微型髮夾式諧振器並聯開路殘段實現具有高頻寬截止帶雙頻帶通濾波器

國 立 交 通 大 學

電信工程學系

碩 士 論 文

微型髮夾式諧振器並聯開路殘段實現

具有高頻寬截止帶雙頻帶通濾波器

Compact Hairpin Resonator Shunt Open Stub Bandpass Filters

with Dual-Band Response and Wide Upper Stopband

研究生：謝秉岳

指導教授：郭仁財教授

微型髮夾式諧振器並聯開路殘段實現

具有高頻寬截止帶雙頻帶通濾波器

Compact Hairpin Resonator Shunt Open Stub Bandpass

Filters with Dual-Band Response and Wide Upper Stopband

研究生

研究生

研究生

研究生：

：

：謝秉岳

：

謝秉岳

謝秉岳

謝秉岳 Student:Ping-Yueh Hsieh

指導教授

指導教授

指導教授

指導教授：

：

：

：郭仁財

郭仁財

郭仁財

郭仁財

博士

博士

博士

博士 Advisor:Dr. Jen-Tsai Kuo

國立交通大學

電信工程學系

碩士論文

A Thesis

Submitted to Department of Communication Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

in Partial Fulfillment of the Requirements

for the Degree of

Master of Science

in

Communication Engineering

June 2009

Hsinchu, Taiwan, Republic of China

微型髮夾式諧振器並聯開路殘段實現

具有高頻寬截止帶雙頻帶通濾波器

研 究 生 : 謝 秉 岳 指 導 教 授 : 郭 仁 財 博 士

國立交通大學電信工程學系

摘要

本論文提出一微型化髮夾式並聯開路殘段諧振器去設計雙頻帶暨寬截

止帶帶通濾波器，其中一開路殘段並聯於中間，而另外兩段則對稱地並聯

在諧振器的兩側。利用調整開路殘段的阻抗及長度和並聯的位置使所有的

諧振器皆有相同的兩個操作頻率，但是彼此的高階諧振頻率會互相錯開。

為了增加壓抑的程度進而拓寬高頻截止帶，由並聯的架構和耦合線段所產

生的傳輸零點來消除存餘的高階諧波。五階濾波器的量測結果顯示小於

28dB 的截止頻帶可以拓展到六倍的基頻頻率。而量測的結果與模擬結果吻

合。

Compact Hairpin Resonator Shunt Open Stub

Bandpass Filters with Dual-Band Response

and Wide Upper Stopband

Student: Ping-Yueh Hsieh

Advisor: Dr. Jen-Tsai Kuo

Department of Communication Engineering

National Chiao Tung University

Abstract

Compact bandpass filters are devised with a dual-passband response and a wide upper stopband based on hairpin resonators with three shunt open stubs. One stub is tapped at the center and two are allocated on both sides symmetrically. By adjusting the stub lengths, all resonators are designed to have two identical leading resonant frequencies but different higher-order ones. To enhance the rejection level and extend the upper stopband, the transmission zeros created by the tap structures and the coupled-line stage are used to eliminate the remaining leading unwanted peaks. Measured results show that the fifth-order filter has a rejection level below 30 dB can be extended to six times the first frequency. Good agreement between measured and simulation is obtained.

Acknowledgement

誌謝

誌謝

誌謝

誌謝

能夠完成本篇論文首先要感謝的是一直指導並細心糾正我錯誤的郭仁

財教授，郭老師不僅在研究上給予我很多很多建議，讓我在研究的瓶頸時

總能看到一絲曙光，而且在為人處世上，郭老師的認真謹慎更是值得我學

習的地方，我要在此致上十二萬分的感謝，郭老師，謝謝您！同時也要感

謝百忙之中抽空來為我的論文作出指導的陳俊雄教授、林祐生教授以及鐘

世忠教授，謝謝！

還有實驗室的成員，多得你們的陪伴，使得我研究生涯很多采多姿。

在我有很多問題時總是可以幫我解答的逸群學長，除了要管理整個實驗室

的大大小小事之外還要做研究，學長的勤奮也是我要學習的典範。還有上

一屆的學長姊，聰明的宇峯，無厘頭的承軒，搞笑的欣穎，認真的慧萍，

多謝你們陪我渡過歡笑快樂的碩一。以及我的同學，邊衝浪邊做研究的正

修，聰明搞笑的評翔，數值分析魔人政良，謝謝你們兩年來的陪伴！還有

下一屆的學弟妹，認真的卓諭，厲害的麒宏，很有種的宣融，很愛模仿別

人的紹展，講究的峻瑜，很麻吉的詩薇，努力的祖偉，因為有你們，實驗

室總有歡笑的聲音，謝謝你們！

Content

Chinese Abstract ...I English Abstract... II Acknowledgement ... III Content ... IV List of Figures... V List of Tables ...VII

Chapter 1 ... 1

Introduction ... 1

Chapter 2 ... 3

Compact Hairpin Resonator Bandpass Filters with Dual-Band Response and Wide Stopband 3 2.1 Hairpin Resonator with Taped Open Stubs... 4

2.2 Coupling Length and Coupling Gap of Each Coupled Stage ... 12

2.3 Transmission Zeros of Coupled Stage ... 21

2.4 Fifth-Order Dual-Band Bandpass Filter with Wide Stopband... 26

Chapter 3 ... 31

Simulation and Measurement ... 31

Chapter 4 ... 34

Conclusion... 34

List of Figures

Fig. 2.1-1. Proposed third-order bandpass filter. ... 3

Fig. 2.1-2. Hairpin resonator with tapped open stubs. ... 4

Fig. 2.1-3. Unfolded resonator with tapped open stubs. ... 5

Fig. 2.1-4. Odd mode analysis of unfolded resonator. ... 5

Fig. 2.1-5. Even mode analysis of unfolded resonator... 7

Fig. 2.1-6. Resonant spectrum with θt = 20°, 40° and 60°. R1 = Zr/Z1 = 0.8, R2 = Zr/Z2 = 1.2, θ1 = 15o. ... 9

