Microwave Circuits Design. Microwave Filters. high pass

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1 Used to control the frequency response at a certain point in a microwave system by providing transmission at frequencies within the passband of the filter and attenuation in the stopband of the filter. Can be found in any type of microwave communication, radar, or test and measurement system. low pass Microwave Filters high pass bandpass bandstop Periodic structures, which consists of a transmission line or waveguide periodically loaded with reactive elements, exhibit the fundamental passband and stopband behavior --- the analysis follows that of the wave propagation in crystalline lattice structures of semiconductor materials. Page of 75

2 . Assume a infinite periodic structure.. Set a unit cell with impedance Z, a length of d and a shunt susceptance b. Periodic Structures Periodic stubs Equivalent circuit of a periodically loaded transmission lines: distributed parameters Periodic diaphragms in a waveguide Page of 75

3 Analysis of periodic structures shows that waves can propagate within certain frequency bands (passbands), but will attenuate within other bands (stopbands). Filter Design By the Image Parameter Method For a reciprocal twoport network on the right, it can be specified by its ABCD parameters. The image impedances are Z i and Z i. Z i = input impedance at port when port is terminated with Z i. Z i = input impedance at port when port is terminated with Z i. Page 3 of 75

4 Using ABCD parameters, we have We can derive Similarly, Z Z in in V I AV CV BI DI V I AV CV AZ CZ BI DI We want to have Z in = Z i and Z in = Z i, which leads to equations for the image impedances: V I DV BI CV AI Zi ( CZi D) AZi Z Solving for Z i and Z i D B B ( A CZ ) i i i Z i Z AB CD i i Z i B D DZ CZ i i BD AC B A Page 4 of 75

5 Then, Zi DZi / A If the network is symmetric, then A=D and Z i = Z i as expected. The voltage transfer ratio is given by ( AD BC ) V V D A The current transfer ratio is given by We can define a propagation factor Where j We can also verify that cosh e I A ( AD BC I D e e AD BC AD Two important types of two-port networks: T and circuits. ) Page 5 of 75

6 ELEC463, Shu Yang, HKUST 6 Page 6 of 75

7 Constant-k Filter Sections (low-pass and high-pass filters) For the T network, we use the results from the image parameters table, and Z jl Z / jc We can derive the image impedance as The cutoff frequency, c, can be defined as A nominal characteristic impedance, R, can be defined as Low-pass constant-k filter sections in T and form. R Z it L C c L C k LC 4 LC constant Page 7 of 75

8 Then, Z it R The propagation factor is c e For =, Z it = R. c c c Frequency response of the low-pass constant-k sections. Slow attenuation rate Page 8 of 75

9 High-pass Constant-k Filter Sections Page 9 of 75

10 m-derived Filter Sections A modification to overcome the disadvantages of slow attenuation after cutoff and frequency-dependent image impedances. Z' mz In order to have the same Z it, we have Z' Z m ( m 4m Basic concept: create a resonator along the shunt path. The resonant frequency here should be slightly high than the cutoff frequency. ) Z Page of 75

11 Cutoff frequency is still c m LC The resonant frequency of the shunt path is m mc 4m m is restricted into to the range of c < m <. L Low-pass High-pass Steep decrease of after > is not desirable. This problem can be solved by cascading with another constant-k section to give a composite response shown in the figure. ELEC463, Shu Yang, HKUST Page of 75

12 The T-section still have the problem of a nonconstant image impedances. m-derived Filter Section Now consider the - equivalent as a piece of an infinite cascade of m- derived T-sections. Then, Z i Z' Z' Z Z Z it R Z ( m / c )/ 4 Infinite cascade of m-derived T-section. A de-embedded -equivalent. Page of 75

13 Since Z Z L / C R and Z L 4 R / c We have Z i ( m ) / / * m provides another freedom to design Z i so that we can minimize the variation of Z i over the passband of the filter. Variation of Z i in the pass band of a low-pass m-derived section for various values of m. A value of m=.6 generally gives the best results --- nearly constant impedance match to and from R. c c R Page 3 of 75

