RF Filters: An Overview www.atlantarf.com
Presentation Content: RF Filters 1. Ideal Filters vs. Actual Filters. 2. Application of RF Filters: A. General Applications. B. Specific Applications. 3. Classifications of RF Filters. 4. Technology of RF Filters. 5. Transmission Media of Filters. 6. Illustration of RF Filters. 7. RF Filter Specifications. 8. The Prototype Filter. 9. Synthesis: Insertion Loss Method. 1.Polynomial Approximations. 11.Prototype Lowpass Circuits. 12.Prototype Element Values: A. Maximally-flat (Butterworth). B. Equi-Ripple (Chebyshev). C. Maximally-flat Time Delay. D. Linear Phase:.5 degree. 13. A Note about Passband Ripple. 14.Frequency Scaling the Prototype. 15.Impedance Scaling the Prototype. 16.Lowpass & Highpass Scaling. 17.Bandpass & Bandstop Scaling. 18.Electrical Circuit Configurations. 19.Summary of Steps: Filter Design. 2.Example: Butterworth Lowpass. V s N-section Filter Circuit 2 Z s P A P inc P L 1 Z L
Attenuation Attenuation Attenuation Attenuation Ideal Filters versus Actual Filters A. Ideal Filter: A linear 2-port networ that provides perfect transmission of signals for frequencies in a certain passband region, infinite attenuation for frequencies in the stopband region, and a linear phase response in the filter s passband region (to reduce signal distortion). B. Goal: The goal of an actual filter s design is to approximate an ideal filter s performance requirements, within acceptable tolerance, using real circuit components: Resistors: R, inductors: L, capacitors: C, transmission line sections, unit elements and resonating structures. Magnitude plot for four ideal filters, along with their circuit symbols: Stop Band Stop Band Low Stop Band Upper Stop Band Stop Band Pass band Pass band Pass band Low Pass band Upper Pass band LowPass Filter: LPF HighPass Filter: HPF BandPass Filter: BPF BandStop Filter: BSF Passes low freq. Passes high freq. Passes freq. band Rejects freq. band Rejects high freq. Rejects low freq. Rejects all other freq. Passes all other freq. 3
4 Importance of RF Filters 1. Frequency spectrum allocation and frequency spectrum preservation. RF signals consist of: A. Desired signals at desired frequencies. B. Undesired signals & undesired noise at unwanted frequencies. 2. Signal interference reduction or elimination: Receiver protection. 3. Elimination of unwanted harmonics & intermodulation products generated from nonlinear devices, lie: A. Frequency multipliers, B. Frequency mixers, C. Power amplifiers. 4. Signal processing & spectrum shaping. 5. Frequency multiplexing.
General Application of RF Filters 1. Receivers: Filters the incoming signal right after reception to remove external white noise and external undesirable & interfering signals, thereby avoiding nonlinear operation of the Receiver s Low Noise Amplifier and/or causing the Receiver to detect & process an unwanted signal. 2. Transmitters: Filters suppress much of the transmitter-generated harmonic frequencies, wide-band noise, intermodulation distortion (IMD) products, and out-of-band conversion frequencies. 3. Communication systems: The various frequency channels are very close, thus requiring bandpass filters with very narrow bandwidth & high out-of-band sirt attenuation/rejection. 4. Diplexers: Filters which provide high isolation (loss) between transmit signal frequencies and receive signal frequencies. 5. Multiplexers: Filters which separate or combine signal frequencies to different signal channels: Frequency channelization. 6. In detector circuits, frequency mixers and frequency multiplier applications, filters are used to attenuate/bloc unwanted high frequency intermodulation products. 5
Specific Applications of RF Filters 1. Communication Systems: A. Terrestrial Microwave Lins: Receiver protection Filters, Transmitter Filters, Channel-dropping Filters, Transmitter harmonic frequency rejection Filters, Local Oscillator Filters, Mixer s image-frequency rejection Filters. B. Satellite Systems: 1) Spacecraft/Satellite: Front-end Receive Filters, Input multiplexer channelization Filters, Output multiplexer channelization Filters, Transmitter harmonic frequency rejection Filters. 2) Earth Stations : LNA s transmit reject Filters, High Power Amplifier s harmonic frequency reject Filters, Up Converter & Down Converter Filters. 2. Mobile and Cellular Systems : A. Base Station Receive protection Filters. B. Base Station Transmitter Filters. C. Subscriber s hand set Diplexer Filters. D. Satellite Mobile Applications: 1) Aeronautical Transmit/Receive Systems. 2) Maritime Satellite Terminals. 3) Land Mobile Satellite Terminals. 3. RADAR Systems: Active & Passive. 4. High RF Power Applications: Various & many. 6
7 RF Filter Applications: Example Transmit and Receive (Tx & Rx) Communication System Radiated RF Signal Rx Bandpass Filter: Suppresses the Power Amplifier s harmonic distortion & limits the LNA input. IF Bandpass Filter: Supplies receive channel selectivity & suppresses mixer image frequency RF signals. Tx Bandpass Filter: Suppresses harmonic distortion from the Power Amplifier. Rx Bandpass Filter: Increases RF selectivity & suppresses LO leaages. LO Bandpass Filter: Suppresses frequency harmonic RF signals generated inside Local Oscillator (LO).
