2 INTRODUCTION What is a Microwave filter ? linear 2-port network controls the frequency response at a certain point in a microwave systemprovides perfect transmission of signal for frequencies in a certain passband regioninfinite attenuation for frequencies in the stopband regiona linear phase response in the passband (to reduce signal distortion).f2
3 Commonly used block Diagram of a Filter INTRODUCTIONThe goal of filter design is to approximate the ideal requirements within acceptable tolerance with circuits or systems consisting of real components.f1f3f2Commonly used block Diagram of a Filter
4 INTRODUCTION Why Use Filters? RF signals consist of: Desired signals – at desired frequenciesUnwanted Signals (Noise) – at unwanted frequenciesThat is why filters have two very important bands/regions:Pass Band – frequency range of filter where it passes all signalsStop Band – frequency range of filter where it rejects all signals
5 INTRODUCTION Categorization of Filters Low-pass filter (LPF), High-pass filter (HPF), Bandpass filter (BPF), Bandstop filter (BSF), arbitrary type etc.In each category, the filter can be further divided into active and passive types.In active filter, there can be amplification of the of the signal power in the passband region, passive filter do not provide power amplification in the passband.Filter used in electronics can be constructed from resistors, inductors, capacitors, transmission line sections and resonating structures (e.g. piezoelectric crystal, Surface Acoustic Wave (SAW) devices, and also mechanical resonators etc.).Active filter may contain transistor, FET and Op-amp.FilterLPFBPFHPFActivePassive
6 INTRODUCTION Types of Filters Low-pass Filter Passes low freq Rejects high freqHigh-pass FilterPasses high freqRejects low freqf1f1f1f2f2f2
7 INTRODUCTION Band-pass Filter Band-stop Filter Passes a small range of freqRejects all other freqBand-stop FilterRejects a small range of freqPasses all other freqf1f1f1f2f2f2f3f3f3
8 INTRODUCTION Filter Parameters Pass bandwidth; BW(3dB) = fu(3dB) – fl(3dB)Stop band attenuation and frequencies,Ripple difference between max and min of amplitude response in passbandInput and output impedancesReturn lossInsertion lossGroup Delay, quality factor
10 INTRODUCTIONFor impedance matched system, using s21 to observe the filter response is more convenient, as this can be easily measured using Vector Network Analyzer (VNA).FilterZcVsa1b2Transmission lineis optionalZcc20log|s21()|0dBArg(s21())Complex value
11 INTRODUCTION Low pass filter response (cont) Transition band Passband A()/dBc31020304050PassbandStopbandTransition bandCut-off frequency (3dB)A FilterH()V1()V2()ZL
12 INTRODUCTION High Pass filter Passband Stopband Transfer function A()/dBc31020304050c|H()|1Transfer functionStopbandPassband
14 INTRODUCTION Figure 4.1: A 10 GHz Parallel Coupled Filter Response Insertion LossPass BW (3dB)Q factorFigure 4.1: A 10 GHz Parallel Coupled Filter ResponseStop band frequencies and attenuation
15 FILTER DESIGN METHODS Filter Design Methods Two types of commonly used design methods:- Image Parameter Method- Insertion Loss MethodImage parameter method yields a usable filterHowever, no clear-cut way to improve the design i.e to control the filter response
16 FILTER DESIGN METHODS Filter Design Methods The insertion loss method (ILM) allows a systematic way to design and synthesize a filter with various frequency response.ILM method also allows filter performance to be improved in a straightforward manner, at the expense of a ‘higher order’ filter.A rational polynomial function is used to approximate the ideal |H()|, A() or |s21()|.Phase information is totally ignored.Ignoring phase simplified the actual synthesis method. An LC network is then derived that will produce this approximated response.Here we will use A() following . The attenuation A() can be cast into power attenuation ratio, called the Power Loss Ratio, PLR, which is related to A()2.
