Synthesis and Design of Four Pole Ultra-Wide Band (UWB)Bandpass Filter (BPF) Employing Multi-Mode Resonators (MMR)

Sohail Khalid∗, Wong Peng Wen∗ and Lee Yen Cheong †∗Department of Electrical and Electronic Engineering Universiti Technologi PETRONAS

Bandar Seri Iskandar, 31750 Tronoh, Malaysia† Department of Fundamental and Applied Science Universiti Technologi PETRONAS

Bandar Seri Iskandar, 31750 Tronoh, Malaysia

Abstract— A complete synthesis and design of fourth orderUWB BPF has been presented in this paper. Based on MMR,equivalent circuit is used to derive transfer function whichis then used to extract the filtering function. For optimalsolution, mathematical formulation is used to calculate char-acteristic impedance of filter in order to have chebyshev typefrequency response. To validate the synthesis a prototypeis designed and fabricated. Experimental results show goodagreement with proposed synthesis.

Index Terms— Ultra-wideband, Bandpass filter.

I. INTRODUCTION

Ultra-wideband technology is the primary candidate forthe development of many modern transmission systemssuch as military and commercial radar systems, gadgetsused in wireless personal area networks (WPANs), medicalimaging and industrial sensing. In 2002, Federal Com-munication Commission (FCC) authorized the unlicenseduse of UWB systems (Band Width: 3.1 GHz–10.6 GHz)for private use [1]. It resulted in rapid growth of aca-demic and industrial research for commercial UWB radiosystem. UWB bandpass filter is the key component inUWB systems. In recent years, several techniques such asMMR, hybrid microstrip/coplaner-waveguide (CPW) andcascaded high-/low-pass filters technique are proposed todesign UWB bandpass filters [2]-[4]. However, the previ-ous work done in UWB BPF is mainly focused on full-wave electromagnetic (EM) simulators and optimizationtools. Secondly, conventional filter theory is based on thenarrowband assumption and hence cannot be used in UWBfilter design. These constraints have raised the need ofexact synthesis theory for UWB bandpass filters.

In this paper, synthesis based design of UWB BPF usingMMR is developed to provide optimum solution, numeri-cal efficiency and physical insight of filter. A generalizedfourth-order filtering function is extracted from an equiva-lent circuit of a filter. The filtering function has frequencydependent term in the denominator, which interferes withgeneralized fourth-order chebyshev polynomial. Proposedsynthesis provides a method to approximate filtering func-tion as a quasi-chebyshev function and compute optimalparameter values for maximum selectivity and equal-ripplefrequency response. To validate proposed synthesis, BPF

Fig. 1. Schematic of MMR UWB BPF and its equivalent circuit.

prototype of 3.2 GHz bandwidth with centre frequency6.85 GHz is fabricated using RT/Duroid 5880. Experi-mental result well correlates with FCC spectrum mask forUWB filters.

II. THEORY OF FILTER SYNTHESIS

The basic MMR filter configuration with its equivalentcircuit model is shown in Fig. 1. The filter consists ofmulti-mode resonator in the middle and a parallel coupledline on each end of MMR of length λg/4. Here, zo isthe characteristic impedance of middle section whereas,zoe and zod are the even and odd-mode characteristicimpedances of parallel coupled line. Due to symmetricalnetwork, an even-odd mode analysis is adopted to deriveoverall transfer function. A total of four transmission poleshave been achieved. The first two resonant modes arederived from MMR while the two additional resonantmodes are introduced by parallel coupled lines. The trans-fer function of MMR filter can be expressed as follows:

|S21(θ)|2 =1

1 + T 2(θ), (1)

where θ is the electrical length of the filter and related tofilter frequency by θ = π

2 ·ffc

, where fc is the centerfrequency. Due to unitary condition [5] the reflection

response is given as

|S11(θ)|2 =T 2(θ)

1 + T 2(θ). (2)