8 RF Filter Classification
Classification of RF Filters Filters are classified according to the following parameters: A. Frequency selection: 1) Lowpass Filter: Low insertion loss below a specified cut-off frequency: f c. 2) Highpass Filter: Low insertion loss above a specified cut-off frequency: f c. 3) Bandpass Filter: Low insertion loss across a specified frequency band. 4) Bandstop Filter: High insertion loss across a specified frequency band. B. Amplitude response : 1) Equi-ripple passband amplitude response: Chebyshev. 2) Maximally-flat passband amplitude response: Butterworth. 3) Elliptic (Cauer), Bessel, Linear Phase, Gaussian, Pascal, etc. C. Technology: Lumped, dielectric, planar, combline, waveguide, SAW, etc. D. Frequency bandwidth: Narrow band (BW < 5%) or broad band (BW > 25%). E. Reflection-type Filter or absorbing-type Filter: 1) The majority of filters achieve out-of-band frequency attenuation by reflection, which results in high insertion loss to those signal frequencies and, also, results in very high Voltage Standing Wave Ration (VSWR). 2) A small class of filters achieve out-of-band frequency attenuation by absorption of specified out-of-band signal frequencies. 9 LPF HPF Filter BPF BPS
Technology & Application of RF Filters Frequency UHF L& S-band C-band X- & Ku-band Ka-band Technology Application Combline Dielectric Helical Planar SAW Waveguide Cellular Satcom Combline Dielectric HTS Planar SAW Waveguide PCS Satcom MMDS Combline Dielectric HTS Planar Waveguide Satcom Combline Dielectric Planar Waveguide Satcom Lin Dielectric Planar Waveguide LMDS Satcom Acronyms for Technology of Filters: 1. SAW: Surface Acoustic Wave. 2. HTS: High Temperature Superconductive. Acronyms for Application of Filters: 1. PCS: Personal Communication System. 2. MMDS: Multichannel Multipoint Distribution Service. 3. LMDS: Local Multipoint Distribution Service. 4. Satcom: Satellite Communication. 1
Relative Bandwidth, % Common Transmission Media of RF Filters 1 Lumped LC Filters 1 Planar: Printed circuit & suspended substrate Filters. 1..1 Coaxial Filters Dielectric Resonator Filters Waveguide Filters.1 P-band L-band S-band C-band X-band K-band Q-band V-band W-band 25 MHz 1 GHz 2 GHz 4 GHz 8 GHz 2 GHz 33 GHz 5 GHz 75 GHz Frequency Band Designation As technology improves, the transmission media of filters can extend across broader frequencies. 11
Features of Transmission Media for RF Filters Transmission Line Media (Construction) Useful Frequency range (GHz) Impedance Range (Ohms) Cross- Sectional Dimensions Quality Factor (Loss) RF Power Rating Active Device Mounting Potential for Low- Cost Production Rectangular <3 1 to 5 Moderate High High Easy Poor Waveguide to large Coaxial Line <5 1 to 12 Moderate Moderate Moderate Fair Poor Stripline <1 1 to 12 Moderate Low Low Fair Good Microstrip 1 1 to 12 Small Low Low Easy Good Suspended 15 2 to 15 Small Moderate Low Easy Fair Stripline Finline 1 2 to 4 Moderate Moderate Low Easy Fair Slotline 6 6 to 2 Small Low Low Fair Good Coplanar 6 4 to 15 Small Low Low Fair Good Waveguide Image guide <3 3 to 3 Moderate High Low Poor Good Dielectric Line <3 2 to 5 Moderate High Low Poor Fair Technology Unloaded Q Microstrip, Stripline & Coplanar 1 to 6 Coaxial cavity & Combline 1 to 6 Rectangular Waveguide 4 to 15 Dielectric Resonator 5 to 5 12
Illustration of planar TEM Bandpass Filters 1. Edge-coupled Band Pass Filter: Input A. Distributed transmission-line bandpass filter. B. Uses quarter wavelength long coupled lines. C. Prone to spurious amplitude response at higher frequencies. 2. Combline Band Pass Filter: A. Physically compact bandpass filter. B. Uses quarter-wave transmission line resonators that are capacitively coupled. C. Good sirt rejection at higher frequencies. 3. Hairpin Band Pass Filter: A. Similar to the edge-coupled bandpass filter, but is considerably shorter. Input 4. Interdigital Band Pass Filter: A. Physically compact bandpass filter. B. Good sirt rejection at high frequencies. C. Short-circuited transmission lines that tae the structure of interlaced fingers. 13 Input Input Output Output Output Output
Illustration of some RF Bandpass Filters 8-pole Helical Filter 4-pole Dielectric-loaded Coaxial Bandpass Filter 4-pole Combline Bandpass Filter 3-pole Parallel Edge-Coupled Stripline Bandpass Filter 14 5-pole Hairpin Bandpass Filter
Illustration of some RF Lowpass Filters Coaxial Lowpass Filter Lumped Element Lowpass Filter Lowpass Highpass Suspended stripline band-pass filter consisting of low-pass filter (left side) and a high-pass filter (right side). 15 Distributed Microstrip Lowpass Filter
