CHAPTER 10 Advanced RF/Microwave Filters There have been increasing demands for advanced RF/microwave filters other than conventional Chebyshev filters in order to meet stringent requirements from RF/mi- crowave systems, particularly from wireless communications systems. In this chap- ter, we will discuss the designs of some advanced filters. These include selective fil- ters with a single pair of transmission zeros, cascaded quadruplet (CQ) filters, trisection and cascaded trisection (CT) filters, cross-coupled filters using transmis- sion line inserted inverters, linear phase filters for group delay equalization, extract- ed-pole filters, and canonical filters. 10.1 SELECTIVE FILTERS WITH A SINGLE PAIR OF TRANSMISSION ZEROS 10.1.1 Filter Characteristics The filter having only one pair of transmission zeros (or attenuation poles) at finite frequencies gives much improved skirt selectivity, making it a viable intermediate between the Chebyshev and elliptic-function filters, yet with little practical difficul- ty of physical realization [1–4]. The transfer function of this type of filter is |S 21 (⍀)| 2 = = (10.1) F n (⍀) = cosh Ά (n – 2)cosh –1 (⍀) + cosh –1 + cosh –1 · where ⍀ is the frequency variable that is normalized to the passband cut-off fre- quency of the lowpass prototype filter, is a ripple constant related to a given return ⍀ a ⍀ + 1 ᎏ ⍀ a + ⍀ ⍀ a ⍀ – 1 ᎏ ⍀ a – ⍀ 1 ᎏᎏ ͙ 1 ෆ 0 ෆ – ෆ L R — 10 ෆ ෆ – ෆ 1 ෆ 1 ᎏᎏ 1 + 2 F n 2 (⍀) 315 Microstrip Filters for RF/Microwave Applications. Jia-Sheng Hong, M. J. Lancaster Copyright © 2001 John Wiley & Sons, Inc. ISBNs: 0-471-38877-7 (Hardback); 0-471-22161-9 (Electronic) loss L R = 20 log|S 11 | in dB, and n is the degree of the filter. It is obvious that ⍀ = ±⍀ a (⍀ a > 1) are the frequency locations of a pair of attenuation poles. Note that if ⍀ a Ǟ ϱ the filtering function F n (⍀) degenerates to the familiar Chebyshev func- tion. The transmission frequency response of the bandpass filter may be determined using frequency mapping, as discussed in Chapter 3, i.e., ⍀ = · – in which is the frequency variable of bandpass filter, 0 is the midband frequency and FBW is the fractional bandwidth. The locations of two finite frequency attenua- tion poles of the bandpass filter are given by a1 = 0 (10.2) a2 = 0 Figure 10.1 shows some typical frequency responses of this type of filter for n = 6 and L R = –20 dB as compared to that of the Chebyshev filter. As can be seen, the improvement in selectivity over the Chebyshev filter is evident. The closer the atten- uation poles to the cut-off frequency (⍀ = 1), the sharper the filter skirt and the higher the selectivity. ⍀ a FBW + ͙ (⍀ ෆ a F ෆ B ෆ W ෆ ) 2 ෆ + ෆ 4 ෆ ᎏᎏᎏ 2 –⍀ a FBW + ͙ (⍀ ෆ a F ෆ B ෆ W ෆ ) 2 ෆ + ෆ 4 ෆ ᎏᎏᎏ 2 0 ᎏ ᎏ 0 1 ᎏ FBW 316 ADVANCED RF/MICROWAVE FILTERS FIGURE 10.1 Comparison of frequency responses of the Chebyshev filter and the design filter with a single pair of attenuation poles at finite frequencies (n = 6). 