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LAPORAN AKHIR
PENELITIAN HIBAH FUNDAMENTAL
Penyaring Lolos Tengah dengan Lebar Pita yang Besar pada Sistim Komunikasi Ultrawideband dan Radar
Tahun ke 2 dari rencana 2 tahun
Prof. Dr.-Ing. MUDRIK ALAYDRUS 0311057101 (Ketua) Ir. SAID ATTAMIMI MT 0307106101 (Anggota)
UNIVERSITAS MERCU BUANA NOVEMBER 2015
III
Ringkasan
Stepped Impedance resonator adalah struktur yang terdiri beberapa jalur
transmisi dengan admittansi yang berbeda. Struktur ini digunakan untuk
mengontrol respon palsu dan kerugian Transmisi filter dengan cara
mengubah rasio impedansi dari stepped impedance. Kondisi modus
resonator ganda, yang diimplementasikan dalam bentuk stepped impedance
resonator, dapat digunakan, baik untuk memperluas pass / stop band atau
untuk memisahkan pass band dan stop band satu sama lain. Dalam
penelitian ini, dipelajari struktur stepped impedance yang terdiri dari dua
segmen saluran transmisi, yang memiliki bentuk umum rasio admittansi dan
dan rasio panjang yang bebas dipilih. Juga diamati penampakan dan
pergerakan resonansi dan memverifikasinya melalui simulasi komputer
struktur ini dalam teknologi microstrip.
Filter yang memiliki resonator tergandeng digunakan untuk mendapatkan
filter yang memiliki dua wilayah kerja (dualbandpass filter). Perhitungan
memberikan hasil dengan frekuensi resonansi pertama pada 1.91 GHz
dengan lebar pita sebesar 90 MHz, atau FBW 4.7% dan pada 2.16 GHz
dengan lebar pita sebesar 110 MHz atau FBW 5.1%.
Hasil pengukuran menunjukkan kemiripan hasil dengan yang didapat dari
simulasi, baik untuk wilayah di sekitar pita lolos dan stop, ataupun jika
diamati secara broadband.
Kata kunci: coupled resonator, dualbandpass filter, mikrostrip, stepped
impedance resonator
IV
PRAKATA
Penelitian tentang filter lolos tengah yang memiliki pita besar bermula dari riset panjang yang
telah peneliti utama lakukan sejak tahun 2003.
Filter adalah komponen sangat penting pada sistim telekomunikasi wireless modern.
Diusulkannya sistim wireless baru memaksa para peneliti untuk memikirkan model baru dari
filter, yang awalnya hanya filter yang bersifat narrow-band, kemudian multiband, broadband
dan ultrawideband.
Di penelitian ini dicari bentuk baru filter yang bisa bekerja secara broadband dan juga bisa
bekerja secara multiband.
Di tahun pertama dari penelitian melalui skema penelitian fundamental ini, dipelajari tingkah
laku dari filter yang berbasiskan Stepped Impedance Resonator (SIR). Dengan tipe filter ini
didapatkan fenomena resonansi pada suatu nilai frekuensi tertentu, yang tergantung dari
parameter yang digunakan bisa bergeser ke nilai frekuensi lain.
Di tahun kedua dikembangkan metoda Metoda perancangan untuk mendapatkan filter dengan
wilayah kerja dua buah yang dipisahkan oleh wilayah stop.
Akhir kata, peneliti mengucapkan puji syukur kepada Allah SWT yang dengan petunjuknya,
kami bisa menyelesaikan penelitian ini, juga kami ucapkan terima kasih kepada Dikti yang
telah menyediakan dana untuk keberlangsungan penelitian ini. Kepada kepala pusat penelitian
UMB kami ucapkan terima kasih atas kerja samanya.
Jakarta, 5 November 2015
Tim Peneliti
V
DAFTAR ISI
halaman
HALAMAN SAMPUL i
LEMBAR PENGESAHAN ii
RINGKASAN iii
PRAKATA iv
DAFTAR ISI v
DAFTAR TABEL vi
DAFTAR GAMBAR vi
DAFTAR LAMPIRAN vi
BAB 1. PENDAHULUAN 1
BAB 2. TINJAUAN PUSTAKA 3
BAB 3. TUJUAN DAN MANFAAT PENELITIAN 8
BAB 4. METODE PENELITIAN 9
BAB 5. HASIL YANG DICAPAI 14
BAB 6. KESIMPULAN DAN SARAN 23
DAFTAR PUSTAKA 24
LAMPIRAN
VI
Daftar Tabel
Tabel 5.1 data perancangan untuk Low pass filter 20
Daftar Gambar
Gambar 4.1 SIR yang tersusun dari dua potong saluran transmisi (mikrostrip) dengan admitansi
gelombang Y1 dan Y2 dengan panjang l1 dan l2 9
Gambar 4.2 (kiri) η = 2 dan (kanan) η = 0.5 10
Gambar 4.3 (kiri) rangkaian mikrostrip dan (kanan) resonansi 10
Gambar 4.4 Faktor transmisi S21 dua buah resonator yang saling berdekatan 11
Gambar 4.5 Phasa dari faktor refleksi pada suatu tapping resonator 11
Gambar 4.6 (kiri) rangkaian resonator open loop, (kanan) resonator open loop dengan central
loading 12
Gambar 4.7 Faktor transmisi (resonansi) kedua struktur di gambar 4.6 12
Gambar 4.8 Dualbandpass Filter N=3 13
Gambar 5.1 Topologi dual passband filter 14
Gambar 5.2 kiri: struktur perhitungan faktor penggandeng (coupling factor) k, kanan: struktur
perhitungan external quality factor Q 14
Gambar 5.3 Faktor kopling dua resonator dengan parameter jarak offset 15
Gambar 5.4 Eksternal faktor kualiti sebagai fungsi dari jarak tapping 15
Gambar 5.5 Model perancangan dan model dalam software Sonnet 16
Gambar 5.6 Faktor refleksi dan transmisi struktur di gambar 5.5 16
Gambar 5.7 (kiri) model filter, (kanan) hasil simulasi 17
Gambar 5.8 Model awal perancangan dan usulan 18
Gambar 5.9 Model awal perancangan dan usulan 18
Gambar 5.10 Model perhitungan dan prototype, hasil perhitungan dan pengukuran. 19
Gambar 5.11Perbandingan hasil perhitungan dan pengukuran secara broadband. 19
Gambar 5.12. Bentuk generik rangkaian low pass fitler dengan fungsi polynomial. 20
Gambar 5.13. Low pass filter ordo N = 3 dengan dua 50 Ω feeding lines. 21
Gambar 5.14. Parameter S dari low pass filter N = 3, garis: perhitungan, putus-putus: pengukuran
dengan network analyzer ZVL13. 21
Gambar 5.15. Low pass filter ordo N = 5 dengan dua 50 Ω feeding lines. 22
Gambar 5.16. Parameter S dari low pass filter N = 5, garis: perhitungan, putus-putus: pengukuran
dengan network analyzer ZVL13. 22
Daftar Lampiran
Publikasi Conference dan Journal
1
1 Pendahuluan
Teknologi pita sangat lebar (Ultra-wideband technology) memainkan peran sangat penting
dalam pengembangan radar modern dan sistem komunikasi berkecepatan tinggi (Arslan et al,
2006, Kaiser, 2010, Li et al, 2008). Solusi pita lebar (wideband) untuk filter gelombang mikro
berarti, melakukan pemilihan sinyal dengan sangat efisien yaitu dengan menekan interferensi
dan derau (noise) pada pita yang tak dperlikan (out-of-band) juga sinyal harmonis yang
diakibatkan oleh komponen tak linier pada ujung akhir pemancar.
Filter lolos tengah yang bekerja pada interval frekuensi yang sangat lebar (ultra
wideband/UWB) mendapatkan perhatian yang cukup luas dewasa ini, baik di kalangan
akademisi dan peneliti, juga di kalangan industri. Ketertarikan di dunia penelitian dan bisnis
ini didukung oleh keluarnya regulasi dari FCC di Amerika Serikat, (FCC, 2002), dan ETSI di
Eropa yang menawarkan inverval frekuensi tanpa berlisensi yang sangat lebar. Regulasi yang
dikeluarkan dua komisi penting dunia ini memiliki aturan/profil frekuensi yang sangat ketat,
karena hanya dengan demikian bisa dihindarinya interferensi dengan sistim lain yang telah ada
di wilayah frekuensi 3.1 – 10.6 GHz. Dengan frekuensi tangah pada 6.85 GHz didapatkan lebar
pita relativnya (fractional bandwidth) FBW = 109.5%.
