Upload
ek-powell
View
214
Download
0
Tags:
Embed Size (px)
DESCRIPTION
tesch
Citation preview
Turk J Elec Eng & Comp Sci
() : 1 { 17
c TUB_ITAKdoi:10.3906/elk-1203-69
Turkish Journal of Electrical Engineering & Computer Sciences
http :// journa l s . tub i tak .gov . t r/e lektr ik/
Research Article
Optimum design of bandpass lters using coupled open- and short-ended
resonators
Homayoon ORAIZI, Mahdi ZOUGHI
Department of Electrical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
Received: 16.03.2012 Accepted: 29.07.2012 Published Online: ..2014 Printed: ..2014
Abstract:In this paper, an optimum design method is developed for bandpass lters composed of open- and short-ended
coupled resonators. The design procedure is based on a matrix representation of the resonator lter conguration for
the derivation of its scattering parameters. They are used for the construction of an error function, which depends on
the geometrical dimensions of the lter. Its minimization leads to the optimum design of the lter. The implementation
of the proposed method, full-wave simulation software results, fabrication, and measurement data indicate that the
proposed method obtains an eective performance with this lter structure, such as the broad band width for the
passband lter, suppression of higher harmonics, low insertion loss in the passband, deep attenuation in the stopbands,
and sharp transition bands. The proposed design procedure also incorporates impedance matching between dierent
arbitrarily specied input and output impedances.
Key words: Harmonic suppression, half wavelength resonators, method of least squares
1. Introduction
Bandpass lters are the basic components in microwave circuits, which are mainly composed of coupled
resonators. Various congurations of resonators have been proposed and investigated for applications in
microwave lters, such as hairpin [1,2] and open-loop [3{7] resonators, which are commonly open-ended.
However, generally, several harmonics appear across their frequency response. Various techniques have been
considered for the suppression of spurious harmonics in their stopbands [8{15], yet few attempts have been made
to apply a combination of open- and closed-loop coupled resonators, such as split- and closed-ring resonators.
Recently, such lter congurations have been proposed, which achieve a notable suppression of the undesired
harmonics [16]. In this design, the coupling regions among split- and closed-loop resonators are deliberately
selected in such a way as to provide the appropriate conditions for harmonic suppression.
In this paper, we develop a design procedure for the achievement of good realizable and optimum
performance from such lters. First, we develop an equivalent circuit for this lter type, for which the
transmission matrix is determined. Second, we construct an error function for the realization of the desired
lter response by the application of scattering parameters. The minimization of the error function gives the
lter geometrical dimensions. Prototype models of such lter designs are fabricated and measured for the
verication of the proposed design procedure, which achieves a drastic reduction of the harmonics in the
stopbands, enhancement of frequency bandwidth, reduction of insertion loss, impedance matching between
the input and output ports, and maximization of return losses.
Correspondence: [email protected]
1
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
1.1. Harmonic suppression by the use of half-wave coupled resonators
We rst review the operation of bandpass lters composed of coupled split- and closed-loop resonators. The
coupling coecient between 2 resonators may be dened as the ratio of coupled energy [17], as depicted in
Figure 1. Therefore, the electric and magnetic coupling coecients (namely ke and km , respectively) between
2 resonators may be dened as:
ke =
RRR" E1: E2dvqRRR
" E12 dv RRR " E22 dv ; (1)
km =
RRR H1: H2dvqRRR
H12 dv RRR H22 dv : (2)
The total coupling coecient is:
k = ke + km: (3)
Coupling
1E
2E
1H
2H
Resonator1 Resonator2
Figure 1. Two dierent coupled resonators having distinct resonant frequencies [17].
Two half-wave coupled resonators (L = g=2) are shown in Figure 2, where the rst line section is short-
circuited at both ends (Figure 2a) and the second is open-circuited at its ends (Figure 2b). The normalized
voltage distributions for the 1st, 2nd, and 3rd harmonics are also shown on the half-wave line sections. The
voltage distributions on the 2 line sections in Figures 2a and 2b are as follows, respectively:
Vi(l) = sin i0l; i = 1; 2; 3; l 2 [0; L] ; (4)
Vj(l) = cos(j 3)0l; j = 4; 5; 6; l 2 [0; L] : (5)
If the 2 lines are coupled in the region of A to C, then the 2nd harmonic will be cancelled, or at least attenuated,
since in the interval A-C relative to the center line B, the voltage V2 is an even function and V5 is an odd
function. Next,
ke =
R lClA
V2(l)V5(l)dlqR lClAjV2(l)j2 dl
R lClAjV5(l)j2 dl
= 0: (6)
Furthermore, the magnetic coupling coecient is also 0. Consequently,
k = ke + km = 0: (7)
2
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Similarly, if the 2 line sections are coupled in the region of D to F, then the 3rd harmonic will be suppressed,
since in this interval relative to the center line E, voltage V3 is an even function and V6 is an odd function.
