Design of RF and Microwave Filters 2009학년도 여파기 설계 Design of RF and Microwave Filters

Contents 1. Introduction ; types of Filters 2. Characterization of Filters 3. Approximate Design Methods 4. Lowpass Prototype Network 5. Impedance Scaling and Frequency Mapping 6. Immittance Inverters

1. Introduction 1.1 Types of Filters A. Lowpass Filters    B. Highpass Filters C. Bandpass Filters   D. Bandstop Filters

2. Filter Characterization(1) Two-port Network ; Fig. 1 Two-port Network

2. Filter Characterization(2) Characteristics of ideal bandpass filters ; Fig. 2 Characteristics of ideal bandpass filter → not realizable → approximation required

2. Filter Characterization(3) Practical specifications ; 1) Passband ; lower cutoff frequency - upper cutoff frequency 2) Insertion loss : ; must be as small as possible 3) Return Loss : ; degree of impedance matching 4) Ripple ; variation of insertion loss within the passband

2. Filter Characterization(4) 5) Group delay ; time to required to pass the filter 6) Skirt frequency characteristics ; depends on the system specifications 7) Power handling capability

3. Approximate Design Methods 1) based on Amplitude characteristics A. Image parameter method B. Insertion loss method                 a) J-K inverters                 b) Unit element - Kuroda identity  2) based on Linear Phase characteristics

3.1 Filter design(the insertion loss method) Definition of Power Loss Ratio (PLR) ; impedance matching as well as frequency selectivity Fig. 3  General filter network ← network synthesis procedures are required

3.1 Filter Design(2) Approximation methods : 1) Maximally Flat (Butterworth) response 2) Chebyshev response 3) Elliptic Function response

3.2 Approximation Methods A. Maximally flat response Where, ; passband tolerance ; order of filter Usually → degree of freedom=1 (order N) Fig. 4 Comparison Between Maximally Flat and Chebyshev response

3.2 Approximation Methods(2) B. Chebyshev response : equal ripple response in the passband : Chebyshev Polynomial of order

3.2 Approximation Methods(3) ; ripple (0.01 dB, 0.1 dB, etc.) ; order of filter → degree of freedom=2 (ripple and order) Fig. 5  Chebyshev and Elliptic Function response

3.2 Approximation Methods(4) C. Elliptic Function response  equal ripple passband in both passband and stopband : stopband minimum attenuation : transmission zero at stopband         degree of freedom=3 (order N, ripple, transmission zero at stopband )

4. Lowpass Prototype Filter ; normalized to 1 Fig. 5  Lowpass prototype

4. Lowpass Prototype Filter(2)    Maximally Flat response ;                                                                     Equal Ripple response ;

4. Lowpass Prototype Filter(3) Element No Butterworth 0.1 dB ripple Chebyshev 0.5 dB ripple 1 0.6180 1.1468 1.7058 2 1.6180 1.3712 1.2296 3 2.0000 1.9750 2.5408 4 5 Table1. Element values for Butterworth and chebyshev filters

5. Impedance and freq. mapping 5.1 Impedance Scaling Impedance level × 50  ; same reflection coefficient maintained        series branch(impedance) elements ;         shunt branch(admittance) elements ; 

5. Impedance and freq. mapping(2) 5.2 Frequency Expansion cutoff frequency 1 → lowpass cutoff frequency mapping function ; series and shunt branch elements ;

5. Impedance and freq. mapping(3) Fig. 6  Various mapping relations derived from lowpass prototype network

5.3 Lowpass to Highpass transformation 5. Impedance and freq. mapping(4) 5.3 Lowpass to Highpass transformation (lowpass cutoff freq. 1 → highpass cutoff freq. )         mapping function ;           series branch(impedance) elements ;        shunt branch(admittance) elements ;  Fig. 7  Highpass filter derived from lowpass prototype

5.4 Lowpass to bandpass transformation 5. Impedance and freq. mapping(5) 5.4 Lowpass to bandpass transformation (low cutoff freq. , high cutoff freq. ) mapping function ;