Selama bertahun-tahun, teori filter bandpass pada sinyal gelombang mikro telah berkembang
dengan baik dan menjadi metoda andalan dalam merancang filter yang memiliki lebar pita
(bandwidth) yang sempit (narrow-band). Tetapi metoda klasik ini tak bisa secara langsung
tanpa modifikasi diaplikasikan untuk sistim filter dengan wilayah kerja yang lebar atau bahkan
sangat lebar. Contoh filter yang bekerja pada wilayah pita yang sempit adalah filter dengan
resonator yang terbuat dari saluran transmisi tergandeng secara paralel (parallel-coupled
resonators) (Cohn, 1958). Jenis filter ini dikembangkan dengan metoda sintesis yang
mendasarkan fenomena elektromagnetika nya pada suatu rangkaian pengganti tertentu
(resistor, kapasitor, induktor dan transformator) yang hanya tepat pada suatu frekuensi tertentu
(yaitu frekuensi resonansinya), sedangkan pada frekuensi lainnya terbentuk suatu
kesalahan/pendekatan. Sehingga dengan mengandaikan suatu kesalahan maksimal tertentu,
filter ini hanya bisa diandalkan pada suatu interval frekuensi yang sempit saja. Sejak awal
dikembangkannya sistim filter yang bekerja secara tergandeng antara saluran transmisinya ini,
tidak begitu akurat untuk situasi broadband. Baru-baru ini, rumus desain yang lebih akurat telah
diusulkan, tapi hanya dibatasi pada suatu ordo tertentu dan hanya filter dengan fungsi transfer
tertentu yang bisa disintesa (Chin et al, 2004, Chin et al, 2005). Selain dari itu ada pembatasan
yang ketat dari filter ini terhadap lebar pita kerja maksimal yang bisa dipakai, yaitu tergantung
2
dari jarak minimal yang memisahkan dua mikrostrip, yang masih bisa difabrikasi dengan
mesin-mesin yang ada di bengkel.
Sebagai akibat dari kesulitan tersebut, upaya yang cukup besar telah dilakukan oleh para
ilmuwan dalam merancang topologi filter bandpass yang baru yang bisa bekerja secara
broadband pada frekuensi yang tinggi. Penelitian dalam mendapatkan filter yang bekerja pada
lebar pita yang besar dan berukuran kecil telah diusulkan pada (Kuo et al, 2002, Hsieh et al,
2003, Soong et al 2005, Görür et al, 2003). Filter yang memiliki faktor penolakan (rejection)
yang tinggi di stop band, walaupun memiliki lebar pita di sana yang lebih kecil dibandingkan
lebar pita pada wilayah lolos. Sehingga menyebabkan filter jenis ini memiliki sedikit kegunaan
pada beberapa aplikasi penting, seperti pada komunikasi pita lebar multichannel atau
komunikasi full-duplex.
Untuk hal ini, teknik sinyal-gangguan digunakan sebagai alternatif untuk merancang filter
menunjukkan lebar spektrum yang luas baik di band lolos ataupun di band tolak (Gomez-
Garcia et al, 2005a, Gomez-Garcia et al, 2005b). Namun demikian, perilaku yang periodis yang
biasa muncul pada spektrum dari struktur saluran transmisi akan membatasi penggunannya
pada nilai lebar pita relatif sebesar 50%. Tantangannya adalah penelitian yang mampu
memenuhi spesifikasi yang diberikan oleh FCC dan ETSI, yaitu filter yang bisa bekerja pada
frekuensi ultrawideband 3,1 GHz sampai 10,6 GHz. Ada beberapa pendekatan yang telah
dilakukan untuk mendapatkan filter bandpass yang bisa bekerja pada frekuensi yang sangat
lebar ini, (Amari et al, 2010) mengusulkan cara penggunaan sintesis langsung untuk
mendapatkan nilai elemen dari rangkaian transversal. Dengan cara ini bisa juga didapatkan
filter yang memiliki lebar frekuensi kerja yang lebar. (Wong et al, 2005) menggunakan
pendekatan penggunaan stub short yang dipasangkan secara parallel dan merancang filter
bandpass dengan 11 pol untuk frekuensi 3.1 – 10.6 GHz. Filter ini direalisasikan dengan
teknologi mikrostrip yang dibuat secara melikuk-likuk (meandered) untuk mereduksi
ukurannya. Pendekatan lainnya dalam merancang filter ultra-wideband adalah dengan
memanfaatkan teknik resonator dengan moda-banyak (multiple-mode resonators/MMR) yang
diperkenalkan pertama kali oleh (Zhu et al, 2005).
3
2 Tinjauan Pustaka
II.1 Filter Bandpass Lebar Pita Sempit
Filter lolos tengah (bandpass filter) adalah salah satu komponen kunci penyusun sistim
pemancar dan penerima. Kinerja sistim pemancar dan penerima nirkabel (wireless) ditentukan
secara signifikan oleh unjuk kerja dari filter lolos tengah ini. Penelitian tentang filter telah
berlangsung sejak 80 tahun, hasil-hasil pentingnya didokumentasikan dalam monograph klasik
seperti, (Zverev, 1965, Matthaei et al, 1980, Young, 1972, dan Malherbe, 1979).
Perkembangan teknologi komputer dua puluh tahun terakhir ini terus mendorong
berkembangnya metoda analisa dan sintesa filter secara lebih akurat, hasil-hasil riset yang baru
ini dihimpun pada monograph berikut, (Hunter, 2001, Cameron et al, 2007, Jarry et al, 2008,
Jarry et al, 2009, Makimoto et al, 2010, Hong, 2011, dan Zhu et al, 2012).
Karakteristik dari pendekatan yang dilakukan oleh penelitian-penelitian tersebut adalah
dibatasinya pembahasan pada wilayah kerja yang sempit (narrow band), (Swanson, 2007).
(Alaydrus, 2010) melaporkan analisa dan perancangan bandpass filter dengan metoda saluran
transmisi yang tergandeng secara paralel dan mendapatkan lebar pita hanya 3%. Metoda lain
yang juga bekerja pada lebar pita yang sempit berbasiskan pada resonator yang saling
tergandeng dengan pendekatan teori matriks (Alaydrus, 2012).
Teori pendekatan untuk filter dengan lebar pita yang sempit ini bisa digunakan sebagai basis
dan bahan acuan untuk menganalisa dan mensintesa filter dengan pita lebar yang lebih besar.
II.2 Filter Bandpass Lebar Pita Besar
Filter lolos tengah yang bekerja pada interval frekuensi yang sangat lebar (ultra
wideband/UWB) mendapatkan perhatian yang cukup luas dewasa ini, baik di kalangan
akademisi dan peneliti, juga di kalangan industri. Ketertarikan di dunia penelitian dan bisnis
ini didukung oleh keluarnya regulasi dari FCC di Amerika Serikat, (FCC, 2002), dan ETSI di
Eropa yang menawarkan inverval frekuensi tanpa berlisensi yang sangat lebar. Regulasi yang
dikeluarkan dua komisi penting dunia ini memiliki aturan/profil frekuensi yang sangat ketat,
karena hanya dengan demikian bisa dihindarinya interferensi dengan sistim lain yang telah ada
di wilayah frekuensi 3.1 – 10.6 GHz. Dengan frekuensi tangah pada 6.85 GHz didapatkan lebar
pita relativnya (fractional bandwidth) FBW = 109.5%. Secara umum teori filter yang telah
berkembang dengan sangat baik digunakan untuk merancang filter yang bekerja pada interval
frekuensi yang sempit (narrow-band filter).
4
(Amari, 2005, Macchiarella, 2008 dan Macchiarella et al, 2012), melaporkan penelitian dengan
di samping resonator juga digunakannya node reaktif yang bisa terbuat dari komponen lumped
(kapasitor atau induktor) atau saluran transmisi atau stup open atau short yang tidak
beresonansi. Dilaporkan didapatkannya performansi stop dan pass yang lebih baik.
(Amari et al, 2010) mengusulkan cara penggunaan sintesis langsung untuk mendapatkan nilai
elemen dari rangkaian transversal. Dengan cara ini bisa juga didapatkan filter yang memiliki
lebar frekuensi kerja yang lebar. (Xiao, 2011) men-gembangkan cara ini dengan
memberikan teknik analitis untuk mendapatkan cara perancangan yang lebih baik.
Ada beberapa pendekatan yang dilakukan untuk meanalisa dan merancang filter bandpass yang
bekerja pada interval frekuensi yang sangat lebar ini, sebagian mengadopsi prototype filter high
pass dengan stub paralel yang ujungnya diberikan hubungan singkat (short) seperti
diperkenalkan di (Levy et al 1968). (Wong et al, 2005) menggunakan pendekatan ini dan
merancang filter bandpass dengan 11 pol untuk frekuensi 3.1 – 10.6 GHz. Filter ini
direalisasikan dengan teknologi mikrostrip yang dibuat secara melikuk-likuk (meandered)
untuk mereduksi ukurannya. Stub yang ujungnya di-short direalisasikan secara via-hole.
(Gomez-Garcia et al, 2006) menunjukkan cara analitis yang sistematis dan konsisten dalam
melakukan sintesa secara eksak filter ultrawideband dengan melakukan sambungan berantai
bagian-bagian low-pass dan high-pass yang saling terisolasi satu dengan lainnya. (Garcia-
Garcia et al, 2006) menggunakan teknik electromagnetic bandgap yang diletakkan pad bagian
bawah rangkaian tepat di bawah dua stub yang saling berdekatan, dengan cara ini dimensi dari
filter bisa direduksi. (Shaman et al, 2007 dan Shaman et al, 2007a) melakukan penggandengan
silang (cross-coupling) antara sisi masukan (input) dan keluaran (output) yang dengan cara ini,
selektivitas filter menjadi meningkat demikian juga kinerja group delay-nya membaik. (Uhm
et al, 2008) merancang filter bandpass dengan lebar kerja yang lebih besar dengan cara
menambahkan sepotong saluran transmisi dengan panjang seperempat panjang gelombang
pada sisi input dan output dari filter dan mengganti sambungan mikrostrip dengan sebuah
struktur low-pass.