14 How to match the constant-k and m-derived T-sections to - section? Using bisected -section. It can be shown that Z Z' 4 i Z' Z' Z it Z i Z' Z' Z' Z' Z' / 4Z' Z it Z i Page 4 of 75

15 Composite Filters The sharp-cutoff section, with m<.6, places an attenuation pole near the cutoff frequency to provide a sharp attenuation response. The constant-k section provides high attenuation further into the stopband. The bisected- sections at the ends match the nominal source and load impedance, R, to the internal image impedances. The composite filter design is obtained from three parameters: cutoff frequency, impedance, and infinite attenuation frequency (or m). Page 5 of 75

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17 Example 8. of Pozar: Low-Pass Composite Filter Design The series pairs of inductors between the sections can be combined. The selfresonance of the bisected p- section will provide additional attenuation. Cutoff freq. Frequency response ELEC463, Shu Yang, HKUST Page 7 of 75

18 Filter Design By the Insertion Loss Method What is a perfect filter? Zero insertion loss in the passband, infinite attenuation in the stopband, and linear phase response (to avoid signal distortion) in the passband. No perfect filters exist, so compromises need to be made. The image parameter method have very limited freedom to nimble around. The insertion loss method allows a high degree of control over the passband and stopband amplitude and phase characteristics, with a systematic way to synthesize a desired response. ELEC463, Shu Yang, HKUST Page 8 of 75

19 Characterization by Power Loss Ratio The power loss ratio and insertion loss of a filter are defined as: P LR Power available from source Power delivered to load P P inc load ( ) IL log P LR When both load and source are matched, PLR = S ^ Since () is an even function of, it can be expressed as a polynomial in. M ( ) ( ) M ( ) N( ) Where M and N are real polynomials in. So the power loss ratio can be given as P LR M ( ) N( ) Page 9 of 75

20 Filter Design by the Insertion Loss Method Several Types of Filter Response: Maximally flat: binomial or Butterworth response Provide the flattest possible passband response. For a low-pass filter, it is specified by N P LR k c Where N is the order of the filter, and c is the cutoff frequency. At the band edge the power loss ratio is +k. Maximally flat means that the the first (N-) derivatives of the power loss ratio are zero at =. Equal ripple: A Chebyshev polynomial is used to represent the insertion loss of an N-order low-pass filter Provide the sharpest cutoff, with ripples in the passband. Page of 75

21 The insertion loss is: P Ripple amplitude: + k For >> c, the insertion loss is, Microwave Circuits Design k LR T N c P LR k 4 c N Insertion loss rises faster in the stopband compared to the binomial filters. Both the maximally flat and equal-ripple responses both have monotonically increasing attenuation in the stopband --- not necessary in applications. Page of 75

22 Elliptic function: Equal ripple responses in the passband and the stopband Specified by the maximum attenuation in the passband, A max, as well as the minimum attenuation in the stopband, A min. See Figure 8.. Better cutoff rate! Adequate attenuation! Page of 75

23 Linear phase: a linear phase response in the passband, to avoid signal distortion, generally incompatible with a sharp cutoff response. A linear phase response: N c p A ) ( Group delay: N c d N p A d d ) ( Where p is a constant. Group delay is a maximally flat function. Microwave Circuits Design Page 3 of 75

24 The process of filter design by the insertion loss method Maximally Flat Low-Pass Filter Prototype (using normalized element values) Assume a source impedance of, and a cutoff frequency c =. For a N=, the desired power loss ratio is. P LR 4 Input impedance Z in jl R( jrc) R C and Z Z in in Page 4 of 75

25 Power loss ratio is P LR [( 4R P LR Solve P LR for R, L, C,, we have Z ( Z in in * Z in R) ( R C L LCR ) L R C P LR Compare this expression with 4 We have, R L C L C This procedure can be extended to find the element values for filters with an arbitrary number of elements. Design Table 8.3 of Pozar gives the component value for N= to. ) 4 ] Page 5 of 75

