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