Pendekatan lainnya dalam merancang filter ultra-wideband adalah dengan memanfaatkan
teknik resonator dengan moda-banyak (multiple-mode resonators/MMR) yang diperkenalkan
pertama kali oleh (Zhu et al, 2005). (Li et al, 2007) merancang filter ultra-wideband ini dengan
menambahkan stub sehingga bisa memperkecil struktur filter tersebut. Sedangkan (Wong et al,
2007, Wong et al, 2008 dan Baik et al, 2008) menggunakan teknik electromagnetic bandgap
untuk mendapatkan ukuran filter yang ter-reduksi.
5
II.2 Filter Dual-Bandpass
Publikasi oleh Kuo(2005) yang memiliki target Merancang filter yang bekerja pada dual band, dengan
tantangan: merancang filter dengan dua passband yang disintesa secara simultan. Metoda yang dipakai adalah
Men-tuning resonansi kedua dari stepped impedance resonator (SIR) dengan konfigurasi tergandeng parallel dan
tersusun vertical.
Kuo(2004) juga merancang dualbandpass filter berbasiskan Hairpin dengan realisasi SIR.
Lebar strip yang berbeda (basis dari SIR) menggeser frekuensi kedua dari 2 x 2.4 GHz = 4.8
GHz ke posisi 5.2 GHz.
6
Lee (2004) menggunakan teknik peletakan pole-zero secara umum pada sebuah pass band, sehingga didapatkan
wilayah tolak (rejection band). Metoda ini memerlukan optimisasi numeris yang bekerja lambat, sehingga
memerlukan waktu perancangan yang lama (konvergensi yang lambat).
Macchiarella (2005) mengusulkan du acara untuk merancang dualbandpass filter: (1) filter yang simetris, fungsi
frekuensi yang simetris dengan frekuensi fo di dalam wilayah stop, (2) untuk filter yang memiliki bandwidth
passband yang bebas, dengan semua TZ berada di wilayah stop. Kedua metoda ini bekerja dengan menggunakan
transformasi frekuensi yang diikuti dengan sintesa prototype low pass filter.
7
8
3 Tujuan dan Manfaat Penelitian
Mengembangkan metoda untuk mendapatkan filter bandpass yang memiliki pita kerja yang
sangat lebar dan atau memiliki dual passband sehingga bisa digunakan pada aplikasi
komunikasi broadband dan radar.
9
4 Metoda Penelitian
4.1 Resonator Impedansi Tangga /Stepped Impedance Resonators (SIR) dan Resonator
Moda-Banyak/Multiple-mode Resonators (MMR)
Resonator impedansi tangga (stepped impedance resonatpr/SIR) yang sederhana ditampilkan
di gambar 4.1, (Alaydrus et al, 2014). Admitansi masukan pada sisi AA’ menjadi
(4.1)
.
Gambar 4.1 SIR yang tersusun dari dua potong saluran transmisi (mikrostrip) dengan
admitansi gelombang Y1 dan Y2 dengan panjang l1 dan l2
Dengan mendefinisikan rasio kedua admitansi karakteristik menjadi η = Y2/Y1 dan rasio
panjang ν = l2/l1. Panjang dari saluran transmisi ini bersama-sama dengan konstanta phasa
akan membentuk phasa yang ada pada fungsi tangen di persamaan (4.1).
Sehingga persamaan itu menjadi
(4.2)
Dengan memvariasikan kedua besaran rasio ini bisa dihasilkan posisi-posisi resonansi, seperti
yang ditampilkan di gambar 4.2.
Verifikasi diberikan di publikasi (Alaydrus et al, 2014), dengan teknologi resonator mikrostrip,
untuk frekuensi resonansi yang pertama 2.3 GHz. Material yang digunakan Rogers TMM10
(εr = 9:56 and tan δ = 0.0022) dengan ketebalan subtrat h = 0.635 mm.
10
Gambar 4.2 (kiri) η = 2 dan (kanan) η = 0.5
Fig. 4.3 sebelah kiri menunjukkan rangkaian mikrostrip yang dirancang pada software Sonnet
v.13. SIR tergandeng secara lemah pada gerbang input dan output. Frekuensi resonansi
pertamanya disetting pada 2.3 GHz. Resonansi kedua terjadi pada phase 3.1415 rad, yang untuk
frekuensi yang digunakan ekuivalen pada 4.588 GHz and yang ketiga pada 6.856 GHz seperti
yang ditunjukkan dengan lingkaran merah di gambar 4.3 sebelah kanan. Perbandingan dengan
perhitungan Sonnet (solid) menunjukkan posisi-posisi resonansi 2.3 GHz, 4.54 GHz (48 MHz
shifted) dan 6.78 GHz (76 MHz shifted dibanding hasil analitis).
Gambar 4.3 (kiri) rangkaian mikrostrip dan (kanan) resonansi
4.2 Filter Dualpassband dengan resonator yang tergandeng
Dalam merancang filter sering dikombinasikan beberapa resonator dalam suatu gandengan
(coupling) tertentu. Resonator yang bekerja pada suatu frekuensi tertentu, jika didekatkan
dengan resonator yang sama, keduanya akan berinteraksi. Tebentuklah moda genap (even
mode) dan moda ganjil (odd mode). Kedua moda ini memiliki frekuensi yang berbeda. Yang
akan menghilang, jika kedua resonator ini dijauhkan. Semakin dekat separasi resonator,
semakin berbeda frekuensi keduanya. Gambar 4.4 menunjukkan faktor transmisi S21 dari
struktur yang mengandung dua resonator yang berdekatan.
11
𝑘 =𝑓𝑝22 − 𝑓𝑝1
2
𝑓𝑝22 + 𝑓𝑝1
2
Gambar 4.4 Faktor transmisi S21 dua buah resonator yang saling berdekatan
Semakin dekat jarak kedua resonator, semakin menjauh kedua posisi resonansi, yang ditandai
dengan puncak nilai S21. Fenomena penggandengan ini dikuantifikasi dengan faktor
gandengan (coupling factor) k, yang bisa dihitung dengan persamaan di gambar 4.4.
Definisi dari external
quality factor
𝑄𝑒 =𝜔0
∆𝜔±90𝑜
Gambar 4.5 Phasa dari faktor refleksi pada suatu tapping resonator
Untuk mendapatkan pengaruh koneksi eksternal ke dalam struktur filter, digunakan Qe yaitu
faktor kualitas ekstenal, yang bertugas untuk mematchingkan pemasangan feed ke filter.
Gambar 4.5 menunjukkan phasa dari faktor refleksi pada posisi pemasangan feed. Dengan
mendapatkan posisi phasa = 0o, didapatkan nilai frekuensi resonansi fo ωo. Kemudian
dengan mendapatkan nilai frekuensi untuk nilai phasa +90o dan -90o didapatkan suatu lebar
interval frekuensi tertentu. Faktor kualitas eksternal bisa dihitung dengan persamaan yang
diberikan di gambar 4.5.
12
Gambar 4.6 (kiri) rangkaian resonator open loop,
(kanan) resonator open loop dengan central loading
Gambar 4.6 menunjukkan dua buah resonator tipe segiempat simpul terbuka (kiri). Resonator
ini diimplementasikan dengan SIR, sehingg posisi resonansi kedua bisa di-tune sesuai dengan
yang diinginkan. Gambar 4.6 sebelah kanan resonator ini memiliki tambahan pembebanan di
tengahnya. Tambahan pembebanan ini menggeser frekuensi harmonis kedua secara signifikan
ke arah lebih rendah. Gambar 4.7 menunjukkan efek penambahan pembebanan ini.
Gambar 4.7 Faktor transmisi (resonansi) kedua struktur di gambar 4.6
13
Gambar 4.8 Dualbandpass Filter N=3
Gambar 4.8 menujukkan contoh perancangan dualbandpass filter ini dengan spesifikasi
perancangan tipe Chebychev N=3 dengan ripple 0.04321 dB, yang mana didapatkan g0=g4=1,
g1=g3=0.8516 dan g2=1.1032. Frekuensi kerja filter pada f1=0.9 GHz (FBW=4%) dan f2=1,4
GHz (FBW=3%)
14
5 Hasil yang dicapai
Di bagian ini ditunjukkan hasil perancangan dengan menggunakan pendekatan resonator yang
tegandeng secara silang.
5.1 Filter tergandeng type 1
Filter tipe 1 dikembangkan dengan merujuk pada data yang diberikan di (Hong, 2011), dengan
topologi yang diberikan di gambar 5.1, beserta faktor penggandeng m dan faktor penggandeng
yang telah dikaitkan dengan FBW, yaitu M.
Gambar 5.1 Topologi dual passband filter
Data-data penggandengan:
ms1 = mL4 = 0.70,
m12 = m34 = 0.86,
m13 = m24 = 0.272,
dan dengan FBW = 0.105
Qs1 = QL4 = 6.667,
M12 = M34 = 0.0903,
M13 = M24 = 0.0286,
Faktor penggandeng ter-norm dengan FBW ini digunakan sebagai basis untuk merancang
filter. Dua resonator jika didekatkan satu dengan lainnya akan terjadi interaksi yang
menyebabkan terbentuknya dua frekuensi resonansi.
Gambar 5.2 kiri: struktur perhitungan faktor penggandeng (coupling factor) k, kanan: struktur
perhitungan external quality factor Q
Gambar 5.2 sebelah kiri menunjukkan dua buah resonator yang digandengkan satu sama lain
dengan suatu jarak s dan offset pergeseran sejauh L. Secara teoretis kita ketahui, semakin besar
jarak s semakin kecil pulang penggandengan keduanya, yang artinya faktor penggandengan
mengecil. Sedangkan gambar 5.2 kanan menunjukkan model untuk perhitungan faktor kualitas
eksternal, di mana posisi ‘tapping’ sehingga terbentuk suatu sambungan yang ‘match’.
