Roy Lewallen wrote:

In circuits involving purely sinusoidal V and I of the same frequency,
the power waveform is actually a true sinusoidal function, except with a
D.C. offset. It doesn't at all resemble the output from a full wave
rectifier. The D.C. offset is the average value, and the frequency of
the sine portion is twice the frequency of V or I.

Yes, thanks Roy. I've had absolute value circuits on the brain all this
week.

Nevertheless, instantaneous power is simply the instantaneous amplitude
at time t of the (sin^2(wt))/2 function.

73, ac6xg

Roy Lewallen, W7EL

Jim Kelley wrote:

Richard Harrison wrote:

What is the value in watts or joules per second when seconds equal
zero? I venture an answer: It is the V x I x cos. theta at that instant,
but since work is power x time, it won`t do anything for you in zero
seconds.


I think you have a slight misconception about the meaning of
instantaneous power. AC power is a pseudo-sinusoidal function with
respect to time, like that of full-wave rectifier. The function has a
value, an instantaneous amplitude, at any time t which represents the
rate at which energy in Joules is moving past a point x at time t. It
may not be a terribly useful thing to know, but it isn't a ficticious
quantity.

73, ac6xg

Jim Kelley wrote:

Roy Lewallen wrote:

In circuits involving purely sinusoidal V and I of the same frequency,
the power waveform is actually a true sinusoidal function, except with a
D.C. offset. It doesn't at all resemble the output from a full wave
rectifier. The D.C. offset is the average value, and the frequency of
the sine portion is twice the frequency of V or I.


Yes, thanks Roy. I've had absolute value circuits on the brain all this
week.

Nevertheless, instantaneous power is simply the instantaneous amplitude
at time t of the (sin^2(wt))/2 function.

73, ac6xg

Only if the voltage and current are in phase. Here's the more general
solution (cosines could be used instead with equal validity):

Given that v = V * sin(wt + phiv)
i = I * sin(wt + phii)

Then p = v * i = VI * sin(wt + phiv) * sin(wt + phii)

The product of the sines can be transformed via a simple trig identity
to give

p = VI * 1/2[cos(phiv - phii) - cos(2wt + phiv + phii)]

The first term in the brackets is D.C. -- it's time-independent. The
second term is a pure sine wave. So the result is a pure sine wave with
a D.C. offset.

I've described the meaning and significance of the power waveform in at
least one earlier posting on this newsgroup. If anyone is interested who
can't find it on Google, I'll look it up and post the subject and date.

Roy Lewallen, W7EL

I apologize if my response seemed argumentative. It wasn't intended that
way. Certainly, sin^2(wt) has the same shape as the power waveform I
derived -- the only difference is its fixed D.C. term. And I certainly
agree that letting delta t approaching zero doesn't make any function of
t become zero at that point. And just as the analysis I've presented is
in your first year college electronics book, so is the point about delta
t in everyone's high school or first semester college calculus book. But
it's evident that some number of participants in this thread have either
forgotten, never seen, or never understood those basic principles. And
quite a few people either don't have any textbooks, don't understand
them, or are unwilling to open and read them. Hence the postings
containing information that you or I could find in moments.

Roy Lewallen, W7EL

Jim Kelley wrote:

You seem to be looking for an argument any way you can, Roy. ;-)
Sin^2(wt)/2 is the general form of any equation with the shape you
described in your previous post. Furthermore, instantaneous power can
be evaluated at any time t, irrespective of relative phase. The point
is simply that instantaneous power isn't necessarily zero as a result of
delta t's approaching zero.


Given that v = V * sin(wt + phiv)
i = I * sin(wt + phii)

Then p = v * i = VI * sin(wt + phiv) * sin(wt + phii)

The product of the sines can be transformed via a simple trig identity
to give

p = VI * 1/2[cos(phiv - phii) - cos(2wt + phiv + phii)]

The first term in the brackets is D.C. -- it's time-independent. The
second term is a pure sine wave. So the result is a pure sine wave with
a D.C. offset.