Thus, the electric and similarly magnetic coupling coecients will be 0.
Coupling Region for 2nd
Harmonic Suppression
ion
Coupling Region for 3rd
Harmonic Suppression
(a)
(b)
D E F A B C
0 L l
V3 V1
V2
V5
V6 V4
Figure 2. Voltage waves along 2 coupled resonators at the fundamental frequency, 2nd, and 3rd harmonics: a) short-
ended resonator, b) open-ended resonator.
1.2. Design procedure for bandpass lters
Consider the coupled square-loop resonator bandpass lter, as shown in Figure 3. The left and right square
loops are the open- and short-circuited resonators, respectively, which cancel the 2nd harmonic. We rst obtain
its equivalent circuit, as shown in Figure 4, which is composed of transmission line sections [18], bends [19],
gaps [20], open-ended lines [21], T-junctions [22], coupled line sections [23,24], and short-circuited vias [25].
The equivalent circuit between the input and output ports, namely between the 2 T-junctions, T(j1)1 and T
(j2)1 ,
may be redrawn as in Figure 5. The voltages and currents at various points on the equivalent circuits are
also denoted. For example, the transmission matrix between points (V (J1); I(J1)2 ) and (V
(C)1 ; I
(C)1 ), denoted by
T (1)22 in Figures 4 and 5, may be obtained as:24 V (J1)
I(J1)2
35 = hT (1)i24 V (C)1
I(C)1
35 ; (8)hT (1)
i=hT(J1)2
ihT (TL1)
ihT (B1)
ihT (TL2)
ihT (B1)
ihT (TL3)
ihT (G)
ihT (TL4)
ihT (B1)
i: (9)
Similarly, the other transmission matricesT (i)
22 ; i = 1; 2; 3; 4 may be obtained. Next, the corresponding
admittance matricesY (i)
22 are derived. Finally, the admittance matrix of the 4-port networks,
Y (l)
44
3
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
andY (r)
44 , as denoted in Figure 5, are obtained.
26666664I(J1)2
I(J1)3
I(C)1I(C)2
37777775 =hY (l)
i44
2666664V (J1)
V (J1)
V(C)1
V(C)2
3777775 ; (10)
where
hY (l)
i44
=
26666664Y(1)11 0 Y
(1)12 0
0 Y(2)11 0 Y
(2)12
Y(1)21 0 Y
(1)22 0
0 Y(2)21 0 Y
(2)22
37777775 ; (11)
hY (r)
i44
=
26666664Y(3)11 0 Y
(3)11 0
0 Y(4)11 0 Y
(4)12
Y(3)21 0 Y
(3)22 0
0 Y(4)21 0 Y
(4)22
37777775 : (12)
The corresponding transmission matrices,T (l)
44 and
T (r)
44 , are then obtained. Moreover, the transmis-
sion matrixT (C)
44 of the coupler may be obtained from its admittance matrix
Y (C)
44 (see Appendix).
The transmission matrix of the 4-port network in Figure 5, namely [T ]44 , may then be derived as:
[T ]44 =hT (l)
i44
hT (C)
i44
hT (r)
i44
; (13)
2666664V (J1)
V (J1)
I(J1)2
I(J1)3
3777775 = [T ]44 2666664
V (J2)
V (J2)
I(J2)2
I(J2)3
3777775 : (14)
The input ports of the 4-port network [T ]44 are connected together and the outputs are also connected. Next,
8
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
L5 L1
W3
L2
L4 L3 L6 L7
L8S1
L9
L10
w1 W2
W4
g1
Figure 3. A 2nd-order bandpass lter with the 2nd harmonic suppression.
TL1 TL5Bend1
TL2
Bend1 TL3 Gap TL4
Bend1
Bend1
Co
up ler
Bend2
Bend2
Bend2
Bend2
ViaTL6
TL8
TL7
TL9
TL10
Port1
Port2
(J1)2T
(J2)2T
(J2)3T
(J1)3T
(J1)1T
(J2)1T
T-Junction1
T-Junction2 [T(1)] [T(2)]
[T(3)]
[T(4)]
.
