powers and the very same people with
whom you are arguing today were trying to convince you that
superposing
power was an invalid operation.

Now you have bought in to the rule and apply it even where it does
not apply.

Just because a sum is being computed does not mean that superposition
is at work.

When you add up the energies to validate that energy is not created
or destroyed, you are not superposing energy.

When I add up the energy flows for the same purpose, that does not
mean that superposition is at work.

From a basic calculus, if the energy balances, so must the energy
flow:

f(t), g(t), etc. are functions that describes the energy in
individual entities of a system with respect to time. Then

f(t) + g(t) + h(t) = k

is the expression that says the sum of all the energies must be
a constant, for energy is neither created nor destroyed.

From basic calculus

d(f(t) + g(t) + h(t))/dt = d(k)/dt
d(f(t) + g(t) + h(t))/dt = 0
d(f(t))/dt + d(g(t))/dt + d(h(t))/dt = 0

d(f(t))/dt is the derivitive with respect to time of the energy
(i.e. energy flow, or power) for the entity.

So it is completely valid to sum these energy flows to
track the energy within the system; as my example does.

You add and subtract powers
willy-nilly as if that mathematical step were valid which it is not.
Two coherent 50w waves do not add up to a 100w wave, even using
average powers, except for the special case of zero interference where
the waves are 90 degrees out of phase with each other.

True. You must always use the actual energy flows in the entity and
not the powers that are not real.

In order to use power as an energy tracking tool, we must be very
careful to ensure that there is a one-to-one correspondence between
energy and power, i.e. every joule passing a point in one second must
result in one watt of power (no VARs allowed).

Also true. And as is done in my example.

If that one-to-one
correspondence doesn't exist, no valid conclusion can be drawn from
tracking the power and any valid conclusion must be based on tracking
the energy which is no small task. The key is that there is no such
thing as imaginary energy. All energy is real. Some "power" is not
real.

Agreed. Especially the power in the waves that are being superposed.
That is why you need a fudge factor when you sum your partial powers
from your waves. I carefully do not sum partial powers from
superposed waves.

A one-to-one correspondence does not exist in a standing wave.
Therefore, tracking power as if it were equivalent to energy in
standing waves is invalid. You have made that error for years.

Please provide an example of such an error, preferably from one
of my expositions.

One-to-one correspondence also does not exist over a fraction of a
wave.

Of course it does. Energy must always balance; not just at the end
of the cycle.

Therefore, instantaneous power is irrevelent in tracking the
energy. That's your latest error which is the same conceptual error as
before. In general, average power in the traveling waves over at least
one complete cycle (or over many cycles) has a one-to-one
correspondence to the average energy in the traveling waves. But that
one-to-one correspondence is more often than not violated within a
fraction of each cycle.

Now you are allowing us to create and destroy energy as long as it
balances at the end of the cycle? Are you sure you want to do this?
It does not align with the well understood principle of conservation
of energy.

If I follow that rule, I just make the cycle long enough, say, my
lifetime, and I am golden.

But I do not hold out any hope for this.

Here's a quote from "Optics", by Hecht, concerning power density
(irradiance).

"If however, the 'T' is now divided out, a highly practical quantity
results, one that corresponds to the average energy per unit area per
unit time, namely 'I'." - where 'I' is the irradiance (*AVERAGE* power
density).

There you go again, assuming that the 'practical' challenges of optics
also apply to transmission lines. On a transmission line, we can
measure the voltage and the current. We do not suffer from the same
constraints as optics.

Use the tools that are available for transmission lines. Don't tie
two hands behind your back just because those tools do not work in
the optical domain.

....Keith

On Jun 30, 6:43*am, Keith Dysart wrote:
You have really latched on to that haven't you. I remember the good
ol' days when you were superposing powers and the very same people with
whom you are arguing today were trying to convince you that
superposing power was an invalid operation.

And I remember the good ol' days when you were still messing your
diapers. Do you still do that? I probably did try to superpose power
back when I was young and foolish but I learned the error of my ways
and corrected it.