Fig. 2.1-7. Resonant spectrum with θt = 40o, 50o and 60o. R1 = Zr/Z1 = 0.8, R2 = Zr/Z2 = 1.2, θ1 = 15o. All resonances are normalized with respect to the fundamental frequency f1... 10

Fig. 2.1-8. Insertion loss |S21| of each resonator by third-order dual-band bandpass filter... 11

Fig. 2.2-1. Frequency Response of gap excitation of two adjacent resonators. ... 13

Fig. 2.2-2. Coupling coefficient with various coupling length or coupling gap for single frequency. Resonator 1: Zr = 50 Ω, Z1 = 52.2 Ω, θ1 = 11.066°, Z2 = 40 Ω, θ2 = 15.7°, θt= 32.421°,Resonator 2: Zr = 50 Ω, Z1 = 41.667 Ω, θ1 = 11.676°, Z2 = 55.556Ω, θ2 = 25.591°, θt= 38.919°. ... 13

Fig. 2.2-3. Third-order bandpass filter fractional bandwidths of central frequency 2.4GHz, and 5.2GHz. ... 15

Fig. 2.2-4. Fractional bandwidths of central frequency 2.4GHz, and 5.2GHz for third-order filter. ... 17

Fig. 2.2-5. Tap point on the resonator. ... 17

Fig. 2.2-6. Third-order bandpass filter Transmission zeros arise from tap in. ... 18

Fig. 2.2-7. Calculate tap in zeros by gap excitation on the other port. ... 18

Fig. 2.2-8. Calculate transmission zeros arise from tapping by making Zin equal to zero...19

Fig. 2.2-9. Spectrum of tapping zeros included tapped in and tapped out resonators...20

Fig. 2.3-1. Hairpin coupled-line section... 21

Fig. 2.3-2(a). Simulation responses of Fig. 2.3-1. with various θt where θ2 = 17.5o. Other circuit size: Zr = 50 Ω, Z2 = 52.2 Ω, θο = 56.1o, and S/h = 0.3937. ... 22

Fig. 2.3-2(b). Simulation responses of Fig. 2.3-1. with various θ2 where θt = 34.6°. Other circuit size: Zr = 50 Ω, Z2 = 52.2 Ω, θo = 56.1°, and S/h = 0.3937. ... 23

Fig. 2.3-3. Resonant spectrum against θ1. All resonances are normalized with respect to the fundamental frequency f1. The physical parameters in Fig. 2.3-3 are as follows; Res. 1: R1 = 1.2, R2 = 1, θ2 = 25°, θt = 40°.Res. 2: R1 = 1, R2 = 1, θ2 = 20°, θt = 30°. Res. 3: R1 = 0.8, R2 = 1.2, θ2 = 22.2°, θt = 37.5°. ... 24

Fig. 2.3-4. Coupled stage frequency response of third-order bandpass filter... 24 Fig. 2.3-5. Design graph for third-order resonators with resonances and transmission zeros. 25

Fig. 2.4-1. Proposed fifth-order bandpass filter... 27 Fig. 2.4-2. Insertion loss |S21| of each resonator by fifth-order dual-band bandpass filter. .... 28 Fig. 2.4-3. Third-order bandpass filter fractional bandwidths of central frequency 2.4GHz, and 5.2GHz. ... 28 Fig. 2.4-4. Fractional bandwidths of central frequency 2.4GHz, and 5.2GHz for fifth-order filter. ... 29 Fig. 2.4-5. Fifth-order bandpass filter transmission zeros arise from tap in. ... 29 Fig. 2.4-6. Coupled stage frequency response of fifth-order bandpass filter. ... 30 Fig. 2.4-7. Design graph for fifth-order resonators with resonances and transmission zeros.. 30 Fig. 3-1. Simulation and measurement results. (a) Third-order, (b) Fifth-order dual-band bandpass filter. ... 32 Fig. 3-2. Circuit photograph of the (a) Third-order, and (b) Fifth-order dual-band bandpass filter. ... 33

List of Tables

Table I... 10

Dimensions of each resonator in third-order dual-band bandpass filter. ... 10

Table II. ... 27

Chapter 1

Introduction

Recent rapid progresses in wireless communication systems have created a need for dual-band operation for RF devices, such as the global systems for mobile communication systems (GSM) at 0.9/1.8 GHz and wireless local area network (WLAN) at 2.4/5.2 GHz. Bandpass filter are a vital device in the RF front ends of both receiver and transmitter. So high-performance microwave/RF bandpass filters are highly required in wireless communication systems. Recently, many planar dual-band bandpass filters have been proposed for synthesis methods [1]-[4], innovative design method [5]-[12], and circuit miniaturization [13]. In [1], a dual-band filter is realized by a bandstop filter and a wideband bandpass filter. In [2], the dual-passband response is designed by stepped-impedance resonators in both inline and parallel-coupled configurations. In [3], a systematic synthesis procedure is proposed to devise a dual-band bandpass filter. Series and parallel open stubs are used as resonators to fulfill the dual-band characteristics. On the basis of the same design concept in [3], a stepped-impedance coupled-line utilized as a dual-band inverter is introduced in[4].

Several structures have been developed for contriving novel dual-band bandpass filters. In [5], compact hairpin resonators are used to carry out two quasi-elliptic function passbands. The cross coupled filter in [6] has a very sharp transition since it possesses a transmission zero on the upper side of each passband. In [7], a resonator with a loaded open stub is adopted to implement a dual-band bandpass filter. In [8], two dual-mode resonators are utilized to obtain the dual-band response. Note that both two operation frequencies can be independently adjusted.

become a popular issue [9]-[12]. In [9] and [10], the substrate suspension technique and dielectric overlay method are proposed for suppressing the second harmonic (2fo). Based on

providing different electric lengths for the even- and odd-modes, corrugated coupled-line microstrip is also an effective way for extending the upper stopband up to 3fo [11]. In [12],

multi-spurious suppression is fulfilled by choosing the constitutive resonators having identical passband frequency but staggered higher order resonances. It is noted that the structures in [9]-[12] are only involved with a single passband.