(J1)1I
(J1)3I
(J1)2I
(J1)V
(J2)3I
(J2)2I
(J2)1I
(J2)V
Figure 4. The equivalent circuit of the bandpass lter in Figure 3.
Port1 Port2
(1)I (1)V
(J1)
1I
(J1)V
(J1)
2I
(J1)
3I
(C)
1I
(C)1V
(C)
2I
(C)
3I
(C)
4I
(C)2V
(C)
3V
(C)
4V
(J2)V (2)I
(2)V
[T]44
(J2)
1I
(J2)
2I
(J2)
3I
[ ](J1)1T [ ](1)T
[ ](2)T
[ ](3)T
[ ](4)T [ ](J2)1T
[ ] [ ] 44)(44)( TY , ll [ ] [ ] 44)(
44
)( TY , rr
[ ] 44(C)T
Figure 5. The equivalent circuit of the bandpass lter in Figure 3 as expressed by the transmission matrices.
5
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Port1 Port2(J1)1I
(J2)1I
(J2)V (J1)V
(2)I
(2)V
[ ](J1)1T
[ ](J2)1T (1)I [ ] 22(M)Y (1)V Figure 6. The 2 port equivalent circuit of the bandpass lter in Figure 3.
24 I(J1)1I(J2)1
35 = hY (M)i22
24 V (J1)
V (J2)
35 (16)Its corresponding transmission matrix
T (M)
22 is used to obtain that of the equivalent circuit of lter
T (T )22 , as shown in Figure 6. Consequently,24 V (1)
I(1)
35 = hT (T )i22
24 V (2)
I(2)
35 ; (17)where
TT22 =
hT(J1)1
i22
hT (M)
i22
hT(J2)1
i22
: (18)
Finally, the transmission matrix may be converted to its scattering matrix,S(T )
22 :
We then specify a desired frequency response for the bandpass lter, as shown in Figure 7, where the
frequency interval is divided into K discrete frequencies and also delineated into the lower stopband (LSB), lower
transition band (LTB), passband (PB), upper transition band (UTB), upper stopband (USB), 2nd harmonic
suppression band (SB2), and 3rd harmonic suppression band (SB3). The desired scattering parameters are
denoted by G(LSB)21 ; G
LTB21 ; G
(PB)21 ; G
UTB21 ; G
(USB)21 , G
(SB2)21 and G
(SB3)21 :
We now construct an error function using the above desired and computed scattering parameters. Note
that since for lossless devices, the scattering parameters are related by jS11j2 + jS21j2 = 1, we need not includethe reection coecients (S11) in the error function. Next,
ef =Wt1NLSBPk=1
(jS21(fk)j G(LSB)21 (fk))2 +Wt2 NLTBP
k=NLSB+1
(jS21(fk)j G(LTB)21 (fk))2
+Wt3NPBP
k=NLTB+1
(jS21(fk)j G(PB)21 (fk))2 +Wt4 NUTBP
k=NPB+1
(jS21(fk)j G(UTB)21 (fk))2
+Wt5NUSBP
k=NUTB+1
(jS21(fk)j G(USB)21 (fk))2 +Wt6 N2NDP
k=NUSB+1
(jS21(fk)j G(SB2)21 (fk))2
;
(19)
where Wt1 , Wt2 , . . . , Wt6 are weighting functions, which enhance one subsection of the frequency interval
relative to the others. The error is a function of the widths (W i), lengths (L i), and gap spacings (S i) of the
line sections, which are determined by locating its minimum point.
We use a combination of the genetic algorithm (GA) and conjugate gradient (CG) method for the
minimization of ef. We rst activate the GA as a global extremum-seeking algorithm, which does not need
the initial values of the parameters, but it is very slow. Accordingly, in order to speed up the minimization of
ef, the GA is aborted prematurely and then it is handed over to the CG, which is a local extremum-seeking
algorithm and needs some initial values for the parameters, but it is quite fast. The stopping criteria for the GA
6
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
may be specied as the maximum number of generations, CPU time limit, tness limit, stall generation, stall
time limit, function tolerance, and nonlinear constraint tolerance [26]. The stopping criteria for the CG may be
specied as the maximum number of iterations, minimum gradient, and minimum value of error function.