26 Ladder circuits for low-pass filter prototypes and their element definitions. Prototype beginning with a shunt element. Prototype beginning with a series element. Page 6 of 75

27 g k definition: Page 7 of 75

28 What to design?. The order (size) of the filter N: decided by a specification on the insertion loss at some frequency in the stopband.. The value of each component. See table 8.3 Page 8 of 75

29 Equal-Ripple Low-Pass Filter Prototype Ripple level has to be specified. Table 8.4 and Figure 8.7. Figure 8.7a (Ed. 4, p. 47) Attenuation versus normalized frequency for equal-ripple filter prototypes. (a).5 db ripple level. Adapted from G.L. Mattaei et al., Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 98) Page 9 of 75

30 Figure 8.7b (Ed. 4, p. 47) Attenuation versus normalized frequency for equal-ripple filter prototypes. (b) 3. db ripple level. Adapted from G.L. Mattaei et al., Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 98) Page 3 of 75

31 Equal-Ripple Low-Pass Filter Prototype Ripple level has to be specified. Table 8.4 and Figure 8.7. Linear Phase Low-Pass Filter Prototypes Table 8.5. Filter Transformations: Scaling in terms of impedance and frequency Conversion to high-pass, bandpass, or bandstop filters. Impedance and Frequency Scaling With a source resistance, R, we have the scaling rule given by L' RL C' C / R R ' R S Yang, HKUST RL' R R L Page 3 of 75

32 Frequency Scaling for Low-Pass Filters Scale the frequency response dependence by the factor / c. P' LR () P LR c Combining impedance and frequency scaling, we have L' k L k L / c ' k RL k c C' k C k / / C C /( R ) ' k k c c Low pass filter for c = Frequency scaling for low-pass Transformation to high-pass response. Page 3 of 75

33 Low-Pass to High-Pass Transformation: c The negative sign is needed to convert inductors (and capacitors) to realizable capacitors (and inductors). The reactance and susceptance become: jx k c j Lk jc' We obtain the conversion rules given by: C' k k /( L ) L' /( C ) c k Microwave Circuits Design k c c jbk j Ck k jl' So the series inductors are replaced with capacitors and shunt capacitors are replaced with inductors. k ELEC463, Shu Yang, HKUST Page 33 of 75

34 After including the impedance scaling, we have C' k /( R clk ) L' k R /( cck ) Example 8.3 of Pozar. Page 34 of 75

35 Conversion Rules for bandpass filters: where Microwave Circuits Design Bandpass and Bandstop Transformation is the fractional bandwidth of the passband. Low-pass filter prototype. Transformation to bandpass response. Transformation to bandstop response. Page 35 of 75

36 If we take the geometric mean for the center frequency, We have, when, when, when, The mapping between the low-pass prototype and the bandpass filter is complete. ELEC463, Shu Yang, HKUST Microwave Circuits Design Page 36 of 75

37 Then the new filter elements are given by performing the conversion k k k k k k C j L j L j L j L j jx ' ' This indicates that, a series inductor, L k, is transformed to a series LC circuit with element values, k k L C ' ' k k L L Similarly, for a shunt susceptance, we have k k k k k k L j C j C j C j C j jb ' ' The shunt capacitor, is transformed to a shunt (parallel) LC circuit with element values, k k C L ' ' k k C C Resonance at ω Resonance at ω Microwave Circuits Design Page 37 of 75

38 Conversion Rules for bandstop filters: Page 38 of 75

39 Example 8.4 of Pozar: Bandpass Filter Design ELEC463, Shu Yang, HKUST 5 Page 39 of 75

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41 Filter Implementation The lumped-element filter design works well at low frequencies, but imposes problems at microwave frequencies. Lumped inductors and capacitors are available only for a limited range of values and are difficult to implement at microwave frequencies which requires smaller inductance and capacitance values. At microwave frequencies, the distances between filter components are not negligible. The conversion from lumped elements to transmission line sections is needed --- Richard s Transformation. Page 4 of 75