10.1.2 Filter Synthesis The transmission zeros of this type of filter may be realized by cross coupling a pair of nonadjacent resonators of the standard Chebyshev filter. Levy [2] has developed an approximate synthesis method based on a lowpass prototype filter shown in Fig- ure 10.2, where the rectangular boxes represent ideal admittance inverters with characteristic admittance J. The approximate synthesis starts with the element val- ues for Chebyshev filters g 1 = g i g i–1 = (i = 1, 2, ···, m), m = n/2 (10.3) ␥ = sinh sinh –1 S = ( ͙ 1 ෆ + ෆ ෆ 2 ෆ + ) 2 (the passband VSWR) J m = 1/ ͙ S ෆ J m–1 = 0 In order to introduce transmission zeros at ⍀ = ±⍀ a , the required value of J m–1 is given by J m–1 = (10.4) –J Ј m ᎏᎏ (⍀ a g m ) 2 – J m Ј 2 1 ᎏ 1 ᎏ n 4 sin ᎏ (2i 2 – n 1) ᎏ sin ᎏ (2i 2 – n 3) ᎏ ᎏᎏᎏ ␥ 2 + sin 2 ᎏ (i – n 1) ᎏ 2 sin ᎏ 2 n ᎏ ᎏ ␥ 10.1 SELECTIVE FILTERS WITH A SINGLE PAIR OF TRANSMISSION ZEROS 317 FIGURE 10.2 Lowpass prototype filter for the filter synthesis. Introduction of J m–1 mismatches the filter, and to maintain the required return loss at midband it is necessary to change the value of J m slightly according to the formu- la J Ј m = (10.5) where J Ј m is interpreted as the updated J m . Equations (10.5) and (10.4) are solved it- eratively with the initial values of J m and J m–1 given in (10.3). No other elements of the original Chebyshev filter are changed. The above method is simple, yet quite useful in many cases for design of selec- tive filters. But it suffers from inaccuracy, and can even fail for very highly selective filters that require moving the attenuation poles closer to the cut-off frequencies of the passband. This necessitates the use of a more accurate synthesis procedure. Al- ternatively, one may use a set of more accurate design data tabulated in Tables 10.1, 10.2, and 10.3, where the values of the attenuation pole frequency ⍀ a cover a wide range of practical designs for highly selective microstrip bandpass filters [4]. For less selective filters that require a larger ⍀ a , the element values can be obtained us- ing the above approximate synthesis procedure. For computer synthesis, the following explicit formulas are obtained by curve fitting for L R = –20 dB: g 1 (⍀ a ) = 1.22147 – 0.35543·⍀ a + 0.18337·⍀ a 2 – 0.0447·⍀ a 3 + 0.00425·⍀ a 4 g 2 (⍀ a ) = 7.22106 – 9.48678·⍀ a + 5.89032·⍀ a 2 – 1.65776·⍀ a 3 + 0.17723·⍀ a 4 J 1 (⍀ a ) = –4.30192 + 6.26745·⍀ a – 3.67345·⍀ a 2 + 0.9936·⍀ a 3 – 0.10317·⍀ a 4 (10.6) J 2 (⍀ a ) = 8.17573 – 11.36315·⍀ a + 6.96223·⍀ a 2 – 1.94244·⍀ a 3 + 0.20636·⍀ a 4 (n = 4 and 1.8 Յ ⍀ a Յ 2.4) J m ᎏᎏ 1 + J m J m–1 318 ADVANCED RF/MICROWAVE FILTERS TABLE 10.1 Element values of four-pole prototype (L R = –20dB) ⍀ a g 1 g 2 J 1 J 2 1.80 0.95974 1.42192 –0.21083 1.11769 1.85 0.95826 1.40972 –0.19685 1.10048 1.90 0.95691 1.39927 –0.18429 1.08548 1.95 0.95565 1.39025 –0.17297 1.07232 2.00 0.95449 1.38235 –0.16271 1.06062 2.05 0.95341 1.37543 –0.15337 1.05022 2.10 0.95242 1.36934 –0.14487 1.04094 2.15 0.95148 1.36391 –0.13707 1.03256 2.20 0.95063 1.35908 –0.12992 1.02499 2.25 0.94982 1.35473 –0.12333 1.0181 2.30 0.94908 1.35084 –0.11726 1.01187 2.35 0.94837 1.3473 –0.11163 1.00613 2.40 0.94772 1.34408 –0.10642 1.00086 g 1 (⍀ a ) = 1.70396 – 1.59517·⍀ a + 1.40956·⍀ a 2 – 0.56773·⍀ a 3 + 0.08718·⍀ a 4 (10.7) g 2 (⍀ a ) = 1.97927 – 1.04115·⍀ a + 0.75297·⍀ a 2 – 0.245447·⍀ a 3 + 0.02984·⍀ a 4 g 3 (⍀ a ) = 151.54097 – 398.03108·⍀ a + 399.30192·⍀ a 2 – 178.6625·⍀ a 3 + 30.04429·⍀ a 4 J 2 (⍀ a ) = –24.36846 + 60.76753·⍀ a – 58.32061·⍀ a 2 + 25.23321·⍀ a 3 – 4.131·⍀ a 4 J 3 (⍀ a ) = 160.91445 – 422.57327·⍀ a + 422.48031·⍀ a 2 – 188.6014·⍀ a 3 + 31.66294·⍀ a 4 (n = 6 and 1.2 Յ ⍀ a Յ 1.6) g 1 (⍀ a ) = 1.64578 – 1.55281·⍀ a + 1.48177·⍀ a 2 – 0.63788·⍀ a 3 + 0.10396·⍀ a 4 (10.8) g 2 (⍀ a ) = 2.50544 – 2.64258·⍀ a + 2.55107·⍀ a 2 – 1.11014·⍀ a 3 + 0.18275·⍀ a 4 g 3 (⍀ a ) = 3.30522 – 3.25128·⍀ a + 3.06494·⍀ a 2 – 1.30769·⍀ a 3 + 0.21166·⍀ a 4 g 4 (⍀ a ) = 75.20324 – 194.70214·⍀ a + 194.55809·⍀ a 2 – 86.76247·⍀ a 3 + 14.54825·⍀ a 4 J 3 (⍀ a ) = –25.42195 + 63.50163·⍀ a – 61.03883·⍀ a 2 + 26.44369·⍀ a 3 – 4.3338·⍀ a 4 J 4 (⍀ a ) = 82.26109 – 213.43564·⍀ a + 212.16473·⍀ a 2 – 94.28338·⍀ a 3 + 15.76923·⍀ a 4 (n = 8 and 1.2 Յ ⍀ a Յ 1.6) 10.1 SELECTIVE FILTERS WITH A SINGLE PAIR OF TRANSMISSION ZEROS 319 TABLE 10.2 Element values of six-pole prototype (L R = –20dB) ⍀ a g 1 g 2 g 3 J 2 J 3 1.20 1.01925 1.45186 2.47027 –0.39224 1.95202 1.25 1.01642 1.44777 2.30923 –0.33665 1.76097 1.30 1.01407 1.44419 2.21 –0.29379 1.63737 1.35 1.01213 1.44117 2.14383 –0.25976 1.55094 1.40 1.01051 1.43853 2.09713 –0.23203 1.487 1.45 1.00913 1.43627 2.0627 –0.20901 1.43775 1.50 1.00795 1.4343 2.03664 –0.18962 1.39876 1.55 1.00695 1.43262 2.01631 –0.17308 1.36714 1.60 1.00606 1.43112 2.00021 –0.15883 1.34103 TABLE 10.3 Element values of eight-pole prototype (L R = –20dB) ⍀ a g 1 g 2 g 3 g 4 J 3 J 4 1.20 1.02947 1.46854 1.99638 1.96641 –0.40786 1.4333 1.25 1.02797 1.46619 1.99276 1.88177 –0.35062 1.32469 1.30 1.02682 1.46441 1.98979 1.82834 –0.30655 1.25165 1.35 1.02589 1.46295 1.98742 1.79208 –0.27151 1.19902 1.40 1.02514 1.46179 1.98551 1.76631 –0.24301 1.15939 1.45 1.02452 1.46079 1.98385 1.74721 –0.21927 1.12829 1.50 1.024 1.45995 1.98246 1.73285 –0.19928 1.10347 1.55 1.02355 1.45925 1.98122 1.72149 –0.18209 1.08293 1.60 1.02317 1.45862 1.98021 1.71262 –0.16734 1.06597 The design parameters of the bandpass filter, i.e., the coupling coefficients and ex- ternal quality factors, as referring to the general coupling structure of Figure 10.3, can be determined by the formulas Q ei = Q eo = M i,i+1 = M n–i,n–i+1 = ᎏ ͙ F g ෆ B i g ෆ W i+ ෆ 1 ෆ ᎏ for i = 1 to m – 1 (10.9) M m,m+1 = ᎏ FB g W m ·J m ᎏ M m–1,m+2 = 10.1.3 Filter Analysis Having obtained the design parameters of bandpass filter, we may use the general formulation for cross coupled resonator filters given in Chapter 8 to analyze the fil- ter frequency response. Alternatively, the frequency response can be calculated by S 21 (⍀) = (10.10) S 11 (⍀) = where Y e and Y o are the even- and odd-mode input admittance of the filter in Figure 10.2. It can be shown that when the filter is open /short-circuited along its symmet- rical plane, the admittance at the two cross admittance inverters are ϯJ m–1 and ϯJ m . Therefore, Y e and Y o can easily be expressed in terms of the elements in a ladder structure such as Y e (⍀) = j(⍀g 1 – J 1 ) + ᎏ j(⍀g 2 1 – J 2 ) ᎏ for n = 4 (10.11a) Y o (⍀) = j(⍀g 1 + J 1 ) + ᎏ j(⍀g 2 1 + J 2 ) ᎏ 1 – Y e (⍀)·Y o (⍀) ᎏᎏᎏ ( 1 + Y e (⍀) ) · ( 1 + Y o (⍀) ) Y o (⍀) – Y e (⍀) ᎏᎏᎏ ( 1 + Y e (⍀) ) · ( 1 + Y o (⍀) ) FBW·J m–1 ᎏᎏ g m–1 g 1 ᎏ FBW 320 ADVANCED RF/MICROWAVE FILTERS M 1,2 Q ei Q eo M m-1,m M m,m+1 