2. Computer implementation for the design and fabrication of bandpass lters
2.1. Second-order bandpass lter with the suppression of the 2nd harmonic
We design a 2nd-order bandpass lter with the suppression of the 2nd harmonic, as shown in Figure 3. The
center frequency is 1 GHz, fractional bandwidth is 25%, and input and output impedances are 50 . The
parameters of the desired frequency response of the lter as denoted in Figure 4 and Eq. (18) are given in Table
1. The substrate RTDuroid 5880 (with dielectric constant "r = 2.2, height h = 31 mil, and loss tangent tan
= 0.0009) is used for the lter design. The optimum geometrical dimensions of the lter (as lengths, widths,
and spacings of the line sections) are given in Table 2. The frequency response of the optimum lter designed
by the proposed method, as the curves of S11 and S21 versus frequency, is drawn in Figure 8. The 3 dB of
bandwidth is from 858 MHz to 1103 MHz and is about 25%. The proposed design procedure has thus achieved
the potential performance of the lter described in [16]. The bandwidth obtained in that reference was about
8%. Note the wide stopband of our lter design, which is about 3 GHz. The 3rd and 4th harmonics are also
attenuated drastically.
(P1)21G
(P2)21G
(P3)21G
PLf 0f 03f
(dB)S21
f(GHz)
0
SLf TLf TUf PL2f PU2f PL3f PU3f 02f PUf SUf
Lower
Stop
band
Lower
transition
band
Pass-
band
Upper
transition
band
Upper
stop
band
2nd Harmonic
suppression band 3rd Harmonic
suppression band
Figure 7. Specied frequency response of the bandpass lter: LSB = lower stopband, LTB = lower transition
band, PB = passband, USB = upper stopband, UTB = upper transition band, HSB = harmonic suppression
band.
A photograph of the fabricated lter is shown in Figure 9. The measurement data and the results of its
full-wave simulation by a high-frequency structural simulator (HFSS) are also shown in Figure 8 for comparison.
Note that the width of the input and output line sections is W3 = W3 = 2.4 mm, to provide an impedance of
50 .
2.2. Second-order bandpass lter with the suppression of the 2nd harmonic, together with
impedance matching
We design a lter with the same characteristics as given in Example 1 and Table 1, except that the input and
output impedances are Z in = 50 and Zout = 75 , respectively. The optimum values of the geometrical
dimensions of the lter are given in Table 3. Its frequency response, as the curves of S11 and S21 versus the
7
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
frequency, is shown in Figure 10, as obtained by the full-wave simulation software HFSS and our proposed design
algorithm denoted by MATLAB. Observe that the achieved 3 dB of bandwidth is about 24%, from 870 MHz
to 1104 MHz. Note also the wide stopband with the deep attenuation across the 2nd, 3rd, and 4th harmonics.
2.3. Third-order bandpass lter with the suppression of the 2nd harmonic
We design a 3rd-order bandpass lter with the geometrical conguration as shown in Figure 11. The center
frequency is 1 GHz, the bandwidth is 20%, and the input and output impedances are 50 . We use the substrate
RTDuroid 5880. The parameters of the desired frequency response of the lter as denoted in Figure 7 and Eq.
(18) are given in Table 4. The circuit conguration of the lter as composed of various components is drawn in
Figure 12 and the resulting equivalent circuit is drawn in Figure 13. The optimum design of the lter by the
proposed method provides the dimensions given in Table 5. The frequency response of the lter, as the curves
of S11 and S21 versus the frequency, is shown in Figure 14. Note that the 3 dB of bandwidth is 20%, from 906
to 1110 MHz. The proposed lter design procedure achieves almost 3 times the bandwidth of the 8% obtained
in [16]. Observe that the 4th harmonic is also considerably attenuated.
2.4. Third-order bandpass lter with the suppression of the 2nd, 3rd, and 4th harmonics
We design a 3rd-order bandpass lter with the geometrical conguration as shown in Figure 15 for the suppres-
sion of the 2nd, 3rd, and 4th harmonics. There are 2 line sections as loads connected to the 2 ends of the 1st
coupler. In this conguration, the 1st coupler (L4) cancels the 2nd and 4th harmonics and the 2nd coupler (L8)
cancels the 3rd harmonic. The center frequency is 1 GHz, the relative 3 dB of bandwidth is to be 45%, and the
input and output impedances are 50 . We use the substrate RTDuroid 5880. The circuit conguration of the
lter composed of various components and its equivalent circuit are drawn in Figures 16 and 17, respectively.
