To Optimally Design Microstrip Nonuniform Transmission Lines as Lowpass Filters
JJOURNAL OF TELECOMMUNICATIONS, VOLUME 2, ISSUE 2, MAY 2010 139 © 2010 JOT http://sites.google.com/site/journaloftelecommunications/
To Optimally Design Microstrip Nonuniform Transmission Lines as Lowpass Filters
M. Khalaj-Amirhosseini and S. A. Akbarzadeh-Jahromi
Abstract — A method is proposed to optimally design the Microstrip Nonuniform Transmission Line (MNTLs) as lowpass filters. Some electrical and physical restrictions are used to design MNTLs. To optimally design the MNTLs, their strip width is expanded as truncated Fourier series, firstly. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. The performance of the proposed structure is studied by design and fabrication of two lowpass filters of cutoff frequency 2.0 GHz.
Index Terms —Nonuniform Transmission Lines, Lowpass Filters, Microstrip Lines. —————————— (cid:1) ——————————
1 I
NTRODUCTION commonly used structure for microstrip lowpass filters (LPFs) is Hi-Z, Lo-Z or stepped-impedance structure in which several very high and very low impedance transmission lines are located alternately [1-3]. However the performance of these popular filters is not as good due to the approximations involved in their de-sign. Moreover, the stepped-impedance filters have dis-continuities, which create some implementation con-straints. On the other hand, Microstrip Nonuniform Transmission Lines (MNTLs) are widely used in micro-wave circuits as resonators [4], impedance matching [4]-[5], delay equalizers [6], wave shaping [7], analog signal processing [8] etc. Some researchers have used MNTLs to design lowpass or bandreject filters with specific response functions such as Butterworth or Chebyshev types [9-11]. The MNTL based lowpass filters have not any discontinu-ity and quasi-TEM approximation is more valid for them than to stepped-impedance filters. This paper presents a method to design optimally the MNTLs as lowpass filters. In the presented method, some arbitrary restrictions are used instead of a known specific response. Moreover, the strip width of MNTLs is considered as a truncated Fouri-er series, instead of a collection of several uniform or cu-bic sections [10]. The optimum values of the coefficients of the Fourier series are obtained through an optimization approach. Finally, the performance of the proposed struc-ture is studied using two experiments.
2 A
NALYSIS OF
MNTL
LPF S In this section MNTLs are considered as lowpass filters. Fig. 1 depicts a typical Microstrip Nonuniform Transmis-sion Line (MNTL) of length d terminated by source and load resistances Z . The relative electric permittivity and the thickness of the substrate are ε r and h , respectively. Also, the width of strip is varying with z as w ( z ). We would like to design MNTLs as lowpass filters specified by arbitrary restrictions shown in Fig. 2. The passband and stopbands are defined in the frequency ranges [0 – f p ] and [ f s – f max ], respectively. The frequency response of designed filter must be kept out of hachuring regions of Fig. 2. To analyze MNTLs, one can use some quasi-TEM based methods such as cascading many short sections [12, 13], finite difference [14], Taylor’s series expansion [15], Fourier series expansion [16], the equivalent sources me-thod [17], the method of Moments [18] and some approx-imated closed form solutions [19]. Of course, the most straightforward method is subdividing MNTLs into many uniform or linear electrically short sections of length ∆ z so that r f cz ελ maxmin =<<∆ (1) where c is the velocity of the light. Then the ABCD matrix of the MNTL will be obtained by multiplying the
ABCD matrices of all short sections. After finding the
ABCD pa-rameters, one can determine the S parameters as follows DZCZBAZ DZCZBAZS +++ −−+= (2) DZCZBAZ ZS +++= (3)
3 S
YNTHESIS OF
MNTL
LPF S In this section a general method is proposed to optimally design MNTLs as lowpass filters. First, we consider the following truncated Fourier series expansion for the nor-malized width function w ( z )/ h . ———————————————— • M. Khalaj-Amirhosseini is with Iran University of Science and Technolo-gy, Tehran, Iran. • S. A. Akbarzadeh-Jahromi is with Iran University of Science and Technolo-gy. A OURNAL OF TELECOMMUNICATIONS, VOLUME 2, ISSUE 2, MAY 2010 140 © 2010 JOT http://sites.google.com/site/journaloftelecommunications/ ∑∑ == += Nn nNn n dnzSdnzChzw )/2sin()/2cos()(ln ππ (4) The optimum values of the unknown coefficients C n and S n in (11) can be obtained through minimizing the follow-ing