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Microwave Filters (8) Periodic Structures

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Microwave Filters (8) Periodic Structures Microwave Circuits 1 Microwave Filters (8) Periodic Structures where the ABCD matrix represents the cascade of a transmission line section of length , a shunt susceptance , and another transmission line section of length . where . Microwave Circuits 2 Assume the port voltages and currents satisfy wave equation Then, or For a nontrivial solution to exist, the following must be satisfied That is . The characteristic impedance of this wave is defined as Also Then Microwave Circuits 3 For symmetrical unit cells, , therefore The solutions correspond to the positively and negatively traveling waves, respectively. Due to reciprocity, , the above equation becomes . Let , then 1. a. Nonattenuating, propagating wave. b. Passband. c. Has multiple solutions only when d. is pure real. 2. a. Attenuating, nonpropagating wave. b. Stopband. c. Microwave Circuits 4 Has two solutions only when d. is pure imaginary. Terminated Periodic Structures if . Example 8.1 , slow wave. Microwave Circuits 5 Filter Design by The Image Parameter Method(8.2) = input impedance at port 1 when port is terminated with . = input impedance at port 1 when port is terminated with . From ABCD matrix, we have Solving and gives and . If symmetric, and . If port 2 is terminated in , Voltage ratio equals Microwave Circuits 6 Current ratio Define propagation factor as or Constant-k Filter Sections Low pass filters From Table 8.1, the image impedance equals where and Microwave Circuits 7 1. : Passband. is real. is imaginary. 2. : Stopband. is imaginary. is real. Attenuation rate for is 40 dB/decade. Disadvantages: 1. Slow cutoff. 2. Image impedance is a function of frequency. High Pass Filters Microwave Circuits 8 Microwave Circuits 9 m-Derived Filter Sections Let In order to keep the image impedance the same For a low pass filter, and . Then where Microwave Circuits 10 If , is real, and for . Stopband. When , , implying infinite attenuation. Disadvantages: 1. Slow attenuation when . 2. Image impedance is a function of frequency. Microwave Circuits 11 Erecta: 1. p. 434, in Table 8.1, the propagation constant should be . 2. p. 434, Eq. 8.36, add absolute value to the term . 3. p. 437, Eq. 8.43, add absolute value to the term . Microwave Circuits 12 m-derived - section Goals: 1. Provide the same form of the propagation factor of an m-derived T-section. 2. Provide a less frequency dependent image impedance. Procedures: 1. Use Table 8.1, to find out the equivalent - section that give the same the propagation constants as the T-section. 2. Compute the image impedance. Choose m=0.6 to minimize the variation. Microwave Circuits 13 3. Now . Split the - section to two parts to match . It can be proved that the image impedances equals to and respectively. 4. Let the propagation constant of the original - section be and the split - section . It is obviously . Composite Filters 1. From the given impedance and the cut-off frequency , determine the values of the inductance and the capacitance by . Then, the constant-k T section can be implemented. 2. Determine the m value from the infinite attenuation pole by Microwave Circuits 14 Then m-derived T section can be implemented. 3. Choose m=0.6 as the m value of the bisected - section. Example 8.2: LOW-PASS COMPOSITE FILTER DESIGN Microwave Circuits 15 From Table 8.2 1. Constant-k T section 2. m-derived T section 3. m=0.6 matching section Microwave Circuits 16 Filter Design by the Insertion Loss Method (8.3) Benefits: can synthesize a desired response systematically. Disadvantage: trade-off between the ideal and real responses. Define: insertion loss or power loss ratio, by Since is an even function, Maximally flat: 1. Order: N 2. Low pass, binomial or Butterworth response. 3. Flattest passband. 4. Cutoff frequency: 5. Passband: . 6. Power loss ratio at : . For , 3-dB loss. 7. When , . 8. Attenuation rate: 9. The first (2N-1) derivatives are zero at Equal ripple: 1. Order: N Microwave Circuits 17 2. Low pass. 3. is the Chebyshev polynomial of order N. 4. Equal ripple of amplitude . 5. When , . 6. Sharper cutoff than maximally flat types by a factor . 7. Attenuation rate: . Linear Phase: Then the group delay is maximally flat. Microwave Circuits 18 Maximally Flat Low-Pass Filter Prototype Consider the above circuit, Since We have In order to fit the above equation to a maximally flat low-pass filter with , , and , that is, , It is required that , , and . Solving for , and , we have Microwave Circuits 19 . Procedures: 1. Referring to Fig. 8.26, determine the order from the required attenuation characteristics. 2. By looking up Table 8.3, determine the normalized component values by the following rules: a. is the generator resistance if the ladder circuit starts with a shunt capacitance, or generator conductance if starts with a series inductors. b. is the inductance for series inductors, or capacitance for shunt capacitors. c. the load resistance if is a shunt capacitor, or load conductance if is a series inductor. 