Its overall transmission and scattering matrices may be obtained in a routine manner. We may then construct
the error function in Eq. (18) according to the lter characteristics denoted in Figure 7 and specied in Table
6. The dimensions of the optimum lter conguration are given in Table 7. Its frequency response, as the vari-
ations of S11 and S21 versus the frequency obtained by the HFSS and our proposed design procedure (denoted
by MATLAB), is drawn in Figure 18. Observe that the 3 dB of bandwidth is about 43%, from 760 MHz to 1171
MHz. Its stopband extends over 3.5 GHz. Note also that the bandwidth of the unoptimized version is about
22%, as reported in [16].
2.5. Third-order bandpass lter with the suppression of the 3rd harmonic
We design a 3rd-order bandpass lter with the suppression of the 3rd harmonic by the geometrical conguration
shown in Figure 19. It is basically similar to the lter in Example 4, except that the 2 loads connected to the 2
ends of the 1st coupler are removed. Accordingly, the equivalent circuit and the computation of the scattering
parameters are the same, with some minor dierences. The center frequency is 1 GHz and the specied 3 dB
of bandwidth is to be 30%. The input and output impedances are 50 . The lter characteristics according
to Figure 7 are specied as in Table 4. We use the substrate RTDuroid 5880. The optimum values of the
dimensions of the geometrical conguration of the lter are given in Table 8. A photograph of the fabricated
prototype model is shown in Figure 20. The frequency response of the lter as S11 and S21 obtained by our
algorithm (denoted by MATLAB), full-wave simulation by the HFSS, and measurement data are drawn in
Figure 21. Good agreement is obtained among the 3 sets of data. The 3 dB of bandwidth of 30%, from 829
MHz to 1124 MHz, is achieved.
8
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Table 1. Desired frequency response for Example 1 with reference to Figure 7.
2nd
HSB
USB UTB UPB LPB LTB LSB
- 1.5 1.25 1.05 0.95 0.75 0.5 f (GHz)
20 20 20 0.5 0.5 20 20 G21 (dB)
2 1 1 10 10 1 1 Wt
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5-70
-60
-50
-40
-30
-20
-10
0
f (GHz)
S (dB
)
MeasurementHFSSCalculation
S
S21
S11
Figure 8. The frequency response of the optimum lter in Example 1.
Figure 9. A photograph of the fabricated lter for Example 1.
9
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Table 2. Optimum geometrical dimensions of the lter in Example 1.
S1 g1 W2 W1 L10 L9 L8 L7 L6 L5 L4 L3 L2 L1
0.51 0.53 0.69 0.21 1 35.625.29 16.82 5.07 9.76 18.35 1 35.6 7.7 Dimension (mm)
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5-70
-60
-50
-40
-30
-20
-10
0
f (GHz)
S (dB
)
MeasurementHFSSCalculation
S
S21
S11
Figure 10. The frequency response of the optimum lter in Example 2.
Table 3. Optimum geometrical dimensions of the lter in Example 2.
L7 L6 L5 L4 L3 L2 L1
14.53 6.45 10.05 18.69 1.845 34.08 8.78 Dimension (mm)
S1 g1 W2 W1 L10 L9 L8
0.7 0.7 0.99 0.25 1.31 34.08 23.52 Dimension (mm)
L5 L1
W4
L2
L4L3 L6
L8 S1 L12 L11
L9 L10
w1 W2 W3
S2
g1 g2
W5
L2
L7
Figure 11. A 3rd-order bandpass lter with the 2nd harmonic suppression in Example 3.
10
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Table 4. Desired frequency response for Example 3 with reference to Figure 7.
2nd
HSB
USB UTB UPB LPB LTB LSB
- 1.5 1.25 1.075 0.925 0.75 0.5 f (GHz)
30 30 0.5 0.5 20 30 G21 (dB)
2 1 1 10 10 1 1 Wt
30
TL1 TL5Bend
1
TL2
Bend
1TL3 Gap1 TL4
Bend
1
Bend
1
Coupler1
Port
1
TL8
TL6
Bend
2
Bend
2Via
[ ](J1)2T [ ](J1)3T[ ](J1)1T
Bend
2
Bend
2
TL7
Bend
3
Bend
3
Bend
3
Bend
3
Gap2TL9
TL12
TL2
TL10
Coupl er2
Port
2
TL11
[ ](J2)1T[ ](J2)3T [ ](J2)2T
T-Junction1
(J1)V
(J1)
3I (J1)
2I
(J1)
1I
[T(1)
] [T(2)
] [T(3)
] [T
(4)]
T-Junction2 [T
(5)]
[T(6)
]
(J2)
3I (J2)
2I
(J2)
1I (J2)V
Figure 12. A block diagram of the 3rd-order bandpass lter in Example 3.