defined error function. += ∑∑ ≤<≤< max )()(1Error ffffff pp fSfSN (5) where N f is total number of frequencies in the range of zero to f max . Moreover, the above defined error function should be restricted by some electrical (as are seen in Fig. 2) and physical constraints like as the followings ( )( ) pp fffS ≤<−≥ α (6) ( )( ) max21 ;)(log20max ffffS ss ≤≤−≤ α (7) ( ) sppps psp ffffffffS <<−−−−−≤ ;)()(log20 ααα (8) maxmin )( ≤≤ hwhzwhw (9) hwhdwhw )()0( == (10) where ( w / h ) min and ( w / h ) max are the minimum and maxi-mum available normalized width, respectively. Also, w / h is desired normalized width at the ends of the mi-crostrip lines corresponding to desired characteristic im-pedance Z . (a) h ε r w ( z ) (b) Figure 1. Typical Microstrip Nonuniform Transmission Line a) Longitudinal view b) The cross section p f s f max f p α s α f ][|| 1 dBS Figure 2. Defined restrictions for a lowpass filter (out of hachuring regions)
4 E
XAMPLES AND R ESULTS
In this section two MNTL lowpass filters are designed and fabricated. The specifications of these filters are con-sidered as the following: 1. LPF No. 1: f p = 2.0 GHz, f s = 3.0 GHz, f max = 6.0 GHz, α p = 0.1 dB, α s = 20 dB, ( w / h ) max = 10, ( w / h ) min = 0.13, d = 10.0 cm and Z = 50 Ω . 2. LPF No. 2: f p = 2.0 GHz, f s = 3.0 GHz, f max = 6.0 GHz, α p = 0.3 dB, α s = 20 dB, ( w / h ) max = 7, ( w / h ) min = 0.1, d = 10.0 cm and Z = 50 Ω . The filters were designed and simulated by HFSS full-wave software and then were fabricated on a substrate with ε r = 3.5 and h = 30 mil = 768 µ m. Figs. 3-5 show the normalized width functions and the pictures of the fabri-cated filters. Also, Table 1 shows the optimum values of the unknown coefficients C n and S n for designed filters. In continuation, Figs. 6 and 7 compare the response of the filters obtained from analysis, simulation and measure-ment. It is seen that the agreement between theoretical and measurement results is good, especially for filter No. 2 whose allowable attenuation in the passband is more than that of filter No. 1. The extra losses in the passband may be due to the losses of substrate and connectors, not considering the thickness of conductive strip, radiation and weak validation of the quasi-TEM approximation at wide regions. It is expected as the length of the filter is chosen larger the required width of conductive strip is decreased and consequently the quasi-TEM approxima-tion becomes more validated. Table 1. Optimum values of the coefficients C n and S n LPF C C C C C C No. 1 −0.0143 −0.1071 −0.4725
No. 2 −0.0637 −0.0078 −0.6005 − S S S S S No. 1 − −0.1593 −0.0968 −0.1729 −0.8906
No. 2 − −0.2200 −1.0636
OURNAL OF TELECOMMUNICATIONS, VOLUME 2, ISSUE 2, MAY 2010 141 © 2010 JOT http://sites.google.com/site/journaloftelecommunications/
Figure 3. The normalized width function w ( z )/ h of filter No. 1 Figure 4. The normalized width function w ( z )/ h of filter No. 2 Figure 5. The picture of fabricated filters a) filter No. 1 b) filter No. 2
5 C
ONCLUSION
A method was proposed to optimally design the Microstrip Nonuniform Transmission Line (MNTLs) as lowpass filters. Some arbitrary electrical and physical restrictions are used to design MNTLs. To optimally design the MNTLs, their strip width is expanded as truncated Fourier series, firstly. Then, the optimum values of the coefficients of the series are obtained through an optimization approach. Two low-pass filters of cutoff frequency 2.0 GHz were designed, fa-bricated and measured. The agreement between theoretical and measurement results was satisfactory, especially for filters whose allowable attenuation in the passband is high. The MNTL based lowpass filters have not any discontinuity. Finally, it is worthy to mention that the proposed method can be used to design MNTLs as Bandreject filters by suita-ble defining the electrical restrictions.
Figure 6. The absolute of S for filter No. 1 Figure 7. The absolute of S for filter No. 2 R EFERENCES [1]
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Mohammad Khalaj Amirhosseini was born in Tehran, Iran in 1969. He received his B.Sc, M.Sc and Ph.D. degrees from Iran University of Science and Technology (IUST) in 1992, 1994 and 1998 respectively, all in Electrical Engineering. He is currently an Associate Professor at College of Electrical Engineering of IUST. His scientific fields of interest are elec-tromagnetic direct and inverse problems including micro-waves, antennas and electromagnetic compatibility.
Abbas Akbarzadeh was born in Jahrom, Iran, on April 13, 1984.