188 5. PLANAR TRANSMISSION LINES
reads:
"
reff
.f / D "
r
"
r
"
reff
.f D 0/
1 C P .f /
: (5.16)
e accuracy of this model depends primarily on the accuracy of "
reff
.f D 0/. e term
P .f / accounts for the frequency dependence (given next in Eqs. (5.22)) and was estimated by
curve fitting of measured or accurate numerical results from electromagnetic simulators, which
accounts for the actual hybrid mode propagation. However, to apply (5.16), the multilayer struc-
ture must be reduced to an equivalent Singler-Layer Reduction, SLR structure. Note that the SLR
approach enables the modeling of losses “dielectric loss” and “conductor loss” through the well-
established closed form expressions for the single layer, e.g., Hoffmann [3].
5.6 EQUIVALENT SINGLE–LAYER MICROSTRIP (SLR)
e SLR initially proposed by Verma and Sadr [15] was further elaborated on by Verma and
Bhupal [17, 18]. e multilayer microstrip line of Figure 5.4a is reduced, for example, to an
equivalent single-layer microstrip shown in Figure 5.4b. e equivalent substrate thickness is
given by the total height of the layers below the strip conductor (measured from the ground
plane):
h
eq
D h D h
M
D
M
X
iD1
d
i
; (5.17)
while the equivalent strip width remains the same, w
eq
D w.
e basic idea for the evaluation of an equivalent complex dielectric constant ."
req
/ ex-
ploits an already available quasi-static effective dielectric constant ("
reff
). For this purpose, the
expressions given in the previous sections, e.g., Eq. (5.16), can be used. Assuming that "
reff
is identical for both the original multilayer and the equivalent single-layer structures [19], the
classical Wheeler’s formulae apply for the latter case. Hence, a microstrip line with substrate
thickness h
eq
D h, complex dielectric constant ."
req
/ and strip width w
eq
D w has the following
complex effective dielectric constant [9]:
"
reff
D 1 Cq
"
req
1
; (5.18)
where q is the filling factor of the multilayer structure that is assumed the same as that of the
equivalent single layer. us, it can be evaluated according to [9] or any microstrip textbook,
e.g., [3]. at is, the equivalent filling factor q is independent of the multi-layer dielectric con-
stants and to a first approximation independent of "
req
. Since the already-given Svacina’s anal-
5.6. EQUIVALENT SINGLE–LAYER MICROSTRIP (SLR) 189
(a)
ε
Ν
ε
Ν-1
ε
Μ+1
ε
Μ
w
ε
2
ε
1
ε
i
h
i
h
d
i
d
1
d
2
(b)
w
ε
eq
h
e
q
Figure 5.4: (a) A multilayer microstrip line and (b) reduction to an equivalent single layer.
ysis [7] is just an extension of that of Wheeler’s [9], it is obvious that q can be obtained from
Eqs. (5.2a) and (5.3a).
q D 1
0:5
Nw
eff
ln
Nw
eff
1
for Nw D w= h 1 (5.19a)
q D 0:5
0:9
ln. Nw/
for Nw D w= h 1; (5.19b)
where Nw
eff
is still given by (5.2c) and w D w
eq
; h D h
eq
.
190 5. PLANAR TRANSMISSION LINES
Alternatively, the original Wheeler’s formulae [9], repeated in [15], reads:
q
D
0:5
C
0:5
1
C
12
Nw
1=2
for
N
w
D
w=h
1
(5.20a)
q D 0:5 C 0:5
1 C
12
Nw
1=2
C 0:02.1 Nw/
2
for Nw D w= h 1: (5.20b)
Instead of following [9], the effective (real part) dielectric constant given in Eq. (5.16) can
estimate the equivalent (real part) dielectric constant. erefore, rewriting (5.18), we have:
"
reff
.f D 0/ D 1 C q
"
req
1
(5.21a)
or
"
req
D 1 C
"
reff
.f D 0/ 1
q
: (5.21b)
An approximation for the frequency dependence of the effective dielectric constant could
be obtained with the aid of the Kirschning and Jansen [16] model presented in Eq. (5.16) by
substituting "
r
D "
req
through (5.21b), where P .f / reads [3, 16]:
P .f / D P
1
P
2
Œ
F H
.
0:1844 C P
3
P
4
/
1:5763
(5.22a)
P
1
D 0:27488 C
0:6315 C
0:525
.1 C 0:157 F H/
20
Nw 0:065683 e
8:7513 Nw
(5.22b)
P
2
D 0:033622
1 e
0:03442"
r
(5.22c)
P
3
D 0:0363e
4:6 Nw
1 e
.
F H
38:7
/
4:97
(5.22d)
P
4
D 1 C2:751
1 e
.
"
r
15:916
/
8
: (5.22e)
e quantities are given as: F D f GHz, H D h mm, "
r
D "
req
, D
"
. Another possi-
bility is to employ Kobayashi’s expression [20]:
"
reff
.f / D "
r
"
r
"
reff
.f D 0/
1 C
.
f =f
50
/
m
(5.23a)
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