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1 EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 4: MICROWAVE FILTERS

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2 INTRODUCTION What is a Microwave filter ? linear 2-port network controls the frequency response at a certain point in a microwave system provides perfect transmission of signal for frequencies in a certain passband region infinite attenuation for frequencies in the stopband region a linear phase response in the passband (to reduce signal distortion). f2

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3 INTRODUCTION The goal of filter design is to approximate the ideal requirements within acceptable tolerance with circuits or systems consisting of real components. f1 f3 f2 Commonly used block Diagram of a Filter

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4 INTRODUCTION Why Use Filters? RF signals consist of: 1. Desired signals – at desired frequencies 2. Unwanted Signals (Noise) – at unwanted frequencies That is why filters have two very important bands/regions: 1. Pass Band – frequency range of filter where it passes all signals 2. Stop Band – frequency range of filter where it rejects all signals

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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. Filter LPFBPFHPF ActivePassiveActivePassive

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6 INTRODUCTION Types of Filters 1. Low-pass Filter Passes low freq Rejects high freq f1 f2 f1 2. High-pass Filter Passes high freq Rejects low freq f1 f2

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7 INTRODUCTION 3. Band-pass Filter Passes a small range of freq Rejects all other freq 4. Band-stop Filter Rejects a small range of freq Passes all other freq f1 f3 f2 f1 f3 f1 f2 f3

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8 INTRODUCTION Filter Parameters Pass bandwidth; BW(3dB) = f u(3dB) – f l(3dB) Stop band attenuation and frequencies, Ripple difference between max and min of amplitude response in passband Input and output impedances Return loss Insertion loss Group Delay, quality factor

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9 INTRODUCTION Low-pass filter (passive). A Filter H( ) V1()V1()V2()V2() ZLZL A( )/dB 0 cc (1.1b) (1.1a) cc |H( )| 1 Transfer function Arg(H( ))

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10 INTRODUCTION For impedance matched system, using s 21 to observe the filter response is more convenient, as this can be easily measured using Vector Network Analyzer (VNA). ZcZc Transmission line is optional cc 20log|s 21 ( )| 0dB Arg(s 21 ( )) Filter ZcZc ZcZc ZcZc VsVs a1a1 b2b2 Complex value

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11 INTRODUCTION A( )/dB 0 cc A Filter H( ) V1()V1()V2()V2() ZLZL Passband Stopband Transition band Cut-off frequency (3dB) Low pass filter response (cont)

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12 INTRODUCTION High Pass filter A( )/dB 0 cc cc |H( )| 1 Transfer function Stopband Passband

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13 INTRODUCTION Band-pass filter (passive). Band-stop filter. A( )/dB 40 1 22 oo 11 |H( )| 1Transfer function 22 oo A( )/dB 40 1 22 oo 11 |H( )| 1 Transfer function 22 oo

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14 INTRODUCTION Figure 4.1: A 10 GHz Parallel Coupled Filter Response Pass BW (3dB) Stop band frequencies and attenuation Q factor Insertion Loss

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15 FILTER DESIGN METHODS Filter Design Methods Two types of commonly used design methods: - Image Parameter Method - Insertion Loss Method Image parameter method yields a usable filter However, no clear-cut way to improve the design i.e to control the filter response

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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 [2]. The attenuation A( ) can be cast into power attenuation ratio, called the Power Loss Ratio, P LR, which is related to A( ) 2.

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17 FILTER DESIGN METHODS P LR large, high attenuation P LR close to 1, low attenuation For example, a low-pass filter response is shown below: P LR large, high attenuation P LR close to 1, low attenuation For example, a low-pass filter response is shown below: ZLZL VsVs Lossless 2-port network 1 ZsZs PAPA P in PLPL P LR (f) Low-Pass filter P LR f 1 0 Low attenuation High attenuation fcfc (2.1a)

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18 P LR and s 21 In terms of incident and reflected waves, assuming Z L =Z s = Z C. ZcZc VsVs Lossless 2-port network ZcZc PAPA P in PLPL a1a1 b1b1 b2b2 (2.1b)

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19 FILTER RESPONSES Filter Responses Several types filter responses: - Maximally flat (Butterworth) - Equal Ripple (Chebyshev) - Elliptic Function - Linear Phase

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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 filter c – cutoff frequency of filter N – order of filter

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21 THE INSERTION LOSS METHOD Equal ripple - also known as Chebyshev. - sharper cutoff - the passband response will have ripples of amplitude 1 +k 2 [8.11] – frequency of filter c – cutoff frequency of filter N – order of filter

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22 THE INSERTION LOSS METHOD Figure 5.3: Maximally flat and equal-ripple low pass filter response.