Table 5. Optimum geometrical dimensions of the lter in Example 3.
L10 L9 L8 L7 L6 L5 L4 L3 L2 L1
6.79 11.59 21.01 9.71 11.3 18.82 10.84 8.48 33.6 0.02 Dimension (mm)
S2 S1 g2 g1 W3 W2 W1 L12 L11
0.58 0.59 2.22 1.89 0.36 1.13 0.35 17.96 0.2 Dimension (mm)
11
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Port2Port1
(1)I
(1)V
(J1)1I (J1)V
(J1)2I
(J1)3I
(C1)1I (C1)
1V
(C1)2I
(C1)3I
(C1)2V
(C1)3V
[T]44
(C1)4I
(C1)4V
(C2)1I
(C2)1V
(C2)2I
(C2)2V
(C2)3I (C2)3V
(C2)4I (C2)4V
(J2)1I (J2)V
(J2)2I
(J2)3I
(2)I
(2)V [ ](J2)1T [ ](1)T
[ ](2)T [ ](J1)1T
[ ] [ ] 44)(44)( TY , ll [ ] [ ] 44)(44)( TY , rr[ ] [ ] 44)(44)( TY , mm
[ ] 44(C1)T [ ] 44(C2)T [ ](3)T
[ ](4)T
[ ](5)T
[ ](6)T
T (1)[ ]
Figure 13. An equivalent block diagram of the lter in Example 3.
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5-120-110-100
-90-80-70-60-50-40-30-20-10
0
f (GHz)
S (dB
)
HFSSCalculation
S11
S21
Figure 14. The frequency response of the optimum lter in Example 3.
L3 L2
W4
L1
L5
L4
L6
L7 S1 L9 L10
L11
W1 W2 W3
S2
W5
L8
Figure 15. The 3rd-order bandpass lter in Example 4.
12
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
TL2 TL3Bend1
TL1
OpenCircuit 1
Bend1
Co
u pler1
Bend3
Bend3TL9
TL11
TL7
Via
Bend2
Co
up ler 2
Bend2
Via
TL10
OpenCircuit 1
OpenCircuit 2
OpenCircuit 2TL5
Bend1 TL6
Bend2
Port1
[ ](J1)2T [ ](J1)3T[ ](J1)1T
Port2
[ ](J2)1T[ ](J2)3T [ ](J2)2T
Z(1) [T(2)]
T-Junction1
[T(3)]
T-Junction2
[T(4)]
(J1)V
(C1)1Z
(C1)3Z
Z(5)
(J2)V
Figure 16. A block diagram of the 3rd-order bandpass lter in Example 4.
Port1
Port2
(J1)1I
(J1)2I
(J1)3I
(J2)3I
(J2)1I
(J2)2I
(1)I
(1)V (J1)V (J2)V
(C1)1I (C1)
1V
(C1)2I
(C1)3I
(C1)4I
(C1)2V
(C1)3V
(C1)4V
[ ](J1)1T
(1)Z
[ ] 22(1)T
[ ](2)T (C1)1Z
(C1)3Z
[ ] 44(C1)T
[ ] 22(C1)T
[ ](3)T [ ](4)T
(C2)1I
(C2)1V
(C2)2I (C2)4I (C2)2V
(C2)3V
(C2)4V
(Via)Z
(OC2)Z
(C2)3I
[ ] 44(C2)T
[ ] 22(C2)T
(5)Z
[ ] 22(5)T
(2)V
(2)I [ ](J2)1T
Figure 17. An equivalent block diagram of the lter in Example 4.
Table 6. Desired frequency response for Example 4 with reference to Figure 7.
