EKT 356 MICROWAVE COMMUNICATIONS CHAPTER 4: MICROWAVE FILTERS

EKT 356 MICROWAVE COMMUNICATIONS CHAPTER 4: MICROWAVE FILTERS 1

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). 2

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

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 4

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 LPF HPF BPF Active Passive Active Passive 5

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

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

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 8

INTRODUCTION Low-pass filter (passive). V 1 (ω) A Filter H(ω) V 2 (ω) Z L 1 H(ω) Transfer function ( ω ) V V 2 ( ω ) ( ω ) H = (1.1a) 1 Arg(H(ω)) ω c ω A(ω)/dB 50 40 ω 30 20 10 3 0 ω c Attenuatio n A ω = 20Log10 V2 V1 ( ω) ( ω) (1.1b) 9

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). a 1 b 2 V s Z c Z c Filter Z c Z c 0dB 20log s 21 (ω) Arg(s 21 (ω)) Transmission line is optional ω c ω ω s 11 = b a b 1 s 2 21 = 1 a a 0 1 2 = a2 = 0 Complex value 10

INTRODUCTION Low pass filter response (cont) 50 40 A(ω)/dB Passband Transition band 30 20 10 3 0 ω c Stopband ω Cut-off frequency (3dB) V 1 (ω) A Filter H(ω) V 2 (ω) Z L 11

INTRODUCTION High Pass filter 1 H(ω) Transfer function 50 40 A(ω)/dB Passband 30 20 10 ω c ω 3 0 ω c ω Stopband 12

INTRODUCTION Band-pass filter (passive). Band-stop filter. A(ω)/dB A(ω)/dB 40 40 30 30 20 20 10 3 0 ω 2 ω 1 ω o ω 10 3 0 ω 1 ω o ω 2 ω H(ω) 1 Transfer function 1 H(ω) Transfer function ω ω ω 1 ω o ω 2 ω 1 ω o ω 2 13

INTRODUCTION Pass BW (3dB) Insertion Loss 0 Filter R esponse Q factor -10-20 7.9024 GHz -3.0057 db 12.124 GHz -3.0038 db -30-40 Input R eturn Loss Insertion Loss -50 6 8 10 12 14 Frequency (G Hz) Figure 4.1: A 10 GHz Parallel Coupled Filter Response Stop band frequencies and attenuation 14

FILTER DESIGN METHODS Filter Design Methods Two types of commonly used design methods: 1. Image Parameter Method 2. 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 15

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(ω), A(ω) or s21(ω). 16

Filter Design Methods 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(ω) following [2]. The attenuation A(ω) can be cast into power attenuation ratio, called the Power Loss Ratio, PLR, which is related to A(ω).

FILTER DESIGN METHODS Z s V s Lossless 2-port network P A P in P L Γ 1 (ω) P Power available from source network LR = Power delivered to Load = Pinc P = A = 1 P Load ( ) 2 P 1 Γ1 ω 1 Γ1 ( ω) 2 A (2.1a) Z L P LR large, LR large, high high attenuation P LR close LR close to to 1, 1, low low attenuation For For example, a low-pass filter filter response is is shown shown below: below: P LR (f) High attenuation 1 Low attenuation Low-Pass filter P LR 0 f c f 18

P LR and s 21 In terms of incident and reflected waves, assuming Z L =Z s = Z C. b 1 a 1 b 2 V s Z c Lossless 2-port network P A P in P L Z c PLR PLR = = PA PL = 1 2 s21 1 2 a 2 1 1 2 b 2 2 = a1 b2 (2.1b) 2 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 P LR = + k ω frequency of filter ω c cutoff frequency of filter N order of filter ω ωc N 2 1 [4.1] 21

THE INSERTION LOSS METHOD Equal ripple - also known as Chebyshev. - sharper cutoff - the passband response will have ripples of amplitude 1 +k 2 2 P = + k T LR N 2 ω ω c 1 4.2] ω frequency of filter ω c cutoff frequency of filter N order of filter 22

THE INSERTION LOSS METHOD Figure 4.2: Maximally flat and equal-ripple low pass filter response. 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. 24

THE INSERTION LOSS METHOD Figure 4.3: Elliptic function low-pass filter response 25

THE INSERTION LOSS METHOD Filter Specification Low-pass Prototype Design Normally done using simulators Scaling & Conversion Optimization & Tuning Filter Implementation Figure 4.4: The process of the filter design by the insertion loss method. 26

THE INSERTION LOSS METHOD Low Pass Filter Prototype Figure 4.5: Low pass filter prototype, N = 2 27

THE INSERTION LOSS METHOD Low Pass Filter Prototype Ladder Circuit Figure 4.6: 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 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. 29

THE INSERTION LOSS METHOD Low Pass Filter Prototype Maximally Flat Figure 4.7: Attenuation versus normalized frequency for maximally flat filter prototypes. 30

THE INSERTION LOSS METHOD Figure 4.8: Element values for maximally flat LPF prototypes 31