So far, it is still a challenge to design a dual-band bandpass filter with wide stopband. Only few relative papers have been published [13]-[14]. In [13], a rigorous design approach for a dual-band filter with broad upper stopband is disclosed. By means of designated coupling and resonator sections, spurious suppression could be readily achieved. In [14], various sizes of stepped-impedance resonators are established to contrive a filter with dual-passband and wide upper rejection. Each resonator is devised to possess two identical leading resonances but different higher order counterparts to make spurious peaks have low levels and small bandwidths. The results show a rejection level of 30 dB up to more than eight times of the fundamental frequency can be obtained.

In this thesis, an alternative design technique is proposed for building a bandpass filter with a dual-passband response with a broad upper stopband. The resonant frequencies of hairpin resonators are adjusted by three tapped open stubs to have two identical frequencies and dispersed higher-order resonances. Through the rejection enhanced by transmission zeros created due to various analytical techniques, a dual-band filter with exceptional performance can be carried out. In the following, Chapter 2 addresses the design idea and investigates the resonant property of a hairpin resonator with tapped open stubs and design procedure of band pass filter, also analyzes the transmission zeros of created by the coupled structure of the circuit, Chapter 3 demonstrates the simulation and measured results of two experimental circuits, and Chapter 4 draws the conclusion.

Chapter 2

Compact Hairpin Resonator Bandpass Filters with

Dual-Band Response and Wide Stopband Procedure

In this chapter, a novel topology of a dual-band filter with broad upper stopband is proposed. Fig. 2.1-1 presents a third-order bandpass filter which constructed by hairpin resonator with tapped open stubs. The idea originates from dissimilar resonators in [12]. Since each resonator of this circuit has different sizes of open stubs, it could simultaneously have two identical operation frequencies and dispersed spurious harmonics. Based on the analogous procedure in [14], a dual-band filter with exceptional performance is finally carried out.

3

2

1

c12 c23

T

2

T

1

S

23 12

S

2.1 Hairpin Resonator with Taped Open Stubs

Fig. 2.1-2 shows the layout of the proposed hairpin resonator with tapped open stubs. It consists of one open stub (Z1, θ1) tapped at its center and two stubs (Z2. θ2) symmetrically

located between its middle and edge. Since this structure is symmetric, we can use even- and odd-mode to analysis. With the aid of even- and odd-mode analysis [19], the resonance conditions can be easily formulated. For this miniaturized Hairpin resonator, the fundamental resonant frequency is odd mode resonance, and the first harmonic frequency is even mode resonance, therefore, analyzing the characteristic of these two modes is important for designing filter.

P

P'

θ

₁

θ

₂

Z

₁ 2

Z

2

θ

Z

2

θ

_t

θ

_o

-

θ

_t r

Z

θ

_o

Fig. 2.1-2. Hairpin resonator with tapped open stubs.

The following, for simplicity to view, we use unfolded resonator Fig. 2.1-3 to formulate resonant mode conditions.

When the odd mode resonance occurs, the plane P-P’ is equivalent to electric wall, means that the central of the resonator is virtual ground, as shown in Fig. 2.1-4. The resistance

looking into the resonator form P-P’ plane to open end is zero. According to the microwave transmission theory [19], we would find:

θ

_t

-2

θ

1

θ

2

θ

Z

1

2

Z

Z

₂

Z

r

P

P'

θ

o

t

θ

θ

_o

Fig. 2.1-3. Unfolded resonator with tapped open stubs.

Z

₂

θ

₂

Z

1

θ

₁

-

θ

_t

θ

_o

θ

_t

_{Short Circuit}

Electric Wall

Z

_in

r

Z

0 tan 1 tan 1 ) tan( ) tan( Z tan 1 tan 1 1 2 2 r 2 2 ₌ + − + − + + = t r t o r t o t r r in Z Z Z j Z j Z j Z Z θ θ θ θ θ θ θ θ (2-1)

Simplify this equation, we get two conditions:

1 tan ₂tan

(

)

tan tan

(

)

0 2 = − − − − r _{o t t o t} Z Z θ θ θ θ θ θ (2-2a)

r + _t+

(

_o− _t

)

=∞ Z Z θ θ θ θ tan tan tan ₂ 2 (2-2b)

Only the 2-2a equation is reasonable to solve. We can rewrite 2-2a as follows; note that the odd mode resonance condition is independent of the center stub (Z1, θ1).

tan ₂tan

(

)

tan tan

(

)

1 0 2 = − − + − _{t t o t} o r Z Z θ θ θ θ θ θ (2-3)

Similarly, when the even mode resonance occurs, the plane P-P’ is equivalent to magnetic wall, means that the central of the resonator is open circuit, as shown in Fig. 2.1-5. The resistance looking into the resonator form P-P’ plane to open end is infinite, so the admittance is zero. According to the microwave transmission theory [19], we would find:

tan 0 2Z 1 ) tan( tan 1 tan 1 1 tan 1 tan 1 ) tan( 1 1 2 2 2 2 _{+ =}             − + + + − + = θ θ θ θ θ θ θ θ θ j jZ Z j Z j Z Z j Z j j Z Y t o r t r r t r t o r in (2-4)

Open Circuit

Magnetic Wall

Z

_in

r

Z

Z

₂

θ

Z

1

2

θ

₁

-

θ

_t

θ

_o

θ

_t

Fig. 2.1-5. Even mode analysis of unfolded resonator.