When I add up the energy flows for the same purpose, that does not
mean that superposition is at work.

But you are NOT adding up the energy flows - you are adding up the
power. That's superposition of power and is a no-no. Power does not
obey any conservation of power principle. The instantaneous maximum
charge on a capacitor contains any amount of energy while power equals
zero. What is it about that concept that you don't understand?

True. You must always use the actual energy flows in the entity and
not the powers that are not real.

What happens when energy = 1 joule, and de/dt = 0 watts. This happens
all the time during an RF cycle so you are not using actual energy
flows. You are using power which goes to zero even when maximum energy
is still present. There is obviously NOT a one-to-one correspondence
between power and energy.

YOU CANNOT USE WATTS TO TRACK ENERGY UNLESS THERE IS A ONE-TO-ONE
CORRESPONDENCE BETWEEN WATTS AND JOULES.

Also true. And as is done in my example.

Actually false, as I have proved above. Please plot the joules, not
the watts.

Please provide an example of such an error, preferably from one
of my expositions.

At a current node in a standing wave, you claim there is zero power
which is true. You claim that since there is zero power, there is also
zero energy which is absolutely false since that is a voltage maximum
point and you will fry yourself if you grab it. The energy in the
voltage maximum is unrelated to the power being zero - just one more
proof that energy does not correspond to power. At the zero power
points, there is a maximum of energy in one of the fields which you
are ignoring.

Of course it does. Energy must always balance; not just at the end
of the cycle.

Energy must balance but power doesn't have to balance and usually does
not balance. Power only balances for special cases. You are using
power in the general case as an example so anything you say is
invalid.

Now you are allowing us to create and destroy energy as long as it
balances at the end of the cycle?

No, energy cannot be created or destroyed. But power is created and
destroyed all the time because there is no conservation of power
principle. Your glaring error is that you are using power as if it
were energy and it is not. I have given numerous examples. In an LC
oscillator, when all of the non-destructable energy is stored in the
capacitor, there is ZERO power, i.e. power has been destroyed. Why do
you refuse to discuss that fact of physics? 90 degrees later in the
cycle, power has been created.

Every time one of your instantaneous power curves crosses the zero
axis, power has been destroyed. Every time one of your instantaneous
power curves reaches a peak, power has been created.

We do not suffer from the same constraints as optics.

You suffer from the delusion that power obeys the same laws of physics
that energy does. You willy-nilly interchange power and energy. Until
you admit the error of your ways, there is little that can be done to
alleviate your ignorance.
--
73, Cecil, w5dxp.com

On Jun 30, 10:03*am, Cecil Moore wrote:
On Jun 30, 6:43*am, Keith Dysart wrote:

You have really latched on to that haven't you. I remember the good
ol' days when you were superposing powers and the very same people with
whom you are arguing today were trying to convince you that
superposing power was an invalid operation.

And I remember the good ol' days when you were still messing your
diapers. Do you still do that?

Warning! Warning! Descent in to scatology.

I probably did try to superpose power
back when I was young and foolish but I learned the error of my ways
and corrected it.

Learning is good. But now you want to over apply the rules.

When I add up the energy flows for the same purpose, that does not
mean that superposition is at work.

But you are NOT adding up the energy flows - you are adding up the
power.

Ummm. Energy flow is power. Joules/s!

If it helps, any place I have written 'power', please replace with
'energy flow'.

That's superposition of power and is a no-no. Power does not
obey any conservation of power principle.

Try the multiple pipe, water container, water flow example to
understand how flows must also be conserved if matter (or
energy) is not to be created or destroyed.

The instantaneous maximum
charge on a capacitor contains any amount of energy while power equals
zero. What is it about that concept that you don't understand?

Excellent concept. There is no conflict with conserving flows, or
energy
for that matter.

True. You must always use the actual energy flows in the entity and
not the powers that are not real.