M m+1,m+2 M n-1,n M m-1,m+2 FIGURE 10.3 General coupling structure of the bandpass filter with a single pair of finite-frequency zeros. Y e (⍀) = j⍀g 1 + for n = 6 (10.11b) Y o (⍀) = j⍀g 1 + 1 Y e (⍀) = j⍀g 1 + _____________________________________ j⍀g 2 + ··· + for n = 8, 10, · · · (m = n/2) (10.11c) 1 Y o (⍀) = j⍀g 1 + _____________________________________ j⍀g 2 + · · · + The frequency locations of a pair of attenuation poles can be determined by impos- ing the condition of |S 21 (⍀)| = 0 upon (10.10). This requires |Y o (⍀) – Y e (⍀)| = 0 or Y o (⍀) = Y e (⍀) for ⍀ = ±⍀ a . Form (10.11) we have j(⍀ a g m–1 + J m–1 ) + = j(⍀ a g m–1 – J m–1 ) + (10.12) This leads to ⍀ a = Ί J m 2 – (10.13) As an example, from Table 10.2 where m = 3 we have g 3 = 2.47027, J 2 = –0.39224, and J 3 = 1.95202 for ⍀ a = 1.20. Substituting these element values into (10.13) yields ⍀ a = 1.19998, an excellent match. It is more interesting to note from (10.13) that even if J m and J m–1 exchange signs, the locations of attenuation poles are not changed. Therefore, and more importantly, the signs for the coupling coefficients M m,m+1 and M m–1,m+2 in (10.9) are rather relative; it does not matter which one is positive or neg- ative as long as their signs are opposite. This makes the filter implementation easier. 10.1.4 Microstrip Filter Realization Figure 10.4 shows some filter configurations comprised of microstrip open-loop resonators to realize this type of filtering characteristic in microstrip. Here the num- bers indicate the sequence of direct coupling. Although only the filters up to eight poles have been illustrated, building up of higher-order filters is feasible. There are J m ᎏ J m–1 1 ᎏ g m 1 ᎏᎏ j(⍀ a g m – J m ) 1 ᎏᎏ j(⍀ a g m + J m ) 1 ᎏᎏᎏᎏ j(⍀g m–1 + J m–1 ) + ᎏ j(⍀g m 1 + J m ) ᎏ 1 ᎏᎏᎏᎏ j(⍀g m–1 – J m–1 ) + ᎏ j(⍀g m 1 – J m ) ᎏ 1 ᎏᎏᎏ j(⍀g 2 + J 2 ) + ᎏ j(⍀g 3 1 + J 3 ) ᎏ 1 ᎏᎏᎏ j(⍀g 2 – J 2 ) + ᎏ j(⍀g 3 1 – J 3 ) ᎏ 10.1 SELECTIVE FILTERS WITH A SINGLE PAIR OF TRANSMISSION ZEROS 321 other different filter configurations and resonator shapes that may be used for the realization. As an example of the realization, an eight-pole microstrip filter is designed to meet the following specifications Center frequency 985 MHz Fractional bandwidth FBW 10.359% 40dB Rejection bandwidth 125.5 MHz Passband return loss –20 dB The pair of attenuation poles are placed at ⍀ = ±1.2645 in order to meet the rejec- tion specification. Note that the number of poles and ⍀ a could be obtained by di- rectly optimizing the transfer function of (10.1). The element values of the lowpass prototype can be obtained by substituting ⍀ a = 1.2645 into (10.8), and found to be g 1 = 1.02761, g 2 = 1.46561, g 3 = 1.99184, g 4 = 1.86441, J 3 = –0.33681, and J 4 = 1.3013. Theoretical response of the filter may then be calculated using (10.10). From (10.9), the design parameters of this bandpass filter are found M 1,2 = M 7,8 = 0.08441 M 2,3 = M 6,7 = 0.06063 M 3,4 = M 5,6 = 0.05375 M 4,5 = 0.0723 M 3,6 = –0.01752 Q ei = Q eo = 9.92027 The filter is realized using the configuration of Figure 10.4(c) on a substrate with a relative dielectric constant of 10.8 and a thickness of 1.27 mm. To determine the physical dimensions of the filter, the full-wave EM simulations are carried out to extract the coupling coefficients and external quality factors using the approach de- scribed in Chapter 8. The simulated results are plotted in Figure 10.5, where the size 322 ADVANCED RF/MICROWAVE FILTERS ( ) c ( ) d () a ( ) b 1 1 1 2 2 2 3 3 3 4 4 4 5 5 6 6 7 8 1 2 3 45 67 8 FIGURE 10.4 Configuration of microstrip bandpass filters exhibiting a single pair of attenuation poles at finite frequencies. 