CHAPTER 6
Highpass and Bandstop Filters
In this chapter, we will discuss some typical microstrip highpass and bandstop fil-
ters. These include quasilumped element and optimum distributed highpass filters,
narrow-band and wide-band bandstop filters, as well as filters for RF chokes. De-
sign equations, tables, and examples are presented for easy reference.
6.1 HIGHPASS FILTERS
6.1.1 Quasilumped Highpass Filters
Highpass filters constructed from quasilumped elements may be desirable for many
applications, provided that these elements can achieve good approximation of de-
sired lumped elements over the entire operating frequency band. Care should be
taken when designing this type of filter because as the size of any quasilumped ele-
ment becomes comparable with the wavelength of an operating frequency, it no
longer behaves as a lumped element.
The simplest form of a highpass filter may just consist of a series capacitor,
which is often found in applications for direct current or dc block. For more selec-
tive highpass filters, more elements are required. This type of highpass filter can be
easily designed based on a lumped-element lowpass prototype such as one shown in
Figure 6.1(a), where g
i
denote the element values normalized by a terminating im-
pedance Z
0
and obtained at a lowpass cutoff frequency ⍀
c
. Following the discus-
sions in the Chapter 3, if we apply the frequency mapping
⍀ = – (6.1)
where ⍀ and
are the angular frequency variables of the lowpass and highpass fil-
= 1.5 GHz (
c
=
2
f
c
). The normalized element values of a corresponding Chebyshev lowpass proto-
type filter are g
0
= g
4
= 1.0, g
1
= g
3
= 1.0316, and g
2
= 1.1474 for ⍀
c
= 1. The high-
pass filter will operate between 50 ohm terminations so that Z
0
= 50 ohm. Using de-
sign equations (6.2) and (6.3), we find
Z
0
ᎏ
g
n
g
n+1
()b
L
n-1
C
n
L
4
C
3
L
2
C
1
Z
0
Z
n+1
FIGURE 6.1 (a) A lowpass prototype filter. (b) Highpass filter transformed from the lowpass
prototype.
(a)
(b)
C
1
= C
3
= = = 2.0571 × 10
c
C
1
. The interdig-
ital capacitor determined by this approach is comprised of 10 fingers, each of which
is 10 mm long and 0.3 mm wide, spaced by 0.2 mm with respect to the adjacent
ones. The dimensions of the short-circuited stub, namely the width W and length l,
can be estimated from
jZ
c
tan
l
= j
c
L
2
(6.4)
where Z
c
is the characteristic impedance of the stub line,
gc
is its guided wave-
length at the cutoff frequency f
c
, and both depend on the line width W on a sub-
× 1.5 × 10
9
× 1.1474
Z
0
ᎏ
c
⍀
c
g
2
1
ᎏᎏᎏᎏ
50 × 2
× 1.5 × 10
9
× 1.0316
1
ᎏᎏ
Z
0
c
⍀
c
g
1
6.1 HIGHPASS FILTERS
properties are not fully utilized. For this reason, we will discuss in this section an-
other type of distributed highpass filter [1].
The type of filter to be discussed is shown in Figure 6.3(a), which consists of a
cascade of shunt short-circuited stubs of electrical length
c
at some specified fre-
quency f
c
(usually the cutoff frequency of high pass), separated by connecting lines
(unit elements) of electrical length 2
c
. Although the filter consists of only n stubs, it
has an insertion function of degree 2n – 1 in frequency so that its highpass response
has 2n – 1 ripples. This compares with n ripples for an n-stub bandpass (pseudo high-
pass) filter discussed in Chapter 5. Therefore, the stub filter of Figure 6.3(a) will have
a fast rate of cutoff, and may be argued to be optimum in this sense. Figure 6.3(b) il-
lustrates the typical transmission characteristics of this type of filter, where f is the
frequency variable and
is the electrical length, which is proportional to f, i.e.,
=
c
(6.5)
For highpass applications, the filter has a primary passband from
c
2
= (6.6)
where
is the passband ripple constant,
is the electrical length as defined in (6.5),
and F
N
is the filtering function given by
F
N
(
) = (6.7)
(1 +
͙
1
ෆ
–
ෆ
x
ෆ
c
2
ෆ
)T
2n–1
ᎏ
2
ᎏ
–
1
ᎏᎏ
1 +
2
F
N
2
(
)
f
ᎏ
f
c
6.1 HIGHPASS FILTERS
165
where n is the number of short-circuited stubs,
x = sin
–
, x
c
n-1
y
0
=1
2
θ
c
θ
c
y
n
y
1,2
y
n-1,n
Short-circuited stub
of electrical length
θ
c
y
0
=1
2
θ
c
(a)
(b)
FIGURE 6.3 (a) Optimum distributed highpass filter. (b) Typical filtering characteristics of the opti-
mum distributed highpass filter.