42 Richard s Transformation and Kuroda s Identities l tan l tan Richard s transformation: This transformation maps the plane to the plane, which repeats with a period of l/v p =. The transformation is used to synthesize an LC network using open- and short-circuited transmission lines. v p If we replace the frequency variable with, the reactance of an inductor can be written as jx L jl jl tan l This is equivalent to a short-circuited stub of length l and characteristic impedance L. Page 4 of 75

43 And the susceptance of a capacitor can be written as jb C jc jc tan l This is equivalent to a open-circuited stub of length l and characteristic impedance /C. For a low-pass filter prototype, cutoff frequency is unity. According to Richard s transformation, we have tan l Which gives a stub size of l = /8, where is the wavelength of the line at cutoff frequency, C. Page 43 of 75

44 Inductors and capacitors can be replaced with /8 lines: The /8 transmission line sections are called commensurate lines, since they are all the same length in a given filter. Page 44 of 75

45 At the frequency = C, the lines will be /4 long, and an attenuation pole will occur. At frequencies away from C, the impedances of the stubs will no longer match the original lumpedelement impedances, and the filter response will differ from the desired prototype response. Also, the response will be periodic in frequency, repeating every 4 C. Page 45 of 75

46 Richard's Transformation and Kuroda's Identities focus on uses of /8 lines, for which the reactance jx = jz o. Richard's idea is to use variable Z o (width of microstrip, for example) to create lumped elements from transmission lines. A lumped low-pass prototype filter can be implemented using /8 lines of appropriate Z o to replace lumped L and C elements. So if we need an inductance of L for a prototype filter normalized to cutoff frequency c = and admittance g o =, we can substitute a /8 transmission line stub that has Z o = L. The last step of the filter design will be to scale the design to the desired c and Z o (typically 5). Page 46 of 75

47 Kuroda's idea is use the redundant /8 line of appropriate Z o to transform awkward or unrealizable elements to those with more tractable values and geometry. As an example, the series inductive stub in the diagram here can be replaced by a shunt capacitive stub on the other end of the /8 line, with different values of characteristic impedance determined by Z Z Kuroda s identities can do the following operations: Physically separate transmission line stubs Transform series stubs into shunt stubs, or vice versa k Change impractical characteristic impedances into more realizable ones n Page 47 of 75

48 L represents short-circuit stub; C represents open-circuit stub 8 Page 48 of 75

49 Illustration of Kuroda identity for stub conversion Page 49 of 75

50 Illustration of Kuroda identity for stub conversion Page 5 of 75

51 Results in (8.8a) and (8.8b) are identical if n = + Z /Z Page 5 of 75

52 (a) (b) (b) is the most commonly used identity, which removes a series stub (difficult to implement in microstrip line form)by transforming it to a shunt stub along with adjustment of characteristic impedances of the /8 lines. Page 5 of 75

53 Low Pass Filter Using Stubs The prototype lowpass LC structure employs series inductors, so a direct conversion to transmission line stubs by Richard's transformation would result in series stubs. However, we can use the Kuroda identity for series inductors to create a structure that has only series transmission line sections and shunt open stubs. In order to do this we must be aware that we should begin by adding unit elements (/8 transmission lines of Z o = ) at each end of the filter, so that there will be structures that are of the form of the Kuroda identities. The filter is designed by the following steps: Lumped element low pass prototype (from tables, typically) Convert series inductors to series stubs, shunt capacitors to shunt stubs Add /8 lines of Z o = at input and output Apply Kuroda identity for series inductors to obtain equivalent with shunt open stubs with /8 lines between them Transform design to 5 and f c to obtain physical dimensions (all elements are /8). Page 53 of 75

54 ELEC463, Shu Yang, HKUST 4 Page 54 of 75

55 (b) Using Richards transformations to convert inductors and capacitors to series and shunt stubs. Page 55 of 75

