Project Cyclops, A Design... - Department of Earth and Planetary ...

APPENDIXK
EFFECTOF DISPERSION IN COAXIALS
The propagation constant of a coaxial line in which
the dielectric loss of the insulating medium may be
neglected
is 7 = a + ifl, with
line, we see that the loss contributes an excess phase of
t_. But t_ is the line loss in nepers, so the excess phase is
one radian per neper of line loss. Since this phase is
proportional to wl/2 rather than co, it does not
represent a constant delay.
v/_o_o( i/a) + ( l/b)
a = £n (b/a)
(DI/2
(KI)
(K2)
The analysis of the phase compensation schemes of
Figure 9-14 and 9-15 shows that with zero offset
frequency (6 = 0) and constant delay lines there is zero
phase error. To determine the errors caused by the
second term of equation (K3) we need merely repeat the
analysis considering this term to be the only phase
where
present.
a = outer radius of inner conductor
b = inner radius of outer conductor
e = dielectric constant of dielectric medium
/a = permeability of dielectric medium and conductors
Let No be the line loss in nepers at co = Wo. Then the
loss at any frequency
is
N = No -- O.2 _1/2
(K4)
, \_o!
o = conductivity of conductors
Equations (1) and (2) cannot be obtained from the
equivalent circuit representation of a transmission line,
which yields "y = x/ (R + iwL ) (G + i_oC), but must be
derived directly from Maxwell's equations (ref. 1).
If the line length is £, then the total phase shift 0 is
For convenience the circuits of Figures 9-14 and 9-15
are reproduced here as Figures K-1 and K-2, with /5
assumed to be zero as a result of phase locking the
remote oscillator. The line losses are from the central
station to the point in question along the line.
error
We see that for the system of Figure K-I the phase
is
0 = -/3£ = -(co V_ £ + c_Q) (K3)
Since the first term in (3) is tile phase shift we would
expect from the delay % = _ £ of a dispersionless
A01
¢.0i._1 _ 1/2 .I/2-IIN0
_o/ x,6°of J
(K5)
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