4th
HSB
3rd
HSB
2nd
HSB USB UTB UPB LPB LTB LSB
- - - 1.5 1.25 1.15 0.85 0.7 0.5 f (GHz)
20 20 20 20 20 0.5 0.5 20 20 G21 (dB)
2 2 2 1 1 10 10 1 1 Wt
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5-90
-80
-70
-60
-50
-40
-30
-20
-10
0
f (GHz)
S (dB
)
HFSSCalculation
S
S21
11
Figure 18. The frequency response of the optimum lter in Example 4.
13
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Table 7. Optimum geometrical dimensions of the lter in Example 4.
L8 L7 L6 L5 L4 L3 L2 L1
39.51 32.38 13.16 13.14 30.34 20.86 7.61 30.29 Dimension (mm)
S2 S1 W3 W2 W1 L11 L10 L9
0.24 0.24 0.39 0.42 0.41 66.54 19.98 9.16 Dimension (mm)
L3 L2
W4
L1
L5 S1 L7 L8
W1
W2 W3
S2
W5
L9 L4 L6
Figure 19. The 3rd-order bandpass lter in Example 5.
Figure 20. A photograph of the fabricated lter for Example 5.
0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5-120
-105
-90
-75
-60
-45
-30
-15
0
f (GHz)
S (dB
)
MeasurementHFSSCalculation
11S
S21
Figure 21. The frequency response of the optimum lter in Example 5.
14
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Table 8. Optimum geometrical dimensions of the lter in Example 5.
L7 L6 L5 L4 L3 L2 L1
25.54 39.33 35.68 41.24 24.49 19.1 18.63 Dimension (mm)
S2 S1 W3 W2 W1 L9 L8
0.26 0.31 0.7 1.07 0.49 17.03 21.07 Dimension (mm)
3. Conclusion
We have amply shown by several examples of computer simulation and actual fabrication and measurement
that the method of least squares is quite suitable for devising optimum design procedures for bandpass lters
composed of coupled split and closed loop resonators. These design methods are capable of achieving the
eective performance realizable from the lter conguration, such as broad passbands, short transition bands,
deep stopbands, and the suppression of harmonics. The study in this paper may also be considered as evidence
for the eectiveness of bandpass lters made of open- and short-ended resonators [16].
Appendix
Transformation of a 4-port admittance matrix to its equivalent transmission matrix
Consider a 4-port network (shown in Figure A1). We can then convert the admittance matrix to a transmission
matrix as:
[Y ] =
2666664Y11 Y12 Y13 Y14
Y21 Y22 Y23 Y24
Y31 Y32 Y33 Y34
Y41 Y42 Y43 Y44
3777775 ="[Y1] [Y2]
[Y3] [Y4]
#; (A1)
[T ] =
24 [Y4] [Y3]1 [Y3]1([Y2] [Y3] [Y1] [Y4]) [Y3]1 [Y1] [Y3]1
35 : (A2)
Port3
Port4
Port1
Port2
(1)I
(1)V
(2)I (2)V
(3)I
(3)V
(4)I (4)V
Four Port Network
Figure A1. A 4-port network.
Determination of the 2-port transmission matrix of a loaded 4-port network
Consider a loaded 4-port network, as shown in Figure A2, where the port loads, currents, and voltages are
indicated. The transmission matrix of the 4-port network is:
15
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
Port1
Port2
Port3
Port4
(1)I
(1)V
(2)I
(2)V
(3)I
(3)V
(4)I
(4)V
(1)Z (3)Z
[ ] 44T Figure A2. The schematic diagram of a loaded 4-port network.
[T ]4 4 =
2666664T11 T12 T13 T14
T21 T22 T23 T24
T31 T32 T33 T34
T41 T42 T43 T44
3777775 : (A3)
Ports 1 and 3 are loaded by Z (1) and Z (3) , respectively. Then,
8
ORAIZI and ZOUGHI/Turk J Elec Eng & Comp Sci
[5] L.H. Hsieh, K. Chang, \Tunable microstrip bandpass lters with two transmission zeros", IEEE Transactions on
Microwave Theory and Technique, Vol. 51, pp. 520{525, 2003.
[6] P. Mondal, M.K. Mandal, \Design of dual-band bandpass lters using stub-loaded open-loop resonators", IEEE
Transactions on Microwave Theory and Technique, Vol. 56, pp. 150{155, 2008.
[7] X.Y. Zhang, J.X. Chen, Q. Xue, S.M. Li, \Dual-band bandpass lters using stub-loaded resonators", IEEE
Microwave and Wireless Components Letters, Vol. 17, pp. 583{585, 2007.