RF Filter Specifications Factors that drive hardware design 1. Amplitude/magnitude response versus frequency characteristics: A. Frequency specifications: f c, f and bandwidth (BW f High f Low ). 1) For Lowpass & Highpass Filters: Cut-off frequency (f c ). 2) For Bandpass & Bandstop Filters: Center frequency (f ) & bandwidth. B. Passband insertion loss & passband amplitude ripple: L ar ; Return Loss. 1) Passband frequency: Frequency range where filter passes all frequencies. 2) Ripple: Difference between max and min of amplitude response in passband. 3) Quality factor: Higher Q lower passband insertion loss. 4) Input & output impedances: Z in & Z out. C. Out-of-band frequency (stopband) amplitude sirt rejection/attenuation. D. Spurious out-of-band frequency response, including higher-order modes. 2. Phase characteristics across passband frequency: A. Passband group delay variation: Affects Inter-symbol interference. 3. RF power handling requirement: A. Continuous Wave (CW) RF power & pea RF power. B. Multipactor effects & voltage breadown. 4.Volume (size & shape) and mass/weight; In & out electrical connectors. 5. Environmental: Temperature, pressure, humidity, shoc, vibration, etc. 6. Cost to manufacture and time to deliver. 16
Attenuation, db Attenuation, db Attenuation, db Filters: Passband Amplitude Response There are four main filter classes with passband amplitude response that approximate the ideal filter s frequency response: 1. Butterworth Filter: Maximally-flat passband amplitude. 2. Chebyshev Filter: Equi-ripple passband amplitude response. A. Inverted Chebyshev amplitude response: Chebyshev Type 2. 3. Bessel Filter: Maximally-flat time delay response. 4. Elliptic-function (Cauer) amplitude response. Amplitude profiles vs Frequency of popular Lowpass Filters 3dB Lar Lar f c freq f c freq f c freq Butterworth Filter (Maximally-flat passband) Chebyshev Filter (Equi-Ripple passband) 17 Elliptic Filter
Insertion Loss, db 3dB Insertion Loss versus Frequency of several types of N 7 Pole Lowpass Filters: f c 1, MHz -1-2 Cutoff Frequency f c 1, MHz -3-4 7PoleChebyshev -5 7PoleButterworth 7PoleBessel 7PoleLinearPhase -6-7 5 1 15 2 25 3 Frequency, MHz 18
The Prototype Filter A. A Prototype filter is an electronic filter design that serves as a template to produce a modified filter design for a particular application. The prototype filter is an example of a filter s design from which the desired filter can be scaled or transformed. B. Filters are required to operate at many different frequencies, bandwidths and impedances. The utility of a prototype filter comes from the property that all these other filters can be derived from it by applying a scaling factor to the components of the prototype filter. Thus, the filter design need only be carried out once in full, with other filters being obtained by simply applying a scaling factor. C.Especially useful is the ability to transform from one frequency passband response to another: Lowpass response, Highpass response, Bandpass response and Bandstop response. D.The prototype filter is most often expressed as a lowpass filter with a cutoff (or corner ) frequency: ω c ' 1 radian/sec and a characteristic impedance set to R ' 1 Ohm. The desired filter is derived from this prototype lowpass filter by frequency scaling & impedance scaling. 19
Filter Synthesis: Insertion Loss Method A. The insertion loss method enables a systematic way to design and synthesize a filter with various frequency responses. B. The insertion loss method also enables a filter s performance to be improved in a straightforward manner, at the expense of a higher order or more complex filter. C. A rational polynomial function: H() is used to approximate the ideal filter s transfer function, but in amplitude only: A() or S 21 (). D. Phase information is totally ignored when using the insertion loss method to synthesize the filter. Ignoring phase simplifies the actual synthesis method. E. An L-C networ is then derived that will produce this approximated amplitude response. F. The attenuation: A() can be cast into a power attenuation ratio, called the Power Loss Ratio: P LR, which is related to A() 2. G. Modern filter synthesis can optimize a filter s electrical circuit to meet both magnitude and phase requirements. 2
21 Filter Approximation Polynomials Insertion Loss Method for Filter Synthesis A. Every physically realizable filter circuit has a transfer function that is a rational polynomial in s ( σ + jω ). B. We want to determine the classes of rational polynomials that approximate the Ideal lowpass filter response. Note: A Lowpass filter s design can be used to derive a Highpass filter, a Bandpass filter or a Bandstop filter. C.Four well nown rational polynomial approximation functions that approach the ideal bric wall filter s response are: 1. Butterworth (193): Britain s Steven Butterworth, 1885 1958. 2. Chebyshev (1854): Russia s Pafnuty Chebyshev, 1821 1894. 3. Elliptic (193): Germany s Wilhelm Cauer, 19 1945. 4. Bessel (1824): Germany s Friedrich Wilhelm Bessel, 1784 1846.