The filtering function, T (θ) extracted from (1), given by

T (θ) =A cos4(θ) +B cos2(θ) + C

sin(θ), (3)

where,

A =[(zcl + zod)

2 − 1](zo + zcl + zod)2

2 zcl2zo, (4)

B ={

[2− (zcl + zod)2]zo

2 + (1− 2zcl2)(zcl + zod)

2

+2(1− zcl2)(zcl + zod)zo

}×{

2 zcl2zo

}−1

, (5)

C =zcl

4 − zo2

2 zcl2zo. (6)

It is require that zo , zcl , zod > 0. The filtering functionhas frequency dependent term in the denominator whichwill distort the equal ripple characteristic of the chebyshevpolynomial. This effect can be nullified by restructuringthe filtering function as shown in the following section.

A. Restructure the Filtering Function

In order to synthesize the filtering function for optimumsolution, the ripple factor is extracted by normalizing thefiltering function in the following form:

T (θ) ≡ ε T̃ (θ)

T̃ (θNorm), (7)

where θNorm is the normalizing electrical length and thenormalization factor is taken as

T̃ (θ) =cos4 (θ) + α cos2 (θ) + ζ

sin (θ), (8)

where

A =ε

T̃ (θNorm), (9)

α =B

A, (10)

ζ =C

A. (11)

The ripple factor ε is related to the return loss by

ε =√

10LR/10 − 1 . (12)

From 8, the value of the parameters A,B and C can beobtained by requiring the transfer function

|S21(θ)|2 =1

1 + ε2∣∣∣∣ T̃ (θ)

T̃ (θNorm)

∣∣∣∣2, (13)

to obey the following conditions: four transmission polesobtained in the passband (for finest selectivity), UWB

bandwidth (5.25-8.45 GHz), equal ripple (approximatingchebyshev frequency response), controlled ripple level byripple factor ε. This is related to using the first derivativesto find the relative maxima and minima of the transferfunction.

B. Finding α and ζ

In order to generate the filter coefficients for equal rippleresponse, let the denominator of (2) equal to H ,

H = 1 + T 2(θ) . (14)

By Chain rule and using (7), the derivative of reflectioncoefficient (2) is

d

dθ|S11(θ)|2 =

1

H2× 2ε2T̃ (θ)

T̃ 2(θNorm)× dT̃ (θ)

dθ. (15)

The vanishing of H−1 corresponds to the reflection polesat θ = 0 and π, and the vanishing of T̃ (θ) corresponds tothe reflection zeros which are located at

θ(±,±)z = arccos

(±1

2

√−2α± 2

√α2 − 4ζ

), (16)

where the ± sign notation stated above is understood. Inorder to ensure the existence of four zeros in the transferfunction, It is require that

α < 0 and ζ > 0 , (17)

and satisfy the inequalities

α2 − 4 ζ > 0 , (18)α+ ζ + 1 > 0 . (19)

Since θ ∈ (0, π), we identify the location of the reflectionzeros in the order from left to right: θ(+,+), θ(+,−), θ(−,−),θ(−,+). The ripple peak frequencies are found by solving

dT̃ (θ)

dθ=

[3 cos4 θ + (α− 4) cos2 θ − (2α+ ζ)] cos θ

sin2 θ.

(20)Equation (20) has a simple root which is located atθ

(c)pk = π

2 (corresponding to center peak). By the standardquadratic formula,

θ(±,±)pk = arccos

(±

√4− α

6±√(4− α

6

)2

+2α+ ζ

3

).