15
Perancangan filter ini dilakukan dengan menggunakan substrate er=3 dengan ketebalan
h=1.016 mm.
Gambar 5.3 Faktor kopling dua resonator dengan parameter jarak offset
Gambar 5.4 Eksternal faktor kualiti sebagai fungsi dari jarak tapping
Untuk faktor kualitas eksternal yang diinginkan sebesar 6,67 diset jarak tapping sejauh 16,7
mm.
Gambar 5.5 menunjukkan model yang digunakan dalam analisa, sedangkan hasil perhitungan
ditunjukkan di gambar 5.6.
16
Gambar 5.5 Model perancangan dan model dalam software Sonnet
Gambar 5.6 Faktor refleksi dan transmisi struktur di gambar 5.5
17
5.2 Filter tergandeng type 2
Filter type kedua yang merupakan basis untuk perancangan prototype adalah filter yang
tergandeng, yang dirancang memiliki dua jalur utama. Kedua jalur utama ini digunakan untuk
mendapatkan pass band, dan bukan untuk mendapatkan TZ. Tapi tetap ada jalur (sebuah) yang
menghasilkan TZ, yaitu dari resonator tempat tap input ke resonator tempat tap output.
Gambar 5.7 menunjukkan model ini dan hasil perhitungannya.
Gambar 5.7 (kiri) model filter, (kanan) hasil simulasi
5.3 Filter tergandeng type 3
Di bagian ini akan dirancang dualbandpass filter dengan frekuensi resonansi pertama di 1.90
GHz dengan FBW sebesar 5% dan frekuensi resonansi kedua di 2.15 GHz dengan FBW sebesar
5%.
Dalam perancangan dualbandpass filter digunakan sebagai basis filter bandpass yang dipakai
di (Astuti,2013) seperti yang ditampilkan di gambar 5.8. Ini adalah Single bandpass filter yang
memiliki 6 buah resonator. Filter ini memiliki dua buah jalur penggandengan silang, yang
menghasilkan empat TZ secara simetris.
Untuk merancang dualbandpass filter topologi filter dimodifikasi menjadi yang tampak di
gambar 5.8, sehingga mirip dengan tipe 2.
18
Gambar 5.8 Model awal perancangan dan usulan
Perancangan dualband pass filter ini dilakukan perancangan dua buah single bandpass flter
secara terpisah pada frekuensi 2.0 GHz dengan TZ pada 1.85 GHz dan 2.44 GHz dan
frekuensi 2.1 GHz dan TZ pada 1.65 GHz dan 2.32 GHz.
Gambar 5.9 Model awal perancangan dan usulan
Langkah selanjutnya adalah menggabungkan kedua filter ini menjadi satu, seperti yang
ditampilkan di gambar 5.10. Hasil perhitungan menunjukkan adanya dua wilayah lolos yang
terpisah. Hasil simulasi dan pengukuran prototype ditunjukkan di gambar 5.10 dengan hasil
terdapat frekuensi resonansi pada 1.91 GHz dengan lebar pita sebesar 90 MHz, atau FBW 4.7%
dan pada 2.16 GHz dengan lebar pita sebesar 110 MHz atau FBW 5.1%.
Gambar 5.10 menunjukkan perbandingan broadband, terlihat adanya kesesuaian antara
perhitungan dan pengukuran secara signifikan.
19
Gambar 5.10 Model perhitungan dan prototype, hasil perhitungan dan pengukuran.
Gambar 5.11Perbandingan hasil perhitungan dan pengukuran secara broadband.
20
5.4 Filter Lolos bawah dengan Metoda Moment
Dalam perancangan filter dualband di bagian 5.3, terlihat adanya passband pada frekuensi-
frekuensi tinggi yang merupakan harmonis dari frekuensi kerja 2 GHz, yaitu pada frekuensi-
frekuensi 4 GHz, 6 GHz dan 8 GHz.
Untuk meredam lolos pada frekuensi tinggi digunakan lowpass filter, yang meloloskan
frekuensi rendah, yaitu 2 GHz dan menolak yang besar, yaitu 4 GHz dan lainnya. Di sini
dirancang low pass filter Chebychev dengan ordo N. Gambar 5.12 menunjukkan model generic
dari low pass filter dengan pendekatan rangkaian LC.
Gambar 5.12. Bentuk generik rangkaian low pass fitler dengan fungsi polynomial.
Nilai-nilai komponen yang digunakan untuk perancangan bisa didapatkan dari banyak buku
teks tentang filter, di penelitian ini dengan oro N = 3, ripple 0.3 dB, fc=1 GHz dan impedansi
referensi Zo=50 ohm, didapatkan
g1 = 1.3713 L1 = 10.91 nH
g2 = 1.1378 C2 = 3.6219 pC
g3 = 1.3713 L3 = 10.91 nH
Implementasi data di atas dilakukan dalam teknologi mikrostrip dengan a TMM10 substrate
dengan permitivitas relatif 9.2, tangent loss 0.0022 dan ketebalan 2.54 mm. Pendekatan yang
diambil adalah memodelkan induktansi dengan mikrostrip yang memiliki impedansi tinggi (Z0L
> Z0) dan kapasitansi dengan impedansi rendah (Z0C < Z0).
Tabel 5.1 memberikan data-data yang dipakai yang didapatkan dari perhitungan dari
(Alaydrus, 2009)
Tabel 5.1 data perancangan untuk Low pass filter
Z0 [Ω] W [mm] εr,eff λg at fo= 1 GHz
[mm] 50 2.62 6.31 119.345
86.7 0.64 5.79 124.627
30 6.34 6.94 113.806
21
Sedangkan panjang dari masing-masing model ini adalah lL = 15.00 mm dan lC = 10.63 mm.
Gambar 5.13 menunjukkan rangkaian mikrostrip ini.
Gambar 5.13. Low pass filter ordo N = 3 dengan dua 50 Ω feeding lines.
Hasil perhitungan dan pengukuran rangkaian low pass filter ini ditampilkan di gambar 5.14.
Diamati adanya deviasi sekitar 0.7 dB pada faktor transmisi (S21) dan osilasi pada faktor
refleksi (S11) yang diukur, yang diduga berasal dari kabel yang digunakan. Diamati juga
adanya minimum pada faktor refleksi pada sekitar 725 MHz.
Gambar 5.14. Parameter S dari low pass filter N = 3, garis: perhitungan, putus-putus:
pengukuran dengan network analyzer ZVL13.
Untuk memperbaiki selektivitas filter dirancang filter ordo N=5, dengan data-data
g1 = 1.4817 L1 = 11.79 nH
g2 = 1.2992 C2 = 4.136 pC
g3 = 2.3095 L3 = 18.38 nH
g4 = 1.2992 C4 = 4.136 pC
22
g5 = 1.4817 L5 = 11.79 nH
Gambar 5.15 menunjukkan rangkaian mikrostrip yang dirancang.
Gambar 5.15. Low pass filter ordo N = 5 dengan dua 50 Ω feeding lines.
Gambar 5.16. Parameter S dari low pass filter N = 5, garis: perhitungan, putus-putus:
pengukuran dengan network analyzer ZVL13.
Karakteristik penyaringan ditampilkan di gambar 5.16. Filter ini memiliki karakteris yang
lebih baik dibandingkan hasil di gambar 5.14. Didapatkan dua minimum pada faktor
refleksi, yang menghasilkan karakter filter di sekitar 1 GHz yang lebih terjal.
23
6 Kesimpulan dan Saran
Perancangan filter yang bekerja pada dua frekuensi yang terpisah (dualbandpass filter) dengan
menggunakan resonator yang tergandeng didapatkn hasil dengan frekuensi resonansi pertama
pada 1.91 GHz dengan lebar pita sebesar 90 MHz, atau FBW 4.7% dan pada 2.16 GHz dengan
lebar pita sebesar 110 MHz atau FBW 5.1%.
Hasil ini cukup akurat, tetapi perancangan dualbandpass filter melalui perancangan dua single
bandpass filter tidak selalu memberikan hasil yang tepat. Harus dilakukan ‘tuning’ pada proses
penggabungan keduannya.
Hasil pengukuran menunjukkan kemiripan hasil dengan yang didapat dari simulasi, baik untuk
wilayah di sekitar pita lolos dan stop, ataupun jika diamati secara broadband.
Filter lowpass filter bisa digunakan untuk menekan fenomena pass pada frekuensi harmonis
dari wilayah kerja band pass filter.
24
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28
LAMPIRAN
Artikel ilmiah (draft, bukti status submission atau reprint), jika ada.
Publikasi di International Conference Quality in Research QiR 2015 Lombok
Said Attamimi, Subiyanto, and Mudrik Alaydrus, Designing Dual-Band Pass Filter By Coupled Resonators.