I've described the meaning and significance of the power waveform in at
least one earlier posting on this newsgroup. If anyone is interested who
can't find it on Google, I'll look it up and post the subject and date.


Yes. It's also in my first year college electronics book.

Thanks and 73,

AC6XG

I apologize if my response seemed argumentative. It wasn't intended that
way. Certainly, sin^2(wt) has the same shape as the power waveform I
derived -- the only difference is its fixed D.C. term. And I certainly
agree that letting delta t approaching zero doesn't make any function of
t become zero at that point. And just as the analysis I've presented is
in your first year college electronics book, so is the point about delta
t in everyone's high school or first semester college calculus book. But
it's evident that some number of participants in this thread have either
forgotten, never seen, or never understood those basic principles. And
quite a few people either don't have any textbooks, don't understand
them, or are unwilling to open and read them. Hence the postings
containing information that you or I could find in moments.

Roy Lewallen, W7EL

Cecil seemed to indicate that he thought delta t going to zero meant that
t was perpetually zero. I know he knows better than that.
73,
Tom Donaly, KA6RUH

GEE!!

If dV/dt = 0 then I must have maximum voltage. This has meaning.

Hmmm ... if dP/dt = 0 then I must have maximum 'PEAK' power. This might
have meaning IF I splatter all over the band. But for SWR purposes it
means nothing ... I think ... am I confused?

Deacon Dave, W1MCE
+ + +

Richard Harrison wrote:

Cecil, W5DXP wrote:
"If dt=0, then time stands still,---."

Yes, it is a spot frozen in time, a snapshot of slope at one instant.

What really interests us is average power over a half cycle or more, not
instantaneous power, energy`s rate of change, or power`s rate of change.

Best regards, Richard Harrison, KB5WZI

Deacon Dave wrote:
"---am I confused?"

If Dave is confused, I suffer the same confusion. The slope at the
maximum of a sine wave is zero. The slope is maximum at zero crossings
of an undistorted sine wave.

For the power sine wave, though the fact that a minus times a minus is a
plus results in 2x the voltage frequency, dP/dt=0 at maxima.

A question raised in this thread is, how can energy, which is joules per
second times seconds, be zero when the number of seconds is zero? The
answer seems obvious. Zero times anything is zero.

Best regards, Richard Harrison, KB5WZI

Tdonaly wrote:
as delta t goes to zero, the quantity dx/dt doesn't necessarily
also go to zero. If it did, no one would ever again have to get a permanent
headache studying calculus.

I am familiar with limits. Some make sense and some don't. The impedance,
frequency, and SWR of a transmission line with an SWR doesn't make sense
as V goes to zero and I goes to zero. Any old piece of transmission laying
in the yard has zero volts and zero amps. What is the SWR? What is the
frequency? What is the Z0 of the line?
--
73, Cecil http://www.qsl.net/w5dxp



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Jim Kelley wrote:
Nevertheless, instantaneous power is simply the instantaneous amplitude
at time t of the (sin^2(wt))/2 function.

And of what benefit is that value to the average ham operator?
--
73, Cecil http://www.qsl.net/w5dxp



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Richard Harrison wrote:

Cecil, W5DXP wrote:
"If dt=0, then time stands still,---."

Yes, it is a spot frozen in time, a snapshot of slope at one instant.

What really interests us is average power over a half cycle or more, not
instantaneous power, energy`s rate of change, or power`s rate of change.

Yeeaaahhhhh! We are not enamored with tits on a boar hog, are we Richard? :-)
--
73, Cecil http://www.qsl.net/w5dxp



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Tdonaly wrote:
Cecil seemed to indicate that he thought delta t going to zero meant that
t was perpetually zero.

If delta-t ever gets to zero, time stands still. All you can allow
delta-t to do is to approach zero. Once it reaches zero the ballgame
is over. Limit delta-t to a minimum of a yoctosecond and everything
will be perfectly OK.
--
73, Cecil http://www.qsl.net/w5dxp



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