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