Reg Edwards wrote:

Anybody who doesn't agree that

rho = ( ZL - Zo ) / ( ZL + Zo )

means something can depart forthwith and forever hold their peace.

source----50 ohm feedline---+---150 ohm feedline---load

What is rho at point '+'? What is s11 at point 'x'?
--
73, Cecil http://www.qsl.net/w5dxp



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"Cecil Moore" wrote in message
...
Reg Edwards wrote:

Anybody who doesn't agree that

rho = ( ZL - Zo ) / ( ZL + Zo )

means something can depart forthwith and forever hold their peace.

source----50 ohm feedline---+---150 ohm feedline---load

What is rho at point '+'? What is s11 at point 'x'?

rho can't be computed unless you know what the load impedance is, and what
the length of the 150 ohm line is in wavelengths.

and no one but you cares what s11 is.

David Robbins wrote:
rho can't be computed unless you know what the load impedance is, and what
the length of the 150 ohm line is in wavelengths.

and no one but you cares what s11 is.

:-) s11 is God's rho! :-) What good is an image rho anyway?
Help! Peter!
--
73, Cecil http://www.qsl.net/w5dxp



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Reg:

[snip]
"Reg Edwards" wrote in message
...
Anybody who doesn't agree that

rho = ( ZL - Zo ) / ( ZL + Zo )

means something can depart forthwith and forever hold their peace.


To assist brain cells a pencil and paper will be found useful
[snip]

Here let me do a little pencil and paper work... er... algebra in ASCII to
expose some of the "complexities" of complex Zo and complex rho.

rho is the ratio of the reflected voltage wave "b" to the incident voltage
wave "a", rho = b/a and rho is in general a complex function of angular
frequency w = 2*pi*f where f is "cyclic" frequency in Hz simply because both
ZL and Zo can themselves be complex. rho may be considered to be a
"transfer function" which gives the reflected voltage "b" or voltage echo,
sometimes known as the "talker echo", resulting from the application of the
incident voltage "a".

b/a = rho(jw) = [(ZL(jw) - Zo(jw)/(ZL(jw) + Zo(jw)] = |rho(w)|
exp[j*phie(w)]

Where phie(w) is known as the "echo phase" and |rho(w)| is the magitude of
the reflection coefficient.

Now taking natural logarithm's of the inverse of rho one finds that:

ln[1/rho(jw)] = -ln|rho(w)| - j*phie(w) = Ae(w) - j*phie(w)

Where Ae(w):

Ae(w) = -ln|rho(jw)|

Measures the echo attenuation in Nepers and is the so-called "Echo
Attenuation" or "Return Loss", in Nepers (Np) Ae(w) measures how much the
talker echo or reflected voltage echo "b" is attenuated below the incident
voltage wave "a".

Aside: Europeans seem to prefer the term Echo Attenuation and the symbol
Ae(w) and Np units, which is also my preference, however North American's
seem to prefer the term "Return Loss" and symbol RL and prefer to use dB's
units rather that Np units. For those who need to convert from Np to dB the
conversion from Np to dB is simply dB = 20 log(e) = 8.686 Np.

phie(w) is the so-called "Echo Phase" or "Return Phase" measured in radians
and measures the phase lag of the voltage echo as compared to the phase
angle of the incident voltage.

Now for some stuff that should really interest Cecil. :-)

Differentiating the echo phase phie(w) with respect to w, one obtains the
well-known "Echo Group Delay", this echo group delay or as it is sometimes
called, the "Return Delay" is always a real function and is given by.

taugre(w) = d[phie(w)]/dw

and has units of seconds.

taugre(w) measures the group delay of the returned voltage echo as a
function of frequency. In a differential bandwidth dw the value of
taugre(w) measures the average delay of a "packet" of energy launched by the
incident wave "a" in that differential frequency band dw and returned to the
source in the voltage echo/

An even more interesting quantity is the echo group velocity or vge(w) which
is simply the reciprocal of taugre(w)

vge(w) = 1/taugre(w)

and has units of inverse seconds. The vge(w) may never be faster than the
speed of light in the Zo media.

vge(w) gives the actual group velocity of packets or groups of frequencies
in the echo as it passes from the source back around to the source again.
It is a function of frequency and illustrates that in waveguides with
complex Zo not all frequencies in the echo travel with the same velocity and
some frequencies arrive back at the source before the others. Of course no
such thing occurs with real Zo lines! Real Zo lines are sooooo
unintersting!

Another interesting quantity which I am sure Cecil will enjoy is the
so-called echo phase delay or tauphe(w)

tauphe(w) = phie(w)/w

The echo phase delay is an interesting function and I will leave it to the
reader to work out its' physical significance. This leads to the curious
echo phase velocity or vpe(w)

vpe(w) = 1/tauphe(w)

It is interesting to note that vpe(w) can actually exceed the speed of light
in the Zo media. In a transmission system with a real Zo of course vpe =
vge. Dull and uninteresting when Zo is real, huh?

Finally the arcane "echo signal front delay" tausfe(w), when it exists
equals the limit.

tausfe(w) = limit as w - infinity of the ration phie(w)/w

And of course the reciprocal of the echo signal front delay gives the echo
signal front velocity vsfe(w) when it exists, thus:

vsfe(w) = 1/tausfe(w) when it exists.

It is left as an exercise for the reader to find out under what physical
circumstances this exotic velocity exists and just what it's physical
meaning might be...

A quantity of great great interest to Cecil will be the value of the echo
group delay evaluated at the origin (DC).

taugre(w) evaluated at w = 0 [DC] gives the first time moment of the echo!

taugre(0) = T*area under the voltage echo = Integral from 0 to infinity of
t*b(t)*dt

taugre(0) when divided by the area under the echo voltage gives the average
time T by which the echo is delayed.

T = taugre(0)/area under b(t).

It turns out that this is an extremely useful value to know if one is
interested in designing or operating an effective and feasible analog or
digital talker echo canceller to suppress echos on the line and allow the
receiver to receive remotely generated signals in the face of extremely
strong incident signals.

The design and operation of talker echo cancellers for transmission systems
with complex Zo is quite a complicated and challenging task.

That's enough for tonight's scribbling...

I'd like some thoughts and comments on all of these functions and about
comparisons between such functions for transmission systems with real Zo and
those with complex Zo.

--
Peter K1PO
Indialantic By-the-Sea, FL.

On Fri, 05 Sep 2003 15:02:17 -0500, Cecil Moore
wrote:

Reg Edwards wrote:

Anybody who doesn't agree that

rho = ( ZL - Zo ) / ( ZL + Zo )

means something can depart forthwith and forever hold their peace.

source----50 ohm feedline---+---150 ohm feedline---load

What is rho at point '+'? What is s11 at point 'x'?

Hi Cecil,

You've been offered an instrument, a method and you can do this at the
bench - but that would settle it wouldn't it?

73's
Richard Clark, KB7QHC

Richard Clark wrote:
You've been offered an instrument, ...

Unfortunately, I know better than to bend over. :-)
--
73, Cecil http://www.qsl.net/w5dxp



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On Sat, 06 Sep 2003 00:01:22 -0500, Cecil Moore
wrote:

Richard Clark wrote:
You've been offered an instrument, ...

Unfortunately, I know better than to bend over. :-)

Would that put your head under water?

73's
Richard Clark, KB7QHC

Reg Edwards wrote:
I've just spent 3 hours on this. Please address any questions to Dr Slick.

Thank you. The effort is appreciated.

....Keith