Simplified this equation, we get two conditions:

0 tan 1 tan 1 ) tan( 1 tan 2Z 1 ) tan( 1 tan 1 tan 1 2 2 1 1 2 2 =             + − − + − + + t r t o r t o r t r Z Z Z Z Z Z θ θ θ θ θ θ θ θ θ (2-5a)

_=∞      + − − _t r t o r Z Z

Z tan(

θ

θ

) 1 tan

θ

1 tan

θ

1 ₂

2

(2-5b)

Only the 2-5a equation is reasonable to solve. We can rewrite 2-5a as follows:

0 tan tan ) tan( 1 tan 2Z ) tan( tan tan 2 2 1 1 2 2 =             + − − + − + + t r t o r r t o t r Z Z Z Z Z Z θ θ θ θ θ θ θ θ θ (2-6)

1 1 Z Z R ₌ r _(2-7)

2 2 Z Z R ₌ r _(2-8)

Using the two resonance conditions, 2-3 and 2-6, we can draw the Fig. 2.1-6. Fig. 2.1-6 plots the resonant frequencies against

θ

2 for the fundamental, first, second, and through to

fifth higher order modes for R1 = 0.8, R2 = 1.2,

θ

1 = 15°,

θ

t = 20°, 40° and 60°. All electrical

lengths are referred at f1. The y-axis is the even- and odd-mode resonant frequencies of tapped

open stubs resonator which is normalized with respect to the fundamental frequency f0 of

uniform impedance resonator. From Fig. 2.1-6, we found that when

θ

2 =0, the odd mode

resonances (independent of center stub Z1,

θ

1) equal to odd numbers. In Fig. 2.1-5, we can

change the value of R1, R2,

θ

1,

θ

2, and

θ

t to appropriately adjust the even- and odd-mode

resonant frequencies.

Fig. 2.1-7 plots f2 through f6 normalized with respect to the fundamental frequency f1 for

various

θ

2 and

θ

t. Note that the data shown in Fig. 2.1-6 form an important basis of our design

for determining the resonator geometry. Each resonator possesses the first two identical resonant frequencies f1 and f2, but has different higher order dispersed to make the spurious

peaks can be made relatively low and then extend the upper stopband. In this thesis, the two operation frequencies f1 and f2 are respectively allocated as 2.4 GHz and 5.2 GHz. So, we

should find a solution for a resonator to resonate at f1 = 2.4GHz, and f2 = 5.2GHz. Based on

the conditions above, and have to make all resonators mutually disperse the high-order resonant frequency; it will restrict the solution to a small range. So as to loosen restrictions, it will be a selection to let f3 = 5.2GHz, not f2, nevertheless, f1 is till 2.4GHz. In this

third-order bandpass filter, the second resonator was chosen to resonate at f1 = 2.4GHz, f2 =

3.9GHz, and f3 = 5.2GHz, that is different from the first and third resonator, it makes the

than the other resonators10% approximately. θ_t= 20o θ_t= 40o θ_t= 60o 0 5 10 15 20 25 30 35 40 45 0.5 θ₂(degree) 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 f1, , 2 f , 3 f , 4 f , 5 f 6 f 1 f 2 f 3 f 4 f 5 f 6 f

Fig. 2.1-6. Resonant spectrum with

θ

t = 20°, 40° and 60°. R1 = Zr/Z1 = 0.8, R2 = Zr/Z2 = 1.2,

θ

1

= 15o.

Fig. 2.1-1 shows the layout of a third-order dual-band bandpass filter where Zr of each

resonator is chosen as 50 Ω, for the sake of avoiding high impedance of open stub that produces less radiation. Other dimensions of each resonator with respect to f1 are listed in

Table I. Fig. 2.1-8 shows the insertion loss |S21| frequency response from 0GHz to 20GHz of

each resonators by gap excitation. Suppose the circuits are designed on a substrate with

ε

r =

10.2 and thickness h = 0.635 mm. From this figure, we can see the resonance frequency

positions of each resonator are mutually dispersed, except for the two operation frequencies f1

θ_t= 60o θ_t= 50o o = 40 t θ f₆ / f₁ / 1 f 5 f f₄ / f₁ / 1 f 3 f / 1 f 2 f , f4 f1 / / / 1 f 3 f , 1 f 2 f / f1 5 f , f6 f1 / , (degree) 2 θ 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 45 40 35 30 25 20 15 10 5 0

Fig. 2.1-7. Resonant spectrum with

θ

t = 40o, 50o and 60o. R1 = Zr/Z1 = 0.8, R2 = Zr/Z2 = 1.2,

θ

1

= 15o. All resonances are normalized with respect to the fundamental frequency f1.

Table I.

Dimensions of each resonator in third-order dual-band bandpass filter.

θ

t (o) Z1 (Ω)

θ

1 (o) Z2 (Ω)

θ

2 (o)

Res. 1 30.7 54.7 13.6 39.6 18.5

Res. 2 65.7 39.6 22.4 54.7 30.0

0 Frequency (GHz) -60 -60 0 0 |S | ( d B ) 2 1 f₀ -60 0 resonator 2 2.5 5 7.5 10 12.5 15 17.5 20 0 2f 3f₀ 4f₀ 5f₀ 6f₀ 7f₀ 8f₀ resonator 1 resonator 3

2.2 Coupling Length and Coupling Gap of Each Coupled

Stage

Next, we determine the coupling scheme of each pair of adjacent resonators to meet the required coupling coefficients for two respective bands.

In synthesis of a bandpass filter with coupled resonators, the coupling coefficient between the jth and (j + 1)th resonators, Kj,j+1, is given by [15]:

1 1 , + + ∆ = j j j j g g K (2-9)

where gj is element value of the low-pass filter prototype, and ∆ is fractional bandwidth of

bandpass filter. When two resonators have a close proximity, the coupling coefficient between two coupled resonators, K, is usually calculated by the approximation [16]:

_{2 2} 2 2 b a b a f f f f K + − = (2-10)

In simulation by the IE3D, the two natural frequencies fa and fb of the two coupled

resonators in Fig. 2.2-1 can be obtained by calculating the transmission response of the circuit with breaking the tap points and leaving a small gap to the input and output ports. Then use E-M simulator IE3D [20] to make electric-magnetic simulations of two adjacent resonators with various coupling length and coupling gap. Therefore, the relationship between coupling coefficient and coupling length or coupling gap is investigated.

