# SWR Question.

Discussion in 'General Technical Questions and Answers' started by KD8GFC, Jul 17, 2008.

Not open for further replies. 1. Tim, you appear to be giving attributes to the phase constant that are simply not there.

The "fundamental" definition of B (usually termed the "phase constant" or the "phase-shift constant") is that it is the imaginary part of the propagation constant (gamma) as follows:

gamma = square root of (R + jwL)(G + jwC) = a + jB

where a, B, and w are usually written as the Greek letters alpha, beta, and omega, respectively.

R, L, G, and C are the per unit length values of the series resistance, series inductance, shunt conductance, and shunt capacitance of the line. For a given transmission line driven by a sinusoidal source at an angular frequency w, the value of B is CONSTANT. (If the line is lossless, as we have been assuming here, then the real part of gamma - the attenuation constant a - is zero.) These various relationships are all derived in terms of the forward and reflected TRAVELING waves, not the total wave discussed by Kraus.

As I mentioned before, the concept of the instantaneous phase velocity of the TOTAL wave appears to be rarely mentioned in the textbooks. It's an interesting mathematical curiosity, but engineers who analyze and design transmission lines can do quite well without it.

73, K5MC

2. Mickey,

Isn't phase velocity equal to omega divided by Beta?

vp = w/B.

If that is so, then B=w/vp

If phase velocity changes how can you say that B doesn't change?

Please note that B = d(phi)/dx -- the very definition of the propagation constant.

Are you saying that B does *NOT* equal d(phi)/dx?

B is considered a constant just like phase velocity is considered a constant -- when in actuality they are both variables and only the average is constant.

The fact that engineers have ignored this for a long time is no different than engineers ignoring the instantaneous values of the forward and reflected waves when calculating the impedance seen at any point on the transmission line, or ignoring that the standing wave is made up of a standing wave and a traveling wave.

tim ab0wr

3. Cecil, if you don't like being called a troll then DON'T BE ONE!

phase velocity = w/B.

B = w/vp = d(phi)/dx where phi is the phase angle.

If phase velocity varies then so does d(phi)/dx and so does B.

Is dy/dx of a second order equation a constant, Cecil?

If not, then does the fact that dy/dx of the second order equation is not a constant cause calculus calculations using dx to yield inaccurate results?

Mickey dismisses this fact by saying engineers have gotten along for a long time while ignoring it. I can't argue with that. But those same engineers have gotten along for a long time using the slotted line without worrying about there being a standing wave and traveling wave both on the line at the same time and, instead, just focusing on the measured standing wave.

Ignoring something doesn't make it go away, however.

tim ab0wr

4. The value of B does not change.

As discussed in Kraus, let's consider the phase velocity of the TOTAL wave on a line having both forward and reflected waves. To be more specific, let's assume the load impedance is purely resistive and equal to twice the characteristic impedance of the line. For this specific case we can write the total instantaneous voltage wave as follows:

v(x, t) = (4/3) cos(wt) cos(Bx) + (2/3) sin(wt) sin(Bx)

After quite a bit of calculus, etc., one can show that the instantaneous phase velocity of v(x, t), denoted by dx/dt, is as follows:

dx/dt = (w/2B) (1/(cos(wt)^2 + 0.25(sin(wt)^2)))

The maximum and minimum values of dx/dt are (2w)/B and w/(2B), respectively; the average (mean) value of dx/dt is equal to w/B. The value of B, however, is simply a constant as I mentioned in my previous post.

When the textbooks write B = w/vp, the phase velocity being assumed is that of the traveling wave, not the total wave. The phase velocity of the traveling wave is constant.

73, K5MC

5. You seem to be missing a very simple mathematical point. x is independent of B and phi. x is an elementary one-dimensional linear independent variable, by definition. Therefore, according to the rules for differential calculus, dx cannot be anything except constant.