56 (c) Adding unit elements at the ends of the filter. ELEC463, Shu Yang, HKUST 6 Page 56 of 75

57 Figure 8.36 (p. 4) (d) Applying the second Kuroda identity. (e) After impedance and frequency scaling. (f) Microstrip fabrication of final filter. Page 57 of 75

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59 Stepped-Impedance Low-Pass Filters Realized with alternating sections of very high and very low characteristics impedance lines. Easier to design and take up less space compared to a similar lowpass stub filter. Easy implementation results in poorer performance such as slow cutoff. Consider the T-section equivalent circuit of a short section (l<</) of transmission line, as determined from conversion of the ABCD parameters to Z parameters to identify the individual elements. X B Z Z l tan sin l Page 59 of 75

60 For high Z o and small l the equivalent circuit becomes For low Z o and small l, the equivalent circuit becomes So the series inductors can be replaced with high-impedance line sections and the shunt capacitors can be replaced with lowimpedance line sections. In order to use this approximation, we need to know the highest and lowest feasible transmission line impedances, Z h and Z l. After considering the impedance scaling, we have LR l Z h X B X B Z βl Y βl (inductor) l CZ R l (capacitor) Page 6 of 75

61 The ratio of Z h /Z l should be as high as possible, so the actual values of Z h and Z l are usually set to the highest and lowest characteristic impedance that can be practically fabricated. Page 6 of 75

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65 Impedance and Admittance Inverters In this process we've uncovered another "magic bullet" comparable to the Kuroda identities, only involving /4 rather than /8 lines. Quarter wave lines can transform series connected element to shunt, and vice versa. Such inverters are especially useful for bandpass or bandstop filters with narrow bandwidths. For the impedance (K) inverter, Z in = K /Z L For the λ/4 line, K = Zo Microwave Circuits Design For the lumped element implementation, K = Zo tan /, K X = -(K/Zo) = -tan- X Zo 6 Page 65 of 75

66 For the admittance inverter, Y in = J /Y L For the λ/4 line, J = Y o For the lumped element implementation, J = Y o tan /, J B = -(J/Y o ) B = -tan- Y o Various implementation schemes * Negative values of, the length of the transmission line sections, poses no problems because they can be absorbed into connecting transmission lines on either side. The same is true for the L and C with negative values ELEC463, Shu Yang, HKUST Page 66 of 75

67 What can be achieved by the impedance and admittance inverter? Form the inverse of the load impedance or admittance. Can be used to transform series-connected elements to shunt-connected elements. Impedance inverters may be used to convert a bandpass-filter network into a network containing only series tuned circuits. Admittance inverters may be used to convert a bandpass-filter network into a network containing only shunt tuned circuits ELEC463, Shu Yang, HKUST Page 67 of 75

68 (a) Impedance inverter used to convert a parallel admittance into an equivalent series impedance. (b) Admittance inverter used to convert a series impedance into an equivalent parallel admittance. ELEC463, Shu Yang, HKUST 9 Page 68 of 75

69 Bandstop and Bandpass Filters Using Quarter- Wave Resonators Quarter-wave opencircuited or shortcircuited transmission line stubs: series or parallel LC resonators, respectively. Page 69 of 75

70 Operating Principle /4 sections between the stubs act as admittance inverters to effectively convert alternate shunt resonators to series resonators. Bandstop filter using opencircuited stubs Page 7 of 75

71 The input impedance of an open-circuited transmission line of characteristic impedance Z n is Near resonance, Therefore, Z jz n tan Z jz n / ( / ) The impedance of a series LC circuit is, jz n cot ( ) L n ω ω Z jωln j jωcn Cn ω ω Ln ( ω ω) j jln( ωω) C ω n Page 7 of 75

72 The characteristic impedance of the stub is then given by, 4 Z L n Equivalent lumped-element bandstop filter n Finding correlation between (b) and (c) ELEC463, Shu Yang, HKUST Page 7 of 75

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