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Microwave Filters Microwave Circuits Design Dr. Vahid Nayyeri Microwave Filters • Used to control the frequency response at a certain point in a microwave system by providing transmission at frequencies within the passband of the filter and attenuation in the stopband of the filter. • Can be found in any type of microwave communication, radar, or test and measurement system. bandpass bandstop low pass high pass • Periodic structures, which consists of a transmission line or waveguide periodically loaded with reactive elements, exhibit the fundamental passband and stopband behavior --- the analysis follows that of the wave propagation in crystalline lattice structures of semiconductor materials. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 1 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Periodic Structures 1. Assume a infinite periodic structure. Periodic stubs 2. Set a unit cell with impedance Z0, a length of d and a shunt susceptance b. Periodic diaphragms in a waveguide Equivalent circuit of a periodically loaded transmission lines: distributed parameters http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 2 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Analysis of periodic structures shows that waves can propagate within certain frequency bands (passbands), but will attenuate within other bands (stopbands). Filter Design By the Image Parameter Method For a reciprocal two- port network on the right, it can be specified by its ABCD parameters. The image Zi1 = input impedance at port 1 when port 2 impedances are Zi1 is terminated with Zi2. and Zi2. Zi2 = input impedance at port 2 when port 1 is terminated with Zi1. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 3 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Using ABCD parameters, we haveV1 AV 2 2 BI I1 CV 2 DI 2 V AV BIAZ B e can deriveW 1 2 2 i2 Zin1 I1 CV2 DI 2 CZi2 D , Similarly V2 DV1 1 BIDZi1 B Zin2 I2 CV1 1AICZi1 A e want to have WZin1 = Zi1 and Zin2 = Zi2 , which leads to equations for the image impedances: Z CZ1i() i 2 D i AZ 2 B Z D1i B Z i 2() A i 1 CZ Solving for Zi1 and Zi2 AB BD Z Z i1 CD i2 AC http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 4 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Then,Zi2 DZ i 1/ A then network is symmetric, If the A=D and Zi1 = Zi2 as expected. V D The voltage transfer ratio is given by 2()AD BC V1 A I A The current transfer ratio is given by 2()AD BC I1 D e can define a propagation factore W AD BC Where j e e e can also verify thatW cosh AD 2 wo important types of two-port networks: TT and circuits. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 5 of 75 Microwave Circuits Design Dr. Vahid Nayyeri ELEC4630, Shu Yang, HKUST 6 http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 6 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Constant-k Filter Sections (low-pass and high-pass filters) For the T network, we use the results from the image parameters table, Low-pass constant-k filter sections in T and form. and Z1 j LZ2 /1 j C L 2LC We can derive the image impedance as Z 1 iT C 4 2 The cutoff frequency, c, can be defined as c LC A nominal characteristic L R0 k constant impedance, R0, can be defined as C http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 7 of 75 Microwave Circuits Design Dr. Vahid Nayyeri 2 Then, ZRiT0 1 2 For = 0, ZiT = R0. c 2 2 The propagation factor is 2 2 e 1 2 2 1 c c c Frequency response of the low-pass constant-k sections. Slow attenuation rate http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 8 of 75 Microwave Circuits Design Dr. Vahid Nayyeri High-pass Constant-k Filter Sections http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 9 of 75 Microwave Circuits Design Dr. Vahid Nayyeri m-Derived Filter Sections A modification to overcome the disadvantages of slow attenuation after cutoff and frequency-dependent image impedances. Z1' mZ 1 In order to have the same ZiT, we have Z ( 1 m2 ) Z'2 Z 2 m 4m 1 Basic concept: create a resonator along the shunt path. The resonant frequency here should be slightly high than the cutoff frequency. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 10 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Cutoff frequency is still 2 Low-pass High-pass c LC The resonant frequency of the shunt path is 1 1 m2 mC L 4m c 1 m2 Steep decrease of after > is not m is restricted into desirable. This problem can be solved by to the range of cascading with another constant-k section to 0 < m < 1. give a composite response shown in the figure. ELEC4630, Shu Yang, HKUST http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 11 of 75 Microwave Circuits Design Dr. Vahid Nayyeri m -Derived Section Filter section -still have The T the problem of a nonconstant image impedances. the Now consider - equivalent as a piece of an cascade eInfinitof m-derived T-section. infinite cascade of m- sections. Then,-derived T ZZ''1 2 Zi ZiT Z Z( Z2 1 2 ) m / 4 1 2 1 de-embedded A-equivalent. 2 R0 1 / c http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 12 of 75 Microwave Circuits Design Dr. Vahid Nayyeri 2 2222 2 SinceZZLCR1 2/ 0 and ZLR10 4/c 1 ( 1m 2 ) /2 R We have Z c 0 i 2 1 / c * m provides another freedom to design Zi so that we can minimize the variation of Zi over the passband of the filter. Variation of Zi in the pass band of a low-pass m-derived section for various values of m. A value of m=0.6 generally gives the best results --- nearly constant impedance match to and from R0. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 13 of 75 Microwave Circuits Design Dr. Vahid Nayyeri How to match the constant-k and m-derived T-sections to - section? Using bisected -section. It can be shown that Z'2 ZZZ'' 1 Z i1 1 2 4 iT ZZ''1 2 ZZ''1 2 Zi2 Zi 1ZZ '1 / 4 2 ' ZiT http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 14 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Composite Filters • The sharp-cutoff section, with m< 0.6, places an attenuation pole near the cutoff frequency to provide a sharp attenuation response. • The constant-k section provides high attenuation further into the stopband. • The bisected- sections at the ends match the nominal source and load impedance, R0, to the internal image impedances. • The composite filter design is obtained from three parameters: cutoff frequency, impedance, and infinite attenuation frequency (or m). http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 15 of 75 Microwave Circuits Design Dr. Vahid Nayyeri http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 16 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Example 8.2 of Pozar: Low-Pass Composite Filter Design The series pairs of inductors Cutoff between the sections can be freq. combined. The self- resonance of the bisected p- section will provide additional attenuation. Frequency response ELEC4630, Shu Yang, HKUST http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 17 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Filter Design By the Insertion Loss Method What is a perfect filter? • Zero insertion loss in the passband, infinite attenuation in the stopband, and linear phase response (to avoid signal distortion) in the passband. No perfect filters exist, so compromises need to be made. The image parameter method have very limited freedom to nimble around. The insertion loss method allows a high degree of control over the passband and stopband amplitude and phase characteristics, with a systematic way to synthesize a desired response. ELEC4630, Shu Yang, HKUST http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 18 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Loss RatioCharacterization by Power Therati oand are as: Power available fromPinc 1source PLR 2 Power delivered toPload 1load ( ) IL 10log PLR matched,are source and load both When PLR |S 21|2 Since |()|2 is an even function of , it can be expressed as a polynomial in 2. 2 2 M() () MN()()2 2 Where M and N are real polynomials in 2power loss . So the ratio can be given as M() 2 P 1 LR N() 2 http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 19 of 75 Microwave Circuits Design Dr. Vahid Nayyeri Filter Design by the Insertion Loss Method Several Types of Filter Response: Maximally flat: binomial or Butterworth response Provide the flattest possible passband response. For a low-pass filter, it is specified by 2N 2 PLR1 k c Where N is the order of the filter, and c is the cutoff frequency. At the band edge the power loss ratio is 1+k2. Maximally flat means that the the first (2N-1) derivatives of the power loss ratio are zero at = 0. Equal ripple: A Chebyshev polynomial is used to represent the insertion loss of an N-order low-pass filter Provide the sharpest cutoff, with ripples in the passband. http://webpages.iust.ac.ir/nayyeri/courses/mcd/ Page 20 of 75 Microwave Circuits Design Dr. Vahid Nayyeri The insertion loss is: P1 k2 2 T LR N Ripple amplitude: 1+ k2 c For >> c, the insertion loss is, 2N k 2 2 PLR 4 c Insertion loss rises faster in the stopband compared to the binomial filters.
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