Fig. 2.2-2 shows the coupling coefficient of two adjacent resonators with various coupling length or coupling gap for single frequency, h is the height of substrate, 0.635 mm.

2 0 -80 |S | (d B ) 2 1 Frequency (GHz) -60 -40 -20 2.2 2.4 2.6 2.8 3.0 a f fb

Fig. 2.2-1. Frequency Response of gap excitation of two adjacent resonators.

0.07 0.09 0.11 0.13 0.15 c Coupling length (mm) 4.823 co u p li n g c o ef fi ci en t ( K ) 0.05 S/h = 0.3149 , 0.472 4,...0.9 449
S 5.823 6.823 7.823 8.823 9.823 10.823

Fig. 2.2-2. Coupling coefficient with various coupling length or coupling gap for single frequency. Resonator 1: Zr = 50 Ω, Z1 = 52.2 Ω,

θ

1 = 11.066°, Z2 = 40 Ω,

θ

2 = 15.7°,

θ

t =

32.421°, Resonator 2: Zr = 50 Ω, Z1 = 41.667 Ω,

θ

1 = 11.676°, Z2 = 55.556 Ω,

θ

2 = 25.591°,

θ

t

If a dual-band bandpass filter is desired, the two bands coupling coefficient need to be satisfied simultaneously:

(

)

(

)

       ∆ = ∆ = + + + + 2 1 2 2 1 , 1 1 1 1 1 , ) ( ) ( j j j j j j j j g g K g g K (2-11)

and the subscript 1 or 2 denotes the first or the second passband. Rewrite 2-11 as:

    × = ∆ × = ∆ + + + + 2 1 2 1 , 2 1 1 1 1 , 1 ) ( ) ( ) ( ) ( j j j j j j j j g g K g g K (2-12)

According to the equations 2-10 and 2-12, we can build up a design graph of each

bandwidth with various coupling length and coupling gap. The gj value is Chenbyshev filter

with a 0.1-dB ripple. As shown in Fig. 2.2.3, the x-axis and y-axis represent the bandwidth of first pass-band 2.4GHz and second pass-band 5.2GHz, respectively. Using the resonators in

Table I. The coupling length and coupling gap of first coupled stage is indicated by
_{c12 and}

S12, to second coupled stage is indicated by
c23 and S23. In this figure, the area that two

seashells overlapped is the fractional bandwidth capable to design dual-band bandpass filter. In order to maintain two passbands have sufficient bandwidth at the same time , we would select
_c12 = 7.25 mm and S12 = 0.325 mm,
c23 = 7.25 mm and S23 = 0.3 mm, fractional

(4,0.2) c23 23 (10,0.4) (10,0.6) 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.06 0.05 0.04 0.03 0.02 0.01 (10,0.2) (
, S ) = (4,0.4) (4,0.6) (10,0.4) (7,0.2) c12 12 (
, S ) = (7,0.2) (10,0.6) S = 0.6 12 S = 0.4 12 (7,0.6) (4,0.6) (10,0.4) (7,0.4) (4,0.2)
= 1c23 0 (10,0.2) ∆ f 0.02 0.18 0.07 0.6 1 ∆ f2 S =23 0.2
_c12= 10 S _{= 0} .2 12
= 7 c12

Fig. 2.2-3. Third-order bandpass filter fractional bandwidths of central frequency 2.4GHz, and 5.2GHz.

To appropriate input and output coupling for the bandpass filter, the tap positions which

is specified by T1, and T2 in Fig. 2.1-1 should be determined by matching the singly loaded Q

(Qsi) of the tapped resonator with the passband specification [17]. The singly loaded Q (Qsi) of

the tapped resonator is defined as

o d dB R Q_{si L} ω

ω

ω

2 0 = (2-13)

where RL is the impedance seen by the resonator looking toward the source/load,

ω

o is the

operating frequency, and B is the input susceptance seen at the tap point looking into the resonator.

FBW g g

Q_{si =} o 1 _(2-14)

Combining equations 2-13 and 2-14, the relationship between RL and tap position is derived

as shown as follows:

o o o in o o in o o in o si L T B FBW g g T B FBW g g T B Q R ω ω ω ω ω ω ω ω ω ∂ ∂ ⋅ ⋅ = ∂ ∂ = ∂ ∂ = ) ( 2 ) ( 2 ) ( 2 ₁ 1 (2-15)

If two different frequencies are concerned, equation 2-15 should be directed against 2.4GHz and 5.2GHz:

         ∂ ∂ ⋅ ⋅ = ∂ ∂ ⋅ ⋅ = 2 1 ) ( 2 ) ( 2 2 2 2 1 2 1 1 1 1 1 ω ω ω ω ω ω in o L in o L T B FBW g g R T B FBW g g R (2-16)

We substitute the fractional bandwidth that described above into equation 2-16, and calculate the differential of susceptance from tap in, and tap out resonators, Fig. 2.2-4 plots the RL versus tap position for input, and output resonators, Tin is the length from tap position

to open end, as shown as Fig. 2.2-5. Since we want to miniaturize the bandpass filter, so it does not like to add an impedance transformer additionally, that increase the overall area of this filter. Instead of, we prefer to tap the position on the resonator that RL is 50 Ω for f1 and f2

both. In Fig. 2.2-4, we can find that when tap at T1 = 6.457 mm for input resonator, and at T2 =

6.448 mm for output resonator, RL is close to 50 Ω for both f1 and f2. Finally, the transmission

simulated by making the other port gap excitation, as Fig. 2.2-7. 0 0 1 2 3 4 5 6 7 8 9 10 400 350 300 250 200 150 100 50 f ₁-Input 2 f -Input 1 f -Output 2 f -Output Tap position T (mm)in RL (Ω )

Fig. 2.2-4. Fractional bandwidths of central frequency 2.4GHz, and 5.2GHz for third-order filter.

