[Antennas] coax 'sweet lenght'

[Bob Nielsen]

On Sun, Oct 10, 2004 at 09:33:48PM -0700, Bob Nielsen wrote:
> On Sun, Oct 10, 2004 at 11:15:04PM -0400, Robert Lay wrote:
> > Dear Bob Nielsen,
> > 
> > > On Sun, Oct 10, 2004 at 06:26:46PM -0700, Dan Richardson wrote:
> > > > At 04:38 PM 10/10/2004, Ron wrote:
> > > > >  If you want to measure the match between an antenna and it's feed
> > line,
> > > > >measure it at the antenna's feed point, to measure it 1/2 wl
> > (electrical)
> > > > >(or multiples of) away from it is the next best place to measure it.
> > > >
> > > > That is correct, however, the impedance magnitude is the same at
> > > > ½-wavelength (or odd multiples thereof) but, if there is a reactive
> > > > component, the sign will be opposite. For it to exactly be the same
> > > > impedance with the correct sign (angle) the feed line must be
> > > > one-wavelength or multiples thereof.
> > >
> > > No, it will be the same for 1/2-wavelength or multiples.  The round trip
> > > path (transmitter-antenna-transmitter) is one-wavelength, so what you
> > > measure (excluding the effect of loss) is the same as if the
> > > transmission line had zero length.  The magnitude (absolute value of
> > > impedance) is the same for any length of transmission line.
> > 
> > You were doing all right until your last sentence. The magnitude of the
> > impedance seen looking into a transmission line is NOT the same for any
> > length of transmission line. For example, take a  transmission line
> > terminated in Za (where Za > Zo) and vary the length of line over the range
> > from 0 length to one half wavelength. You will find that the magnitude of
> > the impedance seen looking into the line will vary from a value of Za  to Zo
> > and then to Za/2, then back to Zo and then to Za (at lengths of 1/8
> > wavelength, 1/4 wavelength, 3/8 wavelength, 1/2 wavelength), etc.
> > 
> > If you start with the load, Za, being smaller than Zo, then the results are
> > the same except at the 1/4 wavelength point the magnitude of the impedance
> > seen looking into the line will be twice Za.
> > 
> > For the case of the lossless and flat line (Za = Zo) the magnitude of the
> > impedance seen looking into the line is independant of the line length.
> > 
> > And yes, Danny was wrong.
> 
> Thanks for the correction, Bob.  What I MEANT to say was the absolute
> value of the VSWR (not the transformed impedance) was constant. 

Reading a bit further, your statement "You will find that the magnitude
of the impedance seen looking into the line will vary from a value of Za
to Zo and then to Za/2, then back to Zo and then to Za (at lengths of
1/8 wavelength, 1/4 wavelength, 3/8 wavelength, 1/2 wavelength), etc."
is also incorrect.

If the line is terminated in Za != Zo, there is no length which will
result in the input impedance being Zo.  The precise mathematics (which
involve hyperbolic functions) are somewhat complex, but the effects can
easily be seen using a Smith chart.  As an example, with Za = Zo/3
(assuming it is purely resistive, this is Ro/3 + j0), The (normalized to
Zo) input impedance as a function of line length is approximately:

Length (wavelengths)       Impedance
-------                    -------
0                           0.33 +j0
.125                        0.6  +j0.8
.250                        3.0  +j0
.375                        0.6  -j0.8
                          
At a line length of 0.168 wavelength, the resistive part of the input
impedance will be equal to 1.0, but there will be a reactive component
of approximately j1.2.  At a length of .332 wavelength, there will be a
resistive component of 1.0 and a reactive component of -j1.2.  At any 
length of line there will be a VSWR of 3:1.

73,
Bob, N7XY

p.s. This is getting into stuff I haven't reallythought about in ~40 
years :-)