323 FIGURE 10.5 Design curve. (a) Magnetic coupling. (b) Electric coupling. (c) Mixed coupling I. (d) Mixed coupling II. (e) External quality factor. (All resonators have a line width of 1.5 mm and a size of 16 mm × 16 mm on a 1.27 mm thick substrate with a relative dielectric constant of 10.8.) of each square microstrip open-loop resonator is 16 × 16 mm with a line width of 1.5 mm on the substrate. The coupling spacing s for the required M 4,5 and M 3,6 can be determined from Figure 10.5(a) for the magnetic coupling and Figure 10.5(b) for the electric coupling, respectively. We have shown in Chapter 8 that both couplings result in opposite signs of coupling coefficients, which is what we need for realiza- tion of this type of filter. The other filter dimensions, such as the coupling spacing for M 1,2 and M 3,4 , can be found from Figure 10.5(c); the coupling spacing for M 2,3 is obtained from Figure 10.5(d). The tapped line position for the required Q e is deter- mined from Figure 10.5(e). It should be mentioned that the design curves in Figure 10.5 may be used for the other filter designs as well. Figure 10.6(a) is a photograph of the fabricated filter using copper microstrip. The size of the filter amounts to 324 ADVANCED RF/MICROWAVE FILTERS FIGURE 10.6 (a) Photograph of the fabricated eight-pole microstrip bandpass filter designed to have a single pair of attenuation poles at finite frequencies. The size of the filter is about 120 mm × 50 mm on a 1.27 mm thick substrate with a relative dielectric constant of 10.8. (b) Measured performance of the fil- ter. (b) (a) [...]... Layout of the microstrip trisection filter designed to have a higher selectivity on low side of the passband on a 1.27 mm thick substrate with a relative dielectric constant of 10.8 (b) Measured performance of the filter 340 ADVANCED RF/ MICROWAVE FILTERS 10.3.4 Microstrip CT Filters It is obvious that the two microstrip trisection filters described above could be used for constructing microstrip CT... bandwidths are 100 MHz and 120 MHz, respectively, which obviously does not meet the rejection requirements Having determined the design parameters, the nest step is to find the physical dimensions for the microstrip CQ filter For reducing conductor loss and increasing power handling capability, wider microstrip would be preferable Hence, the microstrip line width of open-loop resonators used for the... Typical coupling structures of CQ filters n-1 Qen 326 ADVANCED RF/ MICROWAVE FILTERS 10.2.1 Microstrip CQ Filters As examples of realizing the coupling structure of Figure 10.7(a) in microstrip, two microstrip CQ filters are shown in Figure 10.8, where the numbers indicate the sequences of the direct couplings The filters are comprised of microstrip