Z

₁

Z

2

θ

2

in

r

Z

T

1

θ

Frequency (GHz) |S | (d B ) -70 2 1 16 0 Tapped-Input Tapped-Output 14 12 10 8 6 4 2 0 -60 -50 -40 -30 -20 -10

Fig. 2.2-6. Third-order bandpass filter Transmission zeros arise from tapping.

θ

1

θ

2

Z

₁ 2

Z

2

θ

Z

2

Z

r

Port 1

Port 2

Fig. 2.2-7. Simulate tapping zeros by making the other port gap excitation.

The transmission zeros arise from tapping also can be calculated by numerical analysis, as shown as Fig. 2.2-8, the input impedance seen form tapped in point to open end is:

(

)

(

_{Tin t}

)

r t r t Tin t r r r in Z j Z j Z j Z j Z j j Z Z Z θ θ θ θ θ θ θ θ − + − − −       − − + = tan 1 tan 1 tan 1 tan tan 1 tan 1 1 2 2 2 2 (2-15)

When Zin equal to zero, the transmission zeros arise from tapping is occurred. Fig. 2.2-9

depicts the spectrum of tapped zeros included tapped in and tapped out resonators.

If we change the tap position, may alter the transmission zeros to desired frequencies [14], but, for same reason, it will need an extra impedance transformer to match the RL for both f1

and f2.

in

Z

in

T

θ

_t

θ

_o

-

θ

_t

θ

₁

θ

₂

Z

₁ 2

Z

2

θ

Z

₂ r

Z

Input Output fZ1, , Z 2 f , Z 3 f Z 4 f (G H z) Tap position T (mm)in 10.25 9.25 8.25 7.25 6.25 5.25 4.25 0 2.5 5 7.5 10 12.5 15 17.5 20

2.3 Transmission Zeros of Coupled Stage

Since our target is to design a dual-band filter with a wide stopband, the zeros are created by each coupled stage are required to investigate carefully. Fig. 2.3-1. depicts the hairpin coupled-line section with tapped open stubs and its responses of simulation are plotted in Fig. 2.3-2. Note that the horizontal axis is normalized to the first operation frequency f1. Suppose

the circuits are designed on a substrate with εr = 10.2 and thickness h = 0.635 mm. Such a

high dielectric substrate is purposely chosen to demonstrate performance of our approach. The simulation data are obtained by the full-wave software simulator IE3D [20]. Fig. 2.3-2(a) draws the |S21| of a coupled section with various θt. The circuit parameters are listed in the

figure caption. Note that three transmission zeros will be observed from 3f1 to 6f1 and all zeros

can be arbitrarily tuned by altering θt. Fig. 2.3-2(b) demonstrates the insertion results of a

coupled-line stage with various θ2. As θ2 is increased, we expect that zeros would shift to a

lower frequency. It can be deduced that a wide stopband could be readily achieved by adequately selecting each coupled section to suppress dispersed spurious peaks of a bandpass filter. S θ2 2 θ t θ θo Z2 Z2 r Z r Z Port 1

1

θ

_o

= 3.5θ

_t

θ

_o

= 3.0θ

_t

θ

_o

= 2.5θ

_t

θ

_o

= 2.0θ

_t t

θ

= 1.5

o

θ

|S

|

2 1

(d

B

)

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

-50

/f

f

6.0

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

Fig. 2.3-2(a). Simulation responses of Fig. 2.3-1. with various θt where θ2 = 17.5o. Other

circuit size: Zr = 50 Ω, Z2 = 52.2 Ω, θo = 56.1o, and S/h = 0.3937.

Therefore, if there still have some spurious modes in the frequency response, not be suppressed yet, given different θ2 and θt to control the transmission zero arise from the

coupled stage to eliminate that. For desired transmission zero, Fig. 2.3-3 plots the frequency spectrum against θ1 of three different resonators with various θ2, and θt. Nevertheless, alter

the value of θ2 and θt make changes in the distribution of high-order modes, it would turn out

that not to be dispersed with the other resonators’. So the trade off between disperse spurious and control transmission zero should be concerned to optimize the rejection band.

Fig. 2.3-4 plots the coupled stage frequency response of third-order bandpass filter Fig. 2.1-1, this figure shows the transmission zero arise from coupled stage 1, and coupled stage 2. From the stage 1, transmission zeros 6.96GHz, 8.77GHz, and 12.75GHz are introduced, and transmission zeros 6.45GHz, 6.9GHz, and 10.44GHz are caused from the stage 2.

The success of the wide upper stopband relies quite much on Fig. 2.3-5 which is an useful design graph, showing the zeros created by individual coupled-line sections and resonant peaks of the resonators. In the upper part of the plot, the rectangular boxes in the gray area represent the rejection bands produced by coupled-line sections. The widths of the boxes are defined as |S21| smaller than -15 dB. The coupling C01 and C34 match the external Q values of

the end resonators with the two bandwidths by tapping. The tapped input and output can also be designed to create transmission zeros [14]. The lower part of Fig. 2.3-5 shows the resonant frequencies of the resonators. All of them are designed to have resonant frequencies at 2.4 and 5.2 GHz, and also have dispersed resonances, now the purpose of Fig. 2.3-5 becomes clear. To extend the stopband, the stopbaands created by the coupled sections have to cover the spurious peaks.