(21)It can be shown that cos(θ

(±,+)pk ) > 1 for all α and ζ in the

domain defined by (17)–(19). Thus, the ripple peaks cor-responding to these solutions do not exist and are omitted.The remaining roots, θ(+,−)

pk and θ(−,−)pk correspond to the

first and third ripple peak in the passband, respectively.The equal-ripple can be forced in place by matching

the filtering function at cut-off frequency with the firstand second peak, i.e.,

T̃ (θL) = −T̃ (θ(+,−)pk ) = T̃ (π2 ) . (22)

Fig. 2. Frequency Response of |S21(θ)|2 and |S11(θ)|2 using synthe-sized characteristic impedances.

where θL is the angular of the lower cut-off frequency.From these simultaneous equations (22), we obtain

α = 34 [sin(∆BW

2 ) + 13 ]2 − 4

3 , (23)

ζ = 14 cos2(∆BW

2 )[1− sin(∆BW

2 )] , (24)

where ∆BW is the bandwidth. In this particular case, thenormalizing factor is equal to ζ, and from (9) and (11) Cis equal to ε i.e.,

T̃ (θNorm = π2 ) = ζ , (25)

C = ε. (26)

Fig. 2 shows the ideal frequency response of 7.5 GHz and3.2 GHz bandwidth using (23) and (24) with return lossof 20 dB.

III. DESIGN OF MICROSTRIP PROTOTYPE

This section presents physical realization of proposedsynthesis using microstrip line on a substrate (εr =2.2, tan δ = 0.0009 and height h = 787µm). The full-wave simulator HFSS [6] is used to simulate frequencyresponse. A prototype of 3.2 GHz at 20 dB return lossis fabricated. In order to design UWB BPF with passband5.25 to 8.45 GHz, the values α = −0.1298 and ζ = 0.0021are obtained. The ripple factor for 20 dB return loss isε = 0.1005, and using (9)–(11), we find that A = 46.9367,B = −6.0920 and C = 0.1005. The characteristicimpedances are evaluated using (4)–(6) yield zo = 0.8563,zoe = 3.6419 and zoo = 1.6960. Hence the parametervalues labeled in Fig. 1 are determined: l1 = 7507.56µm,w1 = 434.62µm, l2 = 6156.076µm, w2 = 3785.95325µmand s = 369.905875µm. Simulated and measured results

Fig. 3. Fabricated Prototype (filter length = 2.3 cm).

2 3 4 5 6 7 8 9 10 11-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

5

|S11

|2 & |S

21|2 d

BFrequency GHz

S11

Measured S

21 Measured

S21

Simulated S

11 Simulated

Fig. 4. Frequency Response of |S21(θ)|2 and |S11(θ)|2 using Simu-lation and Measurement.

are shown in Fig. 4. The return loss of 17.5 dB is achievedwith approximately equalized ripple level.

IV. CONCLUSION

A synthesis theory is proposed using MMR to fullyunderstand the design procedure. Mathematical formula-tion has been done to achieve characteristic impedancesof UWB filter. Fabricated prototype shows excellent agree-ment with theory, and can further be extended to generalizehigher order UWB BPFs.

REFERENCES

[1] FCC, “Revision of part 15 of the commissions rules regarding ultra-wideband transmission system,” FCC, Washington, D.C., Tech. Rep.ET-Docket 98-153 FCC02-48, Apr. 2002.

[2] L. Zhu, S. Sun, and W. Menzel, “Ultra-wideband (UWB) bandpassfilter using multiple-mode resonator,” IEEE Microwave WirelessCompon.Lett., vol. 15, no. 11, pp. 796-798, Nov. 2005.

[3] H. Wang, L. Zhu, and W. Menzel, “Ultra-wideband (UWB) band-pass filter with hybrid microstrip/CPW structure,” IEEE MicrowaveWireless Compon.Lett., vol. 15, no. 12, pp. 844-846, Dec. 2005.

[4] C. L. Hsu, F. C. Hsu, and J. T. Kuo, “Microstrip bandpass filter forultra-wideband (UWB) wireless communications,” in IEEE MTT-SInt. Microwave Symp. Dig., 2005, pp. 679-682.

[5] I. C. Hunter, Theory and Design of Microwave Filters. London, UK:IEE, 2001, pp. 38-43.

[6] High Frequency Structural Simulator. Ansoft Corp., Pittsburgh, USA.