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Old January 25th 08, 04:33 AM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

It is not too hard to use the concept of traveling waves and reflections
to derive the familiar reflection coefficient to SWR relationship. SWR
is a measurable and useful relationship that most hams are familiar
with. A clear path between SWR and traveling waves should make the
concepts more understandable and believable.

Power placed on a transmission line is placed over time. No matter how
small the time span interval we might want to examine, the span will
always be wide enough to include some quantity of power or energy. If
we desire, we can eliminate the time consideration and just consider
energy, but there is no need to do that. In this derivation, the
distinction between power and energy will be ignored.

We will assume that neither power nor energy can be stored at the
discontinuity in amounts greater than the natural storage capacity of
the lines. This assumption fixes the impedance of any waves to the
impedance of the transmission lines.

Begin the derivation by assuming that power is applied to a transmission
line with impedance Zo. A traveling wave moves down the transmission
line to a discontinuity which is composed of a second transmission line
or resistor with impedance Zl. The junction between the two lines is
like a window or thin plane, with Zo on one side and Zl on the other.

Upon encountering the discontinuity, the lead edge of the wave (and all
following energy levels) follow a "conservation of energy" rule that
requires energy to be preserved at all times. In other words, the
energy that has been conveyed to the junction by some interval of
applied power is not lost to heat, radiation, or storage, but will leave
the junction as fast as it arrives, and can be located, maintaining time
shape.

The following equation will be valid,

Pf = Pl + Pr

where Pf = power forward, Pl = power to load, and Pr = power reflected.

Use the voltage equivalent,

(Vf^2)/Zo = (Vl^2)/Zl + (Vr^2)/Zo

where Vf = forward voltage, Vl = load voltage, and Vr = reflected
voltage. The reflected wave will travel back down the main line with
impedance Zo.

Simplify the equation by rearranging and substitute SWR = Zl/Zo

(Vf^2)/Zo - (Vr^2)/Zo = (Vl^2)/Zl

SWR(Vf^2 - Vr^2) = (Vl^2)

Change the Vl into terms of Vf and Vr. Vl = Vf + Vr. We can do this
because at a reflection, traveling waves double back over one another,
adding voltage. Substitute Vl = Vf + Vr

SWR(Vf^2 - Vr^2) = (Vf + Vr)^2

Factor the polynomial on the left above

SWR(Vf - Vr)(Vf + Vr) = (Vf + Vr)^2

Divide both sides by (Vf + Vr)

SWR(Vf - Vr) = Vf + Vr

Divide both sides by Vf

SWR(1 - Vr/Vf) = 1 + Vr/vf

Vr/Vf = Reflection coefficient Ro, substitute

SWR(1 - Ro) = 1 + Ro

Rearrange to put Ro on one side

Ro + Ro*SWR = SWR - 1

Factor out Ro and rearrange

Ro = (SWR - 1)/(SWR + 1)


We have found the familiar relationship for the Reflection Coefficient
(Ro) and SWR using traveling wave logic.

Using identical logic but using current instead of voltage, the same
relationship can be found from

Zo*If^2 = Zl*Il^2 + Zr*Ir^2

By examining this derivation, the reader can see that power and energy
is reflected when a wave encounters a discontinuity. The reader can
also see that more power is present on the transmission line than is
delivered to the load.

Here is a link to additional information about transmission lines:

http://www.astrosurf.com/luxorion/qs...sion-line2.htm


73, Roger, W7WKB













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Old January 25th 08, 05:54 AM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Roger Sparks wrote in
:

... The reader can
also see that more power is present on the transmission line than is
delivered to the load.


The notion that "power is present" is a different one.

Owen
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Old January 25th 08, 06:07 AM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Owen Duffy wrote:
Roger Sparks wrote in
:

... The reader can
also see that more power is present on the transmission line than is
delivered to the load.


The notion that "power is present" is a different one.

Owen


It's reasonable, though. Looking at demo 4 with the TLVis1 program, you
can see that there's power all along the line except at specific nodal
points (where I or V is always zero), yet there's no power at all being
delivered to the load.

Roy Lewallen, W7EL
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Old January 25th 08, 06:32 AM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Roy Lewallen wrote in news:13pirk5h1cpt4f5
@corp.supernews.com:

Owen Duffy wrote:
Roger Sparks wrote in
:

... The reader can
also see that more power is present on the transmission line than is
delivered to the load.


The notion that "power is present" is a different one.

Owen


It's reasonable, though. Looking at demo 4 with the TLVis1 program, you
can see that there's power all along the line except at specific nodal
points (where I or V is always zero), yet there's no power at all being
delivered to the load.


Roy, my though was that on anything but a lossless line with VSWR=1,
instantaneous power (being the rate of flow of energy) varies with time
and location, so to make the statement that "power is present" and to
quantitatively compare it with the power at a point (being the end of the
line where the load is attached) seems to not be so reasonable.

If the statement is about average power in both cases, then it is
reasonable, obvious even, that power decreases with distance from the
source.