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23 THE INSERTION LOSS METHOD Elliptic function: - have equal ripple responses in the passband and stopband. - 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.

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24 THE INSERTION LOSS METHOD Figure 5.4: Elliptic function low-pass filter response

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25 THE INSERTION LOSS METHOD Filter Specification Low-pass Prototype Design Scaling & Conversion Filter Implementation Optimization & Tuning Normally done using simulators Figure 5.5: The process of the filter design by the insertion loss method.

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26 THE INSERTION LOSS METHOD Figure 5.6: Low pass filter prototype, N = 2 Low Pass Filter Prototype

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27 THE INSERTION LOSS METHOD Figure 5.7: Ladder circuit for low pass filter prototypes and their element definitions. (a) begin with shunt element. (b) begin with series element. Low Pass Filter Prototype – Ladder Circuit

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28 THE INSERTION LOSS METHOD g 0 = generator resistance, generator conductance. g k = inductance for series inductors, capacitance for shunt capacitors. (k=1 to N) g N+1 = load resistance if g N is a shunt capacitor, load conductance if g N 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.

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29 THE INSERTION LOSS METHOD Figure 4.8: Attenuation versus normalized frequency for maximally flat filter prototypes. Low Pass Filter Prototype – Maximally Flat

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30 THE INSERTION LOSS METHOD Figure 4.9: Element values for maximally flat LPF prototypes

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31 THE INSERTION LOSS METHOD For an equal ripple low pass filter with a cutoff frequency ω c = 1, The power loss ratio is: [5.12] Where 1 + k 2 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 + k 2 at ω = 0 for N even. Low Pass Filter Prototype – Equal Ripple

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32 THE INSERTION LOSS METHOD Figure 4.10: Attenuation versus normalized frequency for equal-ripple filter prototypes. (0.5 dB ripple level)

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33 THE INSERTION LOSS METHOD Figure 4.11: Element values for equal ripple LPF prototypes (0.5 dB ripple level)

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34 THE INSERTION LOSS METHOD Figure 4.12: Attenuation versus normalized frequency for equal-ripple filter prototypes (3.0 dB ripple level)

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35 THE INSERTION LOSS METHOD Figure 4.13: Element values for equal ripple LPF prototypes (3.0 dB ripple level).

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36 FILTER TRANSFORMATIONS [8.13a] [8.13b] [8.13c] [8.13d] Low Pass Filter Prototype – Impedance Scaling

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37 FILTER TRANSFORMATIONS The new element values of the prototype filter: Frequency scaling for the low pass filter: [8.14] [8.15a] [8.15b]

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38 FILTER TRANSFORMATIONS The new element values are given by: [8.16a] [8.16b]

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39 FILTER TRANSFORMATIONS Low pass to high pass transformation The frequency substitution: The new component values are given by: [8.17] [8.18a] [8.18b]

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40 BANDPASS & BANDSTOP TRANSFORMATIONS Where, The center frequency is: [8.19] [8.20] [8.21] Low pass to Bandpass transformation

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41 BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, L k, is transformed to a series LC circuit with element values: The shunt capacitor, C k, is transformed to a shunt LC circuit with element values: [8.22a] [8.22b] [8.23a] [8.23b]

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42 BANDPASS & BANDSTOP TRANSFORMATIONS Where, The center frequency is: [8.24] Low pass to Bandstop transformation

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43 BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, L k, is transformed to a parallel LC circuit with element values: The shunt capacitor, C k, is transformed to a series LC circuit with element values: [8.25a] [8.25b] [8.26a] [8.26b]

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44 BANDPASS & BANDSTOP TRANSFORMATIONS

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45 EXAMPLE 5.1 Design 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.