impedance levels for short-circuited stubs. Nevertheless, practical stub filter de-
c
can be found from
– 1
f
c
= 6.5
This gives
c
= 0.589 radians or
c
= 33.75°. Assume that the filter is designed with
six shorted-circuited stubs. From Table 6.1 we could choose the element values for
n = 6 and
c
= 30°, which will gives a wider passband, up to 7.5 GHz, because the
smaller the electrical length at the cutoff frequency, the wider the passband. Alter-
natively, we can find the element values for
c
= 33.75° by interpolation from the el-
ᎏ
c
35° 0.30755 1.08967
3 25° 0.19690 1.12075 0.18176
30° 0.28620 1.09220 0.30726
35° 0.40104 1.05378 0.48294
4 25° 0.22441 1.11113 0.23732 1.10361
30° 0.32300 1.07842 0.39443 1.06488
35° 0.44670 1.03622 0.60527 1.01536
5 25° 0.24068 1.10540 0.27110 1.09317 0.29659
30° 0.34252 1.07119 0.43985 1.05095 0.48284
35° 0.46895 1.02790 0.66089 0.99884 0.72424
6 25° 0.25038 1.10199 0.29073 1.08725 0.33031 1.08302
30° 0.35346 1.06720 0.46383 1.04395 0.52615 1.03794
35° 0.48096 1.02354 0.68833 0.99126 0.77546 0.98381
ement values presented in the table. As an illustration, for n = 6 and
c
= 33.75°, the
element value y
1
is calculated as follows:
y
1
= 0.35346 + × 3.75 = 0.44909
In a similar way, the rest of element values are found to be
y
1,2
= 1.03446, y
2
= 0.63221, y
2,3
= 49.8 ohms, and Z
3,4
= 50.1 ohms.
The filter is realized in microstrip on a substrate with a relative dielectric con-
stant of 2.2 and a thickness of 1.57 mm. The initial dimensions of the filter can be
easily estimated by using the microstrip design equations discussed in Chapter 4 for
realizing these characteristic impedances and the required electrical lengths at the
cutoff frequency, namely,
c
= 33.75° for all the stubs and 2
c
= 67.5° for all the
connecting lines. The final filter design with all the determined dimensions is
shown in Figure 6.4(a), where the final dimensions have taken into account the ef-
fects of discontinues, and have been slightly modified to allow all the connecting
lines to have a 50 ohm characteristic impedance. The design is verified by full-wave
EM simulation. Figure 6.4(b) is the simulated performance of the filter; we can see
that the filter frequency response does show eleven or 2n – 1 ripples in the designed
passband, as would be expected for this type of optimum highpass filter with only
n = 6 stubs.
6.2 BANDSTOP FILTERS
6.2.1 Narrow-Band Bandstop Filters
Figure 6.5 shows two typical configurations for TEM or quasi-TEM narrow-band
bandstop filters. In Figure 6.5(a), a main transmission line is electrically coupled to
half-wavelength resonators, whereas in Figure 6.5(b), a main transmission line is
magnetically coupled to half-wavelength resonators in a hairpin shape. In either
case, the resonators are spaced a quarter guided wavelength apart. If desired, the
half-wavelength, open-circuited resonators may be replaced with short-circuited,
0
and FBW are the midband frequency and fractional bandwidth of the
2
–
1
ᎏ
0
⍀
c
FBW
ᎏᎏ
(
/
0
–
0
/
)
6.2 BANDSTOP FILTERS
169
150
30
Via hole
U
are the characteristic impedance and admittance of immittance
inverters, and all the circuit parameters including inductances L
i
and capacitances C
i
can be defined in terms of lowpass prototype elements [2]. For the circuit in Figure
6.7(a):
ᎏ
Z
Z
U
0
ᎏ
2
=
ᎏ
g
0
g
1
n+1
ᎏ
(6.11a)
x
i
=
FBW
170
HIGHPASS AND BANDSTOP FILTERS
FIGURE 6.5 TEM or quasi-TEM narrow-band bandstop with (a) electric couplings and (b) magnetic
couplings.
(b)
(a)
6.2 BANDSTOP FILTERS
171
FIGURE 6.6 Bandstop filter characteristics defining midband frequency and band-edge frequencies.
(a) Chebyshev characteristic. (b) Butterworth characteristic.
FIGURE 6.7 Equivalent circuits of bandstop filters with (a) shunt series-resonant branches and (b) se-
ries parallel-resonant branches.
()b
L
n
C
n
L
3
C
3
Y
0
Y
0
L
2
C
2
o
Z
U
90
o
Z
U
L
1
C
1
where g
i
are the element values of lowpass prototype, and x
i
are the reactance slope
parameters of shunt series resonators. For series branches in Figure 6.7(b):
ᎏ
Y
Y
U
0
ᎏ
2
=
ᎏ
g
0
2
ᎏ
g
i
⍀
c
g
F
0
BW
ᎏ
for i = 1 to n
where b
i
are the susceptance slope parameters of series parallel resonators.
It is obvious that for a chosen lowpass prototype, with known element values, the
desired reactance/susceptance slope parameters can easily be determined using
(6.11). The next step is to design microwave bandstop resonators such as those in
Figure 6.5 so as to have prescribed slope parameters. A practical and general tech-
nique that allows one to extract slope parameters of microwave bandstop resonators
using EM simulations or experiments is discussed next.
Consider a two-port network with a single shunt branch of Z = j
L + 1/(j
C),
such as the one in Figure 6.7(a). The shunt branch resonates at
0
= 1/
Z Ϸ j
0
L
(6.13)
in which the approximation (
/
0
–
0
/
) Ϸ 2⌬
/
0
has been made. By substitu-
tion, we can obtain from (6.12)
|S
21
| = (6.14)
This is at resonance when
=
΄
ᎏ
4
(x
1
/Z
0
)
ᎏ
ᎏ
⌬
0
ᎏ
΅
2
2⌬
ᎏ
Copyright: Tài liệu đại học ©