Publikasi di Jurnal International : Journal of ICT Research and Applications
Said Attamimi, Mudrik Alaydrus, Design of Chebychev’s Low Pass Filters Using Nonhomogeneous Transmission Lines
Produk penelitian
Prototype
Designing Dual-band Pass Filterby Coupled Resonators
Said Attamimi, Subiyanto, and Mudrik AlaydrusDepartment of Electrical Engineering
Universitas Mercu Buana, Jakarta, IndonesiaEmail: mudrikalaydrus@yahoo.com
Abstract—Dual-band pass filters pass signals at two distinctfrequency regions with a specified bandwidth. In this work adual-band pass filter based on coupled open-loop rectangularresonators is designed. The microstrip filter is designed to passsignals at uplink and downlink frequencies of the 3G cellularnetworks, 1920 MHz and 2110 MHz respectively. The originaldesign of the dual-band filter is the sixth order filter by shiftingthe feed positions so that the signals get two different paths dueto the parallel circuit of the filter. The designed filter works atthe first frequency at 1.91 GHz with a bandwidth of 90 MHz andreflection and insertion loss of 23.9 dB and 1.09 dB, respectively.The second frequency of the filter is located at 2.16 GHz witha bandwidth of 110 MHz and reflection and insertion loss of 17dB and 1.24 dB, respectively. The measurement results verify thedesigned filter very well.
Keywords—coupled resonator, dual-band filter, microstrip, openloop rectangular resonator
I. INTRODUCTION
The demand on higher data rate in wireless communicationsystems leads to utilization of several frequency regions. Theuse of a microwave component in two different frequency in-tervals becomes an efficient choice in modern communicationsystems [1]. Dual-band filters are passive components whichare able to separate wanted from unwanted signals in such twodifferent frequency ranges. The accurate design of dual-bandfilters is essential to guarantee the overall performance of thesystems.
In [2] a structure to design a dual-band bandpass filter(BPF) by using a dual feeding structure and embedded uniformimpedance resonator (UIR) is presented. Here, two passbandscan be designed distinctly and transmission zeros can beintroduced to enhance filter performance. The first passbandis determined by the dual feeding structure with the suitablephysical length and the second passband is determined bythe UIR of half-wavelength embedded in the dual feedingstructure. Using the inter coupling in the UIR, the performanceof the second passband can be tuned without degrading thefirst passband. Two dual-band filters 0.9/1.575 GHz for multi-services communication and 2.4/5.7 GHz for wireless localarea network (WLAN), are designed.
Open-loop resonators loaded by shunt open stubs areproposed in [3], which are used to design compact dual-band bandpass filters with improved out-of-band rejectioncharacteristics. The second passband of the dual-band filter canbe tuned by the stub length and position. Required externalcoupling is obtained by adjusting the tapping position and
dimension of the stub-loaded resonator. A lossless transmissionline model is used to determine the resonance properties ofthe resonator and the external quality factor. Three dua-bandfilters 2.78/4.38 GHz, 2.44/5.3 GHz and 2.44/5.75 GHz aredesigned and fabricated for experimental verification. A similarapproach is introduced in [4], here, through modification ininput and output tapping, additional transmission zones can beobtained.
Chang et al [5] presented a step-impedance bandpass filterfor multimode wireless LANs. The filter has a new dual-bandfeature of two tunable passbands at desired frequencies andhigh out-of-band suppression, generated by incorporating step-impedance resonators in a comb-filter topology. It saves morethan half the circuit size compared with the switch-type dual-band topology. The simulation and measurement results showthe dual-band feature of two passbands at 2.45 and 5.75 GHzwith 85 dB suppression at 3.5 GHz.
In this work a dual-band pass filter based on coupled open-loop rectangular resonators is designed. The microstrip filteris designed to pass signals at uplink and downlink frequenciesof the 3G cellular networks, 1920 MHz and 2110 MHzrespectively. The challenge of the design is the close frequencyseparation of both pass bands. The original design of the dual-band filter is the sixth order filter by shifting the feed positionsso that the signals get two different paths due to the parallelcircuit of the filter.
II. BANDPASS FILTERS BASED ON OPEN LOOP COUPLEDRECTANGULAR RESONATORS
Open loop rectangular resonators are used extensively asbasic structure to construct bandpass filters [6], [7]. The theoryof open loop rectangular resonators is described exhaustedly in[8]. A single open loop resonator shows a resonant characteris-tic, i.e. the transmission factor has a maximal value at a certainresonant frequency fres, if it is fed weakly with an input andoutput port. In the case of microstrip implementation, the exactposition of this resonant frequency depends on the length ofthe loop, the thickness and the permittivity of the substrate.The thickness and the permittivity of the substrate reduce thefree space wavelength λo to a shorter guided wavelength λg , sothat for a given resonant frequency fres, the total geometricallength of the loop becomes L = c/(2fres
√εr,eff ), whereas
c and εr,eff are the light velocity in free space and effectiverelative permittivity of the substrate, respectively.
By arranging two open loop resonators close to each other,the resonators could interact either in electric, magnetic or
schematic of the coupling structure, and (c) is the prototypeof the filter.
Fig. 8. Comparison simulated (solid lines) and measured (dashed lines)reflection and transmission factors.
Fig.8 shows the calculated (solid lines) and measured(dashed lines) reflection and transmission factors in the fre-quency interval of 1 GHz and 3 GHz. The fabricated filterworks at the first frequency at 1.91 GHz with a bandwidth of90 MHz and reflection and insertion loss of 23.9 dB and 1.09dB, respectively. The second frequency of the filter is locatedat 2.16 GHz with a bandwidth of 110 MHz and reflectionand insertion loss of 17 dB and 1.24 dB, respectively. Somedeviations between simulation and measurement results are ob-served; position of rejection in the middle is shifted, probablydue to inaccuracies of the striplines during the fabrications.And we see that the transmission zeros about 1.6 GHz andabout 2.5 GHz in measurement are not so convincing, probablydue to more losses inside the filter should be considered.Overall, the measurement results verify the designed filter verywell.
Fig. 9. Wideband comparison simulated (solid lines) and measured (dashedlines) reflection and transmission factors.
The wideband characteristics of the filter are depicted inFig.9, we see the similarities between the calculated resultsand measurements in the interval 1 GHz to 8 GHz. We canalso recognize the spurious harmonics at about 4 GHz, 6 GHz
and 8 GHz, which could be rejected by circuiting a low passfilter.
IV. CONCLUSION
The designed filter works at the first frequency at 1.91 GHzwith a bandwidth of 90 MHz and reflection and insertion lossof 23.9 dB and 1.09 dB, respectively. The second frequencyof the filter is located at 2.16 GHz with a bandwidth of 110MHz and reflection and insertion loss of 17 dB and 1.24 dB,respectively. The measurement results verify the designed filtervery well. The wideband observation gives also very goodsimilarities between calculations and measurements.
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[2] R.-Y. Yang, K. Hon, C.-Y. Hung, C.-S. Ye, Design of Dual-bandbandpass Filters using a Dual Feeding Structure and Embedded UniformImpedance resonators, Progress In Electromagnetics Research, Vol. 105,pp. 93-102, 2010
[3] Priyanka Mondal, Mrinal K. Mandal, ”Design of Dual-Band BandpassFilters Using Stub-Loaded Open-Loop Resonators”, IEEE Trans. Mic.Theory and Tech., Vol.56, No. 1, Jan. 2008, pp.150 - 155.
[4] Xiu Yin Zhang, Jian-Xin Chen, Quan Xue, and Si-Min Li,”Dual-BandBandpass Filters Using Stub-Loaded Resonators”,IEEE Microwave andWireless Components Letters, Vol. 17, No.8, August 2007, pp. 583-585.
[5] Sheng-Fuh Chang, Yng-Huey Jeng and Jia-Liang Chen, ”Dual-band step-impedance bandpass filter for multimode wireless LANs”, ElectronicsLetters, Vol. 40, No. 1, January 2004, pp.38-39.
[6] M. Alaydrus, D. Widiastuti and T. Yulianto, Designing Cross-CoupledBandpass Filters with Transmission Zeros in Lossy Microstrip, Infor-mation Technology and Electrical Engineering (ICITEE), InternationalConference on, pp. 277 - 280, 2013.
[7] D.W. Astuti, Juwanto, M. Alaydrus, A bandpass filter based on squareopen loop resonators at 2.45 GHz, Instrumentation, Communications,Information Technology, and Biomedical Engineering (ICICI-BME), 3rdInternational Conference on, pp. 147-151 , 2013.
[8] Hong, Jia-Seng, Microstrip Filters for RF/Microwave Applications, Sec-ond Edition. New Jersey, John Wiley & Sons, 2011.
[9] Richard J. Cameron, ”Advanced Coupling Matrix Synthesis Techniquesfor Microwave Filters”, IEEE Trans. Mic. Theory and Tech., Vol.51, No.1, Jan. 2003, pp.1 - 10.
[10] www.sonnetsoftware.com (verified on January 15, 2014)
J. ICT Res. Appl. Vol. XX, No. X, 20XX, XX-XX
1
Received ________, Revised _________, Accepted for publication __________
Design of Chebychev’s Low Pass Filters Using
Nonuniform Transmission Lines
Said Attamimi & Mudrik Alaydrus
Department of Electrical Engineering, Universitas Mercu Buana,
Jalan Meruya Selatan, Jakarta 11650
Email: mudrikalaydrus@mercubuana.ac.id
Abstract. Transmission lines are utilized in many applications to convey energy
as well as information. Nonuniform transmission lines (NTL) are obtained through
variation of the characteristic quantities along the axial direction. Such NTL can
be used to design network element, like matching circuits, delay equalizers, filters,
VLSI interconnections, etc. In this work, NTL is analyzed with a numerical
method based on the implementation of method of moment. In order to
approximate the voltage and current distribution along the transmission line, a sum
of basis functions with unknown amplitudes is introduced. As basis function, a
constant function is used. In this work, we observed several cases such as lossless
and lossy uniform transmission lines with matching and arbitrary load. These cases
verified the algorithm developed in this work. The second example consists of
nonuniform transmission lines in form of abruptly changing transmission lines.