Filter Classifications by Response Type Insertion Loss Method for Filter Synthesis Popular characteristic polynomials used to define the insertion loss of an N-section lowpass prototype filter are: 1. Butterworth or Maximally-flat passband amplitude response: P LR 1+ / o 2n 2. Chebyshev or equal-ripple passband amplitude response: P LR 1+ e 2 C 2 n (/ o, where C 2 n Chebyshev cosine polynomial. 3. Inverse Chebyshev maximally-flat passband & equal-ripple stopband response: P LR 1+1/ [e 2 C 2 n (/ o]. 4. Elliptic function or Quasi-elliptic function response (equal-ripple in both pass band and stopband): P LR 1+ e 2 U 2 n (/ o. 5. Bessel-Thompson function response (Maximally-flat passband Phase/Group Delay). Insertion Loss 1log 1 (P LR ) 22
Prototype Low Pass Filter Ladder Circuits Insertion Loss Method for Filter Synthesis R s g 1Ω The prototype filter is a lowpass filter with a normalized cut-off frequency of ω c 1 radian/second and 1 ohm terminations at both the input/source: g o and output/load: g n+1. Shown below are two N-section L-C ladder circuits for Lowpass filter prototypes and their electrical circuit element definitions: L 2 g 2 L 4 g 4 Identical Dual Circuits L 1 g 1 L 3 g 3 L 5 g 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω (a) Prototype Lowpass Filter with series inductor input: L 1. (b) Prototype Lowpass Filter with shunt capacitor input: C 1. Feb-215 where: N Order of the filter Number of reactive elements in the filter. g Generator s source resistance or generator s source conductance. g i Inductance for series inductors or capacitance for shunt capacitors. g N+1 Load resistance if g N is a shunt capacitor or load conductance if g N is a series inductor. www.atlantarf.com
Attenuation, db Lowpass Filter Prototype Circuit Element Values Maximally-flat passband amplitude Butterworth Filter When a Butterworth polynomial is used to define the insertion loss of an N-section lowpass filter, its power loss ratio is: P 2 n 1 2 LR( Butterwort h ) + e c where: e 1 for a -3dB cutoff point. n Order of the filter. c Cutoff frequency, radians/sec. Calculate the number of sections: n needed in a Butterworth filter as: n log 1 2log 1 1 A / 1 1 / 1 where: A is the attenuation in db at some out-of-band frequency: 1 ( 1 > c ). 24 c Butterworth lowpass prototype filter circuit element values: g g n+1 1 g i 2 sin 2i 1 2n Series R s R L Z o Shunt G s G L 1/Z o f c 3dB Butterworth Filter, i 1,2,3.n freq
Attenuation, db 7 6 Attenuation versus Normalized Frequency for an N-section Butterworth Filter N 1 N 2 N 3 N 4 7 6 5 N 5 N 6 5 N 7 4 N 8 N 9 N 1 4 3 N 11 N 12 N 13 3 2 N 14 N 15 2 1 1.1.2.3.4.5.7 1. 2. 3. 5. 7. 1. Normalized Frequency: [ω/ω C - 1] 25
Attenuation, db 3dB -1 Attenuation versus Normalized Frequency for an N-section Butterworth Filter -1-2 Normalized Cutoff Frequency: f c 1-2 -3-3 -4-4 -5-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω C 26
Lowpass Filter Prototype Circuit Element Values Maximally-flat passband amplitude Butterworth Filter Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 R s g 1Ω 1 2. 1. 2 1.41421 1.41421 1. 3 1. 2. 1. 1. 4.76537 1.84776 1.84776.76537 1. 5.6183 1.6183 2. 1.6183.6183 1. 6.51764 1.41421 1.93185 1.93185 1.41421.51764 1. 7.4454 1.24698 1.8194 2. 1.8194 1.24698.4454 1. 8.3918 1.11114 1.66294 1.96157 1.96157 1.66294 1.11114.3918 1. 9.3473 1. 1.5329 1.87939 2. 1.87939 1.5329 1..3473 1. 1.31287.9798 1.41421 1.7821 1.97538 1.97538 1.7821 1.41421.9798.31287 1. L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g 4 ω c 1 radian/sec g i Inductance for series inductors. Capacitance for shunt capacitors. L 1 g 1 L 3 g 3 g i 2 sin L 5 g 5 2i 1 2n R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 27 g Source resistance. Atlanta Source conductance. RF
Lowpass Filter Prototype Circuit Element Values Equi-ripple passband amplitude Chebyshev Filter When a Chebyshev polynomial is used to define the insertion loss of an N-section lowpass prototype filter, its power loss ratio is: 2 2 PLR( Chebyshev) 1+ e CN c where: C N : Chebyshev polynomial of order N. c : Cutoff frequency, radians/sec. P LR oscillates between 1 & 1+e 2. Calculate the number of sections needed in a Chebyshev filter as: N cosh 1 A / 1 Lar (1 1) /(1 1 cosh ( 1 ) c / 1 1) where: A is the attenuation in db at some out-of-band frequency: 1 ( 1 > c ). Chebyshev lowpass prototype circuit element values: where: 28 g 1 ; g 1 1 for n odd + 1 coth for n 4 even gn 2 2a 1 ; g Lar ln(coth ) 17.34 ( 2i 1) ai sin 2N 4a b i1 i1 g a i1 2 2 i bi + sin ; i 1,2,...,N N N : Order of the filter. L ar ; : Maximum passband ripple, db. ; sinh( ) 2n i 1,2,... N i
Attenuation, db 7 6 Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.1dB passband ripple N 2 N 3 N 4 Case of L AR.1dB Passband Ripple; Return Loss > 26 db 7 6 N 5 5 N 6 N 7 5 4 N 8 N 9 N 1 4 3 N 11 N 12 3 N 13 2 N 14 N 15 2 1 1..2.3.5.1.2.3.5 1. 2. 3. 5. 1. Normalized Frequency: [ω/ω C - 1] 29
