Complex Z0 [Corrected]
 

Complex Z0 [Corrected]
-
Maybe it has already been noticed,
but anyway,
it seems [this time] that,
given a uniform transmission line with
complex characteristic impedance,
the magnitude of the reflection coefficient
for any passive terminal load
is lower or equal than
-
Sqrt([1 + Sin(Abs[t0])]/[1 - Sin(Abs[t0])]),
-
where t0 is the argument of Z0.
-
Sincerely,
-
pez,SV7BAX
&
yin,SV7DMC

Complex Z0 [Corrected]
-
Maybe it has already been noticed,
but anyway,
it seems [this time] that,
given a uniform transmission line with
complex characteristic impedance,
the magnitude of the reflection coefficient
for any passive terminal load
is lower or equal than
-
Sqrt([1 + Sin(Abs[t0])]/[1 - Sin(Abs[t0])]),
-
where t0 is the argument of Z0.
-
===============================

It is true the formula gives the greatest possible
magnitude of the reflection corfficient, Rho, for
any given value of the angle of Zo.

As t0 approaches -45 degrees, Rho approaches
1+Sqrt(2) = 2.414

As you must know, the formula is obtained by
differentiating Rho with respect to the angle of
Zo and then equating to zero.

It provides proof that values of Rho greater than
unity do exist. But a worship of mathematical logic
is not part of any religion on this newsgroup.
---
Reg, G4FGQ

Reg Edwards wrote:

I challenge anyone to find a reflectometer
calculator that shows rho 1.

First of all, please define precisely what is a
'reflectometer calculator' ?

Is it hardware or is it software?

Heh, heh, if it's software, Reg will have one by tomorrow. :-)
--
73, Cecil http://www.qsl.net/w5dxp



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"Reg Edwards" wrote in message ...
I challenge anyone to find a reflectometer
calculator that shows rho 1.


First of all, please define precisely what is a
'reflectometer calculator' ?

Is it hardware or is it software?


Hehe, well, this really is an amateur group,
isn't it! Sorry, i keep forgetting.

If you work in the RF field long enough,
you eventually come across these cardboard slide-rules
that will give you SWR versus rho versus mismatch loss.

Every one that i have seen (HP, Roos, etc.) have a
scale for rho that goes from zero to one.

Also, i have never see a negative SWR in my life.

Besser does mention that when you have an active
device, that you can have a rho 1, and actually
a Return GAIN instead of a Return Loss.

Some people here seem to incorrectly think you can
have a return gain with a passive network...


Slick

Dr. Slick wrote:
Some people here seem to incorrectly think you can
have a return gain with a passive network...

Does anyone remember what is the absolute value of a
complex number?
--
73, Cecil http://www.qsl.net/w5dxp



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Cecil Moore wrote:
Some people here seem to incorrectly think you can
have a return gain with a passive network...

Does anyone remember what is the absolute value of a
complex number?

Found the answer in, "Higher Mathematics for Engineers and Physicists".
I suspect the square of the absolute value of the voltage reflection
coefficient is the volt-amp reflection coefficient, not the power
reflection coefficient.

With a complex characteristic impedance, what is being reflected is
volt-amps. I suspect the reflected volt-amps can be higher than the
incident volt-amps. I seriously doubt that the reflected watts
can be higher than the incident watts. The correct *power* reflection
coefficient therefore may be something like |Re(rho)|^2 where 'Re'
means "the real part of". The simpler |rho|^2 may be the volt-amp
reflection coefficient when Z0 is complex.

Using deductive reasoning, since the real part of the voltage
reflection coefficient cannot be greater than 1.0, it seems to
me that |1.0|^2 may be the maximum power reflection coefficient.
The complex voltage reflection coefficient squared may be the
volt-amp reflection coefficient which can be greater than 1.0.