[8] W.H. Tu, K. Chang, \Compact microstrip bandstop lter using open stub and spurline", IEEE Microwave and
Wireless Components Letters, Vol. 15, pp. 268{270, 2005.
[9] J.T. Kuo, W.H. Hsu, W.T. Huang, \Parallel coupled microstrip lters with suppression of harmonic response",
IEEE Microwave and Wireless Components Letters, Vol. 12, pp. 383{385, 2002.
[10] T. Lopetegi, M.A.G. Laso, J. Hernandez, M. Bacaicoa, D. Benito, M.J. Garde, M. Sorolla, M. Guglielmi, \New
microstrip `wigglyline' lters with spurious passband suppression", IEEE Transactions on Microwave Theory and
Technique, Vol. 49, pp. 1593{1598, 2001.
[11] I.K. Kim, N. Kingsley, M. Morton, R. Bairavasubramanian, J. Papapolymerou, M.M. Tentzeris, J.G. Yook, \Fractal-
shaped microstrip coupled-line bandpass lters for suppression of second harmonic", IEEE Transactions on Mi-
crowave Theory and Technique, Vol. 53, pp. 2943{2948, 2005.
[12] B.S. Kim, J.W. Lee, M.S. Song, \An implementation of harmonic-suppression microstrip lters with periodic
grooves", IEEE Microwave and Wireless Components Letters, Vol. 14, pp. 413{415, 2004.
[13] M. Moradian, M. Tayarani, \Spurious-response suppression in microstrip parallel-coupled bandpass lters by
grooved substrates", IEEE Transactions on Microwave Theory and Technique, Vol. 56, pp. 1707{1713, 2008.
[14] C.F. Chen, T.Y. Huang, R.B. Wu, \Design of microstrip bandpass lters with multiorder spurious-mode suppres-
sion", IEEE Transactions on Microwave Theory and Technique, Vol. 53, pp. 3788{3793, 2005.
[15] S.C. Lin, P.H. Deng, Y.S. Lin, C.H. Wang, C.H. Chen, \Wide-stopband microstrip bandpass lters using dissimilar
quarter-wavelength stepped-impedance resonators", IEEE Transactions on Microwave Theory and Technique, Vol.
54, pp. 1011{1018, 2006.
[16] G.L. Dai, X.Y. Zhang, C.H. Chan, Q. Xue, M.Y. Xia, \An investigation of open- and short-ended resonators
and their applications to bandpass lters", IEEE Transactions on Microwave Theory and Technique, Vol. 57, pp.
2203{2210, 2009
[17] J.S. Hong, M.J. Lancaster, Microstrip Filters for RF/Microwave Applications, New York, Wiley, 2001.
[18] E. Hammerstad, O. Jensen, \Accurate models for microstrip computer-aided design", IEEE MTT-S International
Microwave Symposium Digest, pp. 407{409, 1980.
[19] M. Kirschning, R.H. Jansen, N.H.L. Koster, \Measurement and computer-aided modeling of microstrip discontinu-
ities by an improved resonator method", IEEE MTT-S International Microwave Symposium Digest, pp. 495{497,
1983.
[20] R.K. Homann, Handbook of Microwave Integrated Circuits, Norwood, MA, USA, Artech House, 1987.
[21] K.C. Gupta, R. Garg, I. Bahl, P. Bhartia, Microstrip Lines and Slotlines, 2nd ed., Norwood, MA, USA, Artech
House, 1996.
[22] E. Hammerstad, \Computer-aided design of microstrip couplers with accurate discontinuity models", IEEE MTT-S
International Microwave Symposium Digest, Vol. 81, pp. 54{56, 1981.
[23] M.K. Amirhosseini, \Determination of capacitance and conductance matrices of lossy shielded coupled microstrip
transmission lines", Progress in Electromagnetics Research, Vol. 50, pp. 267{278, 2005.
[24] V. Tripathi, \Asymmetric coupled transmission lines in an inhomogeneous medium", IEEE Transactions on Mi-
crowave Theory and Technique, Vol. MTT-23, pp. 734{739, 1975.
[25] M. Goldfarb, R. Pucel, \Modeling via hole grounds in microstrip", IEEE Microwave and Guided Wave Letters, Vol.
1, pp. 135{137, 1991.
[26] MATLAB-R2009a Help, \How the genetic algorithm works?", Natick, MA, USA, MathWorks, 2009.
17