wrote in message ...
Cecil Moore wrote:

Dr. Slick wrote:
If you agree that the Pref/Pfwd ratio cannot be greater than 1
for a passive network, then neither can the [Vref/Vfwd]= rho be
greater
than 1 either.

Sqrt(Pref/Pfwd) cannot be greater than one. (Z2-Z1)/(Z2+Z1) can be
greater than one. Both are defined as 'rho' but they are not
always equal.

On the other hand...

if
Vrev = rho * Vfwd
then
rho^2 = (Vrev^2/Z0) / (Vfwd^2/Z0)

So
(Vrev^2/Z0) / (Vfwd^2/Z0)
can be greater than one.

If Sqrt(Pref/Pfwd) can not be greater than 1
then either
Pfwd is not equal to (Vfwd^2/Z0)
or
Prev is not equal to (Vrev^2/Z0)

Intriguing result, is it not?

indeed... and directly to the crux of the whole problem. there are a few
too many powers being tossed around here without proper definition. i was
just looking at section 7.1 in 'Basic Circuit Theory' by Desoer and Kuh that
kind of sums up all the problems this discussion has been having in its
title "Instantaneous, Average, and Complex Power". for some reason this
section seems to have many more highlighted formulas than much of the rest
of the book.

I think many of our problems come from mis-applying power formulas to the
wrong cases. for example, the well known P=VI=V^2/R=I^2R takes a bit of
modification to work for general complex impedances.

lets look at the simple one first:
Instantaneous power: p(t)=v(t)i(t)
aha! you say, there it is, nice and simple.... but not so fast. this is in
the time domain. the Vfwd and Vrev we throw around so easily in phasor
notation when talking about transmission lines aren't the same thing and
can't be so easily converted... in fact, in going to the phasor notation you
intentionally throw away time information.... so in this case when you write
v(t) and i(t) in their full form they look like:
v(t)=|V| cos(wt+/_V)
i(t)=|I| cos(wt+/_I)
where w=omega, the angular rate, and /_ is used to denote the relative angle
at t=0....
thus expanding out this power formula you get and ugly thing:
p(t)=.5 |V| |I| cos(/_V-/_I) + .5 |V| |I| cos(2wt+/_V+/_I)
which when averaged becomes:
Pav=.5 |V| |I| cos(/_V - /_I)
which hopefully looks familiar to people out there who deal with power
factors and such.

Now, the odd one...
Complex Power:
P=.5 V I*
where V and I are the phasor representations used in sinusoidal steady state
analysis, and I* is the conjugate of I.
now here is where it gets weird...substituted in the exponential forms of
the phasors:
P=.5 |V| |I| e^(j(/_V-/_I))
oh for a good way to represent equations in plain text... but anyway, here
is a similar notation to the instantaneous power above, but there is no time
in it... only the magnitudes of the voltage and current multiplied by a
complex exponential from the difference in their phase angles. this can be
expanded into a sin/cos expression to separate the real and imaginary parts
like this:
P=.5 |V| |I| cos(/_V-/_I) + j.5 |V| |I| sin(/_V-/_I)
and from this you can show that the real part of this is the average power,
so:
Pav=.5 |V| |I| cos(/_V-/_I)
but since we are in phasor notation we can transform this one more time
using
V=ZI and I=YV (Y=1/Z, the complex admittance)
to get:
Pav= .5 |I|^2 Re[Z] = .5 |V|^2 Re[Y]

there are also a couple of important notes to go with this. in a passive
network Re[Z] and Re[Y] are both =0 which also constrains cos(/_V-/_I)
=0... essentially, no negative resistances in passive networks... which of
course then results in Pav always being positive(or zero).

now wait a minute you may say.. we have two different Pav equations... what
happens if we equate them??? (well, actually we have 4 different equations,
so lets play around a bit)

from time domain:
Pav=.5 |V| |I| cos(/_V - /_I)

from sinusoidal steady state:
Pav=.5 |V| |I| cos(/_V-/_I)
Pav= .5 |I|^2 Re[Z]
Pav= .5 |V|^2 Re[Y]

well, what do you know, the two methods give the same result (the first
formula of the sss method is the same as the time domain formula). but what
do the other two with the impedance and admitance do for us.... they show
that power in phasor calculations is not quite as simple as the
P=VI=V^2/R=I^2R we are used to... in fact, what are those equations good
for? basically, just for resistive circuits where I and V are in phase...
see section 7.3 of that reference for derivation of the I^2R power formula
for resistive loads... and also for how the rms value gets rid of that pesky
factor of .5 in all those equations to make I^2R work for sinusoidal
voltages... so I^2R and V^2/R are REALLY only good for rms voltages and
resistive loads. so you can't just use P=V^2/Z0 in the general case! but
didn't we all know that? after all, why is there a power factor added to
the calculation when working with reactive loads?? and why can you have
LOTS more reactive power in a circuit than real power??? don't believe in
reactive power? wait till the report about the blackout comes out, it was
basically run away loss of control of reactive power that probably resulted
in circulating currents that brought the grid to its knees.

so you are right:

Pfwd is not equal to (Vfwd^2/Z0)
and
Prev is not equal to (Vrev^2/Z0)

but by definition:

Vrev = rho * Vfwd

now, who really read and understood that???