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

f /f

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

(d

B

)

2 1

|S

|

_θ

2

= 10

o o

= 15

2

θ

o

= 20

2

θ

o

= 25

2

θ

o

= 30

2

θ

1

f₆ / f₁ / 1 f 5 f f₄ / f₁ / 1 f 3 f / 1 f 2 f , f4 f1 / / / 1 f 3 f , 1 f 2 f / f1 5 f , f6 f1 / , (degree) 1 θ 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 45 40 35 30 25 20 15 10 5 0 Res. 3 Res. 2 Res. 1

Fig. 2.3-3. Resonant spectrum against θ1. All resonances are normalized with respect to the

fundamental frequency f1. The physical parameters in Fig. 2.3-3 are as follows; Res. 1: R1 =

1.2, R2 = 1, θ2 = 25°, θt = 40°.Res. 2: R1 = 1, R2 = 1, θ2 = 20°, θt = 30°. Res. 3: R1 = 0.8, R2 = 1.2, θ2 = 22.2°, θt = 37.5°. Frequency (GHz) |S | ( d B ) -60 2 1 16 -45 -30 -15 0 1st stage 2st stage 14 12 10 8 6 4 2 0

1

i

C

o

u

p

le

d

S

ta

g

e

th

C

₃₄

C

₂₃

C

₁₂ 01

C

th

i

R

eso

n

at

o

r

3

2

1

f

/

f

6

5

4

3

2

1

0

2.4 Fifth-Order Dual-Band Bandpass Filter with Wide

Stopband

This section we will summarize the filter design procedure, and introduce the second circuit, fifth-order dual-band bandpass filter.

At beginning, solve the two resonance conditions, 2-3 and 2-6, to figure out the resonator that resonates at two identical operation frequencies, and alsodisperses the high-order modes mutually. Fig. 2.4-1 shows the layout of a fifth-order dual-band bandpass filter where Zr of

each resonator is chosen as 50 Ω, prevent the radiation from high impedance of open stub. Dimensions of the five stub-tapped hairpin resonators are listed in Table II, where the electric lengths are evaluated at f1. Two operational frequencies are chosen as 2.4GHz, and 5.2GHz.

Fig. 2.4-2 shows the insertion loss |S21| of each resonator by fifth-order dual-band bandpass filter.

Second, according to equations 2-10 and 2-12, draw the figure of two bands’ fractional bandwidth with various coupling length and coupling gap, the g value is Chenbyshev filter with a 0.1-dB ripple, Fig 2.4-3 depicts the four coupled stage,
cij and Sij present the coupling

length and coupling gap between ith and jth resonators. The area of four seashells overlapped is a quite small range in the center of this figure. We choose S12 = 0.275 mm, S23 = 0.46 mm,

S34 = 0.44 mm, S45 = 0.265 mm,
c12 =
c45 = 7.25 mm, and
c23 =
c34 = 7.075 mm for ∆f1 =

12.12%, and ∆f2 = 6.84%. Then use equation 2-16 determine the tap position Tin for input and

output resonators, Fig. 2.4-4 plots the RL against the tap position with two concerned

frequencies, we can figure out that intersection point of f1 and f2 is on T1 = T2 = 6.75 mm by

RL = 63 Ω for input and output resonators both, not 50 Ω. In order to obtain a better response

without adding an extra impedance transformer, we adjust the tap position to T1 = T2 = 7.2

Finally, investigate the coupled section zeros, tuning θ2 and θt to control the coupled-line

zeros to suppress unexpected peaks. Fig 2.4-6 plots the coupled stage frequency response of fifth-order bandpass filter Fig. 2.4-1, this figure shows the transmission zero arise from coupled stage 1 to coupled stage 4.

Fig. 2.4-7 summarizes the overall transmission zeros on the upper part, the lower part of Fig. 2.4-7 shows the resonant frequencies of the resonators. All of them are designed to have resonant frequencies at 2.4 and 5.2 GHz, but have dispersed resonances.

1 2 3 4 5 12 S S₂₃ S₃₄ S₄₅ c12
c23

_c34 c45
T1 T2

Fig. 2.4-1. Proposed fifth-order bandpass filter.

Table II.

Dimensions of each resonator in fifth-order dual-band bandpass filter.

θt (o) Z1 (Ω) θ1 (o) Z2 (Ω) θ2 (o) Res. 1 27.4 51.9 10.4 39.6 14.8 Res. 2 35.1 41.4 10.9 54.7 25.7 Res. 3 57.3 54.7 20.0 61.6 33.9 Res. 4 34.1 54.7 20.0 54.7 25.7 Res. 5 31.0 61.4 10.4 49.4 16.7

0 Frequency (GHz) -60 -60 0 0 |S | ( d B ) 2 1 resonator 1 f₀ -60 0 -60 0 -60 0 resonator 2 resonator 3 resonator 4 resonator 5 2.5 5 7.5 10 12.5 15 17.5 20 0 2f 3f₀ 4f₀ 5f₀ 6f₀ 7f₀ 8f₀

Fig. 2.4-2. Insertion loss |S21| of each resonator by fifth-order dual-band bandpass filter.

_{= 8} c34
= 4c12
= 4c45 (8,0.2) (10,0.2) (10,0.3) (6,0.2) (
, S ) = (10,0.5)c23 23 (10,0.6) (4,0.5) (8,0.5) 0.10 0.08 0.06 0.04 0.02 (10,0.2) (10,0.3) (10,0.4) (10,0.5) (4,0.4) S = 012 .2 (4,0.3) (4,0.4) (10,0.2) ∆ f 0.03 0.08 0.13 0.18 0.23 0.28 0.12 6 1 ∆ f2 S ₌ 0 .5 34 S ₄₅= 0.2 10
= 6c12 S ₌ 0 .2 3₄

Fig. 2.4-3. Third-order bandpass filter fractional bandwidths of central frequency 2.4GHz, and 5.2GHz.

