elk-1203-69

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


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


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


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


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


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


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


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


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

)


S

S21

S11



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



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.


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


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


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


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.


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


13



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

)


11S

S21


14



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


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


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17

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