4.3 Bandpass Filter Miniaturization for Microstrip Structures
4.3.2 New Planar Filter without Using Cross-Coupled Effect
4.3.2.1 Intermediate, Dual Mode Microstrip Resonator
Fig. 33 shows a typical measured response of a squared open-loop resonator, centered at 7 GHz, with two gaps at the off-diagonal corners. This squared open-loop resonator, which is a key element for the development of our new dual mode microstrip resonator filters, has a length of λg/4 and a loop conductor width of 10 mils (which corresponds to half of a 50 line width). The filter is realized on a Duroid substrate with a permittivity of 10.2 and a thickness of 25 mils. It is fed on both sides at the centre of the loop with 50 line-width transmission lines (See Fig. 33). This intermediate proposed structure is very different from that proposed in [161] as the two gaps are placed at the off-diagonal corners instead of at the centre, and to achieve a smaller estate area, we have intentionally adopted a square loop instead of the rectangular loop reported in [161]. A coupled feed approach is utilized to clearly demonstrate the dual mode phenomenon in Fig. 33.
Ω
Ω
Fig. 33 A typical measured response of a squared open-loop resonator
As shown from the measured result in Fig. 33, it is noted that with the incorporation of
the two gaps at the off-diagonal corners, it gives rise to two split resonant modes as observed in the figure. By adjusting the line width of the squared open-loop resonator to the 50Ω line-width (=20 mil) while keeping the overall dimension the same, the resultant resonant peaks are shifted only in frequency and with no change in the general shape response. The degree of coupling greatly depends on the gap widths and for the ease of proper fabrication, a 5 mils gap is adopted in the present design. Also noted from the figure, there is no transmission zero generated.
To achieve a sharp roll-off at the edges of the pass-band, the feed line in Fig. 33 is shifted to the diagonal corners and connected parallel to the squared loop as in Fig. 34 (indicated as solid line). From the measured response shown in Fig. 34, it is noted that the inclusion of the parallel direct feed structure merely introduces two added transmission zeros at positions very close to the two prominent resonant modes. Fig.
34 also depicts the case when the input and output feed lines are placed orthogonal to each other. The dimensions of the resonators are kept the same. From the figure, there is no added transmission zeros in the measured response (indicated as dash line) when the input and output feed lines are orthogonal. The transmission zeros are caused by coupling of the feedline and the filter structure.
Fig. 34 Comparison of shifted feed lines
1, 1
Z θ
1, 1
Z θ
p1
C
1
Cp
Cg
Cg
2
Cp
2
Cp
1, 1
Z θ Z1, θ1
Fig. 35 Equivalent circuit for the parallel direct feed structure
To better understand this and be able to control the presence of these transmission zeros, Fig. 35 shows the lossless equivalent circuit and its parameters for the parallel direct feed structure. The capacitancesCp1, Cp2and Cgare taken from [163] and are not reproduced in here. One should take note that our intermediate proposed circuit is completely different from the 0° feed structure proposed in [161] where identical values for capacitances, Cp1and Cp2 were adopted. In our circuit, the capacitance
2
Cp represents the open-end transmission line shunt capacitance whereas capacitance
1
Cp is the shunt capacitance contributed by a microstrip gap. With the aid of the circuit theory, the scattering parameters for the lower and upper signal paths are respectively given as
