Wave Propagation in Linear Media | re-examined

z
y
x
l
2d
3Wave propagation in electromagnetic transmission lines
Figure 3.2: Structure of the delay line.
as well as the corresponding electric eld components
Ey = kC
!" sinh y ; Ey = kC
!"
Ex = C
j!" cosh y ; Ex = C
j!"
cosh y ; (3.21)
sinh y : (3.22)
As for the boundary conditions, the conducting combs act like a shorted strip line of length
l, and therefore the elds see at y = d the corresponding well-known impedance. This
structure was quite commonly used and is known as `inductive wall'. The boundary condition
then becomes
, Ex
r
= jX(!) =j
Hz y=d
" tan !l
: (3.23)
c
With the normalisation K = kd, = !d=c, and L = l=d we nally obtain the implicit
dispersion relations for the antisymmetric mode,
p p
K2 , 2 coth K2 , 2 = tan( L) (3.24)
and for the symmetric mode
p p
K2 , 2 tanh K2 , 2 = tan( L) : (3.25)
The numerical evaluation of these relations for the base band is given in g. 3.3 . Note that
the boundary condition (3.23) relies on the fact that the teeth of the combs are in nitely close
together. If this distance is not small compared to the wavelength, then spatial harmonics
would have tobetaken into account.
For the calculation of the stored energy it is important to consider not only the space jyj d
where the propagation occurs, but also the inductive walls d jyj d + l. It was this
component that Borgnis [38] disregarded in his rst analysis and that led to the false result
ve = vp. The average energy stored in the upper wall is given by
W w = 1
4
jHzj 2
36
y=d
dX
d!
(3.26)