200 5. PLANAR TRANSMISSION LINES
5.11.1 SINGLE–LAYER CPW
e effective permittivity of an ordinary single-layer CPW can be obtained from Eqs. (5.30)
and (5.31) by just allowing only one substrate .M D 1/ and no superstrate layers .N D 0/. Also,
the infinitely extending layer above and below the structure must have air characteristics ("
l
r2
D
"
u
r1
D 1). Moreover, for a symmetrical single-layer CPW, the result becomes identical to that of
Ghione and Naldi [29]. For the effective dielectric constant, (5.31) reads:
"
reff
D q
i1
"
l
r1
C q
i2
"
l
r2
C q
j1
"
u
r1
D q
i2
C q
j1
C q
i1
"
r
: (5.40)
For the first bottom layer i D 1, q
i1
is given by (5.37), with k
"
defined in (5.32e), and
k
i"
by (5.32a) with h
i
D h
1
D h. For the upper semi-infinite layer where j D 1, q
j1
is de-
fined by (5.38). e second term is zero according to (5.36b), since k
j 1
D 0, thus, q
j1
D 1=2.
Likewise, for the bottom semi-infinite layer i D 2, q
i2
is given by (5.38) with i 1 D 1 or
k
i1;"
D k
i"
, which yields:
q
i2
D
1
2
q
i1
: (5.41)
In turn,
"
reff
D
1
2
C
1
2
q
i1
C q
i1
"
r
D 1 C
.
"
r
1
/
q
i1
or (5.42a)
"
reff
D 1 C
"
r
1
2
K
k
0
"
K
.
k
"
/
K
.
k
i"
/
K
k
0
i"
: (5.42b)
e above result is identical to that of [29].
5.12 MULTILAYER CPW WITH FINITE GROUND PLANES
e characteristics of CPW with infinite ground planes are presented in the previous section,
and the analysis given is mainly based on the work of Svacina [12]. e CPW with finite ground
plane (Figure 5.7a), as well as the CPS transmission line (Figure 5.7b) is analyzed by Chen and
Chou [30]. is work is also based on conformal transformation for its quasi-static analysis,
but a slightly different approach is followed. at is, [30] extends the “superposition of partial
capacitances” approximation proposed by Veyres and Fouad Hanna [31]. A careful comparison
of the final expression of [29] for the effective dielectric constant against that of the infinite
ground plane given in Eqs.(5.30) and (5.31) reveals that they are identical. Actually, [30] gives
an expression for five layers. Hence, this can be extended to .N C 1/ plus .M C 1/ layers with a
careful construction of an expression like Eqs.(5.30) and (5.31). e information regarding the
finite ground planes is included in k
i
expressions corresponding to those of Eqs (5.29) or (5.32)
for the symmetric geometry.
5.12. MULTILAYER CPW WITH FINITE GROUND PLANES 201
(a)
h
j
h
N
u
1N
ε
+
u
N
ε
u
1
ε
ℓ
1M
ε
+
ℓ
M
ε
ℓ
i
ε
u
j
ε
ℓ
1
ε
ℓ
2
ε
h
M
h
i
h
1
h
2
-x
c
x
c
x
a
-x
a
x
b
-x
b
(b)
h
j
h
N
u
1N
ε
+
u
N
ε
u
1
ε
ℓ
1M
ε
+
ℓ
M
ε
ℓ
i
ε
u
j
ε
ℓ
1
ε
ℓ
2
ε
h
M
h
i
h
1
h
2
-x
a
x
a
x
b
-x
b
Figure 5.7: Generalized multilayer transmission lines with finite ground planes. (a) Coplanar
waveguide (CPW) with finite ground planes and (b) coplanar strips (CPS).
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