CHAPTER 5
Lowpass and Bandpass Filters
Conventional microstrip lowpass and bandpass filters such as stepped-impedance
filters, open-stub filters, semilumped element filters, end- and parallel-coupled
half-wavelength resonator filters, hairpin-line filters, interdigital and combline fil-
ters, pseudocombline filters, and stub-line filters are widely used in many RF/mi-
crowave applications. It is the purpose of this chapter to present the designs of these
filters with instructive design examples.
5.1 LOWPASS FILTERS
In general, the design of microstrip lowpass filters involves two main steps. The
first one is to select an appropriate lowpass prototype, such as one as described in
Chapter 3. The choice of the type of response, including passband ripple and the
number of reactive elements, will depend on the required specifications. The ele-
ment values of the lowpass prototype filter, which are usually normalized to make a
source impedance g
0
= 1 and a cutoff frequency ⍀
c
= 1.0, are then transformed to
the L-C elements for the desired cutoff frequency and the desired source imped-
ance, which is normally 50 ohms for microstrip filters. Having obtained a suitable
lumped-element filter design, the next main step in the design of microstrip lowpass
filters is to find an appropriate microstrip realization that approximates the lumped-
element filter. In this section, we concentrate on the second step. Several microstrip
realizations will be described.
5.1.1 Stepped-Impedance, L-C Ladder Type Lowpass Filters
Figure 5.1(a) shows a general structure of the stepped-impedance lowpass mi-
crostrip filters, which use a cascaded structure of alternating high- and low-
impedance transmission lines. These are much shorter than the associated guided-
109
Microstrip Filters for RF/Microwave Applications. Jia-Sheng Hong, M. J. Lancaster
must not allow any transverse resonance to oc-
cur at operation frequencies.
ț A higher Z
0L
leads to a better approximation of a lumped-element inductor,
but Z
0L
must not be so high that its fabrication becomes inordinately difficult
as a narrow line, or its current-carrying capability becomes a limitation.
In order to illustrate the design procedure for this type of filter, the design of a
three-pole lowpass filter is described in follows.
The specifications for the filter under consideration are
Cutoff frequency f
c
= 1 GHz
Passband ripple 0.1 dB (or return loss Յ –16.42 dB)
Source/load impedance Z
0
= 50 ohms
110
LOWPASS AND BANDPASS FILTERS
FIGURE 5.1 (a) General structure of the stepped-impedance lowpass microstrip filters. (b) L-C ladder
type of lowpass filters to be approximated.
(a)
(b)
A lowpass prototype with Chebyshev response is chosen, whose element values are
g
0
= g
4
2
= 3.652 × 10
–12
F
The filter is to be fabricated on a substrate with a relative dielectric constant of 10.8
and a thickness of 1.27 mm. Following the above-mentioned considerations, the
characteristic impedances of the high- and low-impedance lines are chosen as Z
0L
=
93 ohms and Z
0C
= 24 ohms. The relevant design parameters of microstrip lines,
which are determined using the formulas given in Chapter 4, are listed in Table 5.1,
where the guided wavelengths are calculated at the cutoff frequency f
c
= 1.0 GHz.
Initially, the physical lengths of the high- and low-impedance lines may be found
by
l
L
= sin
–1
(5.2)
l
C
= sin
–1
(
tan
ᎏ
g
l
C
C
ᎏ
(5.3)
c
C =
ᎏ
Z
1
0C
ᎏ
sin
ᎏ
2
g
l
C
C
gL
ᎏ
2
⍀
c
ᎏ
2
f
c
g
0
ᎏ
Z
0
⍀
c
ᎏ
2
f
c
Z
0
ᎏ
g
0
5.1 LOWPASS FILTERS
C
, resulting in l
L
= 9.81 mm
and l
C
= 7.11 mm.
A layout of this designed microstrip filter is illustrated in Figure 5.2(a), and its
performance obtained by full-wave EM simulation is plotted in Figure 5.2(b).
5.1.2 L-C Ladder Type of Lowpass Filters Using Open-Circuited Stubs
The previous stepped-impedance lowpass filter realizes the shunt capacitors of the
lowpass prototype as low impedance lines in the transmission path. An alternative
realization of a shunt capacitor is to use an open-circuited stub subject to
112
LOWPASS AND BANDPASS FILTERS
FIGURE 5.2 (a) Layout of a three-pole, stepped-impedance microstrip lowpass filter on a substrate
with a relative dielectric constant of 10.8 and a thickness of 1.27 mm. (b) Full-wave EM simulated per-
formance of the filter.