This structure is used to design a Chebychev’s low pass filter. The calculated
reflection and transmission factor of the filters show some coincidences with
measurements.
Keywords: filter; lossy line; method of moment; nonuniform transmission line; standing
wave; VSWR.
1 Introduction
Transmission lines are the most used components in electrical engineering. In
many applications, transmission lines are utilized to convey energy and
information from one position to others. At higher frequencies, radio frequency
or microwave applications, they are exploited as network elements; they are
designed as signal processing components, for examples as filters. In the theory
of transmission lines several characteristic quantities (R', L', C’ and G’) are
defined. Nonuniform transmission lines, which have position varying quantities,
can be used to design matching circuit [1], delay equalizer [2], filters [3], wave
shaping [4], processing of analog signals [5] and VLSI interconnections [6].
From the theory of transmission lines, two coupled differential equations of first
order can be derived from the circuit model. These equations give the relationship
2 Said Attamimi & Mudrik Alaydrus
between voltage and current. Eliminating one of these electric quantities leads to
a differential equation of second order. This differential equation for nonuniform
case can be solved analytically without approximation just for a few special types
of NTLs, linear [7], exponential [8], power-law [9, 10], binomial [11],
exponential power law [12] and hermite [13] types. For other general varying
characteristic quantities approximation methods are introduced. These methods
are based on expansion of some describing functions with unknown amplitudes,
i.e. expansion of Taylor’s series [14], expansion of Fourier’s series [15], and
application of numerical computation such as method of moment [16] and finite
element method [17].
In this work we implement the same approach as described in [16]. The method
of moment is a numerical approach for solving an integral equation, in which we
have an unknown integrand. In the method of moment, the unknown function is
approximated by a series of simple basis functions with unknown amplitudes. By
sampling the equation in several positions, the integral equation can be converted
into a system of linear equations. By inverting this matrix we can get the
distribution of voltage and current along the transmission line. In this work, we
observe several simple cases such as lossless and lossy uniform transmission lines
with matching and arbitrary load. These canonical cases should verify the
algorithm developed in this work. The second example concerns with nonuniform
transmission lines in form of abruptly changing transmission lines. This structure
is used to design a low pass filter. A computer code based on MATLAB is
developed to calculate the reflection and transmission factor of such nonuniform
transmission lines.
2 Related Works on Nonuniform Transmission Lines
Khalaj-Amirhosseini in [1] introduced a method to synthesize microstrip
Nonuniform Transmission Lines (NTLs) for matching between two arbitrary
complex frequency dependent impedances in a wideband or multi-band
frequency range. In synthesizing the structure, he used Fourier series [8]. The
characteristic impedance function of the microstrip NTL is expanded by means
of a truncated Fourier series. The unknown coefficients are obtained by
optimizing certain frequency responses for the matching circuit. The usefulness
of the proposed method is verified using wideband and dual-band matching
between resistors and capacitors. It is observed that the solutions yield a good
impedance matching and as the length of matcher is chosen larger its efficiency
is increased.
In [18], the contribution develops formulations for reflection and transmission
factors of nonuniform transmission lines having unequal reference impedances.
By using the ABCD matrix, the reflection and transmission factors are expressed
Design of Chebychev’s Low Pass Filters Using NTLs 3
as polynomial ratios in Z transforms. These formulations, in conjunction with
techniques in digital signal processing (autoregressive moving average process)
and a reconstruction method, lead to the realization of nonuniform lines which
satisfy prescribed scattering characteristics in frequency domain. A predefined
low pass filter with a step-wise transmission factor is designed and verified by
measurements.
Juric-Grgic et al in [19] presents a finite-element frequency domain model for
numerical solution of the coupled nonuniform transmission line problem. Based
on the finite-element method, a novel numerical procedure for solution of a
system of the nonuniform multi-conductor transmission line equations in the
frequency domain is presented. The results obtained by the proposed method have
been compared to the analytical solution.
Khalaj-Amirhosseini [16] proposed a method for the analysis of arbitrarily loaded
lossy and dispersive single or coupled nonuniform transmission lines (NTLs). In
this work, Based on the available differential equations for the voltage and
current, the governing integral equations for the NTLs are derived and solved
using the method of moments. It is assumed that per-unit-length matrices are
known at all or even some points along the length of the coupled NTLs. The
method of moment in this work used rectangular pulse function expansion with
point matching. A coupled transmission line pair is observed and the calculated
voltage distribution along the excited line and along the unexcited line is
compared with the exact results.
Lu [7] gives the analytical solution of an ideal linear varied non-uniform
transmission line (LNTL) including the exact linear two-port ABCD matrix of
LNTL. Based on this result, he cascaded the LNTL sections to approximate an
arbitrary characteristic impedance profile and presented a technique for analyzing
an arbitrary non-uniform transmission line (NTL). The technique is better than
the conventional technique in terms of the computational accuracy and intensity
since it uses a piecewise linear characteristic impedance profile in place of the
stepped profile used by the conventional technique.
In other publications [20, 21] non-uniform transmission lines (NTL) are used for
designing dual band and multiband power divider. In this paper, the NTLs are
subdivided into small sections, which are described by ABCD matrices, and the
ABCD matrix of the whole structure can be obtained just by multiplication of all
sub ABCD matrices. For modeling the normalized characteristic impedance, the
truncated Fourier series expansion is used. The optimum values of the Fourier
coefficients can be obtained through minimization a certain error function, which
quantifies the difference between the obtained ABCD matrix and desired ones.
4 Said Attamimi & Mudrik Alaydrus
The Wilkinson power dividers designed here can be set to work in dual band or
multiband depending on the setting ABCD matrix target.
3 Wave Equation of Nonuniform Transmission Lines and Its
Solution
In two-conductor transmission lines the voltage and current can be defined
uniquely. If the cross-section of the transmission lines is small enough compared
to the wavelength, the voltage and the current are related to each other by the
following equations
𝑑𝑉(𝑧)
𝑑𝑧= −𝑍′(𝑧)𝐼(𝑧) (1)
𝑑𝐼(𝑧)
𝑑𝑧= −𝑌′(𝑧)𝑉(𝑧) (2)
with arbitrarily position-varying parameters Z’(z) = R’(z) + jωL’(z) and Y’(z) =
G’(z) + jωC’(z), and z is the direction of the propagation. R’, L’, G’ and C’ are
the primary parameters of the transmission lines.
By eliminating the current, eq. (1) and (2) lead to a nonhomogeneous differential
equation of second order
𝑑2𝑉(𝑧)
𝑑𝑧2 − 𝑓(𝑧)𝑑𝑉(𝑧)
𝑑𝑧− 𝛾2(𝑧)𝑉(𝑧) = 0 (3)
with f (z) = (dZ’/dz) Z’ and γ2 = Z’Y’.
The solution of Eq. (3) is not simple, and is available analytically only for some
special functions of Z’ and Y’. To give a solution for the problem, this paper takes
the integrals of eq. (1) and (2) in respect to z leading to
𝑉(𝑧) = − ∫ 𝑍′(𝑧′)𝐼(𝑧′)𝑑𝑧′ + 𝐶1𝑧
0 (4)
𝐼(𝑧) = − ∫ 𝑌′(𝑧′)𝑉(𝑧′)𝑑𝑧′ + 𝐶2𝑧
0 (5)
The integration constants, C1 and C2, can be derived from the boundary conditions
given in Figure 1, which are
𝐶1 =𝑍𝐿
𝑍𝑖+𝑍𝐿𝑉𝑆 +
𝑍𝑖
𝑍𝑖+𝑍𝐿∫ [𝑍′(𝑧′)𝐼(𝑧′) − 𝑍𝐿𝑌′(𝑧′)𝑉(𝑧′)]𝑑𝑧′𝑑
0
𝐶2 =1
𝑍𝑖+𝑍𝐿𝑉𝑆 −
1
𝑍𝑖+𝑍𝐿∫ [𝑍′(𝑧′)𝐼(𝑧′) − 𝑍𝐿𝑌′(𝑧′)𝑉(𝑧′)]𝑑𝑧′𝑑
0
If we set as value for z = 0 in eq. (4), the integral vanishes and the equation
becomes V (z = 0) = C1. Physically it means, C1 is the voltage measured at the
position z = 0 and analogous, if eq. (5) is used for z = 0, it leads to I (z = 0) = C2,
so C2 is the current at the position z = 0.
Design of Chebychev’s Low Pass Filters Using NTLs 5
Figure 1 Nonuniform transmission line of length d with source VS and internal
impedance Zi and load ZL.
Eqs. (4) and (5) are integrals of unknown current and voltage along the
transmission line. These equations are coupled to each other, so that solving the
problem must include the equations simultaneously. In order to approximate the
unknown voltage and current distribution, it is worthy to express the voltage and
current as a linear combination of the so-called basis functions with unknown
amplitudes,
𝑉(𝑧) = ∑ 𝑉𝑛 𝑓𝑛(𝑧)𝑁𝑛=1 (6)
𝐼(𝑧) = ∑ 𝐼𝑛 𝑓𝑛(𝑧)𝑁𝑛=1 (7)
fn are simple known basis functions, Vn and In are the unknown constants, and N
is the number of approximating functions. In this case, we have 2 × N unknowns.