THE INSERTION LOSS METHOD Low Pass Filter Prototype Equal Ripple For an equal ripple low pass filter with a cutoff frequency ω c = 1, The power loss ratio is: P = 1+ k 2 T 2( ω) [4.3] LR N Where 1 + k 2 is the ripple level in the passband. Since the Chebyshev polynomials have the property that T N ( ω) 0 = 1 [4.3] 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 : two cases to consider depending on N 32

THE INSERTION LOSS METHOD Figure 4.9: Attenuation versus normalized frequency for equal-ripple filter prototypes. (0.5 db ripple level) 33

THE INSERTION LOSS METHOD Figure 4.10: Element values for equal ripple LPF prototypes (0.5 db ripple level) 34

THE INSERTION LOSS METHOD Figure 4.11: Attenuation versus normalized frequency for equal-ripple filter prototypes (3.0 db ripple level) 35

THE INSERTION LOSS METHOD Figure 4.12: Element values for equal ripple LPF prototypes (3.0 db ripple level). 36

EXAMPLE 4.1 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. 37

FILTER TRANSFORMATIONS Impedance scaling: In the prototype design, the source and load resistance are unity (except for equal ripple filters with even N, which have non unity load resistance). 38

FILTER TRANSFORMATIONS Low Pass Filter Prototype Impedance Scaling L C R R ' ' ' s ' L = R L [4.4a] = = 0 C R R = R 0 0 0 R [4.4b] [4.4c] [4.4d] L 39

FILTER TRANSFORMATIONS Low Pass Filter Prototype Frequency Scaling Frequency scaling: To change the cut-off frequency of a LP prototype from unity to ω c requires to scale the frequency dependence of the filter by the factor 1/ ω c which is accomplished by replacing ω by ω/ω c Frequency scaling for the low pass filter: ω ω ω c 40

FILTER TRANSFORMATIONS The new Power Loss Ratio, P LR P LR (ω) = P LR (ω/ω c ) [4.5] Cut off occurs when ω/ωc = 1 or ω = ωc The new element values of the prototype filter: jx jb k k = = j j ω ω ω ω c c L C k k = = jω L jω C ' k ' k [4.6] [4.7] 41

FILTER TRANSFORMATIONS The new element values are given by: L C ' k ' k L R L = k = 0 k [4.8a] = ω C k ω = ω R c C 0 k ω c [4.8b] 42

FILTER TRANSFORMATIONS Low pass to high pass transformation The frequency substitution: C L ' k ωc ω ω The new component values are given by: ' 1 k = R 0ω c = R 0 ω C c k L k [4.9] [4.10a] [4.10b] 43

BANDPASS & BANDSTOP TRANSFORMATIONS Low pass to Bandpass transformation ω 0 ω ω0 1 ω ω0 ω = ω2 ω1 ω0 ω ω0 ω Where, = The center frequency is: ω ω 2 ω 0 1 [4.11] Centre freq. [4.12] Edges of passband ω = ω ω 0 1 2 [4.13] 44

BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, L k, is transformed to a series LC circuit with element values: ' L k L k = [4.14a] ω C ' k = ω L 0 0 The shunt capacitor, C k, is transformed to a shunt LC circuit with element values: ' [4.15a] L C k ' k = ω 0 k C C k = ω 0 k [4.14b] [4.15b] 45

BANDPASS & BANDSTOP TRANSFORMATIONS Low pass to Bandstop transformation Where, ω ω ω0 = 2 ω 0 ω 0 ω ω ω 1 1 [4.16] The center frequency is: ω = ω ω 0 1 2 46

BANDPASS & BANDSTOP TRANSFORMATIONS The series inductor, L k, is transformed to a parallel LC circuit with element values: ' Lk Lk = [4.17a] ω0 ' 1 Ck = ω L [4.17b] 0 The shunt capacitor, C k, is transformed to a series LC circuit with element values: ' 1 [4.18a] L C k ' k = = 0 C ω 0 k ω C k k [4.18b] 47

BANDPASS & BANDSTOP TRANSFORMATIONS 48

THE INSERTION LOSS METHOD Filter Specification Low-pass Prototype Design Normally done using simulators Scaling & Conversion Optimization & Tuning Filter Implementation 49

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) 50

SUMMARY OF STEPS IN FILTER DESIGN (cont) B. Low Pass Prototype Design 1. Min Insertion Loss level, Number of Filter Order/Elements by using IL values 2. Determine whether shunt capacitance model or series inductance model to use 3. Draw the low-pass prototype ladder diagram 4. Determine elements values from Prototype Table 51

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 52

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 53

EXAMPLE 4.2 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. 54

EXAMPLE 4.3 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? 55

Filter Realization Using Distributed Circuit Elements (1) Lumped-element filter realization using surface mounted inductors and capacitors generally works well at lower frequency (at UHF, say < 3 GHz). At higher frequencies, the practical inductors and capacitors loses their intrinsic characteristics. A limited range of component values : available from manufacturer difficult design at microwave freq. Therefore for microwave frequencies (> 3 GHz), passive filter is usually realized using distributed circuit elements such as transmission line sections. 56