Attenuation, db Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.1dB passband ripple -1-2 Normalized Cutoff Frequency: f c 1-1 -2-3 -4 Case of L AR.1dB Passband Ripple; Return Loss > 26 db -3-4 -5-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω c 3
Lowpass Prototype Circuit Element Values.1 db equi-ripple passband amplitude Chebyshev Filter Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 R s g 1Ω 1.9611 1. 2.4491.4796 1.184 3.62941.9747.62941 1. 4.7139 1.25 1.32156.64777 1.184 5.75655 1.354 1.57755 1.354.75655 1. 6.78157 1.3611 1.68989 1.5359 1.49727.7997 1.184 7.79716 1.39251 1.74833 1.63316 1.74833 1.39251.79716 1. 8.8749 1.41317 1.78263 1.68335 1.85311 1.61933 1.55568.73352 1.184 9.81467 1.42714 1.8454 1.71254 1.9595 1.71254 1.8454 1.42714.81467 1. 1.81986 1.4373 1.81944 1.73111 1.93638 1.759 1.9568 1.65277 1.58195.74476 1.184 L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g 4 Passband Ripple: L ar.1 db; Passband Return Loss > 26dB ω c 1 radian/sec g i Inductance for series inductors. Capacitance for shunt capacitors. L 1 g 1 L 3 g 3 L 5 g 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 31 g Source resistance. Atlanta Source conductance. RF
Attenuation, db 7 Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.4dB passband ripple 7 6 N 1 N 2 N 3 Case of L AR.4dB Passband Ripple; Return Loss > 2 db 6 N 4 5 N 5 N 6 5 N 7 4 N 8 N 9 4 N 1 3 N 11 N 12 3 N 13 2 N 14 N 15 2 1 1..2.3.5.1.2.3.5 1. 2. 3. 5. 1. Normalized Frequency: [ω/ω C - 1] 32
Attenuation, db Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.4dB passband ripple -1-2 Normalized Cutoff Frequency: f c 1-1 -2-3 -4 Case of L AR.4dB Passband Ripple; Return Loss > 2 db -3-4 -5-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω C 33
Lowpass Prototype Circuit Element Values.4 db equi-ripple passband amplitude Chebyshev Filter Passband Ripple: L ar.4 db; Passband Return Loss > 2dB Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 C 11 1.19256 1. 2.65114.53725 1.21199 3.83768 1.9767.83768 1. 4.91772 1.28897 1.56222.7572 1.21199 5.95791 1.3743 1.78785 1.3743.95791 1. 6.9863 1.41214 1.8883 1.55185 1.7115.8911 1.21199 7.99466 1.43632 1.9279 1.62455 1.9279 1.43632.99466 1. 8 1.389 1.4516 1.95513 1.6675 2.1282 1.61315 1.75933.8283 1.21199 9 1.128 1.46188 1.97239 1.68152 2.5451 1.68152 1.97239 1.46188 1.128 1. 1 1.1489 1.46914 1.9846 1.69462 2.7819 1.71469 2.5387 1.6373 1.7858.83737 1.21199 g i Inductance for series inductors. Capacitance for shunt capacitors. R s g 1Ω L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 11 L 2 g 2 L 4 g 4 ω L 1 g 1 L 3 g 3 L 5 g c 1 radian/sec 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 34 g Source resistance. Atlanta Source conductance. RF
Attenuation, db 7 Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.1dB passband ripple 7 6 N 1 N 2 N 3 Case of L AR.1dB Passband Ripple; Return Loss > 16 db 6 N 4 5 N 5 N 6 5 N 7 4 N 8 N 9 N 1 4 3 N 11 N 12 N 13 3 2 N 14 N 15 2 1 1..2.3.5.1.2.3.5 1. 2. 3. 5. 1. Normalized Frequency: [ω/ω C - 1] 35
Attenuation, db Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.1dB passband ripple -1-2 Normalized Cutoff Frequency: f c 1-1 -2-3 -4 Case of L AR.1dB Passband Ripple; Return Loss > 16.4 db -3-4 -5-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω c 36
Lowpass Prototype Circuit Element Values.1 db equi-ripple passband amplitude Chebyshev Filter Passband Ripple: L ar.1 db; Passband Return Loss > 16dB Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 1.3552 1. 2.84349.62216 1.35574 3 1.321 1.14745 1.321 1. 4 1.1923 1.3617 1.7783.81818 1.35574 5 1.14726 1.37117 1.97544 1.37117 1.14726 1. 6 1.16855 1.4391 2.5663 1.51698 1.9333.86193 1.35574 7 1.18162 1.42273 2.977 1.57325 2.977 1.42273 1.18162 1. 8 1.1919 1.43457 2.1229 1.684 2.1733 1.56394 1.94491.87789 1.35574 9 1.19611 1.44252 2.13494 1.61655 2.2574 1.61655 2.13494 1.44252 1.19611 1. 1 1.236 1.44811 2.14482 1.62641 2.22569 1.64168 2.2499 1.5823 1.96326.88539 1.35574 g i Inductance for series inductors. Capacitance for shunt capacitors. R s g 1Ω L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g 4 ω L 1 g 1 L 3 g 3 L 5 g c 1 radian/sec 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 37 g Source resistance. Atlanta Source conductance. RF
Attenuation, db 7 Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.5dB passband ripple 7 6 N 1 N 2 N 3 Case of L AR.5dB Passband Ripple; Return Loss > 9.6 db 6 N 4 5 N 5 N 6 5 N 7 4 N 8 N 9 N 1 4 3 N 11 N 12 N 13 3 2 N 14 N 15 2 1 1..2.3.5.1.2.3.5 1. 2. 3. 5. 1. Normalized Frequency: [ω/ω C - 1] 38
Attenuation, db Attenuation versus Normalized Frequency for an N-section Chebyshev Filter:.5dB passband ripple -1-2 Normalized Cutoff Frequency: f c 1-1 -2-3 -4 Case of L AR.5dB Passband Ripple; Return Loss > 9.6 db -3-4 -5-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω C 39