By inserting eq. (6) and (7) into eq. (4) and (5) we get
∑ 𝑉𝑛 [𝑓𝑛 +𝑍𝑖𝑍𝐿
𝑍𝑖 + 𝑍𝐿
∫ 𝑌′𝑓𝑛𝑑𝑧′
𝑑
0
]
𝑁
𝑛=1
+ ∑ 𝐼𝑛 [∫ 𝑍′𝑓𝑛𝑑𝑧′
𝑧
0
−𝑍𝑖
𝑍𝑖 + 𝑍𝐿
∫ 𝑍′𝑓𝑛𝑑𝑧′
𝑑
0
]
𝑁
𝑛=1
=𝑍𝐿
𝑍𝑖 + 𝑍𝐿
𝑉𝑆
and
∑ 𝑉𝑛 [∫ 𝑌′𝑓𝑛𝑑𝑧′
𝑧
0
−𝑍𝐿
𝑍𝑖 + 𝑍𝐿
∫ 𝑌′𝑓𝑛𝑑𝑧′
𝑑
0
]
𝑁
𝑛=1
+ ∑ 𝐼𝑛 [𝑓𝑛 +1
𝑍𝑖 + 𝑍𝐿
∫ 𝑍′𝑓𝑛𝑑𝑧′
𝑑
0
]
𝑁
𝑛=1
=𝑉𝑆
𝑍𝑖 + 𝑍𝐿
These equations can be given in a compact matrix form as follows
6 Said Attamimi & Mudrik Alaydrus
(𝑨 𝑩𝑪 𝑫
) (𝑽𝑰
) =𝑉𝑆
𝑍𝑖+𝑍𝐿(
𝑍𝐿
1) (8)
which V = [V1 V2 … VN]T and I = [I1 I2 … IN]T are the unknown vectors and
four known 1 × N matrices A, B, C and D, whose elements are
𝐴𝑛(𝑧) = 𝑓𝑛(𝑧) +𝑍𝑖𝑍𝐿
𝑍𝑖+𝑍𝐿∫ 𝑌′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′
𝑑
0 (9)
𝐵𝑛(𝑧) = ∫ 𝑍′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′𝑧
0−
𝑍𝑖
𝑍𝑖+𝑍𝐿∫ 𝑍′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′
𝑑
0 (10)
𝐶𝑛(𝑧) = ∫ 𝑌′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′𝑧
0−
𝑍𝐿
𝑍𝑖+𝑍𝐿∫ 𝑌′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′
𝑑
0 (11)
𝐷𝑛(𝑧) = 𝑓𝑛(𝑧) +1
𝑍𝑖+𝑍𝐿∫ 𝑍′(𝑧′)𝑓𝑛(𝑧′)𝑑𝑧′
𝑑
0 (12)
The evaluation of the integrals in eq. (9) to (12) in computer can be performed
effectively by dividing the transmission line into small segments, i.e. the
transmission line is discretized into N segments. In case of uniform discretization,
the segment length Δz is equal to d/N. The complexity of the integral calculation
can be substantially reduced, if we use as the basis function a constant function
in each segment, which is
𝑓𝑛 = {1 for (𝑛 − 1)∆𝑧 ≤ 𝑧 ≤ 𝑛∆𝑧
0 otherwise (13)
The choice of the basis function in eq. (13) leads to simplification of the integral
along 0 to d and along 0 to z. For example the integral in eq. (9) delivers results
different than zero just along the small segment (𝑛 − 1)∆𝑧 ≤ 𝑧 ≤ 𝑛∆𝑧.
Furthermore, if the transmission line is discretized fine enough, i.e. N is large, Δz
becomes small enough, we can approximate the integration just by a simple
multiplication between the segment length Δz and the mean value of Z’ or Y’, or
just taking the value of Z’ or Y’ at the midpoint of the segment zn. These results
are valid for all 0 ≤ z ≤ d. However, in order to solve the problem with 2 × N
unknowns uniquely, we must choose N special positions to get totally 2 × N
equations. These special positions are called testing points. A simple procedure
is to use the collocation method. In this method, we choose the middle point of
each segment zm, so that the approximate values of An, Bn, Cn and Dn at the
position zm become
𝐴𝑛(𝑧𝑚) = 𝛿𝑚𝑛 +𝑍𝑖𝑍𝐿
𝑍𝑖+𝑍𝐿𝑌′(𝑧𝑛)∆𝑧 (14)
𝐵𝑛(𝑧𝑚) = 𝑍′(𝑧𝑛)𝑈𝑚𝑛∆𝑧 −𝑍𝑖
𝑍𝑖+𝑍𝐿𝑍′(𝑧𝑛)∆𝑧 (15)
𝐶𝑛(𝑧𝑚) = 𝑌′(𝑧𝑛)𝑈𝑚𝑛∆𝑧 −𝑍𝐿
𝑍𝑖+𝑍𝐿𝑌′(𝑧𝑛)∆𝑧 (16)
Design of Chebychev’s Low Pass Filters Using NTLs 7
𝐷𝑛(𝑧𝑚) = 𝛿𝑚𝑛 +1
𝑍𝑖+𝑍𝐿𝑍′(𝑧𝑛)∆𝑧 (17)
δmn is the Kronecker function, and
𝑈𝑚𝑛 = {1 for 𝑚 > 𝑛
1/2 for 𝑚 = 𝑛0 for 𝑚 < 𝑛
Which means, if the observation position on the right side of the integration
boundary (m > n), we get the full integration. If the observation position is located
exactly at the middle of the integration range, it yields a half of the result. And if
the observation position on the left side of the integration range (m < n), the
integration gives the value zero.
The collocation method is a type of the method of moment, whose basis function
uses pulse function and as test function is delta function used.
From the information on the voltage distribution along the structure, we can
extract the standing wave along the connecting line on the source side, from
which we can calculate the voltage standing wave ratio by
𝑉𝑆𝑊𝑅 =𝑉𝑚𝑎𝑥
𝑉𝑚𝑖𝑛 (18)
Vmax and Vmin are the maximal and minimal value of the voltage along that
connecting line, respectively. From these values, we can calculate the reflection
factor S11 and incident as well as reflected voltages.
From the voltage distribution along the connecting line on the load side, we can
also obtain the transmitted voltage. By dividing this value by the incident voltage
we can calculate the transmission factor S21.
4 Simulation Results
In this work, firstly we observe uniform transmission lines with matching and
arbitrary loadings. As additional parameter we use lossless and lossy transmission
lines. We design and analyze two low pass filters as practical implementation of
nonuniform transmission lines.
4.1 Voltage Distribution along Uniform Transmission Lines
A 60 cm length uniform transmission line is observed. The transmission line has
a constant capacitance per unit length C’ = 66.7 pF/m and inductance per unit
length L’ = 0.167 μH/m. By ignoring any losses, a wave impedance of about Zo
8 Said Attamimi & Mudrik Alaydrus
= 50 Ω for lossless case is obtained independent on the frequency. The
transmission line is connected with a load of ZL = 50 Ω, and excited by a voltage
source VS = 1 V with a frequency of f = 1 GHz and an internal impedance of Zi =
50 Ω. In this example, we set G’ = 0 and make a variation on the resistance per
unit length R’.
Figure 2 shows voltage distribution along the transmission line for different
losses. For lossless case, we see a constant curve, which means there is no
standing wave. The value of voltage standing wave ratio (VSWR) is 1, which
means we do not have any reflected waves. For lossy cases (R’ > 0), the
propagating waves experience attenuation along the transmission line. Figure 2
illustrates, the larger the resistance per unit length, the smaller the amplitude of
the voltage apart from the source. Interestingly, in case of higher losses, for this
’matching’ condition, we observe standing wave in form of small ripples. Such
ripples must originate from superposition between incident and reflected waves.
In this case, reflections indeed happen. With losses (R’≠0), the value of the wave
impedance is no longer 50 Ω. For example, with R’ = 250 Ω/m, we get a complex
wave impedance of (50.3864 - j5.9196) Ω. This yields together with the load ZL
= 50 Ω a reflection factor of |r| = 0.059, or a VSWR of 1.1254.
Figure 2 Voltage distribution along transmission line with matching loading in
dependence on resistance per unit length R’.
In order to study the effect of losses to standing wave, we replace now the load
by an impedance ZL = 20 Ω. Connecting this load to a transmission line with a
wave impedance of 50 Ω, leads to a reflection factor r = −0.4286 or VSWR= 2.5.
For the lossless case in Figure 3, the standing wave pattern does not change along
the transmission line. The maximal voltage of Vmax = 0.7143 V and the minimal
Design of Chebychev’s Low Pass Filters Using NTLs 9
voltage of Vmin = 0.2857 V can be read, whose ratio is exactly equal to the value
of VSWR given above. At the load position, we have a minimum, because the
load impedance is smaller than the wave impedance of the transmission line, as
verified by the theory.
Figure 3 Voltage distribution along transmission line with ZL = 20 Ω in
dependence on resistance per unit length R’.
By including losses in calculation, the standing wave pattern changes along the
structure substantially. We observe smaller VSWR at positions near to the source
than near to the load. From the theory of lossy transmission lines, we learnt, the
incident wave from the source to the load is attenuated, and after reflected by the
load back to the source, the reflected wave is again attenuated. This makes the
contribution of the reflected wave to the standing wave pattern near the source
small as compared to the lossless case. So that for very lossy cases, near to the
source the value of Vmax is practically equal to the value of Vmin and any reflections
even total reflection can be neglected.