What happens when energy = 1 joule, and de/dt = 0 watts. This happens
all the time during an RF cycle so you are not using actual energy
flows. You are using power which goes to zero even when maximum energy
is still present.

Yes, indeed. That is a fundamental possibility and occurs on
transmission lines with infinite VSWR.

There is obviously NOT a one-to-one correspondence
between power and energy.

Correct. Power is the time derivitive of energy. They are related
but definitely not one-to-one.

YOU CANNOT USE WATTS TO TRACK ENERGY UNLESS THERE IS A ONE-TO-ONE
CORRESPONDENCE BETWEEN WATTS AND JOULES.

This is quite incorrect. Energy flows must balance, otherwise energy
is being created or destroyed to sustain a difference in flow.

Also true. And as is done in my example.

Actually false, as I have proved above. Please plot the joules, not
the watts.

If it makes you happier, since it is a discrete simulation, you can
simply substitute joules per degree wherever there is a curve labeled
Watts. You need to scale, of course, between degrees and seconds, but
the shape and sums will be identical.

Please provide an example of such an error, preferably from one
of my expositions.

At a current node in a standing wave, you claim there is zero power
which is true.

Excellent. I am glad you have come to accept that.

You claim that since there is zero power, there is also
zero energy

I have never claimed that. In fact, the minima and maxima are
where the energy peaks will be found. I have claimed there is
no energy flow across (i.e. power) these points.

which is absolutely false since that is a voltage maximum
point and you will fry yourself if you grab it. The energy in the
voltage maximum is unrelated to the power being zero

Not quite. It follows from the functions for each. Since power is
the derivitive of energy, it should come as no surprise that
maximum energy occurs at the point of minimum power. In basic
calculus looking for the zero in the derivitive is how you
locate the maximum value of the curve.

- just one more
proof that energy does not correspond to power. At the zero power
points, there is a maximum of energy in one of the fields which you
are ignoring.

Of course it does. Energy must always balance; not just at the end
of the cycle.

Energy must balance but power doesn't have to balance and usually does
not balance. Power only balances for special cases. You are using
power in the general case as an example so anything you say is
invalid.

Unfortunately wrong. Energy flows must balance as well. Otherwise,
energy is coming from nowhere to sustain the flow.

Now you are allowing us to create and destroy energy as long as it
balances at the end of the cycle?

No, energy cannot be created or destroyed. But power is created and
destroyed all the time because there is no conservation of power
principle. Your glaring error is that you are using power as if it
were energy and it is not. I have given numerous examples. In an LC
oscillator, when all of the non-destructable energy is stored in the
capacitor, there is ZERO power, i.e. power has been destroyed.

Yes, indeed. At that instant, zero energy is flowing from the inductor
to the capacitor. But very soon, energy will be flowing from the
capacitor to the inductor. The balance is that the energy flowing
out of the capacitor is always and exactly equal to the energy flowing
in to the inductor. That is the energy flow balance. The only way for
this not to be true is for energy to be created or destroyed.

Why do
you refuse to discuss that fact of physics? 90 degrees later in the
cycle, power has been created.

One way of thinking about it, I suppose. But not too useful.

Instead, think that at every instant, the energy flow between the
entities in the experiment must balance.

Every time one of your instantaneous power curves crosses the zero
axis, power has been destroyed. Every time one of your instantaneous
power curves reaches a peak, power has been created.

I think you may be confused because you are only looking at the
flow in and out of a single entity. This is clearly not conserved.
Nor for that matter is the energy within that entity. It is the
total energy within the system that is conserved, just as it is
the total of the flows of energy between the entities within the
system that must be conserved.

Put more strictly: The sum of all the energy flows in to all of
the entities within the system must equal the energy flow in to
the system.

We do not suffer from the same constraints as optics.

You suffer from the delusion that power obeys the same laws of physics
that energy does. You willy-nilly interchange power and energy. Until
you admit the error of your ways, there is little that can be done to
alleviate your ignorance.

An intriguing accusation, but you do need to provide a concrete
example of where my analysis has gone wrong.