Perhaps "power is present" is an avoidance of the somewhat tautological
form "power flows to the load".

Owen
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Old January 25th 08, 07:14 AM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Owen Duffy wrote:
Perhaps "power is present" is an avoidance of the somewhat tautological
form "power flows to the load".


Want to muddy the waters even more? Ramo & Whinnery say:
"Another very important case is that of a perfect conductor,
which by definition must have a zero tangential component
of electric field at its surface. Then ^P^ [Poynting vector]
can have no component normal to the conductor and there can
be no power flow through the perfect conductor."
--
73, Cecil http://www.w5dxp.com


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Old January 25th 08, 05:08 PM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

On Fri, 25 Jan 2008 05:32:33 GMT
Owen Duffy wrote:

Roy Lewallen wrote in news:13pirk5h1cpt4f5
@corp.supernews.com:

Owen Duffy wrote:
Roger Sparks wrote in
:

... The reader can
also see that more power is present on the transmission line than is
delivered to the load.

The notion that "power is present" is a different one.

Owen


It's reasonable, though. Looking at demo 4 with the TLVis1 program, you
can see that there's power all along the line except at specific nodal
points (where I or V is always zero), yet there's no power at all being
delivered to the load.


Roy, my though was that on anything but a lossless line with VSWR=1,
instantaneous power (being the rate of flow of energy) varies with time
and location, so to make the statement that "power is present" and to
quantitatively compare it with the power at a point (being the end of the
line where the load is attached) seems to not be so reasonable.

If the statement is about average power in both cases, then it is
reasonable, obvious even, that power decreases with distance from the
source.

Perhaps "power is present" is an avoidance of the somewhat tautological
form "power flows to the load".

Owen


Nothing mysterious was hinted with the words "power is present".

As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored.

73, Roger, W7WKB


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Old January 25th 08, 05:15 PM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Roger Sparks wrote:
As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored.


The amount of energy existing in a transmission
line is exactly the amount required to support
the measured forward power and reflected power.

If the steady-state forward power is 200 watts,
the reflected power is 100 watts, and the lossless
transmission line is one microsecond long, it
contains 300 microjoules of energy. I don't
think that is a sheer coincidence. :-)
--
73, Cecil http://www.w5dxp.com
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Old January 25th 08, 07:23 PM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Roger Sparks wrote:

Nothing mysterious was hinted with the words "power is present".

As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored.


A better reason to avoid "stored" is that power isn't stored at all,
anywhere. Anyone who believes so should be able to tell us how many
watts of power are stored in a 50 Ah, 12 volt battery.

Roy Lewallen, W7EL
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Old January 25th 08, 08:22 PM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

On Fri, 25 Jan 2008 16:15:33 GMT
Cecil Moore wrote:

Roger Sparks wrote:
As I finished writing the post, I wanted to call attention to the assumption that the reflected power is true power and adds to the amount of energy "stored" on the transmission line. But "stored" is a word that implies static conditions, and static conditions are not found on a transmission line. So I substituted "present" for "stored.


The amount of energy existing in a transmission
line is exactly the amount required to support
the measured forward power and reflected power.

If the steady-state forward power is 200 watts,
the reflected power is 100 watts, and the lossless
transmission line is one microsecond long, it
contains 300 microjoules of energy. I don't
think that is a sheer coincidence. :-)
--
73, Cecil http://www.w5dxp.com


Yep, and if we quickly replaced the source with a termination having the impedance of the transmission line, 100 watts of power would continue to be delivered to the load for one microsecond, delivering 100 microjoules of energy. 100 watts of power would be delivered to the reflected wave termination for two microseconds, delivering 200 microjoules of energy.

The transmission line was a dynamic power storage device for two microseconds after the power source was disconnected.

73, Roger, W7WKB




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Old January 25th 08, 08:33 PM posted to rec.radio.amateur.antenna
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Default Derivation of Reflection Coefficient vs SWR

Roger Sparks wrote in
:

On Fri, 25 Jan 2008 05:32:33 GMT
Owen Duffy wrote:

Roy Lewallen wrote in news:13pirk5h1cpt4f5
@corp.supernews.com:

Owen Duffy wrote:
Roger Sparks wrote in
:

... The reader can
also see that more power is present on the transmission line than
is delivered to the load.

The notion that "power is present" is a different one.

Owen

....
Nothing mysterious was hinted with the words "power is present".

As I finished writing the post, I wanted to call attention to the
assumption that the reflected power is true power and adds to the
amount of energy "stored" on the transmission line. But "stored" is
a word that implies static conditions, and static conditions are not
found on a transmission line. So I substituted "present" for "stored.


Roger,

If you were wanting to mean "stored", perhaps it is energy that is stored
(over a non-zero length of line) rather than power. In that sense, energy
is "present" on the line, and the load may store energy (only if it has
reactive elements, and irrespective of whether it looks resistive at its
terminals).

Owen


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