(a)
(b)
C = tan
l
for l <
g
/4 (5.4)
where the term on the left-hand side is the susceptance of shunt capacitor, whereas
–1
(
c
CZ
0C
) = 8.41 mm
To compensate for the unwanted susceptance resulting from the two adjacent high-
impedance lines, the initial l
C
should be changed to satisfy
c
C = tan
+ 2 × tan
(5.5)
which is solved for l
C
and results in l
C
= 6.28 mm for this example. Furthermore, the
open-end effect of the open-circuited stub must be taken into account as well. Ac-
cording to the discussions in Chapter 4, a length of ⌬l = 0.5 mm should be compen-
sated for in this case. Therefore, the final dimension of the open-circuited stub is l
C
= 6.28 – 0.5 = 5.78 mm.
The layout and EM-simulated performance of the designed filter are given in
Figure 5.3. Comparing to the filter response to that in Figure 5.2, both filters show a
ᎏ
2
c
L
ᎏ
Z
0L
gL
ᎏ
2
2
ᎏ
g
1
ᎏ
Z
0
5.1 LOWPASS FILTERS
113
cuited stub is about a quarter guided wavelength so as to almost short out a trans-
mission, and cause the attenuation peak.
To obtain a sharper rate of cutoff, a higher degree of filter can be designed in the
same way. Figure 5.5(a) is a seven-pole, lumped-element lowpass filter with its mi-
crostrip realization illustrated in Figure 5.5(b). The four open-circuited stubs, which
5
= 6.6737 pF
L
4
= 12.52 nH
The microstrip filter design uses a substrate having a relative dielectric constant
r
=
10.8 and a thickness h = 1.27 mm. To emphasize and demonstrate that the mi-
crostrip realization in Figure 5.5(b) can only approximate the ideal lumped-element
filter in Figure 5.5(a), two microstrip filter designs that use different characteristic
impedances for the high-impedance lines are presented in Table 5.2. The first design
(Design 1) uses the high-impedance lines that have a characteristic impedance Z
0L
=
110 ohms and a line width W
L
= 0.1 mm on the substrate used. The second design
(Design 2) uses a characteristic impedance Z
0L
= 93 ohms and a line width W
L
= 0.2
mm. The performance of these two microstrip filters is shown in Figure 5.5(c), as
compared to that of the lumped-element filter. As can be seen, the two microstrip
filters behave not only differently from the lumped-element one, but also differently
from each other. The main difference lies in the stopband behaviors. The microstrip
filter (Design 1) that uses the narrower inductive lines (W
L
7
L
2
C
1
C
5
Z
0
Z
0
l
2
l
1
l
3
l
5
W
C
W
L
l
7
l
4
l
6
(a)
r
= 10.8, h = 1.27 mm) l
1
= l
7
l
2
= l
6
l
3
= l
5
l
4
W
C
= 5 mm (mm) (mm) (mm) (mm)
Design 1 (W
L
= 0.1 mm) 5.86 13.32 9.54 15.09
Design 2 (W
L
= 0.2 mm) 5.39 16.36 8.67 18.93
FIGURE 5.6 (a) An elliptic-function, lumped-element lowpass filter. (b) Microstrip realization of the
elliptic function lowpass filter.
L
3
L
4
g
L1
= g
1
= 0.8214 g
C4
= g
4
= 0.9077
g
L2
= gЈ
2
= 0.3892 g
L5
= g
5
= 1.1170
g
C2
= g
2
= 1.0840 g
C6
= g
6
= 1.1360
g
L3
= g
=
ᎏ
2
1
f
c
ᎏ
Z
0
g
Li
(5.6)
C
i
=
ᎏ
2
1
f
c
ᎏ
ᎏ
Z
1
0
ᎏ
g
Ci
1
L
ෆ
4
C
ෆ
4
ෆ
ᎏ
= 1.219 GHz
(5.8)
f
p2
= = 1.540 GHz
For microstrip realization, a substrate with a relative dielectric constant of 10.8 and
a thickness of 1.27 mm is assumed. All inductors will be realized using high-imped-
ance lines with characteristic impedance Z
0L
= 93 ohms, whereas the all capacitors
1
ᎏᎏ
2
͙L
ෆ
2
C
ෆ
2
ෆ
Z
L
0
i
L
ᎏ
(5.9)
l
Ci
= sin
–1
(2
f
c
Z
0C
C
i
)
Substituting the corresponding parameters from (5.7) and Table 5.3 results in
l
L1
= 8.59 l
L2
= 3.96
l
L3
= 13.01 l
L
5
= Z
0L
sin
+ Z
0C
tan
(5.10)
2
f
c
C
6
=
ᎏ
Z
1
0C
ᎏ
sin
+
ᎏ
Z
1
0L
C6
ᎏ
gC
( f
c
)
2
l
L5
ᎏ
gL
( f
c
)
gc
( f
c
)
ᎏ
2
5.1 LOWPASS FILTERS
119
TABLE 5.3 Microstrip design parameters for an elliptic function lowpass filter
Characteristic impedance (ohms) Z
0C
gC
(f
p1
) = 83
gL
(f
p1
) = 97
Guided wavelength (mm) at f
p2
gC
(f
p2
) = 66
gL
(f
p2
) = 77
inductive line elements for L
1
, L
2
, and L
3
as well as at the junction of the line ele-
ments for L
2
2
and C
2
, is given
by
1
B
2
(f) =
Z
0L
sin
ᎏ
2
gL
l
(
L
f
2
)
ᎏ
+ Z
0C
tan
L2
and l
C2
. The solutions are found to be l
L2
= 2.98
mm and l
C2
= 5.61 mm.