....Keith

Keith obviously understands the requirements for energy flow, but a
casual reader might draw the wrong conclusions. . .

Keith Dysart wrote:
. . .

This is quite incorrect. Energy flows must balance, otherwise energy
is being created or destroyed to sustain a difference in flow.

On the average, yes. But not moment-by-moment. Energy can be stored and
retrieved from storage, resulting in unequal energy flow (power) into
and out of a point. For a while. This was explicitly stated earlier when
discussing a capacitor, but I think it's important to make the
distinction here between instantaneous and average requirements. In
steady state, the average condition (energy flow balance) must be met
each cycle. That is, the total energy into a node over a cycle has to
equal the energy out of a node over a cycle.

. . .

Unfortunately wrong. Energy flows must balance as well. Otherwise,
energy is coming from nowhere to sustain the flow.

Ditto.

. . .

Yes, indeed. At that instant, zero energy is flowing from the inductor
to the capacitor. But very soon, energy will be flowing from the
capacitor to the inductor. The balance is that the energy flowing
out of the capacitor is always and exactly equal to the energy flowing
in to the inductor. That is the energy flow balance. The only way for
this not to be true is for energy to be created or destroyed.

Ditto.

. . .

Instead, think that at every instant, the energy flow between the
entities in the experiment must balance.

No, it doesn't, unless I'm misunderstanding the statement. At a given
instant, more energy can flow into a component (e.g., a capacitor or
inductor) than is flowing out, or vice-versa. But in steady state,
whatever flows in during one part of the cycle must flow out during the
remainder of the cycle.

Every time one of your instantaneous power curves crosses the zero
axis, power has been destroyed. Every time one of your instantaneous
power curves reaches a peak, power has been created.

I think you may be confused because you are only looking at the
flow in and out of a single entity. This is clearly not conserved.
Nor for that matter is the energy within that entity. It is the
total energy within the system that is conserved, just as it is
the total of the flows of energy between the entities within the
system that must be conserved.

Put more strictly: The sum of all the energy flows in to all of
the entities within the system must equal the energy flow in to
the system.

Again, only on an average or steady-state cycle-by-cycle basis. Great
inequalities can exist for shorter periods.

. . .

Like Keith, I firmly believe that an instantaneous time-domain analysis
is essential in understanding what really happens to the energy in an AC
system. Averaging reduces the amount of information you have -- if all
you know is the average value of a waveform, you have no way of going
back and finding out what the waveform was, out of an infinite number of
possibilities. If averaging is to be done, it should be done after you
calculate and understand what's going on at each instant, not before you
begin the analysis.

But it's also essential to make absolutely clear what conditions must be
met every instant, such as p(t) = v(t) * i(t), and which must be met
only on the average, such as energy in = energy out.

Roy Lewallen, W7EL

Considering the steady state...

If we accept the P(t) is the product of instantaneous voltage and
current, then there will be some points on any mismatched line where P(t)
is always positive. In between those points, P(t) will have positive and
negative excursions.

I think that it is a reasonable interpretation that at those points where
P(t) is always positive, then there is never at any instant, a flow of
energy away from the load, energy is never exchanged during a cycle
across those points, it always flows from source to load.

It may be that energy is exhanged during a cycle at the load end of the
line, and it may be that energy is exchanged during a cycle at the source
end of the line, but if the line is sufficiently long, there will exist
points where instantanous power is always positive, and therefore, energy
always flows in the load to source direction at those points.

The notion that a reflected wave in general conveys power over the entire
path from load to source is not consistent with the above. This notion is
emboddied in common language when talking about 'reflected power', but
the language belies the actual phenomena.

Owen

Owen Duffy wrote:
Considering the steady state...

If we accept the P(t) is the product of instantaneous voltage and
current, then there will be some points on any mismatched line where P(t)
is always positive. In between those points, P(t) will have positive and
negative excursions.