4.2 Chebychev’s Low Pass Filters based on Stepwise Nonuniform
Transmission Lines
Microwave filters play a significant role in modern communication systems [22].
Designing microwave filters becomes an interesting combination between
creativity and possibility. One of the most basic design procedures is
approximating the ideal characteristics of low pass filters with Chebychev’s
functions. Approximation of filtering characteristics with Chebychev’s functions
relies on the number of components used, i.e. the order of the filter N, and on the
10 Said Attamimi & Mudrik Alaydrus
ripple allowed in the pass band. Figure 4 shows the generic form of low pass
filter designed by this approximation.
Figure 4 Generic form of low pass filter approximated by polynomial functions.
The value of each component in Figure 4 can be obtained in many text books,
which is given in tabularized form as shown in Table 1. The data given in this
table is the standard information valid for any reference impedance and
frequency. For special load or internal impedance, for example 50 ohms, and
special reference frequency, for example ωc = 2π . 109 rad/s, we must multiply
the inductance value by the impedance and divide by the radial frequency ωc, and
divide the capacitance value by the impedance and again by the radial frequency.
Table 1 Element values for Chebychev low pass prototype for normed
impedance Zi = 1, frequency Ω = 1 and ripple 0.3 dB for filter order up to
5.
N g1 g2 g3 g4 g5 g6
1 0.5349 1.0000
2 1.1805 0.6957 1.6967
3 1.3713 1.1378 1.3713 1.0000
4 1.4457 1.2537 2.1272 0.8521 1.6967
5 1.4817 1.2992 2.3095 1.2992 1.4817 1.0000
In this paper, a low pass filter with order N = 3 and ripple 0.3 dB is designed with
fc = 1 GHz and reference impedance of Z0 = 50 ohms. The element values given
in table 1 become the following values of inductances and capacitances
g1 = 1.3713 L1 = 10.91 nH
g2 = 1.1378 C2 = 3.6219 pC
g3 = 1.3713 L3 = 10.91 nH
Design of Chebychev’s Low Pass Filters Using NTLs 11
In microstrip technology, inductances can be modeled by high impedance
microstrip line (Z0L > Z0) and capacitances by low impedance microstrip line (Z0C
< Z0). The choice of the impedance values Z0L and Z0C are rather arbitrary,
however it is restricted to, that the value of Z0L might be not to high, which could
yield a very thin strip line that is difficult to be fabricated, and that the value of
Z0C might be not to low, which leads to a wide strip line, that can cause a
transversal propagation of wave in this microstrip segment. Here, a TMM10
substrate [23] with a relative permittivity of 9.2, a tangent loss of 0.0022 and a
thickness of 2.54 mm is used. Table 2 gives an overview for the relationship
between the set impedance values and the widths of the microstrip lines, which
are designed by standard design equation given for example in [24].
Table 2 Design data for Low pass filter
Z0 [Ω] W [mm] εr,eff λg at fo= 1 GHz [mm]
50 2.62 6.31 119.345
86.7 0.64 5.79 124.627
30 6.34 6.94 113.806
In order to get each values of inductances and capacitances given above, the
microstrip lines must have certain lengths, which fulfill the following equations
[24]
2𝜋𝑓𝑜𝐿 = 𝑍0𝐿 sin (2𝜋𝑙𝐿
𝜆𝑔𝐿) + 𝑍0𝐶 tan (
𝜋𝑙𝐶
𝜆𝑔𝐶) (19)
2𝜋𝑓𝑜𝐶 =1
𝑍0𝐶sin (
2𝜋𝑙𝐶
𝜆𝑔𝐶) +
2
𝑍0𝐿tan (
𝜋𝑙𝐿
𝜆𝑔𝐿) (20)
Applying these coupled equations and data given in table 1, with an iterative
method we get the values lL = 15.00 mm and lC = 10.63 mm. Figure 5 shows a
microstrip circuit for this low pass filter.
Figure 5 Low pass filter of order N = 3 with two 50 Ω feeding lines.
12 Said Attamimi & Mudrik Alaydrus
On the source and load sides, two connecting transmission lines with a wave
impedance of 50 Ω are connected, and we connect an internal impedance ZS = 50
Ω and a load impedance ZL = 50 Ω, so that both sides are in matching condition.
In order to extract the voltage distribution along the connecting lines, we use 150
mm and 10 mm long lines on the source and load sides, respectively. Our target
here is to analyze the low pass filter structure in the frequency range between 100
MHz and 1.5 GHz. We calculate the reflection factor (S11) and transmission factor
(S21) of the filter. The reflection emerges not due to the load, but rather due to the
nonuniform structure of the transmission line used (here abrupt changes of the
impedances). This can be verified later, that we have a constant voltage
distribution along the connecting line on the load side. It is indeed, because we
have there just a wave propagating to the load. However along the connecting
line on the source side, we expected a standing wave pattern, having a maximum
and minimum voltage. From this pattern we can calculate the VSWR, and then
the reflection factor.
Figure 6 shows the voltage distribution along the transmission line at the
frequency 1.5 GHz. The curve on the load side (190 mm < z < 200 mm) is
constant, this is the voltage wave towards the load with the value Vt = 0.2822 V.
On the source side (z < 150 mm), we have a standing wave pattern with Vmax =
0.9096 V and Vmin = 0.0904 V, which yields a VSWR of 10.0619 or a reflection
factor of 0.8192 (= −1.732 dB). On the source side, we can calculate the voltage
wave propagating to the input side of the filter with Vinc = 0.5(Vmax+Vmin) = 0.5 V,
so that the transmission factor can be calculated to t = 0.2822/0.5 = 0.5643
(=−4.970 dB).
Figure 6 Voltage distribution inside the low pass filter and along its connecting
lines at the frequency 1.5 GHz.
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Vo
lta
ge
[V
]
inside the filter
Design of Chebychev’s Low Pass Filters Using NTLs 13
In Figure 6, the maximal voltages on the source side are about 40 mm apart from
each other’s, which is exactly the half of the guided wavelength from the theory
of transmission lines.
At lower frequencies, i.e. 600 MHz, we expected the maxima are separated by
longer distances. With the same procedure used for table 2, we get a guide
wavelength of about 199.66 mm for 600 MHz. The connecting line with the
length 150 mm is still enough for observing the standing wave as shown in Figure
7. The maximal voltage of 0.5460 V and the minimal voltage of 0.4540 V deliver
a VSWR of 1.2026 or a reflection factor of 0.092 (=−20.725 dB). With the voltage
at the load side Vt = 0.4951 V and the incident voltage on the source again Vinc =
0.5(Vmax + Vmin) = 0.5 V, so that the transmission factor can be calculated to t =
0.4951/0.5 = 0.9901 (=−0.0863 dB).
Figure 7 Voltage distribution inside the low pass filter and along its connecting
lines at the frequency 600 MHz.
By varying the frequency from 100 MHz to 1.5 GHz, we can calculate the
reflection and transmission factor over this frequency range. The result is
depicted in Figure 8. As comparison, a prototype of the low pass filter is
fabricated and measured. We observe a deviation of about 0.7 dB in transmission
factor and some oscillation in the measured reflection factor, which could come
from the cable of the network analyzer. We find the minimum of the reflection
factor at about 725 MHz. In general, we see a very good similarity between both
results.
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [mm]
Vo
lta
ge
[V
]
inside the filter
14 Said Attamimi & Mudrik Alaydrus
Figure 8 Scattering parameters of low pass filter N = 3, solid lines: this work,
dashed lines: measured by network analyzer ZVL13.
In order to enhance the selectivity of the filter, a second filter with N = 5 is
considered. With the same impedance and frequency reference as the previous
filter, and the element values given in table 1, we get the following values of
inductances and capacitances
g1 = 1.4817 L1 = 11.79 nH
g2 = 1.2992 C2 = 4.136 pC
g3 = 2.3095 L3 = 18.38 nH
g4 = 1.2992 C4 = 4.136 pC
g5 = 1.4817 L5 = 11.79 nH
Figure 9 shows a microstrip circuit for this low pass filter of order N = 5. The
substrate used is the same as before. The dimensions are calculated again by
solving the nonlinear equations (19) and (20) iteratively.
Figure 9 Low pass filter of order N = 5 with two 50 Ω feeding lines.
Design of Chebychev’s Low Pass Filters Using NTLs 15
Figure 10 Scattering parameters of low pass filter N = 5, solid lines: this work,
dashed lines: measured by network analyzer ZVL13.
The characteristic filtering is shown in Figure 10, this filter has sharper curve
between pass and stop regions compared to the low pass filter with N = 3. The
comparison of the result found here and measurements shows also
coincidences. We see the minimums of reflection factor at frequencies 510
MHz and 845 MHz.
5 Conclusion
The implementation of method of moment as solution for integral equation in
nonuniform transmission lines yields very good results. Some canonical
problems, such as matching and unmatched load connected to uniform
transmission lines with and without losses, verify this computer simulation. The
numerical results coincide with that yielded by theoretical approach. At last,
abruptly changing nonuniform transmission lines, which presents itself as
Chebychev’s low pass filters, was considered. These results were compared with
measurements. We observe at each filter design the minimum of reflection factor
and sharper transition between pass and stop regions. In general, the comparison
shows very good coincidences.
Acknowledgement
This research is supported by the Directorate of Higher Education, Republic of
Indonesia under the scheme of Decentralization Research Grant 2014. The
authors would like to express their thanks for the financial support.
16 Said Attamimi & Mudrik Alaydrus
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