CHAPTER 4
Transmission Lines
and Components
In this chapter, basic concepts and design equations for microstrip lines, coupled
microstrip lines, discontinuities, and components useful for design of filters are
briefly described. Though comprehensive treatments of these topics can be found in
the open literature, they are summarized here for easy reference.
4.1 MICROSTRIP LINES
4.1.1 Microstrip Structure
The general structure of a microstrip is illustrated in Figure 4.1. A conducting strip
(microstrip line) with a width W and a thickness t is on the top of a dielectric sub-
strate that has a relative dielectric constant
r
and a thickness h, and the bottom of
the substrate is a ground (conducting) plane.
4.1.2 Waves in Microstrips
The fields in the microstrip extend within two media—air above and dielectric be-
low—so that the structure is inhomogeneous. Due to this inhomogeneous nature,
the microstrip does not support a pure TEM wave. This is because that a pure TEM
wave has only transverse components, and its propagation velocity depends only on
the material properties, namely the permittivity
and the permeability
. However,
with the presence of the two guided-wave media (the dielectric substrate and the
air), the waves in a microstrip line will have no vanished longitudinal components
of electric and magnetic fields, and their propagation velocities will depend not
only on the material properties, but also on the physical dimensions of the mi-
crostrip.
C
d
a
ᎏ
(4.1)
Z
c
=
in which C
d
is the capacitance per unit length with the dielectric substrate present,
C
a
is the capacitance per unit length with the dielectric substrate replaced by air, and
c is the velocity of electromagnetic waves in free space (c Ϸ 3.0 × 10
8
m/s).
1
ᎏ
c͙C
ෆ
a
C
ෆ
d
ෆ
78
TRANSMISSION LINES AND COMPONENTS
Ground plane
Conducting strip
(4.2b)
where
= 120
ohms is the wave impedance in free space.
For W/h Ն 1:
re
= +
1 + 12
–0.5
(4.3a)
Z
c
=
Ά
+ 1.393 + 0.677 ln
+ 1.444
·
–1
(4.3b)
Hammerstad and Jensen [3] report more accurate expressions for the effective
dielectric constant and characteristic impedance:
re
= +
+
2
΅
(4.5)
where u = W/h,
= 120
ohms, and
F = 6 + (2
– 6)exp
΄
–
0.7528
΅
The accuracy for Z
c
͙
ෆ
re
ෆ
1
ᎏ
18.7
u
4
+
ᎏ
5
u
2
ᎏ
2
ᎏᎏ
u
4
+ 0.432
1
ᎏ
49
10
ᎏ
u
r
– 1
ᎏ
2
W
ᎏ
h
8h
ᎏ
W
ᎏ
2
͙
ෆ
re
ෆ
W
ᎏ
h
h
ᎏ
W
r
– 1
ᎏ
2
r
+ 1
ᎏ
8
m/s) in free space.
The electrical length
for a given physical length l of the microstrip is defined by
=

l (4.9)
Therefore,
=
/2 when l =
g
/4, and
=
when l =
g
/2. These so-called quarter-
wavelength and half-wavelength microstrip lines are important for design of mi-
crostrip filters.
4.1.6 Synthesis of W/h
Approximate expressions for W/h in terms of Z
c
and
Z
c
ᎏ
60
8 exp(A)
ᎏᎏ
exp(2A) – 2
W
ᎏ
h
c
ᎏ
͙
ෆ
re
ෆ
ᎏ

2
ᎏ
g
300
ᎏᎏ
f(GHz)
͙
Nevertheless, its effect on the characteristic impedance and effective dielectric con-
stant may be included [5].
For W/h Յ 1:
Z
c
(t) = ln
Ά
+ 0.25
·
(4.12a)
For W/h Ն 1:
Z
c
(t) =
Ά
+ 1.393 + 0.667 ln
+ 1.444
·
–1
(4.12b)
where
ᎏ
W
h
ᎏ
+ ᎏ
1.
25
h
t
ᎏ
1 + ln
ᎏ
2
t
h
ᎏ
(W/h Ն 0.5
)
re
(t) =
re
– (4.13b)
In the above expressions,
re
is the effective dielectric constant for t = 0. It can be
observed that the effect of strip thickness on both the characteristic impedance and
effective dielectric constant is insignificant for small values of t/h. However, the ef-
fect of strip thickness is significant for conductor loss of the microstrip line.
t/h
ᎏ
͙W
W
e
(t)
ᎏ
h
8
ᎏ
W
e
(t)/h
ᎏ
2
͙
ෆ
re
ෆ
60
2
ᎏ
Z
c
͙
ෆ
r
ෆ
re
are
obtained based on the quasi-TEM or quasistatic approximation, and therefore are
rigorous only with DC. At low microwave frequencies, these expressions provide a
good approximation. To take into account the effect of dispersion, the formula of
re
( f ) reported in [6] may be used, and is given as follows:
re
( f ) =
r
– (4.14)
where
f
50
= (4.15a)
f
TM
0
= tan
–1
r
Ί
5f
ᎏ
·
for W/h Յ 0.7
(4.16c)
m
c
=
Ά
1 for W/h Ն 0.7
where c is the velocity of light in free space, and whenever the product m
0
m
c
is
greater than 2.32 the parameter m is chosen equal to 2.32. The dispersion model
shows that the
re
( f ) increases with frequency, and
re
( f ) Ǟ
r
as f Ǟ ϱ. The accu-
racy is estimated to be within 0.6% for 0.1 Յ W/h Յ10, 1 Յ
r
Յ 128 and for any
1 + ͙W
ෆ
/h
ෆ
1
ᎏᎏ
1 + ͙W
ෆ
/h
ෆ
re
– 1
ᎏ
r
–
re
c
ᎏᎏ
2
h͙
ෆ
r
ෆ
–
ෆ
␥
=
␣
+ j