I think that it is a reasonable interpretation that at those points where
P(t) is always positive, then there is never at any instant, a flow of
energy away from the load, energy is never exchanged during a cycle
across those points, it always flows from source to load.

I'd make a small addition, that . . .there is never at any instant a
*net* flow of energy away from the load. . .

The problem is that I don't know of any way to keep track of a
particular bundle of energy -- it gets mixed together. So you could have
energy constantly flowing both ways through a point while maintaining a
net flow (power) in one direction and it would look just the same as
energy going only one way. Keith's DC thought experiments illustrate
these different approaches and some of their logical -- and illogical --
consequences.

Quite some time ago I wrote and made available a little graphic program
showing the voltage, current, power, and energy on lines under several
conditions. When a complete standing wave exists, there are points of
zero voltage and current and hence zero power. For one half the cycle
you can see energy moving into those points equally from both directions
(obviously being stored at the node), and during the other half, energy
is moving out of those points in both directions (being retrieved from
storage). One interpretation is that the energy arriving from the left
exits to the right, and vice-versa, and that fits neatly into the
concept of waves of energy simultaneously moving in both directions. Or
you can decide that the energy which came in from the right exits to the
right, and in from the left exits to the left. If that's your
interpretation, then you conclude that no energy ever crosses the
boundary. I think this is the genesis of Cecil's view that energy waves
somehow bounce off the standing wave node. Both interpretations fit
equally well with the observed net flow of energy but, like Keith's DC
thought experiments and Cecil's writings show, take you down quite
different paths when trying to divine some concept of what's
fundamentally happening.

. . .

Roy Lewallen, W7EL

On Jun 30, 5:55*pm, Roy Lewallen wrote:
Keith obviously understands the requirements for energy flow, but a
casual reader might draw the wrong conclusions. . .

Keith Dysart wrote:
. . .
This is quite incorrect. Energy flows must balance, otherwise energy
is being created or destroyed to sustain a difference in flow.

On the average, yes. But not moment-by-moment. Energy can be stored and
retrieved from storage, resulting in unequal energy flow (power) into
and out of a point. For a while. This was explicitly stated earlier when
discussing a capacitor, but I think it's important to make the
distinction here between instantaneous and average requirements. In
steady state, the average condition (energy flow balance) must be met
each cycle. That is, the total energy into a node over a cycle has to
equal the energy out of a node over a cycle.

. . .
Unfortunately wrong. Energy flows must balance as well. Otherwise,
energy is coming from nowhere to sustain the flow.

Ditto.

. . .
Yes, indeed. At that instant, zero energy is flowing from the inductor
to the capacitor. But very soon, energy will be flowing from the
capacitor to the inductor. The balance is that the energy flowing
out of the capacitor is always and exactly equal to the energy flowing
in to the inductor. That is the energy flow balance. The only way for
this not to be true is for energy to be created or destroyed.

Ditto.

. . .
Instead, think that at every instant, the energy flow between the
entities in the experiment must balance.

No, it doesn't, unless I'm misunderstanding the statement. At a given
instant, more energy can flow into a component (e.g., a capacitor or
inductor) than is flowing out, or vice-versa. But in steady state,
whatever flows in during one part of the cycle must flow out during the
remainder of the cycle.

I think you are misunderstanding, possibly because I am not expressing
as clearly as could be. I find it difficult to pick a vocabulary that
will not be confusing due to prior associations with the words. But
then words like 'entity' are too fuzzy.

The system I have in mind has ports through which energy can flow in
or out of the system and components inside the system which can
store energy. For such a system, the energy flowing in to ports
of the system minus the energy flowing out of ports must
equal the increase in energy being stored in the system.

This must be true at all times, or energy is being created or
destroyed; a bit of a no-no.

This system can be subdivided in to sub-systems for which this
energy flow balance must also hold.

As such, if the energy stored in one of the entities (e.g.
capacitor) is increasing, either net energy is flowing in to
the ports of the system, or the energy stored in some other
entity in the system is decreasing (or both). But the sum
of all the flows entering or leaving the ports, plus
the flows between the internal entities must balance on
a moment by moment basis.