1 2 f -Input 1 f -Output 2 f -Output Tap position T (mm)in RL (Ω ) 250 200 150 100 50 2 4 6 8 10 0 0 300 f -Input

Fig. 2.4-4. Fractional bandwidths of central frequency 2.4GHz, and 5.2GHz for fifth-order filter. -60 -50 -40 -30 -20 -10 Frequency (GHz) |S | ( d B ) -70 2 1 16 0 Tapped-Input Tapped-Output 14 12 10 8 6 4 2 0

Frequency (GHz) |S | (d B ) -50 2 1 16 0 1st stage 14 12 10 8 6 4 2 0 2st stage 3st stage 4st stage -40 -30 -20 -10

Fig. 2.4-6. Coupled stage frequency response of fifth-order bandpass filter.

1

i

C

o

u

p

le

d

S

ta

g

e

th

C

₅₆

C

₄₅

C

₃₄

C

₂₃

C

₁₂ 01

C

th

i

R

eso

n

at

o

r

₅

4

3

2

1

f

/

f

6

5

4

3

2

1

0

Chapter 3

Simulation and Measurement

Two circuits are realized to validate our design idea. Fig. 3-1(a) plots the simulation and measured responses of the third-order experimental circuit. In both passbands, |S11| and |S21|

are approximately -18 dB and -2.2 dB, respectively. The isolation between two passbands is approximately -40 dB. The -3 dB bandwidths are 12.19 % and 6.07% at the two designated bands. A stopband which is better than -23 dB up to 6f1 can be observed.

Fig. 3-1(b) demonstrates the performance of a trial fifth-order bandpass filter. The design procedure is similar to that of the previous. Fig. 3-1(b) compares the simulation and measured responses. The insertion loss and return loss in both passbands are about -3 dB and -17 dB, respectively. The -3 dB bandwidths at f1 and f2 are 9.86 % and 5.26 %, respectively. Since

more coupled-line sections can be used to eliminate more spurious peaks, the rejection performance of this filter in the stopband is better than that of the third-order circuit. The measured results show that a rejection level better than -28 dB can be reached up to 6f1.

Reasonably good agreement between simulation and measured results can be observed. In spite of obtaining a better rejection band, as trade off, higher order circuitsusually accompany an increase of circuit area and more loss due to metal. Fig. 3-2 is the photograph of the measured filter.

Measurement Simulation >6f₁ 0 Frequency (GHz) 0 |S |, | S | ( d B ) 1 1 2 1 -20 -40 -60 -80 2 4 6 8 10 12 14 16 -10 -30 -50 -70 (a) 0 Frequency (GHz) 0 |S |, | S | ( d B ) 1 1 2 1 -20 -40 -60 -80 2 4 6 8 10 12 14 16 -10 -30 -50 -70 Simulation Measurement >6f₁ (b)

Fig. 3-1. Simulation and measurement results. (a) Third-order, (b) Fifth-order dual-band bandpass filter.

(a)

(b)

Fig. 3-2. Circuit photograph of the (a) Third-order, and (b) Fifth-order dual-band bandpass filter.

Chapter 4

Conclusion

Dual-band bandpass filter is equipped with a wide upper stopband by using hairpin resonators with tapped open subs. Those open stubs have different geometric parameters so that all modified resonators have two identical designated resonant frequencies and different higher-order resonances. To enhance the filter performance in the stopband, the transmission zeros created by the coupling structures are carefully configured to achieve multi-spurious suppression. Design graph of resonant peaks and stopband is plotted to facilitate the wide stopband design. Measurement indicates that rejection of better than -28 dB can be obtained for extension up to over six times the first resonance frequency. The measured results have good agreement with the simulation.

References

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[5] J.-T. Kuo and H.-S. Cheng, “Design of quasi-elliptic function filters with a dual-passband response,” IEEE Microwave Wireless Compon. Lett., vol. 14, no. 10, pp. 472-474, Oct. 2004.

[6] C.-C. Chen, “Dual-band bandpass filter using coupled resonator pairs,” IEEE Microwave

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[7] P. Mondal and M. K. Mandal, “Design of dual-band bandpass filters using stub-loaded open-loop resonators,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 1, pp. 150-155, Jan. 2008.

[9] J.-T. Kuo, M. Jiang, and H.-J. Chang, “Design of parallel-coupled microstrip filters with suppression of spurious resonances using substrate suspension,” IEEE Trans. Microwave

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[10] J.-T. Kuo and M. Jiang, “Enhanced microstrip filter design with a uniform dielectric overlay for suppressing the second harmonic response,” IEEE Microwave Wireless Comp.

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[11] J.-T. Kuo, W.-H Hsu, and W.-T. Huang “Parallel coupled microstrip filters with suppression of harmonic response,” IEEE Microwave Wireless Comp. Lett., vol. 12, no. 10, pp. 383-385, Oct. 2002.

[12] C.-F. Chen, T.-Y. Huang and R.-B. Wu, “Design of microstrip bandpass filters with multi-order spurious-mode suppression,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 12, pp. 3788-3793, Dec. 2005.

[13] K.-K. M. Cheng and C. Law, “A new approach to the realization of a dual-band microstrip filter with very wide upper stopband”,” IEEE Trans. Microwave Theory Tech., vol. 56, no. 6, pp. 1461-1467, Jun. 2008.

[14] J.-T. Kuo and H.-P. Lin, “Dual-band bandpass filter with improved performance in extended upper rejection band,” submitted to IEEE Trans. Microwave Theory Tech., May 2008.

[15] G. L. Matthaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance-Matching

Network, and Coupling Structures. Norwood, MA: Artech House, 1980

[16] J. S. Hong and M. J. Lancaster, “Design of highly selective microstrip bandpass filters with a single pair of attenuation poles at finite frequencies,” IEEE Trans. Microwave

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[17] J. S. Wong, “Microstrip tapped-line filter design,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, pp. 44–50, Jan. 1979

[18] G. L. Mattaei, L. Young and E. M. T. Jones, Microwave Filters, Impedance-Matching

Network, and Coupling Structures, Norwood MA: Artech House, 1980. [19] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [20] IE3D Simulator, Zeland Software Inc., Jan. 1997.