In a transmission line with a complex characteristic impedance, the
reflected voltage and reflected current would not be in phase.
Therefore, their product would be volt-amps, not watts. Reflected
watts could be obtained from Vref*Iref*cos(theta) which would always
be less than (or equal to) Vref*Iref.
--
73, Cecil http://www.qsl.net/w5dxp



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In the numerical example I posted, I calculated the average real power
incident at the load (that is, the power calculated from the forward
voltage and current waves), and the average real reflected power at the
load (that is, the power calculated from the reverse voltage and current
waves). The "reflected power" is greater than the "incident power".
However, the net power exiting the line and entering the load is a
positive value. That's because the net power isn't equal to the "forward
power" minus the "reverse power" at that point. I gave the equation for
total power in that analysis, and if you plug in the numbers, you'll see
that the total power is correct.

If you are interested in calculating the "reactive power" for some
reason, you can easily do so from the complex voltages and currents
which have been calculated for you.

And for those who are wondering about your question, the absolute value
of a complex number is the magnitude of that number. In the example I
gave, all the complex values were given in polar form, with the first
part being the magnitude.

Roy Lewallen, W7EL

Cecil Moore wrote:
Cecil Moore wrote:

Some people here seem to incorrectly think you can
have a return gain with a passive network...


Does anyone remember what is the absolute value of a
complex number?


Found the answer in, "Higher Mathematics for Engineers and Physicists".
I suspect the square of the absolute value of the voltage reflection
coefficient is the volt-amp reflection coefficient, not the power
reflection coefficient.

With a complex characteristic impedance, what is being reflected is
volt-amps. I suspect the reflected volt-amps can be higher than the
incident volt-amps. I seriously doubt that the reflected watts
can be higher than the incident watts. The correct *power* reflection
coefficient therefore may be something like |Re(rho)|^2 where 'Re'
means "the real part of". The simpler |rho|^2 may be the volt-amp
reflection coefficient when Z0 is complex.

Using deductive reasoning, since the real part of the voltage
reflection coefficient cannot be greater than 1.0, it seems to
me that |1.0|^2 may be the maximum power reflection coefficient.
The complex voltage reflection coefficient squared may be the
volt-amp reflection coefficient which can be greater than 1.0.

In a transmission line with a complex characteristic impedance, the
reflected voltage and reflected current would not be in phase.
Therefore, their product would be volt-amps, not watts. Reflected
watts could be obtained from Vref*Iref*cos(theta) which would always
be less than (or equal to) Vref*Iref.

Nope.

I'm glad you're finding the time to look over the example. I see you've
stumbled into the first problem with assigning a power to each
individual wave. I'm afraid you'll encounter additional dilemmas as you
dig deeper into it.

Roy Lewallen, W7EL

Cecil Moore wrote:
Roy Lewallen wrote:

The "reflected power" is greater than the "incident power".


So if the load is put into a black box, there is more power coming
out of the box than is going in?

Roy Lewallen wrote:
I see you've stumbled into the first problem with assigning
a power to each individual wave.

I've stumbled upon the first problem in your solution. :-)
What are Z0 and ZLoad again? Is Z0 physically possible? Is
ZLoad physically possible?
--
73, Cecil http://www.qsl.net/w5dxp



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Roy Lewallen wrote:
No, the average Poynting vector points toward the load.

That automatically says Pz- is not larger than Pz+. There are only
two component Poynting vectors, 'Pz+' forward and 'Pz-' reflected.

If so,
surely you came up with the same result, including the third power term.
If you haven't done the derivation, or if you'd like to compare your
derivation of total average power with mine, I'll be glad to post it.

Assuming coherent waves, all wave components flowing toward the load
superpose into the forward wave and all wave components flowing away
from the load superpose into the reflected wave. Since there are only
two directions, there cannot exist a third wave. If your average Poynting
vector points toward the load, Pz- cannot possibly be larger than Pz+.
But feel free to post the derivation.
--
73, Cecil http://www.qsl.net/w5dxp



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