17 FILTER DESIGN METHODS (2.1a) ZL Vs Lossless 2-port network 1 Zs PA PinPLPLR large, high attenuationPLR close to 1, low attenuationFor example, a low-passfilter response is shownbelow:PLR(f)(2.1a)f1LowattenuationHighfcLow-Pass filter PLR
18 PLR and s21In terms of incident and reflected waves, assuming ZL=Zs = ZC.ZcVsLossless2-port networkPAPinPLa1b1b2(2.1b)
19 FILTER RESPONSES Filter Responses Several types filter responses: - Maximally flat (Butterworth)- Equal Ripple (Chebyshev)- Elliptic Function- Linear Phase
20 THE INSERTION LOSS METHOD Practical filter response:Maximally flat:- also called the binomial or Butterworth response,- is optimum in the sense that it provides the flattest possible passband response for a given filter complexity.- no ripple is permitted in its attenuation profile[8.10] – frequency of filterc – cutoff frequency of filterN – order of filter
21 THE INSERTION LOSS METHOD Equal ripple- also known as Chebyshev.- sharper cutoff- the passband response will have ripples of amplitude 1 +k2[8.11] – frequency of filterc – cutoff frequency of filterN – order of filter
22 THE INSERTION LOSS METHOD Figure 5.3: Maximally flat and equal-ripple low pass filter response.
23 THE INSERTION LOSS METHOD Elliptic function:- have equal ripple responses in the passband andstopband.- maximum attenuation in the passband.- minimum attenuation in the stopband.Linear phase:- linear phase characteristic in the passband- to avoid signal distortion- maximally flat function for the group delay.
24 THE INSERTION LOSS METHOD Figure 5.4: Elliptic function low-pass filter response
25 THE INSERTION LOSS METHOD Low-pass Prototype DesignFilter SpecificationScaling & ConversionNormally done using simulatorsOptimization & TuningFilter ImplementationFigure 5.5: The process of the filter design by the insertion loss method.
26 THE INSERTION LOSS METHOD Low Pass Filter PrototypeFigure 5.6: Low pass filter prototype, N = 2
27 THE INSERTION LOSS METHOD Low Pass Filter Prototype – Ladder CircuitFigure 5.7: Ladder circuit for low pass filter prototypes and their element definitions. (a) begin with shunt element. (b) begin with series element.
28 THE INSERTION LOSS METHOD g0 = generator resistance, generator conductance.gk = inductance for series inductors, capacitance for shunt capacitors.(k=1 to N)gN+1 = load resistance if gN is a shunt capacitor, load conductance if gN is a series inductor.As a matter of practical design procedure, it will be necessary to determine the size, or order of the filter. This is usually dictated by a specification on the insertion loss at some frequency in the stopband of the filter.
29 THE INSERTION LOSS METHOD Low Pass Filter Prototype – Maximally FlatFigure 4.8: Attenuation versus normalized frequency for maximally flat filter prototypes.
30 THE INSERTION LOSS METHOD Figure 4.9: Element values for maximally flat LPF prototypes
31 THE INSERTION LOSS METHOD Low Pass Filter Prototype – Equal RippleFor an equal ripple low pass filter with a cutoff frequency ωc = 1, The power loss ratio is:[5.12]Where 1 + k2 is the ripple level in the passband. Since the Chebyshev polynomials have the property that[5.12] shows that the filter will have a unity power loss ratio at ω = 0 for N odd, but the power loss ratio of 1 + k2 at ω = 0 for N even.