Of course the expressions written to describe this will
be dependent on the details of the system. One must also
not forget to account for energy that leaves the system
as heat courtesy resistors. These can be thought of as
ports which only remove energy from the system.

The requirement for moment-by-moment balance is more stringent
than the requirement for average balance. The former inevitably
leads to the latter, but the converse is not true.

From Wikipedia, I have just learned that the concept I am attempting
to describe is known as a "Continuity equation".

Every time one of your instantaneous power curves crosses the zero
axis, power has been destroyed. Every time one of your instantaneous
power curves reaches a peak, power has been created.

I think you may be confused because you are only looking at the
flow in and out of a single entity. This is clearly not conserved.
Nor for that matter is the energy within that entity. It is the
total energy within the system that is conserved, just as it is
the total of the flows of energy between the entities within the
system that must be conserved.

Put more strictly: The sum of all the energy flows in to all of
the entities within the system must equal the energy flow in to
the system.

Again, only on an average or steady-state cycle-by-cycle basis. Great
inequalities can exist for shorter periods.

*. . .

Like Keith, I firmly believe that an instantaneous time-domain analysis
is essential in understanding what really happens to the energy in an AC
system. Averaging reduces the amount of information you have -- if all
you know is the average value of a waveform, you have no way of going
back and finding out what the waveform was, out of an infinite number of
possibilities. If averaging is to be done, it should be done after you
calculate and understand what's going on at each instant, not before you
begin the analysis.

But it's also essential to make absolutely clear what conditions must be
met every instant, such as p(t) = v(t) * i(t), and which must be met
only on the average, such as energy in = energy out.

Roy Lewallen, W7EL

On Jul 4, 8:08*pm, Keith Dysart wrote:
The system I have in mind has ports through which energy can flow in
or out of the system and components inside the system which can
store energy. For such a system, the energy flowing in to ports
of the system minus the energy flowing out of ports *must
equal the increase in energy being stored in the system.

This must be true at all times, or energy is being created or
destroyed; a bit of a no-no.

But you are not tracking energy - you are tracking power. As Roy has
said, there is no requirement that instantaneous power must balance.
Where are the stored energy terms in any of your instantaneous power
equations? How do you handle the difference in dimensions between
energy and power? The only condition for which NET power must balance
is during a time interval in which there is zero NET stored power,
e.g. during one cycle.

I have rev'ed my zero interference article to include the following
statement:

"Over a time period of many cycles, e.g. one second at MHz
frequencies, the net average energy and the net average power are
related by joules/second. Thus, if certain conditions are met, net
average power can be used to track the net average energy flow based
on the conservation of energy principle. However, at time intervals of
less than one cycle, as exists for instantaneous power, power cannot
be used to track energy because energy is often stored in a reactance,
is not moving at that instant, and is therefore technically not power.
In fact, unlike energy, power often appears and disappears. There are
special cases where average power in joules/second can be used to
track average energy in joules but instantaneous power is not one of
those special cases."
--
73, Cecil, w5dxp.com

Keith Dysart wrote:

I think you are misunderstanding, possibly because I am not expressing
as clearly as could be. I find it difficult to pick a vocabulary that
will not be confusing due to prior associations with the words. But
then words like 'entity' are too fuzzy.

. . .

The problem I had was the use of "energy flow balance" which implied
equal energy flow into and out of any point at any time. Your more
detailed description explicitly includes stored and dissipated energy,
which as we've both said, makes it possible for energy flow into and out
of a point to be unequal at times, while obeying conservation of energy
by being equal on the average (when dissipated energy is accounted for
as removed from the system). The more detailed detailed description
should help alleviate misunderstandings some readers might have had.

. . .

From Wikipedia, I have just learned that the concept I am attempting
to describe is known as a "Continuity equation".

I can't recall ever having come across that term, but then I'm not a
physicist.

Roy Lewallen, W7EL