International Journal of Infrared and Millimeter Waves, Vol. 27, No. 3, March 2006 (© 2006) DOI: 10.1007/s10762-006-9056-0

FILTERING CHARACTERISTICS OF DOUBLY SINUSOIDAL PERIODIC MEDIA* Tian-lin Dong and Ping Chen** Department of electronics and information, Huazhong University of Science and Technology Wuhan, Hubei 430074, P. R .China Received 11 November 2005 Abstract Dispersion and filtering characteristics of doubly sinusoidal periodic (DSP) medium is investigated.

Based on its feature different from singly sinusoidal

periodic medium, a novel dual-band filter model is realized and measured. The results show that even a single unit cell of DSP medium can provide rather good filtering performance. And the filter is of perfect compatibility with regular waveguide and substrate integrated waveguide technology. Keywords: Periodic structure, Doubly sinusoidal periodic medium, Recurrence relation, Dual-band filtering. 1. Introduction In modern communication and electronic systems, filters are always basic components. Their particular characteristics are of the most importance in high-performance systems including local multipoint distribution services, advanced collision avoidance sensors, and compact millimeter systems. Though most researches thus far have focused mainly on the filters for a single *Supported in part by 60171014 and 50577029 NSFC, P. R. China. **Corresponding author. 435

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frequency band, the filters of dual or multiple frequency bands attract active research in recent years [1-7]. The concept and approach of the filters for a single frequency band can be used to realize filters for dual or multiple frequencies.

However, recently we

found that doubly sinusoidal periodic (DSP) medium can be used to describe a series of practical interesting complicated devices and here we present the results in dual frequency filtering research. 2.

Doubly periodic media Analogy to singly sinusoidal periodic medium

H(x)= Ha [1+2G cos(2Sx/d1)] , permitivity distribution of a DSP medium under investigation is defined by

H(x)= Ha [1+4G cos(2Sx/d1) cos(2Sx/d2)] Here G is modulation factor, d1 and d2 are two different periods, and Ha is the average permitivity.

Suppose A is common period for both functions

cos(2Sx/d1) and cos(2Sx/d2).

Mathematically, A is the least common multiple

of d1 and d2, and it is held that p1d1= p2d2=A, p1, p2 are integers. Because of double periodicity, propagation constants of space harmonics should contain both 2Sm1 /d1 and 2Sm2 /d2 terms, where m1 and m2 are arbitrary integers that kxm=kx+2Sm1 / d1+2Sm2 /d2 =kx+2Sm1 p1 /A+2Sm2 p2 /A =kx+2S (m1 p1+ m2 p2) /A =kx+2Sm / A

(1)

with m = m1 p1+ m2 p2, Correspondingly, permitivity of DSP medium can be expressed as

H(x)= Ha [1+4G cos(2Sp1x/A) cos(2Sp2x/A)]

(2)

Obviously, it is a special case of general periodic medium. When no y-associated variation (w/wy {0) and concentrate on the y-polarized TE-fields normal propagation case (ky{0), the E-field in this medium can be represented according to Floquet theorem that

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Filtering Characteristics of Doubly Sinusoidal Periodic Media

E(x, y, z) =y0 Ey(x, y, z) = y0exp(-jkzz)

¦

Vm exp(-jkxmx)

(3)

m

For the time harmonic case, and with no variation in y-direction, the governing equation for electrical fields can be derived from the Maxwell equations that ’2E = -k02H (x)E

(4)

in which k0 = Z(P0H 0)1/2 is free-space wave number. Substituting (3) into (4), it follows that

¦

(k2xm+ k2z )Vm exp(-jkxmx) =k02

m

¦

H(x)Vm exp(-jkxmx)

(5)

m

Multiplying exp(jkxnx) then integrating on the both side of equation (5) in the period A, a five-term recurrence relation can be obtained that Dm Vm+ Vm+p1+p2+Vm+p1-p2+Vm-p1+p2+Vm-p1-p2=0

(6)

where Dm = (k2 k2xmk2z)/( k2G ) 2

and k =

(7)

k02 a.

H

This recurrence relation solely determines the behaviours of waves in DSP medium including the filtering characteristics. 3.

Filtering characteristics For the problem in hand, consider p1=1, p2=3.

Now the recurrence

relation becomes Vm+4+Vm+2+Dm Vm+ Vm-2+Vm-4=0

(8)

This recurrence relation can convert to a homogenous equation system. And the sufficient and necessary condition for the existence of its nontrivial solutions is the determinant of its coefficient matrix vanishes.

Thus the

dispersion relation can be obtained. Numerically, only finite number of space harmonics can be taken into account. Therefore, emphasis of the dominant space harmonics in the concerned frequency range is important.

The calculated dispersion curves are shown

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Dong and Chen

in Fig. 1.

Here kz=S/a, a

is the width of waveguide. We can see DSP medium has a distinguish feature quite different from singly sinusoidal medium: different

periodic It

has

stop

two bands

corresponding to the space harmonics

Fig. 1. Dispersion curves of DSP for p1=1, p2=3, and Ha =2.04, G =0.05, O=2S/k0.

interactions

between m=0 and m=-2 and between m=0 and m=-4, however, no stop band corresponding to that

between m=0 and m=-1.

It is reasonable because relation (8) tells that lowest

order space harmonic m=0 does not couple with first higher order space harmonics m=r1, instead, it directly couples with higher order space harmonics m=r2, r4. Based on the above feature of DSP medium, a filtering scheme can be set up. It is basically a rectangular waveguide filled with DSP medium.