Cont d Distributed elements : open cct TL stubs or short cct TL stubs. At microwave freq, the distance between filter components is not negligible. Richard s transformation: Can be used to convert lumped elements to TL sections Kuroda s identities: Can be used to physically separate the filter elements by using TL sections. The four kuroda s identities operations: Physically separate transmission line stubs Transform series stubs into shunt stubs or vice versa Change impractical characteristic impedances into more realizable values. 57

Filter Realization Using Distributed Circuit Elements (2) Recall in the study of Terminated Transmission Line Circuit that a length of terminated Tline can be used to approximate an inductor and capacitor. This concept forms the basis of transforming the LC passive filter into distributed circuit elements. l Z c, β L l Z c, β C Z o Z o Z o Z o 58

Filter Realization Using Distributed Circuit Elements (3) This approach is only approximate. There will be deviation between the actual LC filter response and those implemented with terminated Tline. Also the frequency response of distributed circuit filter is periodic. Other issues are shown below. How do we implement series Tline connection? (only practical for certain Tline configuration) Z o Connection physical length cannot be ignored at microwave region, comparable to λ Thus Thus some some theorems are are used used to to facilitate the the transformation of of LC LC circuit circuit into into stripline microwave circuits. Chief Chief among among these these are are the the Kuroda s Identities (See (See Appendix) Z o 59

More on Approximating L and C with Terminated Tline: Richard s Transformation l Z in ( ) Z c, β L ( βl) = jωl jlω Zin = jzc tan = tan βl = ω Z c = L (3.1.1a) Z in l Z c, β For LPP design, a further requirment is that: tan ( βl ) ω = 1 = c C tan ( βl) = jωc jcω ( βl) = ω Yin = jyc tan = tan Y = 1 C (3.1.1b) 2π λc l c = Zc = 1 l = λc 8 Wavelength at cut-off frequency (3.1.1c) 60

More on Approximating L and C with Terminated Tline: Richard s Transformation 61

Kuroda s Identities As taken from [2]. l n 2 = 1+ Z2 Z1 l Note: The inductor represents shorted Tline while the capacitor represents open-circuit Tline. 1 Z2 Z 1 β Z 2 /n 2 l l β Z1 n Z 1 Z 2 β n 2 Z 1 β 1 n 2 Z2 l l 1: n 2 Z 1 Z 2 β Z 2 /n 2 β Z1 n 2 l l n 2 : 1 1 Z2 Z 1 β n 2 Z 1 β 1 n 2 Z2 62

Example LPF Design Using Stripline Design a 3rd order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc = 1.5GHz. Step 1 & 2: LPP Z o =1 g 1 1.000H g 3 1.000H g 2 2.000F g 4 1 1 = 2.000 0.500 Length = λ c /8 for all Tlines at ω = 1 rad/s 63

Example LPF Design Using Stripline Design a 3rd order Butterworth Low-Pass Filter. Rs = RL= 50Ohm, fc = 1.5GHz. Step 3: Convert to Tlines using Richard s Transformation Length = λ c /8 for all Tlines at ω = 1 rad/s 1 = 2.000 0.500 64

Example Cont Step 4: Add extra Tline on the series connection Extra T-lines Length = λ c /8 for all Tlines at ω = 1 rad/s 65

Example Cont Step 5: apply Kuroda s 1 st Identity. Step 6: apply Kuroda s 2 nd Identity. Similar operation is performed here 66

Example Cont After applying Kuroda s Identity. Length = λ c /8 for all Tlines at ω = 1 rad/s Since Since all all Tlines Tlineshave similar similar physical length, length, this this approach to to stripline filter filter implementation is is also also known known as as Commensurate Line Line Approach. 67

Example Cont Step 5: Impedance and frequency denormalization. Microstrip line using double-sided FR4 PCB (ε r = 4.6, H=1.57mm) Z c /Ω λ/8 @ 1.5GHz /mm W /mm 50 13.45 2.85 25 12.77 8.00 100 14.23 0.61 Length = λ c /8 for all Tlines at f = f c = 1.5GHz 68

Example Cont Step 6: The layout (top view) 69

Example 2 Design a low pass filter for fabrication using microstrip lines. The specifications are: cutoff freq of 4 GHz, third order, impedance of 50 ohms and a 3dB equal ripple characteristics g 1 = 3.3487 = L 1 g 2 = 0.7117 = C 2 g 3 = 3.3487 = L 3 g 4 = 1.0000 = R L 70

Example 2: Richard s Transformation 71

Example 2: Cont d UE =1 (Z2 =1) Z1 =3.3487 Z1 =3.3487 UE =1 (Z2 =1) 72

Cont d UE UE 73

Example 5.8 (cont) Kuroda s Identity 74

Example 5.8 (cont) 75

Kuroda Identities 76