3. Impedance and frequency scaling. Let and be the real generator resistance and cutoff frequency respectively, then Microwave Circuits 20 Example 8.3 Design a maximal flat low-pass filter with a cutoff frequency of 2 GHz, impedance of , and at least 15 dB insertion loss at 3 GHz Microwave Circuits 21 Microwave Circuits 22 Equal Ripple Low-Pass Filter Prototype Procedures: 1. Referring to Fig. 8.27, determine the order from the required attenuation characteristics and choose the desired ripple level. 2. Decide the component values by Table 8.4 and follow the same procedure in the maximal flat prototype. Note when N is even, the generator resistance and the load resistance are not equal. 3. Impedance and frequency scaling. Use the same procedure in the maximal flat prototype. Microwave Circuits 23 Microwave Circuits 24 Linear Phase Low-Pass Filter Prototypes Procedure: Same as previous prototypes except using Table 8.5. Filter Transformation (8.4) Goals: 1. Low pass high pass. 2. Low pass band pass. 3. Low pass band stop. Low Pass High Pass Let , then the low pass response becomes a high pass Microwave Circuits 25 response. Procedure: 1. Convert the inductances in the low pass prototypes to capacitances as follows 2. Convert the capacitances in the low pass prototypes to inductances as follows 3. Include the effect of impedance scaling Low Pass Band Pass Let , where and . Then, which is a band pass response with cutoff at and . An inductance in the low pass prototype will converts to a series inductance and a series capacitance as follow Microwave Circuits 26 An capacitance in the low pass prototype will converts to a parallel inductance and a parallel capacitance as follow Notice that both pairs of inductances and capacitance have the same resonant frequency . Low Pass Band Stop Similarly, Let . Convert an inductance in the low pass prototype to a parallel inductance and a parallel capacitance as follow Convert an capacitance in the low pass prototype to a series inductance and a series capacitance as follow Microwave Circuits 27 To sum up Microwave Circuits 28 Example 8.4: BANDPASS FILTER DESIGN N=3, 0.5 dB equal ripple, , , . Microwave Circuits 29 Filter Implementation (8.5) Richard’s Transformation Choose at such that and . A zero occur at . Kuroda’s identities • Physically separate transmission line stubs. • Transform series stubs into shunt stubs, or vice versa. • Change impractical characteristic impedance into more realizable ones. Microwave Circuits 30 Consider Table 8.7(a), Microwave Circuits 31 where . Since the left and the right ABCD matrix must be the same, we have Example 8.5 LOW-PASS FILTER DESIGN USING STUBS , 3 dB equal ripple. Microwave Circuits 32 Impedance and Admittance Inverters Microwave Circuits 33 Stepped-Impedance Low-Pass Filters (8.6) Convert the ABCD matrix of a short transmission line to Z- parameters, we have Using T equivalent circuit, we have If is large . If is small To sum up, Microwave Circuits 34 Inductor: Capacitor: Example 8.7 STEPPED IMPEDANCE FILTER DESIGN , 20 dB attenuation at 4 GHz. Maximal flat. Substrate: Microwave Circuits 35 Filters Using Coupled Resonators (8.8) Bandstop and Bandpass Filters Using Quarter-wave Resonators For a open-circuit transmission line, where for . Let , where . Assume ideal transmission line, then . Now For a series LC circuit near resonance Thus Microwave Circuits 36 For the circuit Microwave Circuits 37 Which should equal to the bandstop prototype To make the two values the same, we need Solving for the above equations, we have Then, where . To sum up, bandstrop filter: open stub with , Microwave Circuits 38 bandpass filter: short stub with Example 8.8 BANDSTOP FILTER DESIGN , N=3, 0.5 dB equal ripple. n 1 1.5963 265.9 2 1.0967 387.0 3 1.5963 265.9 Microwave Circuits 39 Coupled Line Filters Even-Odd Mode Analysis Denote the left side as port 1 and the right side port 2. Let be the even-odd mode voltages and currents. Goal: find out the impedance matrices 1. Even mode: Microwave Circuits 40 2. Odd mode: To sum up, Since We have Microwave Circuits 41 Let port 2 and 4 open, we have Computing the image impedance If , which is real and positive since . Microwave Circuits 42 . Cutoff frequency Propagation constant Passband is real for The image impedance of the above is At , . The propagation constant is Comparing to the original equations, we have with . Solve for the even and odd mode impedance to give Microwave Circuits 43 Microwave Circuits 44 The shunt impedance For a parallel LC circuit near resonance Matching to the bandpass prototype, we have In general: Microwave Circuits 45 Microwave Circuits 46 Example 8.7 COUPLED LINE BANDPASS FILTER DESIGN , N=3, 0.5 dB equal ripple.
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