"Why is the length of coax between the balun and the shack an important electrical factor?"
When a feedline is NOT
matched to the load, the impedance varies along the feedline.
It's actually rare in modern ham usage for an antenna to be matched to open wire line or
twinlead. (The most common example is a folded dipole fed with 300 ohm line).
If you are feeding a single antenna (such as a dipole) on multiple bands, that means that
not only does the impedance vary from band to band, but it also varies with the feedline
length. It is often worthwhile looking for a combination of wire and feedline lengths that
give a relatively benign impedance on multiple bands (examples are the G5RV or ZS6BKW
antennas.) Otherwise we end up with bands where the SWR on the coax section so very
high, and much of our RF gets dissipated as heat in the coax before it reaches the antenna.
(Like the G5RV when used on 10m.)
We can take a specific example to illustrate the point: let's say we have an 80m doublet
and we are using it on 40m with Wireman 551 ladder line. Depending on the wire size,
the feedpoint impedance might be 3000 ohms or so. We can use VK1OD's online transmission
line loss calculator to examine the results: http://vk1od.net/calc/tl/tllc.php
With no ladder line, the SWR would be 3000/50 = 60 : 1. That's not too good. If we add
20' of ladder line the impedance at the bottom is now 75-j250 ohms, and the SWR has dropped
to only 18 : 1. If instead we use 50' of ladder line the impedance is 170+j540, for an SWR of
38 : 1. But a length of 31 feet gives us 55+j1 ohms, so the SWR in 50 ohm coax is 1.1 : 1.
Which of those will have the lowest loss in the coax?
And it gets more complicated because what the tuner in the shack sees is the impedance
at the end of the ladder line transformed through the coax: that initial 3000 ohm impedance
if connected to the 50 ohm coax means the impedance the tuner sees could be 3000 ohms,
or 1.2 ohms, or somewhere between those values with some amount of reactance (either
positive or negative), depending on the coax length.
That's why it generally works best to arrange for the impedance at the end of the ladder line
to be a low impedance point, as that usually corresponds to minimum SWR on the coax, lower
losses, and less extreme impedances for the tuner to match. One way to do this on multiple
bands is to use a relay to switch extra lengths of ladder line in and out - these can run up
and down the tree trunk along with the main feeder (as long as they aren't too close
Not that you can't get it to work otherwise, but remember that, just because your tuner
can match the impedance presented by the coax, that doesn't mean that your RF is
actually making it to the antenna. It isn't difficult to waste 50% or more of your power
heating the coax even when the tuner can find a match.