, where the real part
␣
in nepers per unit length is the
attenuation constant, which is the sum of the attenuation constants arising from
each effect. In practice, one may prefer to express
␣
in decibels (dB) per unit length,
which can be related by
␣
(dB/unit length) = (20 log
10
e)
␣
(nepers/unit length)
Ϸ 8.686
␣
(nepers/unit length)
A simple expression for the estimation of the attenuation produced by the conductor
loss is given by [9]
␣
c
= dB/unit length (4.18)
in which Z
c
is the characteristic impedance of the microstrip of the width W, and R
dB/unit length (4.19)
where tan
␦
denotes the loss tangent of the dielectric substrate.
Since the microstrip is a semiopen structure, any radiation is either free to propa-
gate away or to induce currents on the metallic enclosure, causing the radiation loss
or the so-called housing loss.
tan
␦
ᎏ
g
r
ᎏ
re
re
– 1
ᎏ
r
– 1
0
ᎏ
2
8.686 R
f
c
= (4.21)
In practice, the lowest value (the worst case) of the two frequencies given by
(4.20) and (4.21) is taken as the upper limit of operating frequency of a microstrip
line.
4.2 COUPLED LINES
Coupled microstrip lines are widely used for implementing microstrip filters.
Figure 4.2 illustrates the cross section of a pair of coupled microstrip lines under
consideration in this section, where the two microstrip lines of width W are in the
parallel- or edge-coupled configuration with a separation s. This coupled line
structure supports two quasi-TEM modes, i.e., the even mode and the odd mode,
as shown in Figure 4.3. For an even-mode excitation, both microstrip lines have
the same voltage potentials or carry the same sign charges, say the positive ones,
c
ᎏᎏ
͙
ෆ
r
ෆ
(2W + 0.8h)
c tan
–1
r
ᎏᎏ
͙
2
ෆ
e
and C
o
may be expressed as [11]
C
e
= C
p
+ C
f
+ C
Ј
f
(4.22)
C
o
= C
p
+ C
f
+ C
gd
+ C
ga
(4.23)
In these expressions, C
p
denotes the parallel plate capacitance between the strip and
the ground plane, and hence is simply given by
C
c
) – C
p
(4.25)
The term C
Ј
f
accounts for the modification of fringe capacitance C
f
of a single line
due the presence of another line. An empirical expression for C
Ј
f
is given below
C
Ј
f
= (4.26)
where
A = exp[–0.1 exp(2.33 – 2.53W/h)]
For the odd-mode, C
ga
and C
gd
represent, respectively, the fringe capacitances for
the air and dielectric regions across the coupling gap. The capacitance C
gd
may be
found from the corresponding coupled stripline geometry, with the spacing between
the ground planes given by 2h. A closed-form expression for C
h
W/h
ᎏ
(4.28b)
kЈ =
͙
1
ෆ
–
ෆ
k
ෆ
2
ෆ
and the ratio of the elliptic functions is given by
ᎏ
1
ᎏ
ln
2
ᎏ
1
1
+
–
͙
͙
k
r
Ն 1.
ᎏᎏ
ln
2
ᎏ
1
1
+
–
͙
͙
k
ෆ
k
ෆ
ᎏ
K(kЈ)
ᎏ
K(k)
1
ᎏ
r
2
0.02
͙
can be obtained
from the capacitances. This yields
Z
ce
= (c͙C
ෆ
e
a
C
ෆ
e
ෆ
)
–1
(4.29)
Z
co
= (c
͙
C
ෆ
o
a
C
ෆ
o
ෆ
)
–1
(4.30)
(4.31)
and
o
re
= C
o
/C
a
o
(4.32)
4.2.3More Accurate Design Equations
More accurate closed-form expressions for the effective dielectric constants and the
characteristic impedances of coupled microstrip are available [12]. For a static ap-
proximation, namely, without considering dispersion, these are given as follows:
e
re
= +
1 +
–a
e
b
e
(4.33)
With
v = + g exp(–g)
a
r
+ 1) –
re
+ a
o
]exp(–c
o
g
do
) (4.34)
r
– 0.9
ᎏ
r
+ 3
v
ᎏ
18.1
1
ᎏ
18.7
v
4
+ (v/52)
2
ᎏᎏ
= 0.7287[
re
– 0.5(
r
+ 1)][1 – exp(–0.179u)]
b
o
=
c
o
= b
o
– (b
o
– 0.207)exp(–0.414u)
d
o
= 0.593 + 0.694 exp(–0.526u)
where
re
is the static effective dielectric constant of single microstrip of width W as
discussed previously. The error in
o
re
is stated to be on the order of 0.5%.
The even- and odd-mode characteristic impedances given by the following
Q
4
= ·
Z
co
= (4.36)
with
Q
5
= 1.794 + 1.14 ln
΄
1 +
΅
Q
6
= 0.2305 + ln
΄΅
+ ln(1 + 0.598 g
1.154
)
Q
7
=
Q
8
= exp[–6.5 – 0.95 ln(g) – (g/0.15)
5
]
10 + 190g
2
ෆ
ᎏᎏᎏ
1 – Q
10
͙
ෆ
re
ෆ
· Z
c
/377
1
ᎏᎏᎏᎏ
u
Q3
exp(–g) + [2 – exp(–g)]u
–Q3
2Q
1
ᎏ
Q
2
g
10
ᎏᎏ
1 + (g/3.4)
10
1
ᎏ
r
ᎏ
0.15 +
r
88
TRANSMISSION LINES AND COMPONENTS
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