5. Impedance and freq. mapping(6) series branch element : impedance shunt branch element : admittance Fig. 8 Bandpass filter derived from the lowpass prototype

5. Impedance and freq. mapping(7) Example : Design a bandpass filter having a 0.5dB equal-ripple response, with N=3. The f0 is 1GHz, bandwidth is 10%, and the input and output impedance 50Ω. step 1 : from the element values of lowpass prtotype (0.5dB ripple Chebyshev) step 2 : apply impedance scaling                 

5. Impedance and freq. mapping(8) step 3 : apply bandpass transformation

5.5 Lowpass to bandstop transformation 5. Impedance and freq. mapping(9) 5.5 Lowpass to bandstop transformation (low cutoff freq. , high cutoff freq. ) mapping function ; inverse of bandpass mapping function

5. Impedance and freq. mapping(10) series branch element : admittance shunt branch element : impedance Fig. 9 Bandstop network derived from the lowpass prototype

5. Impedance and freq. mapping(11) 5.6 Immitance Inverters Fig. 10 Immitance inverter K ; impedance inverter  →    J ; admittance inverter  →  ex. simplest form of inverter : λ/4 transformer     series LC  →  J-inverter + shunt LC     shunt LC  →  K-inverter + series LC

5.7 Bandpass filters using J-, K-inverters 5. Impedance and freq. mapping(12) 5.7 Bandpass filters using J-, K-inverters Fig. 11 Equivalent Network for lowpass prototype and bandpass network Reflection coefficient ; lowpass : bandpass : If (mapping relation)

5. Impedance and freq. mapping(13) Fig. 12  Lowpass network and bandpass network

5. Impedance and freq. mapping(14) From the partial fraction expansion including bandpass mapping relation                           : fractional bandwidth,  : center freq. In the same manner, J-inverter values are derived as

5. Impedance and freq. mapping(15) Typical immittance inverters ; Fig. 13  Impedance(K-) inverters

5. Impedance and freq. mapping(16) Fig. 14  Admittance(J-) inverters

6. LC filters, Distributed filters A. C-coupled bandpass filters Fig. 14  Bandpass filter network using ideal J-inverters Fig. 15 Bandpass filter network containing practical inverters

6. LC filters, Distributed filters(2) Fig. 16  Inverter of first and last stages By equating the real and imaginary part of and

6. LC filters, Distributed filters(3) Fig.17 C-coupled   Bandpass filter B. L-coupled bandpass filter Fig.18 L-coupled   Bandpass filter

6. LC filters, Distributed filters(4) Design a LC bandpass filter. The f0 is 2.8 GHz, bandwidth is 500 MHz, and the input and output impedance 50Ω. step 1 : from the element values of lowpass prototype step 2 : apply impedance scaling      step 3 : apply bandpass transformation using J-inverters Step 4 : simulation

6. LC filters, Distributed filters(3) Simulated results:

Step 5 : Realization Insertion loss < 3.1 dB Retrun loss > 15.5 dB Attenuation @ 3.3 GHz : 15 dB

Step 6. improvement C-coupling LC filter + L-coupling LC filter =

Measured results 27 dB

6. LC filters, Distributed filters(3)   At microwave frequencies :      Resonators made of Lumped elements are lossy(low Q) or bulky → Distributed Resonators      Distributed resonators ; quarter-wavelength or half-wavelength transmission lines such as microstrip lines, coaxial lines and waveguides

6. LC filters, Distributed filters(3) A. Combline filters : cellular base stations as well as portable phone Fig. 19 (a) Top view of Combline Filter Fig. 19 (b) Side view of Combline Filter

Instead of lumped element inductors distributed inductors (L < λ/4) are used. Overall equivalent circuit : Fig. 20 Coupled line Fig. 21 Equivalent circuit of Fig.20 Fig. 22 Equivalent circuit of Fig. 19