The compensation for the unwanted reactance/susceptance at the junction of the
inductive line elements for L
3
, L
4
, and L
5
as well as at the junction of the line ele-
ments for L
4
and C
4
can be done in the same way as the above. This results in the
corrected lengths l
L4
= 6.49 mm and l
C4
= 4.24 mm.
To correct for the fringing capacitance at the ends of the line elements for C
2
and C
Z
0L
l
L2
ᎏ
gL
(f)
1
ᎏ
Z
0L
l
L1
ᎏ
gL
(f)
1
ᎏ
Z
0L
1
ᎏᎏᎏᎏ
ᎏ
Z
1
0C
(
2
f)
ᎏ
1
ᎏᎏᎏ
(2
fL
2
) – 1/(2
fC
2
)
120
LOWPASS AND BANDPASS FILTERS
5.2 BANDPASS FILTERS
5.2.1 End-Coupled, Half-Wavelength Resonator Filters
The general configuration of an end-coupled microstrip bandpass filter is illustrated
in Figure 5.8, where each open-end microstrip resonator is approximately a half
guided wavelength long at the midband frequency f
0
of the bandpass filter. The cou-
pling from one resonator to the other is through the gap between the two adjacent
open ends, and hence is capacitive. In this case, the gap can be represented by the
5.2 BANDPASS FILTERS
121
(a)
fined in Chapter 3. The J
j,j+1
are the characteristic admittances of J-inverters and Y
0
is the characteristic admittance of the microstrip line.
Assuming the capacitive gaps act as perfect, series-capacitance discontinuities of
susceptance B
j,j+1
as in Figure 3.22(d)
= (5.13)
and
j
=
–
΄
tan
–1
+ tan
–1
΅
radians (5.14)
where the B
j,j+1
and
j
are evaluated at f
1 –
ᎏ
J
Y
j,j
0
+1
ᎏ
2
B
j,j+1
ᎏ
Y
0
FBW
ᎏ
2g
n
g
n+1
J
n,n+1
ᎏ
Y
0
1
ᎏ
Y
0
122
LOWPASS AND BANDPASS FILTERS
FIGURE 5.8 General configuration of end-coupled microstrip bandpass filter.
C
g
j,j+1
= (5.15)
where
0
= 2
f
0
is the angular frequency at the midband. The physical lengths of
resonators are given by
l
j
=
j
– ⌬l
j
e1
– ⌬l
j
e2
(5.16)
= 6 GHz. A
three-pole (n = 3) Chebyshev lowpass prototype with 0.1 dB passband ripple is cho-
sen, whose element values are g
0
= g
4
= 1.0, g
1
= g
3
= 1.0316, and g
2
= 1.1474.
From (5.12) we have
= =
Ί
×
= 0.2065
= = = 0.0404
The susceptances associated with the J-inverters are calculated from (5.13)
= = = 0.2157
= = = 0.0405
The electrical lengths of the half-wavelength resonators after absorbing the negative
electrical lengths attributed to the J-inverters are determined by (5.14)
0.0404
ᎏᎏ
ෆ
.0
ෆ
3
ෆ
1
ෆ
6
ෆ
×
ෆ
1
ෆ
.1
ෆ
4
ෆ
7
ෆ
4
ෆ
× 0.028
ᎏᎏ
2
J
2,3
ᎏ
Y
0
j, j+1
ᎏ
Y
0
g0
ᎏ
2
0
C
p
j–1, j
ᎏ
Y
0
g0
ᎏ
2
B
j,j+1
ᎏ
0
5.2 BANDPASS FILTERS
123
C
g
0,1
= C
g
3,4
= 0.11443 pF
(5.19)
C
g
1,2
= C
g
2,3
= 0.021483 pF
For microstrip implementation, we use a substrate with a relative dielectric constant
r
= 10.8 and a thickness h = 1.27 mm. The line width for microstrip half-wave-
length resonators is also chosen as W = 1.1 mm, which gives characteristic imped-
ance Z
0
= 50 ohm on the substrate. To determine the other physical dimensions of
the microstrip filter, such as the coupling gaps, we need to find the desired coupling
capacitances C
g
j,j+1
given in (5.19) in terms of gap dimensions. To do so, we might
have used the closed-form expressions for microstrip gap given in Chapter 4. How-
ever, the dimensions of the coupling gaps for the filter seem to be outside the para-
FIGURE 5.9 Layout of a microstrip gap for EM simulation.
