184 5. PLANAR TRANSMISSION LINES
5.4 MULTIPLE DIELECTRIC LAYER MICROSTRIP LINE
Svacina in [12] extended his conformal mapping method to multilayer microstrips and coplanar
waveguides. e N -dielectric layer cross-section with the strip conductor printed on the top of
the M th layer is shown in Figure 5.3a. Figure 5.3b presents the conformal transformation of a
multilayer strip.
(a)
ε
Ν
ε
Ν-1
ε
Μ+1
ε
Μ
w
ε
2
ε
1
ε
i
h
i
h
h
(b)
j
j
Ν
U
U
U
U
U
U
Ν-1
j-1
0
i
i-1
1
w
ef/2
1
2
i
MM+
1
Figure 5.3: Multiple-dielectric-layer microstrip line: (a) the geometry and (b) the conformal
transformation [
12].
e filling factor expressions .q
i
; q
j
/ are quite similar to Eqs. (5.2), (5.3), (5.5), and (5.6),
where again the bar above each symbol represents normalization with respect to the strip con-
5.4. MULTIPLE DIELECTRIC LAYER MICROSTRIP LINE 185
ductor height .h/ from the ground plane. e strip is assumed on the top of the M th layer
.h D h
M
/ and the quantities involved are:
Nw
eff
D Nw C
2
ln
Œ
17:08.0:5 Nw C 0:92/
(5.9a)
N
U
j
D
2
tan
1
2
Nw
eff
4
N
h
j
1
(5.9b)
N
b
j
D
N
h
j
C 1
N
h
j
1 C 0:25 Nw
(5.9c)
Na
i
D
N
h
i
C 1
1
N
h
i
C 0:25 Nw
: (5.9d)
Filling factors for wide microstrips, Nw D w=h 1:
• Layers below the strip conductor i D 1 to M .
q
1
D q
k
N
h
k
D
N
h
1
(5.10a)
q
i
D q
k
N
h
k
D
N
h
i
q
k
N
h
k
D
N
h
i1
for i D 2; 3; : : : ; M 1; (5.10b)
where
q
k
N
h
k
D 0:5
N
h
k
(
1 C
4
1
Nw
eff
ln
"
2 Nw
eff
sin
0:5
N
h
k
N
h
k
C cos
0:5
N
h
k
#)
(5.10c)
q
M
D 1
1
2 Nw
eff
ln
Nw
eff
1
q
k
N
h
k
D
N
h
M 1
: (5.10d)
• Layers above the strip conductor j D M C 1 to N :
q
M C1
D q
n
N
h
n
D
N
h
M
;
N
U
h
D
N
U
M
(5.11a)
q
j
D q
n
N
h
n
D h
j
;
N
U
n
D
q
N
U
j
q
n
N
h
n
D h
j 1
;
N
U
n
D
q
N
U
j 1
: (5.11b)
For j D M C 2; M C 3; : : : to N 1.
186 5. PLANAR TRANSMISSION LINES
q
n
N
h
n
;
N
U
n
D
1
2 Nw
eff
˚
ln
Nw
eff
1
1
N
U
n
ln
"
2 Nw
eff
cos
0:5
N
U
n
2
N
h
n
1 C
N
U
n
C sin
0:5
N
U
n
#)
(5.11c)
and
q
N
D 1
N 1
X
vD1
q
v
: (5.11d)
Filling factors of the narrow microstrips, Nw D w=h 1:
• Layers below the strip conductor i D 1 to M .
q
1
D q
k
N
h
k
D
N
h
1
; N˛
k
D N˛
1
(5.12a)
q
i
D q
k
N
h
k
D
N
h
i
; N˛
k
D N˛
i
q
k
N
h
k
D
N
h
i1
; N˛
k
D N˛
i1
for i D 2; 3; : : : ; M 1; (5.12b)
where
q
k
N
h
k
; Na
k
D
ln
.
Na
k
/
2 ln
.
0:125 Nw
/
1 C
4
0:5 cos
1
Nw
8
N
h
k
p
Na
k
(5.12c)
q
M
D 0:5
0:9
ln
.
0:125 Nw
/
q
k
N
h
k
D
N
h
M 1
; Na
k
D Na
M 1
: (5.12d)
• Layers above the strip conductor j D M C 1 to N :
q
M C1
D q
n
N
h
n
D
N
h
M C1
;
N
b
n
D
N
b
M C1
(5.13a)
q
j
D q
n
N
h
n
D
N
h
j
;
N
b
n
D
N
b
j
q
n
N
h
n
D
N
h
j 1
;
N
b
n
D
N
b
j 1
for j D M C 2; M C 3; : : : ; N 1; (5.13b)
where
q
n
N
h
b
;
N
b
n
D 0:5 C
1
ln
.
0:125 Nw
/
(
0:9 C
4
ln
N
b
n
cos
1
"
p
N
b
n
N
h
n
N
h
n
1 C 0:125 Nw
#)
: (5.13c)
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