Examining

the field distribution of TE10 mode shown in Fig. 2, it can be seen that this field distribution is exactly the same as that in unbounded DSP medium with ky{0 and kzA/2S = 1.254. The key point of design view is to realize the DSP medium filled in the waveguide.

The desire permitivity

distribution curve is shown in Fig.3 a). Fig. 2. TE10 field distribution

It has four valley values at x=A/6, A/3, 2A/3, and 5A/6, thus the width of

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Filtering Characteristics of Doubly Sinusoidal Periodic Media

dielectric bar filled in the waveguide should

gradually

be

reduced

corresponding to the curve at the right places, as shown in Fig.3 b).

The

reduction can also be made by drilling holes, as shown in c), this shape is much easier to be machined.

Then

metal wires can be placed into the holes to make the modulation stronger, as shown in d) in the same figure. To avoid disturb the field distribution too much, the holes, especially the metal Fig. 3. Permitivity distribution. (a)

wires, should not locate on the centre

- - -: cos(2Sx/A),

line of the width of the dielectric bar,

- · - : cos(6Sx/A)

here they locate apart from the centre

__ : cos(2Sx/A) cos(6Sx/A)

line a quarter width away as shown in

(b) Sinus dielectric bar. (c) Dielectric bar with holes.

the figure. The fundamental parameter is the

(d) Wires placed in the holes

common period A. It should be equal

labelled with W-s.

to Og, the wavelength of the waveguide.

Og can in turn be decided according to the centre wavelength of the filter that

O0 =O1+(O2-O1)/2 where O2, O1 are centre wavelengths of two stop bands. The centre wavelengths depend on the wavelength at which Bragg conditions are satisfied.

Therefore,

the unperturbed dispersion line shown in Fig.1 by thin solid line can be used to calculate Og that (2S/O)2Ha +(2S/Og) 2+(S/a)2=0

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A series of experimental model are made and used to verify the theory and principle of design presented here. The experimental set-up is shown in Fig. 4. Short-cut connection

Isolator

Power source

Variable attenuator

Frequency meter

Filter under test

Power meter

Fig. 4. Block diagram of experimental set-up On each measurement, first use a short-cut waveguide to directly connect the frequency meter and power meter as shown by the dashed line in the figure, and adjust the attenuator to get a suitable reading of power meter. Then use the filter under test to replace the short-cut waveguide and adjust the attenuator again to get the same reading of power meter. The second adjustment value of attenuator is the attenuation of the filter under test. The models shown in Fig. 3 b) and c) have relatively small effective modulation factor, therefore, they would be too long to be a millimetre wave filter with practical attenuation.

However, the

models shown in Fig. 3 d) have much stronger effective modulation factor. In fact, only a single unit cell of DSP medium can produce considerable strong filtering effect.

The measured attenuation curve of a typical model is shown in

Fig. 5. It has parameters A=17.81 mm, a=7.1 mm, Ha =2.04, and height of waveguide is 3.5 mm and diameter of the metal wire is taken to be 0.044 mm.

Filtering Characteristics of Doubly Sinusoidal Periodic Media

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Fig. 5. Measured attenuation of a DSP filter model

4.

Discussion and conclusion In a straightforward way the novel dual-band bandstop filter can be built

up based on the filtering characteristics of DSP medium.

The main benefits

are including extremely conformal type with the regular waveguide and completely compatibility with the substrate integrated waveguide technology [8], by which even the sinus shape of waveguide as shown in Fig 3 b) can easily be realized. Though dual-band bandstop filtering is treated here, the multi-band is also possible by extending the modulation fold, and bandpass filtering is possible invoking the microwave network or amplifier techniques based on the bandstop block. References 1. L.C. Tsai, and C.W. Hsue, “Dual-band bandpass filters using equallength coupled-serial-shunted lines and Z-transform technique,” IEEE Trans. Microw. Theory

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Tech., 2004, 52, (4), pp. 1111–1117. 2. S.F. Chang, Y.H. Jeng, and J.L Chen, “Dual-band step-impedance bandpass filter for multimode wireless LANs,” Electron. Lett., 2004, 40, (1), pp. 38–39. 3. S.F. Chang, et al., “A dual-band RF transceiver for multistandard WLAN applications,” IEEE Trans. Microw. Theory Tech., 2005, 53, (3), pp. 1048–1055. 4. V. Palazzari, S. Pinel, J. Laskar, L. Roselli, and M.M. Tentzeris, “Design of an asymmetrical

dual-band

WLAN

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liquid

crystal

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(LCP)

system-on-package technology,” Microwave and Wireless Components Letters, 2005, 15(3), pp.165 – 167. 5. M.L. Chuang, “Cascaded dual band microstrip open-loop filter,” Microw. Opt. Technol. Lett., 2005, 45, (6), pp. 519–522. 6. Sheng Sun and Lei Zhu, “Compact dual-band microstrip bandpass filter without external feeds,” Microwave and Wireless Components Letters, IEEE, Volume 15, Issue 10, Oct. 2005, pp.644 – 646. 7. M. L. Chuang, “Concurrent dual band filter using single set of microstrip open-loop resonators,” Electronics Letters, 2005, 41(18), pp.1013 – 1014. 8. L.Yan, W. Hong, K. Wu, and T.J. Cui, “Investigations on the propagation characteristics of the substrate integrated waveguide based on the method of lines,” Microwaves, Antennas and Propagation, IEE Proceedings, 2005, 152(1), pp. 35 – 42.