C
g
=
–
ᎏ
Im
(Y
0
21
)
ᎏ
(5.20)
C
p
=
ᎏ
Im(Y
1
1
0
+ Y
21
)
ᎏ
where
0
p
1,2
= C
p
2,3
= 0.0457 pF
At the midband frequency, f
0
= 6 GHz, the guided-wavelength of the microstrip line
resonators is
g0
= 18.27 mm. The effective lengths of the shunt capacitances are
calculated using (5.17)
⌬l
1
e1
= ⌬l
3
e2
= = 0.0269 mm
⌬l
1
e2
= ⌬l
3
e1
= = 0.2505 mm
⌬l
2
(1/50)
5.2 BANDPASS FILTERS
125
TABLE 5.4 Characterization of microstrip gaps with line width W = 1.1 mm on the substrate
with
r
= 10.8 and h = 1.27 mm
Y
11
= Y
22
(mhos) Y
12
= Y
21
(mhos)
s (mm) at 6 GHz at 6 GHz C
g
(pF) C
p
(pF)
0.05 j0.0045977 –j0.004434 0.11762 0.00434
0.1 j0.0039240 –j0.003604 0.09560 0.00849
0.2 j0.0032933 –j0.0026908 0.07138 0.01598
0.5 j0.0026874 –j0.0014229 0.03774 0.03354
0.8 j0.0025310 –j0.00081105 0.02151 0.04562
1.0 j0.0024953 –j0.00055585 0.01474 0.05145
1.5 j0.0024808 –j0.0001876 0.00498 0.06083
Finally, the physical lengths of the resonators are found by substituting the above ef-
1.1
Unit: mm
FIGURE 5.10 (a) Layout of the three-pole microstrip, end-coupled half-wavelength resonator filter on
a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm. (b) Full-wave EM sim-
ulated frequency response of the filter.
(b)
(a)
5.2.2 Parallel-Coupled, Half-Wavelength Resonator Filters
Figure 5.11 illustrates a general structure of parallel-coupled (or edge-coupled) mi-
crostrip bandpass filters that use half-wavelength line resonators. They are posi-
tioned so that adjacent resonators are parallel to each other along half of their
length. This parallel arrangement gives relatively large coupling for a given spacing
between resonators, and thus, this filter structure is particularly convenient for con-
structing filters having a wider bandwidth as compared to the structure for the end-
coupled microstrip filters described in the last section. The design equations for this
type of filter are given by [1]
=
Ί
(5.21a)
= j = 1 to n – 1 (5.21b)
=
Ί
(5.21c)
where g
o
, g
ෆ
j
g
ෆ
j+
ෆ
1
ෆ
FBW
ᎏ
2
J
j,j+1
ᎏ
Y
0
FBW
ᎏ
g
o
g
1
ᎏ
2
J
01
ᎏ
Y
type are best illustrated by use of an example. Let us consider a design of five-pole
(n = 5) microstrip bandpass filter that has a fractional bandwidth FBW = 0.15 at a
midband frequency f
0
= 10 GHz. Suppose a Chebyshev prototype with a 0.1-dB rip-
ple is to be used in the design. The desired n = 5 prototype parameters are
g
0
= g
6
= 1.0 g
1
= g
5
= 1.1468
g
2
= g
4
= 1.3712 g
3
= 1.9750
The calculations using (5.21) and (5.22) yield the design parameters, half of which
are listed in Table 5.5 because of symmetry of the filter, where the even- and odd-
mode impedances are calculated for Y
0
= 1/Z
0
and Z
0
ᎏᎏᎏ
4(
͙
(
ෆ
re
ෆ
)
j
ෆ
×
ෆ
(
ෆ
ෆ
ro
ෆ
)
j
ෆ
)
1/2
J
j, j+1
ᎏ
Y
0
J
(Z
0e
)
j, j+1
(Z
0o
)
j, j+1
0 0.4533 82.9367 37.6092
1 0.1879 61.1600 42.3705
2 0.1432 58.1839 43.8661
where ⌬l
j
is the equivalent length of microstrip open end, as discussed in Chapter 4.