Lowpass Prototype Circuit Element Values.5 db equi-ripple passband amplitude Chebyshev Filter Passband Ripple: L ar.5 db ; Passband Return Loss > 9.6dB Prototype g Elements: 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 R s g 1Ω 1.6993 1. 2 1.438.779 1.98533 3 1.5972 1.9649 1.5972 1. 4 1.67122 1.1923 2.3671.84179 1.98533 5 1.7669 1.22933 2.54179 1.22933 1.7669 1. 6 1.72628 1.24756 2.6731 1.31329 2.47681.86952 1.98533 7 1.73821 1.25792 2.63923 1.34395 2.63923 1.25792 1.73821 1. 8 1.74599 1.26439 2.65735 1.35864 2.69735 1.3385 2.5122.87945 1.98533 9 1.75135 1.26871 2.66872 1.36692 2.72483 1.36692 2.66872 1.26871 1.75135 1. 1 1.7552 1.27175 2.67635 1.3728 2.7414 1.382 2.7243 1.3487 2.52484.8849 1.98533 L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g 4 ω c 1 radian/sec g i Inductance for series inductors. Capacitance for shunt capacitors. L 1 g 1 L 3 g 3 L 5 g 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 4 g Source resistance. Atlanta Source conductance. RF
Attenuation, db 3dB -1 Attenuation versus Normalized Frequency for N-section Bessel Filters -1-2 Cutoff Frequency f c 1, MHz -2-3 -4 N 2 N 3 N 4 N 5 N 6 N 7-3 -4-5 N 8 N 9-5 -6-6 -7-7..5 1. 1.5 2. 2.5 3. Normalized Frequency: ω/ω c 41
Lowpass Prototype Circuit Element Values Maximally-flat Time Delay Bessel Filter Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 1 2. 1. 2.5755 2.1486 1. 3.33742.9751 2.2341 1. 4.23342.67252 1.8152 2.2438 1. 5.17432.5724.841 1.1113 2.25822 1. 6.13649.419.63916.85379 1.11264 2.26452 1. 7.1156.32589.52489.721.8693 1.1516 2.2659 1. 8.9191.27191.4492.59357.7326.8695 1.9556 2.26561 1. 9.7797.23129.37699.5178.636.7473.86387 1.8628 2.26488 1. 1.6716.21.3277.44544.55282.64934.7422.8567 1.789 2.26413 1. g i Inductance for series inductors. Capacitance for shunt capacitors. R s g 1Ω L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g ω 4 c 1 radian/sec L 1 g 1 L 3 g 3 L 5 g 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 42 g Source resistance. Atlanta Source conductance. RF
Lowpass Prototype Circuit Element Values Linear Phase Filter with equi-ripple passband phase error of.5 o Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 g 9 g 1 g 11 Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 L 8 C 9 L 1 1 2. 1. 2.648 2.185 1. 3.4328 1.427 2.2542 1. 4.3363.7963 1.1428 2.2459 1. 5.2751.6541.8892 1.134 2.2873 1. 6.2374.5662.7578.876 1.1163 2.2448 1. 7.285.4999.6653.7521.8749 1.671 2.2845 1. 8.1891.4543.631.675.759.8427 1.91 2.2415 1. 9.1718.4146.5498.6132.6774.7252.845 1.447 2.2834 1. 1.161.3867.5125.572.6243.6557.7319.8178 1.767 2.2387 1. g i Inductance for series inductors. Capacitance for shunt capacitors. R s g 1Ω L 1 C 2 L 3 C 4 L 5 C 6 L 7 C 8 L 9 C 1 L 2 g 2 L 4 g ω 4 c 1 radian/sec L 1 g 1 L 3 g 3 L 5 g 5 R L g N+1 C 1 g 1 C 3 g 3 C 5 g 5 g 1Ω C 2 g 2 C 4 g 4 R L g N+1 1Ω g n+1 Load resistance if g n is a shunt capacitor. Load conductance if g n is a series inductor. Dual Circuits 43 g Source resistance. Atlanta Source conductance. RF
Comparison of some Lowpass Prototype Circuit Element Values vs. Response Type Prototype Elements: g 1 g 2 g 3 g 4 g 5 g 6 g 7 g 8 Filter's Response: Order : N C 1 L 2 C 3 L 4 C 5 L 6 C 7 Butterworth 3 1. 2. 1. 1..1dB Chebyshev 3.62941.9747.62941 1..5dB Chebyshev 3 1.5972 1.9649 1.5972 1..5 o Linear Phase 3.4328 1.427 2.2542 1. Bessel Filter 3.33742.9751 2.2341 1. Butterworth 5.6183 1.6183 2. 1.6183.6183 1..1dB Chebyshev 5.75655 1.354 1.57755 1.354.75655 1..5dB Chebyshev 5 1.7669 1.22933 2.54179 1.22933 1.7669 1..5 o Linear Phase 5.2751.6541.8892 1.134 2.2873 1. Bessel Filter 5.17432.5724.841 1.1113 2.25822 1. Butterworth 7.4454 1.24698 1.8194 2. 1.8194 1.24698.4454 1..1dB Chebyshev 7.79716 1.39251 1.74833 1.63316 1.74833 1.39251.79716 1..5dB Chebyshev 7 1.73821 1.25792 2.63923 1.34395 2.63923 1.25792 1.73821 1..5 o Linear Phase 7.285.4999.6653.7521.8749 1.671 2.2845 1. Bessel Filter 7.1156.32589.52489.721.8693 1.1516 2.2659 1. L 1 C 2 L 3 C 4 L 5 C 6 L 7 For R(source) R(load) g o g n +1 1 Ohm ; ω c 1 radian/second. 44
A Note about Passband Amplitude Ripple Filters synthesized using the Insertion Loss Method produce a passband VSWR that is related to the passband s amplitude ripple: 1. The passband s amplitude ripple is directly related to the filter s input/output VSWR as: 1+ 1 Lar 1+ VSWR 1 / 1 As such, a higher passband amplitude ripple produces a higher passband VSWR, as shown in the table. Often, higher passband VSWR can adversely effect the performance in communication systems and, therefore, it seems silly to publish any filter s design whose ripple exceeds L ar >.5dB passband ripple. Butterworth 45 Passband Reflection Return Ripple, Coefficient Loss, VSWR Lar, db Γ db.1.151734 36.38 1.3.5.33929 29.39 1.7.1.479576 26.38 1.1.2.677834 23.38 1.15.3.829696 21.62 1.18.4.9575 2.38 1.21.5.16992 19.41 1.24.6.1171346 18.63 1.27.7.1264472 17.96 1.29.8.13512 17.39 1.31.9.1432132 16.88 1.33.1.158734 16.43 1.36.25.2365145 12.52 1.62.5.3297712 9.64 1.98.75.3982523 8. 2.32 1..453515 6.87 2.66 2..674888 4.33 4.1 3..762668 3.2 5.81