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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:

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47 EXAMPLE 5.1 (Cont) The ladder diagram of the LPF prototype to be used is as follow:

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48 EXAMPLE 5.1 (Cont) pF nH pF nH pF LPF prototype for maximally flat filter

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49 EXAMPLE 5.1 (Cont) pF nH pF nH pF LPF prototype for equal ripple filter:

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50 THE INSERTION LOSS METHOD Filter Specification Low-pass Prototype Design Scaling & Conversion Filter Implementation Optimization & Tuning Normally done using simulators

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51 SUMMARY OF STEPS IN FILTER DESIGN A.Filter Specification 1.Max Flat/Equal Ripple, 2.If equal ripple, how much pass band ripple allowed? If max flat filter is to be designed, cont to next step 3.Low Pass/High Pass/Band Pass/Band Stop 4.Desired freq of operation 5.Pass band & stop band range 6.Max allowed attenuation (for Equal Ripple)

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52 SUMMARY OF STEPS IN FILTER DESIGN (cont) B.Low Pass Prototype Design 1.Min Insertion Loss level, No of Filter Order/Elements by using IL values 2.Determine whether shunt cap model or series inductance model to use 3.Draw the low-pass prototype ladder diagram 4.Determine elements’ values from Prototype Table

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53 SUMMARY OF STEPS IN FILTER DESIGN (cont) C.Scaling and Conversion 1.Determine whether if any modification to the prototype table is required (for high pass, band pass and band stop) 2.Scale elements to obtain the real element values

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54 SUMMARY OF STEPS IN FILTER DESIGN (cont) D.Filter Implementation 1.Put in the elements and values calculated from the previous step 2.Implement the lumped element filter onto a simulator to get the attenuation vs frequency response

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55 EXAMPLE 5.2 Design 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.

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56 EXAMPLE 5.2 (Cont) Solution: The low pass filter (LPF) prototype ladder diagram is shown as follow: = 0.1N = 3 = 1 GHz RSRS L1L1 L3L3 C2C2 RLRL

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57 EXAMPLE 5.2 (Cont) From the equal ripple filter table (with 0.5 dB ripple), the filter elements are as follow;

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58 EXAMPLE 5.2 (Cont) Transforming the LPF prototype to the BPF prototype RSRS RLRL C1C1 C2C2 C3C3 L1L1 L2L2 L3L3

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59 EXAMPLE 5.2 (Cont)

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60 EXAMPLE 5.2 (Cont)

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61 EXAMPLE 5.3 Design 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?

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62 EXAMPLE 5.3 (Cont) Solution: The high pass filter (HPF) prototype ladder diagram is shown as follow: N = 5 = 1 GHz At c = 0.6 GHz, ; referring back to Fig 4.12 The attenuation for N = 5, is about 41 dB RSRS RLRL L1L1 L3L3 L5L5 C2C2 C3C3

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63 EXAMPLE 5.3 (Cont) From the equal ripple filter table (with 3.0 dB ripple), the filter elements are as follow;

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64 EXAMPLE 5.3 (Cont) Impedance and frequency scaling:

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65 EXAMPLE 5.3 (Cont)

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66 EXAMPLE 5.4 Design a 4th order Butterworth Low-Pass Filter. R s = R L = 50Ohm, f c = 1.5GHz. L 1 =0.7654HL 2 =1.8478H C 1 =1.8478F C 2 =0.7654F R L = 1 g 0 = 1 L 1 =4.061nHL 2 =9.803nH C 1 =3.921pF C 2 =1.624pF R L = 50 g 0 =1/50 Step 1&2: LPP Step 3: Frequency scaling and impedance denormalization

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67 EXAMPLE 5.5 Design a 4th order Chebyshev Low-Pass Filter, 0.5dB ripple factor. R s = 50Ohm, f c = 1.5GHz. L 1 =1.6703HL 2 =2.3661H C 1 =1.1926F C 2 =0.8419F R L = g 0 = 1 L 1 =8.861nHL 2 =12.55nH C 1 =2.531pF C 2 =1.787pF R L = 99.2 g 0 =1/50 Step 1&2: LPP Step 3: Frequency scaling and impedance denormalization

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68 EXAMPLE 5.6 Design a bandpass filter with Butterworth (maximally flat) response. N = 3. Center frequency f o = 1.5GHz. 3dB Bandwidth = 200MHz or f 1 =1.4GHz, f 2 =1.6GHz.

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69 EXAMPLE 5.6 (cont) From table, design the Low-Pass prototype (LPP) for 3rd order Butterworth response, c =1. Z o =1 g F g F g H g41g41 2<0 o Step 1&2: LPP

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70 EXAMPLE 5.6 (cont) LPP to bandpass transformation. Impedance denormalization. 50 VsVs pF pF79.58nH nH pF 50 RLRL Step 3: Frequency scaling and impedance denormalization

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