The final filter layout with all the determined dimensions is illustrated in Figure
5.12(a). The EM simulated frequency responses of the filter are plotted in Figure
5.12(b).
5.2.3 Hairpin-Line Bandpass Filters
Hairpin-line bandpass filters are compact structures. They may conceptually be ob-
tained by folding the resonators of parallel-coupled, half-wavelength resonator fil-
ters, which were discussed in the previous section, into a “U” shape. This type of
“U” shape resonator is the so-called hairpin resonator. Consequently, the same de-
sign equations for the parallel-coupled, half-wavelength resonator filters may be
used [4]. However, to fold the resonators, it is necessary to take into account the re-
duction of the coupled-line lengths, which reduces the coupling between resonators.
Also, if the two arms of each hairpin resonator are closely spaced, they function as a
pair of coupled line themselves, which can have an effect on the coupling as well.
To design this type of filter more accurately, a design approach employing full-wave
EM simulation will be described.
For this design example, a microstrip hairpin bandpass filter is designed to have
B
0
g
W
1
ᎏ
, Q
en
=
ᎏ
g
F
n
B
g
n
W
+1
ᎏ
(5.24)
M
i,i+1
=
ᎏ
͙
F
g
ෆ
B
i
j
(
ro
)
j
1 and 6 0.385 0.161 6.5465 5.7422
2 and 5 0.575 0.540 6.7605 6.0273
3 and 4 0.595 0.730 6.7807 6.1260
For this design example, we have
Q
e1
= Q
e5
= 5.734
M
1,2
= M
4,5
= 0.160 (5.25)
M
2,3
= M
3,4
= 0.122
We use a commercial substrate (RT/D 6006) with a relative dielectric constant of
6.15 and a thickness of 1.27 mm for microstrip realization. Using a parameter-ex-
traction technique described in Chapter 8, we then carry out full-wave EM simula-
tions to extract the external Q and coupling coefficient M against the physical di-
mensions. Two design curves obtained in this way are plotted in Figure 5.13. It
= 50 ohms. Hence, the tapped line is 1.85 mm wide on the substrate.
Also in Figure 5.13(a), the tapping location is denoted by t, and the design curve
gives the value of external quality factor, Q
e
, as a function of t. In Figure 5.13(b),
the value of coupling coefficient M is given against the coupling spacing (denoted
by s) between two adjacent hairpin resonators with the opposite orientations as
shown. The required external Q and coupling coefficients as designed in (5.25) can
be read off the two design curves above, and the filter designed.
The layout of the final filter design with all the determined dimensions is illus-
trated in Figure 5.14(a). The filter is quite compact, with a substrate size of 31.2
mm by 30 mm. The input and output resonators are slightly shortened to compen-
sate for the effect of the tapping line and the adjacent coupled resonator. The EM
simulated performance of the filter is shown in Figure 5.14(b).
An experimental hairpin filter of this type has been demonstrated in [5], where a
design equation is proposed for estimating the tapping point t as
t = sin
–1
Ί
(5.26)
in which Z
r
is the characteristic impedance of the hairpin line, Z
0
is the terminating
4.7
1.85
2.0
1.0
18.4
FIGURE 5.14 (a) Layout of a five-pole, hairpin-line microstrip bandpass filter on a 1.27-mm-thick
substrate with a relative dielectric constant of 6.15. (b) Full-wave simulated performance of the filter.
(b)
(a)
nores the effect of discontinuity at the tapped point as well as the effect of coupling
between the two folded arms. Nevertheless, it gives a good estimation. For instance,
in the filter design example above, the hairpin line is 1.0 mm wide, which results in
Z
r
= 68.3 ohm on the substrate used. Recall that L = 20.4 mm, Z
0
= 50 ohm, and the
required Q
e
= 5.734. Substituting them into (5.26) yields a t = 6.03 mm, which is
close to the t of 7.625 mm found from the EM simulation above.
5.2.4 Interdigital Bandpass Filters
Figure 5.15 shows a type of interdigital bandpass filter commonly used for mi-
crostrip implementation. The filter configuration, as shown, consists of an array of
n TEM-mode or quasi-TEM-mode transmission line resonators, each of which has
an electrical length of 90° at the midband frequency and is short-circuited at one
end and open-circuited at the other end with alternative orientation. In general, the
physical dimensions of the line elements or the resonators can be different, as indi-
cated by the lengths l
1
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