( )
( )
, , , ,
11 11 22 11
11 22 , , , , ,
11 11 22 11
1 1
x l x x x l x x x
x t x b
x l x l x x x
S S S
S S
S S S S
S ,
π π π
π π π
= = ∆ ∆ − + − ∆
= =
⎡ ⎤
+ ⎣ ∆ − + ⎦ , (4.27)
( ) ( )
, 2 ,
21 21
, ,
12 21 , , , , ,
11 11 22 11
1
x l x
x t b x t b
x l x l x x x
S S
S S
S S S S
π
π π π
= = = =
⎡ ⎤
+ ⎣ ∆ − + ⎦
, (4.28)
( )
( )
, , , ,
11 11 11 22
22 11 , , , , ,
11 11 22 11
1 1
x l x x x l x x x
x t x b
x l x l x x x
S S S
S S
S S S S
S ,
π π π
π π π
= = ∆ ∆ − + − ∆
= =
⎡ ⎤
+ ⎣ ∆ − + ⎦
, (4.29)
where∆ = ∆ = ∆ = ∆x t b ,∆ = ∆ = ∆ = ∆x,π t,π b,π π are respectively the determinants of the matrix Sxand Sx,π,
, upper path, , lower path, x t
b
= ⎨⎧
⎩ (4.30)
( ) ( )( )
( )( ) ( ( ) )
2
1 2 2 1
, ,
11 22
1 2 2 1
g p p o p o p
t b
o p o p g o p p
s C C C Y sC Y sC
S S
Y sC Y sC sC Y s C C
π π − + + + −
= =
+ + + + + 2
)
,
(4.31)
( )( ) ( ( )
, , , ,
12 21
2 1 1
2 ,
2
o g
x t b x t b
o p o p g o p p
S S sY C
Y sC Y sC sC Y s C C
π π
= = = =
+ + + + + 2
(4.32)
( ) ( )( )
( )( ) ( ( ) )
2
1 2 1 2
, ,
22 11
2 1 1
2 ,
g p p o p o p
t b
o p o p g o p p
s C C C Y sC Y sC
S S
Y sC Y sC sC Y s C C
π π − + + + −
= =
+ + + + + 2
(4.33)
( 2− 2) θ
( )
, , , , 1
12 21 12 21 2 2
1 1 1
2
sin 2 cos
t l t l b l b l o
o o
S S S S jZ Z
Z Z θ jZ Z θ1
= = = =
+ − , (4.35)
and finally, s= jω. As noted from equations (4.28), (4.31) to (4.35), except for the transmission zero at GHz, there is no other transmission zeros observed at other higher frequencies for both the individual lower and upper signal paths. Thus, one can conclude that the presence of the transmission zeros are mainly caused by the coupling between the feedline and the filter structure. This fact is not clearly highlighted in [161] and their equation (4.32) seems to imply that the junction has no effect on the overall transmission zeros. By using the network theory [164], the overall S- parameters of Fig. 35 are given as
0 f =
[ ] [ ] [ ]
( ) ( ) ( ) ( ) ( ) [ ]
12 13
11
2 2 2
11 22 12 12 12 13 11 22 13 12 22 22 23 13 12 11 22 23 13
12
0 0
0 0 0 0
1 ,
j j
j
t t j t j j t t j j t j j j j t j j j
t
S S
S S I P
S S S S S S S S S S S S S S S S S S S P
S π π π π
⎡ ⎤
= + ⎢ ⎥
⎣ ⎦
⎧⎡ ⎤⎡ ⎤⎫
⎪⎢ ⎥ ⎪
+ ⎨⎪⎩⎢⎣− + ⎡⎢⎣ + ⎤⎥⎦ − ⎥⎢⎦⎢⎣ − ∆ − ∆ − ∆ − ∆ ⎥⎥⎦⎬⎪⎭
(4.36) where the superscript j denotes the microstrip junction S-parameters. Clearly, only the second term of equation (4.36) contributes to the transmission zeros. To simplify the
analysis, one can assume that and . Since , the
transmission zeros can be evaluated from the matrix P through
12 13 23
j j
S =S =Sj S22j =S33j S12j =S13j ≠0
12 22 0
P +P = . (4.37)
By grouping the terms in equation (4.37), it results in two equations, namely:
( ) ( )
( ) ( )
2
12 22 12 11
22 11 22 22 12 11
1
t j j t
j j t t j j t
S S S S
S S S S S S
⎡∆ + − ⎤
⎣ ⎦ =
⎡ ⎤ ⎡
∆ ∆∆ −⎣ + ⎦ ⎣∆ − − ⎤⎦ , (4.38)
( )
( )
12 12 22 12 22
11 11 22 22
2 1
j t t j j
t j t t j
S S S S S
S S S S
⎡ + + ∆ − ⎤
⎣ ⎦ = −
⎡∆∆ − + ⎤
⎣ ⎦
, (4.39)
where . By dividing equation (4.38) with equation (4.39), we obtain the necessary and sufficient equation for determining Z
( ) ( )22 2 12
j j j
S S
∆ = − 2
1 and θ1, i.e.
( ) ( )
( ) ( )
2
11 12 22 12 11
12 22 12 11 12 22 12 22
2 1.
t t j j t
j j j t t t j j
S S S S S
S S S S S S S S
⎡ ⎤
∆⎣ + ⎦− = −
⎡ ⎤
⎡ ⎤
∆ ∆⎣ − ⎦− ⎣ + + ∆ − ⎦
(4.40)
Z1 in equation (4.40) is usually selected based on matching criteria and in here, 70Ω, and θ1 is subsequently solved from equation (4.40). In the process of evaluating θ1, we can usually proceed initially with ideal junction S-parameters. The various capacitances are computed from [163] once Z1 has been fixed.