32 THE INSERTION LOSS METHOD Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter prototypes. (0.5 dB ripple level)
33 THE INSERTION LOSS METHOD Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple level)
34 THE INSERTION LOSS METHOD Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter prototypes (3.0 dB ripple level)
35 THE INSERTION LOSS METHOD Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple level).
37 FILTER TRANSFORMATIONS Frequency scaling for the low pass filter:[8.14]The new element values of the prototype filter:[8.15a][8.15b]
38 FILTER TRANSFORMATIONS The new element values are given by:[8.16a][8.16b]
39 FILTER TRANSFORMATIONS Low pass to high pass transformationThe frequency substitution:[8.17]The new component values are given by:[8.18a][8.18b]
40 BANDPASS & BANDSTOP TRANSFORMATIONS Low pass to Bandpass transformation[8.19]Where,[8.20]The center frequency is:[8.21]
41 BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, Lk, is transformed to a series LC circuit with element values:[8.22a][8.22b]The shunt capacitor, Ck, is transformed to a shunt LC circuit with element values:[8.23a][8.23b]
42 BANDPASS & BANDSTOP TRANSFORMATIONS Low pass to Bandstop transformation[8.24]Where,The center frequency is:
43 BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, Lk, is transformed to a parallel LC circuit with element values:[8.25a][8.25b]The shunt capacitor, Ck, is transformed to a series LC circuit with element values:[8.26a][8.26b]
45 EXAMPLE 5.1Design a maximally flat low pass filter with a cutoff freq of 2 GHz, impedance of 50 Ω, and at least 15 dB insertion loss at 3 GHz. Compute and compare with an equal-ripple (3.0 dB ripple) having the same order.
46 EXAMPLE 5.1 (Cont) Solution: First find the order of the maximally flat filter to satisfy the insertion loss specification at 3 GHz.We can find the normalized freq by using:
47 EXAMPLE 5.1 (Cont)The ladder diagram of the LPF prototype to be used is as follow:
48 EXAMPLE 5.1 (Cont) pF nH pF nH pF LPF prototype for maximally flat filterpFnHpFnHpF
49 EXAMPLE 5.1 (Cont) pF nH pF nH pF LPF prototype for equal ripple filter:pFnHpFnHpF
50 THE INSERTION LOSS METHOD Low-pass Prototype DesignFilter SpecificationScaling & ConversionNormally done using simulatorsOptimization & TuningFilter Implementation
51 SUMMARY OF STEPS IN FILTER DESIGN Filter SpecificationMax Flat/Equal Ripple,If equal ripple, how much pass band ripple allowed? If max flat filter is to be designed, cont to next stepLow Pass/High Pass/Band Pass/Band StopDesired freq of operationPass band & stop band rangeMax allowed attenuation (for Equal Ripple)
52 SUMMARY OF STEPS IN FILTER DESIGN (cont) Low Pass Prototype DesignMin Insertion Loss level, No of Filter Order/Elements by using IL valuesDetermine whether shunt cap model or series inductance model to useDraw the low-pass prototype ladder diagramDetermine elements’ values from Prototype Table
53 SUMMARY OF STEPS IN FILTER DESIGN (cont) Scaling and ConversionDetermine whether if any modification to the prototype table is required (for high pass, band pass and band stop)Scale elements to obtain the real element values
54 SUMMARY OF STEPS IN FILTER DESIGN (cont) Filter ImplementationPut in the elements and values calculated from the previous stepImplement the lumped element filter onto a simulator to get the attenuation vs frequency response
55 EXAMPLE 5.2Design a band pass filter having a 0.5 dB equal-ripple response, with N = 3. The center frequency is 1 GHz, the bandwidth is 10%, and the impedance is 50 Ω.
56 EXAMPLE 5.2 (Cont) RS L1 L3 C2 RL Solution: The low pass filter (LPF) prototype ladder diagram is shown as follow:= 0.1 N = 3 = 1 GHzRSL1L3C2RL
57 EXAMPLE 5.2 (Cont)From the equal ripple filter table (with 0.5 dB ripple), the filter elements are as follow;
58 EXAMPLE 5.2 (Cont) RS RL C1 C2 C3 L1 L2 L3 Transforming the LPF prototype to the BPF prototypeRSRLC1C2C3L1L2L3
61 EXAMPLE 5.3Design a five-section high pass lumped element filter with 3 dB equal-ripple response, a cutoff frequency of 1 GHz, and an impedance of 50 Ω. What is the resulting attenuation at 0.6 GHz?
62 EXAMPLE 5.3 (Cont) RS RL L1 L3 L5 C2 C3 Solution: The high pass filter (HPF) prototype ladder diagram is shown as follow:N = 5 = 1 GHzAt c = 0.6 GHz, ; referring back to Fig 4.12The attenuation for N = 5, is about 41 dBRSRLL1L3L5C2C3
63 EXAMPLE 5.3 (Cont)From the equal ripple filter table (with 3.0 dB ripple), the filter elements are as follow;
64 EXAMPLE 5.3 (Cont)Impedance and frequency scaling:
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