Penyaring Lolos Tengah dengan Lebar Pita yang Besar...

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.

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


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


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: [email protected]

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.

REFERENCES

[1] Richard J. Cameron, Chandra M. Kudsia, Raafat R.Mansour, Microwavefilters for communication systems, Wiley, 2007.

[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.

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[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: [email protected]

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.


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.


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


(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)


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


= 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


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


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


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.


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


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


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.


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.


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