Frequency Scaling the Prototype Lowpass Filter 1. The basis for frequency normalization of filters is: A. A prototype filter s response can be scaled (shifted) to a different frequency range by dividing the reactive elements: L & C, by a frequency scale factor. B. The frequency scale factor is the ratio of the desired filter s reference frequency to the prototype filter s reference frequency (often ω c 1 rad/sec): Frequency Scale Factor Desired Existing Frequency Frequency C. The cutoff frequency: f c is selected as the desired frequency for lowpass & highpass filters, while the center frequency: f and fractional bandwidth is selected as the desired frequency factor for bandpass & bandstop filters. 2. Bandpass & Bandstop Filters: It is sometimes desirable to compute two geometrically related frequencies that correspond to a given bandwidth. When given the center frequency: f and the bandwidth: BW, the lower cutoff frequency: f 1, and upper cutoff frequency: f 2 are computed as: 46
47 Frequency Mapping the Lowpass Prototype The Insertion Loss Method for Filter Synthesis Frequency mapping the lowpass prototype filter into the desired filter type: Frequency mapping to a lowpass filter: c Frequency mapping to a highpass filter: ω c Cutoff frequency c 1 Frequency mapping to a bandpass filter: BW Frequency mapping to a bandstop filter: BW ω Center frequency; BW Bandwidth ω 2 ω 1 Electrical circuit components realized after frequency mapping: 1 Series Circuit Components Shunt Circuit Components
Impedance Scaling the Prototype Lowpass Filter 1. The basis for impedance normalization of filters is: A. A prototype filter s impedance level can be scaled to a different value by adjusting the prototype element values using an impedance scale factor. B. The impedance scale factor is the ratio of the desired filter s impedance level to the prototype filter s impedance level (often: Z 1 ohm): Impedance Scale Factor Desired Prototype C. To impedance-transform the prototype filter: 1) Multiply all resistances & inductances by the impedance scale factor. 2) Divide all capacitors by the impedance scale factor. C. For many RF filters, the filter s desired impedance is often: Z 5 ohms. Impedance Impedance Load Impedance 2. Since the filter is a linear circuit, we can multiply all impedances by some factor without changing the transfer function of the filter. This leaves the frequency response unchanged, but impedance-scales the desired filter to its required impedance level. Its frequency response remains as a lowpass, highpass, bandpass or bandstop filter. Feb-215 www.atlantarf.com L C ' desired R R ' desired Z ' L Z Z Source R L C L prototype Z ' s prototype Impedance
49 Frequency Transformation and Impedance Scaling of Low Pass Prototype 1. Once the Lowpass Prototype Filter is designed, the cut-off frequency: c can be transformed to other frequencies. 2. Furthermore, the Lowpass Prototype Filter can be frequency mapped into other filter types, such as: Highpass Filter, Bandpass Filter, and Bandstop Filter. 3. This frequency scaling and transformation entails changing the value and configuration of the electrical circuit elements of the Lowpass Prototype Filter. 4. Finally, the impedance presented by the filter can also be scaled, from unity to another impedance value, lie: Z o 5 ohms. Filter Specifications Lowpass Prototype Design Frequency Mapping Impedance Scaling Desired Filter Implementation
Lowpass & Highpass Filter Transformations Frequency-scaling & Impedance-scaling circuit components Frequency & Impedance scaling to a Lowpass Filter: 1. The series inductor: g L is transformed into a series inductor: L with a value: L ' Z o g c Frequency & Impedance scaling to a Highpass Filter: 1. The series inductor: g L, is transformed into a series capacitor: C with a value: ' 1 C Z g o c 2. The shunt capacitor: g C is transformed into a shunt capacitor: C with a value: 2. The shunt capacitor: g C, is transformed into a shunt inductor: L with a value: C ' Z g o c L ' o gz c Feb-215 www.atlantarf.com ω c 2πf c Desired cut-off frequency. Z Impedance of the system, Ohms. 5
Bandpass & Bandstop Filter Transformations Frequency-scaling & Impedance-scaling circuit components Frequency & Impedance scaling to a Bandpass Filter: 1. The series inductor: g L is transformed into a series LC circuit with element values: L ' Z g 2. The shunt capacitor: g C is transformed into a shunt LC circuit with element values: C ' Z g Frequency & Impedance scaling to a Bandstop Filter: 1. The series inductor: g L, is transformed into a parallel LC circuit with element values: L ' Zg 2. The shunt capacitor: g C, is transformed into a series LC circuit with element values: C ' 1 Z g C ' g Z Feb-215 L o Z ' g +2 2 www.atlantarf.com Center frequency: 1 or 1 2 Fractional bandwidth: 21 o 51 L C ' ' Z Z g g
Frequency-Scaling & Impedance-Scaling Circuit Components of Low Pass Prototype (LPP) Filter L (g ) LPP to Lowpass Z o L c Series Component LPP to Highpass c 1 L Z o L Zo o LPP to Bandpass o L Z o LPP to Bandstop o L 1 Z o L Z o o Shunt Component C (g ) Z C o c Z c o C C o Z o Z o o C Zo o C C o Z o Center frequency: +2 o or 2 1 1 2 Fractional bandwidth: 1 2 52 o Note: The inductor always multiplies with Z o while the capacitor divides with Z o.