Last edited: Jul 30, 2008
6. The last time I checked, Beta was the inverse coefficient of the negative latitued of the muffler bearings, divided by the bidirectional propagation factor Gama cosign of the past quater wave of the inverse of the q subband deflatulator constant. Now, if you divide the anticosign of pahse vector omega by the parasitic downblow variable (which is .5 times the square root of the root cause) then that would negate the 2.645 dB loss due to phase bolonial parsecting. What this means is the effect is insignificant, and the operator can ignore any influence it may have on operations.

7. I have not changed my position so I have not "come around".

Consider the following lossless steady-state configuration. Source power = 100w. Forward power = 200 watts. Reflected power = 100 watts. Power delivered to the load = 100 watts.

During steady-state, the source supplies a 100 watt traveling wave and the load absorbs a 100 watt traveling wave.
Power In = Power Out

During steady-state, 100 watts of the 200 watts of forward traveling wave energy is delivered to the load. Rho^2=0.5 so 100 watts of the forward traveling wave energy is reflected in a rearward-traveling wave.

100 watts of the 200 watts of forward energy is contained in the standing wave. All of the 100 watts of reflected energy is contained in the standing wave. The energy in the standing wave is 100w+100w = 200 watts multiplied by the length in seconds of the transmission line, e.g. for a 1 microsecond long transmission line, 200 microjoules of energy are stored in the standing wave. That energy was supplied during the key-down transient state and has no effect on Power In or Power Out during steady-state. That 200 microjoules of energy, needed to support the steady-state existence of the standing wave, will be delivered to the load as a traveling wave during the key-up transient state.

Last edited: Jul 30, 2008
8. Minus some heat dissipated in the feedline, etc. <-- sorry, couldn't help it. But seriously... Cecil, what I fail to understand is why this (how the energy gets into, is stored in, and delivered from the feedline) isn't more obvious to the radio folks - at least at the RF tech level. I recognize that there's decades of science and numerous books on the subject, as well as some amount of classroom instruction on it (at least in my case), but the math behind it and the concepts themselves just don't seem that hard. "SWR" is so ubiquitous it just seems like this should be more of a common knowledge thing. (shakes head)

73 de KI6NNO

Last edited: Jul 30, 2008
9. Forgot to say that my above transmission line example is lossless.

Energy is a scalar quantity measured here in joules. Anyone capable of tracking "gallons of water" is capable of tracking "joules of energy". Watts do not have to be conserved but joules must be conserved. Another advantage of scalar energy is that it has no phase and therefore, no phase velocity. Last edited: Jul 30, 2008
10. I somewhat disagree with what you are saying. As Kraus says:
He also notes:
Note the words "resultant wave", "phase velocity", and "varies as a function of position"

As Kraus notes d(phi)/dt = (-2pi/T) so dt = (-T/2pi)d(phi)

if vp = dx/dt then

vp = -dx/(T/2pi)d(phi) = -w/[d(phi)/dx] and

vp = dx/dt = (-w)/[d(phi)/dx]

Therefore the phase velocity of the resultant wave *IS* a function of of position.

This is the exact form of vp= w/B.

I think the main problem here is that B is usually considered to be a constant, i.e. "phase constant", when in fact it is just an average over the transmission line.

That doesn't mean that the actual "phase constant" and phase velocity is a constant everywhere on the transmission line. In fact, he also defines "p" as being a relative phase velocity which is a function of position as given by:

p = B/[d(phi)/dx]

where B *is* the average phase constant over the transmission line.

When you are working on a transmission line trying to impact the impedance matching and such you are only interested in the *resultant* wave, not the traveling wave. It is the standing wave which is a combination of the forward wave and the reflected wave that determines the impedance seen at any point on the line, not the traveling wave. While the phase velocity as a function of position may be constant for the traveling wave it is not a constant for the resultant of the forward wave and the reflected wave.

While engineers may have gotten along for a long time in ignoring this fact it is the same kind of thing as assuming the first two terms of a binomial expansion are always "good enough" to work with. That may or may not always be the case.

tim ab0wr