open-loop resonators; each has a perimeter about a... response Having obtained the required design parameters for the bandpass filter, the physical dimensions of the microstrip trisection filter can be determined using full-wave EM simulations to extract the desired coupling coefficients and external quality factors, as described in Chapter 8 Figure 10.14(a) shows the layout of the designed microstrip filter with the dimensions on a substrate having a relative... Unit: mm 12.0 1.4 4.6 1.0 1.1 17.4 0.7 1.2 12.0 1.8 (a) (b) FIGURE 10.14 (a) Layout of the microstrip trisection filter designed to have a higher selectivity on high side of the passband on a 1.27 mm thick substrate with a relative dielectric constant of 10.8 (b) Measured performance of the filter 338 ADVANCED RF/ MICROWAVE FILTERS 10.3.3.2 Trisection Filter Design: Example Two The filter is designed... 1.75 1.4 (a) (b) FIGURE 10.10 (a) Layout of the designed microstrip CQ filter with all the dimensions on the 1.27 mm thick substrate with a relative dielectric constant of 10.8 (b) Measured performance of the microstrip CQ filter 1 |S21| = ᎏᎏ 2 ͙ෆෆෆ2F ෆෆෆ 1 + ෆn (⍀) Fn = cosh ⍀ – 1/⍀ai Αcosh–1 ᎏᎏ 1 – ⍀/⍀ai i=1 ΄ n ΅ (10.16) 330 ADVANCED RF/ MICROWAVE FILTERS M3,4 Mn-2,n-1 M4,5 Mn-1,n M1,3 Qe1 Qen... Radio) for minimization of linear distortion while prescribed channel selectivity is being maintained Although only the eight-pole microstrip CQ filters are illustrated, the building up of filters with more poles and other configurations is feasible 10.2.2 Design Example For the demonstration, a highly selective eight-pole microstrip CQ filter with the configuration of Figure 10.8(a) has been designed,... 0.07063 For a 60 MHz passband bandwidth, the required 50 dB and 65 dB rejection bandwidths set the selectivity, of 6 1 8 3 (a) 1 4 2 4 2 5 6 7 5 7 8 3 (b) FIGURE 10.8 Configurations of two eight-pole microstrip CQ filters 10.2 CASCADED QUADRUPLET (CQ) FILTERS 327 the filter To meet this selectivity the filter was designed to have two pairs of attenuation poles near the passband edges, which correspond... the midband frequency This size is evidently very compact Figure 10.14(b) shows the measured results of the filter As can be seen, an attenuation pole of finite frequency on the upper side of the passband leads to a higher selectivity on this side of the passband The measured midband insertion loss is about –1.15dB, which is mainly due to the conductor loss of copper microstrip 10.3 TRISECTION AND CASCADED... 0.29 g0 The measured performance is shown in Figure 10.6(b) The midband insertion loss is about 2.1dB, which is attributed to the conductor loss of copper The two attenuation poles near the cut-off frequencies of the passband are observable, which improves the selectivity High rejection at the stopband is also achieved 10.2 CASCADED QUADRUPLET (CQ) FILTERS When high selectivity and/or other requirements . filter having only one pair of transmission zeros (or attenuation poles) at finite frequencies gives much improved skirt selectivity, making it a viable. copper microstrip. The size of the filter amounts to 324 ADVANCED RF/ MICROWAVE FILTERS FIGURE 10.6 (a) Photograph of the fabricated eight-pole microstrip