Electrical Circuit Configuration Lowpass Filter & Highpass Filter after Freq. & Impedance Scaling L 1 L 3 L n-1 C 2 C 4 C n Z Z n+1 Lowpass filter derived from the lowpass prototype filter. Highpass filter derived from the lowpass prototype filter. 53
Electrical Circuit Configuration Bandpass Filter & Bandstop Filter after Freq. & Impedance Scaling R... L 5 C 5 L 3 C 3 L 1 C 1 L N L 4 C N R L 1 C 4 L 2 C 2... Bandpass filter derived from the prototype lowpass filter. R L N... L 4 L 2 L 5 L 3 C N C C4 2 L 1 R L 1... C 5 C 3 C 1 Bandstop filter derived from the prototype lowpass filter. 54
Summary of Steps in Filter Design & Synthesis The Insertion Loss Method for Filter Synthesis Step 1: Filter Specifications A. Filter Type: Lowpass or Highpass or Bandpass or Bandstop. B. Frequency Response: Butterworth, Chebyshev or Bessel. C. Desired frequency of operation: f c or f o + bandwidth. D. Passband & stopband frequency range. E. Maximum allowed attenuation (for Equal ripple). Step 2: Low Pass Prototype Design A. Minimum Insertion Loss level, Number of Filter Order/Elements by using insertion loss values. B. Circuit configuration: Shunt capacitor model or Series inductance model. C. Draw the low-pass prototype circuit ladder diagram. D. Determine circuit elements values from the Lowpass Prototype Table. Step 3: Scaling and Conversion Feb-215 A. Determine if any modification to the prototype table is required (for high pass, band pass and band stop). B. Frequency scale & impedance scale prototype lowpass circuit element values to obtain the desired filter s real circuit element values.... then implement. www.atlantarf.com 55 Filter Specs Lowpass Prototype Design Frequency Mapping Impedance Scaling Desired Filter Implementation
Example: Butterworth Lowpass Filter 1. Specifications: A. Maximally-flat lowpass filter; f c 2. GHz ; Z 5 Ohms. B. Out-of-band sirt attenuation > 15 db at 3. GHz. 2. Filter Design Solution: A. Normalized frequency at 3 GHz: (ω/ω c 1) (3GHz/2GHz 1).5. B. Refer to graph titled: Attenuation vs Normalized Frequency for N-section Butterworth filter. Find best curve that is above the point located at Attenuation 15dB when (ω/ω c 1).5. Result: N 5. Alternately, you can calculate the number of circuit elements needed using: A / 1 log1 1 1 n 2log11 / c C. For N 5 circuit elements, calculate prototype element values for Butterworth filter or use element values from Butterworth filter table: g g 6 1., g 1.6183 ; g 2 1.6183 ; g 3 2. ; g 4 1.6183 ; g 5.6183. D. The prototype lowpass ladder circuit is shown below: R s g 1Ω L 2 g 2 L 4 g 4 R L g 6 1Ω C 1 g 1 C 3 g 3 56 C 5 g 5
Example: Butterworth Lowpass Filter (cont.) 3. Frequency-scale & impedance-scale lowpass prototype filter s circuit elements to: f c 2 GHz and Z o 5 ohms: C C C g 9 5 2 2 1 4. Implement the final (desired) Butterworth Lowpass Filter: R s 5Ω.618 1 1 Zc g 2. 3 3 Zc g 9 5 2 2 1.618 5 5 Zc 9 5 2 2 1 L 2 6.438nh.984 pf 3.183 pf.984 pf L 4 6.438nh L L Z g 5 1.618 2 2 9 c 2 2 1 Z g 5 1.618 4 4 9 c 2 2 1 6.438nh 6.438nh R L 5Ω Feb-215 www.atlantarf.com C 1.984pf C 3 3.183pf C 5.984pf 57
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