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( )( )
Y Y Y Y YY
where Y0 is the characteristic admit-
tance of the input port and the out-
The condition for production
of the transmission zeros can be
simplified by setting YT 21=0, which yields the relationship
θ−π θ θ
cos 4 fC sin cos / Y
2 2 22
The relevant graph of Equation 5
is shown in Figure 2. The solid line
and dashed line represent bU21 and -bL 21, respectively, and the intersec- tion points in the curves reveal the
approximate locations of the stopband transmission zeros.
As shown in Figure 2, the first
transmission zero positioned at fz1
is generated at the lower side of f1
since f2 < f1 is satisfied in this design. Another two transmission zeros fz2 and fz3 are provided between f1 and 2f1 when
b21 U 3f1 2 ⎛⎝⎜ ⎞⎠⎟>−b21 L 3f1 2 ⎛⎝⎜ ⎞⎠⎟ .
With the choice of appropriate parameters, the structure produces
three transmission zeros for a BSF
Note that the transmission zero
distribution in this design is not inherently symmetrical about the
central transmission zero, while the
transmission zero distributions of
almost all the previous reported
WBSFs are symmetrical. In fact, by
tuning C or Y1, different transmission
zero distributions can provide either
symmetric or asymmetric BSF responses. The asymmetric bandstop
filter response can also eliminate the
restriction of a fixed central transmission zero, resulting in better adjustability and flexibility compared with
the previous reported WBSFs.
the upper passband. The structure
includes two bilateral transmission
lines and one embedded capacitor
to reduce signal interference. This
generates three adjustable transmission zeros in the controllable
stopband for sharp rejection and
expands the upper passband tremendously compared to the similar
7-11 through the use of
the embedded capacitor.
The modified transmission line
model (see Figure 1) can be decomposed into two parallel sections. One is the upper transmission
line having a characteristic admittance Y1 and electrical length θ1; the
other is the cascaded structure consisting of two bilateral transmission
lines with characteristic admittance
Y2, electrical length θ2 and one embedded capacitor.
Considering the entire structure
to be lossless, the overall Y matrix is
Y jb jY csc (1)
cos 4 fC sin cos / Y
θ−π θ θ
If θ1 = π when f = f1, then YU21=+∞ at frequency f1.
For the lower section, at frequency,
= +∞ f Y
4 C tan
Obviously, there is a series resonance for the lower section when f
= f2. For this design, f2 is always set
lower than f1 to generate three
transmission zeros. Then the total
parallel Y matrix of the structure
can be expressed as
⎡⎣ ⎤⎦=⎡⎣ ⎤⎦+⎡⎣ ⎤⎦ Y Y Y ( 3)
21 T 21 U 21 L
The BSF creates transmission zeros at the frequencies where |S21| =
0, and the relationship between the
admittance matrices and scattering
matrices can be given by
s Fig. 1 Bandstop filter structure.
C Y2 θ2Y2 θ2
s Fig. 2 Graphical view of Equation 5.
U2 1 ,
b U 21
–b L 21