Even-degree elliptic function monolithic crystal filters

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<ul><li><p>Even-degree elliptic function monolithiccrystal filters</p><p>S.K.S. Lu</p><p>Indexing terms: Bandpass filters. Crystal filters</p><p>Abstract: In monolithic crystal filter design even-degree elliptic function lowpass prototypes of the typesdesignated b and c have been used. However, monolithic crystal filters designed from the original even-degree elliptic function are not only feasible but also more selective. An example of the filter, using cas-cade synthesis, is given.</p><p>1 Introduction</p><p>The application of elliptic functions to the design ofelectrical filters was first suggested by Cauer in 1931; hencesuch filters are often referred to as Cauer-parameter filters.These filters have equiripple characteristics in both the pass-band and stopband. Darlington and Piloty independentlysolved the design problems for the general case in 1939.Saal and Ulbrich1 simplified the filter design with a usefulset of tables of the element values in the LC lowpass ladderfilters. Later, Holt and Gray2 described a method by whichcascade lattice crystal bandpass filters may be designed by asimple transformation of the lowpass ladder. Recently, theauthor3'4 extended the design of Holt and Gray to mono-lithic crystal filters having symmetrical as well as single-sideband responses.</p><p>Bandpass filters are commonly encountered in communi-cation engineering, and are usually designed from the lowpass prototypes. If the degree of the elliptic-function filteris even, however, no attenuation pole is located at infinitefrequency, a property not realisable for the ladder network.A suitable frequency transformation is needed to shift theoutermost transmission zero to infinity, and the filters ob-tained are identified by the designation b. If, furthermore,it is required that the two termination resistances be equalor that there is no attenuation at zero frequency, such afilter is designated as type c. Both types b and c are not asselective as the original of the same degree as shown in Fig.1.</p><p>In the design of narrowband bandpass filters, as in thecase of monolithic crystal filters, it is not essential for thelowpass prototypes to be realisable. Indeed, the lowpassprototypes may include the two fictitious positive andnegative frequency-invariant imaginary components, aconcept developed by R.F. Baum5 for narrowband band-pass filters. A table, similar to that of Saal,1 of the elementvalues in the lowpass lattice prorotypes may be producedusing the cascade synthesis theory of Scanlan and Rhodes.6</p><p>2 Even-degree elliptic function filters</p><p>Consider a normalised even-degree in) lossless lowpasselliptic filter whose power-gain function is given as</p><p>2 _ 1 0)where e is the ripple constant, and where for a given para-meter K which is a measure of the steepness of the gain</p><p>characteristic in the transitional band, the normalisedcharacteristic function is</p><p>Fn(oj) = sn nK</p><p>Lsn~1</p><p>AFn (GO) may alternatively be expressed as</p><p>2 2IT</p><p>= Hn/2</p><p>1 K</p><p>(2)</p><p>(3)where</p><p>com = sn[(2m-l)K/n,K] m = 1, 2,. . . , - (4)</p><p>The transmission zeros of the filter are located at</p><p>1(5)</p><p>140</p><p>130</p><p>120m"110</p><p>^ 1 0 0</p><p>* 90o0 80</p><p>1 70a</p><p>6070</p><p>40</p><p>30</p><p>20</p><p>86</p><p>1 03 1 06 111-04 108</p><p>12 1-31-4 1-61-8 2</p><p>Fig. 1 Attenuation verses normalised angular frequency of theelliptic filterFor a passband ripple, substract the corresponding constant given inthe tableRipple Ap</p><p>Paper T226 E, received 5th May 1978Mr. Lu is with the Department of Electrical Engineering, Universityof Malaya, Kuala Lumpur, Malaysia</p><p>ELECTRONIC CIRCUITS AND SYSTEMS, SEPTEMBER 1978, Vol. 2, No. 5</p><p>dB1 00-50-250 10 0 50 0 1</p><p>5-879 1 4</p><p>12-2716-3319-3626-37</p><p>143</p><p>0308-6984/78/050143 + 04 $01-50/0</p></li><li><p>It can also be shown that</p><p>m</p><p>(6)</p><p>a parameter which determines the level of the ultimateattenuation in the stopband. Fig. 2 shows a plot of thecharacteristic function Fn(co).</p><p>From eqn. 1, we obtain the input reflection coefficientSn (s) as</p><p>Su(s)Sn(-s) = e2F2(s)</p><p>or</p><p>(7)</p><p>(8)where P(s) is formed by the zeros of Fn(s), while Q(s) isformed by the l.h.s. zeros of the equation</p><p>l+e2F2(s) = 0That is</p><p>nil</p><p>where</p><p>with</p><p>= /sn [(2m - 1 )A/ /a, K</p><p>K</p><p>andX = 1</p><p>1/2</p><p>The input impedance of the filter may be obtained as</p><p>(9)</p><p>(10)</p><p>01)</p><p>(12)</p><p>(13)</p><p>(14)</p><p>from which a cascade network may be synthesised.</p><p>3 Cascade synthesis</p><p>Scanlan and Rhodes6 have discussed a cascade synthesistheory in which the basic zero-producing sections (Brune,Darlington C and D) are extracted. Recently, Dillon andLind7 employed the theory to obtain explicit expressionsfor the element values in the lowpass lattice prototypes.</p><p>In the case of the elliptic-function filters, only the</p><p>F(u&gt;)</p><p>Brune sections need to be considered, as all the transmissionzeros occur on the real-frequency axis.</p><p>The transmission matrix of the Brune section is</p><p>4Wo</p><p>1 + as2 bs</p><p>cs +ds2 (15)</p><p>where the transmission zeros of the section are located ats = jcj0, and the polynomial coefficients a and c arefound to be</p><p>a =</p><p>c =</p><p>X(U)0)o {CJQX'(U)O)-X(CJO)}</p><p>2(16)</p><p>X(CJ0)}</p><p>The element values of the Brune lowpass lattice sectionmay then be evaluated by</p><p>L = a</p><p>_</p><p>c</p><p>.1/2X =</p><p>B =and</p><p>A =</p><p>1/2</p><p>(17)</p><p>Fig. 2 Plot of the characteristic function F4 (u&gt;) as a function of</p><p>Using the above cascade synthesis theory, a set of tablesof the element values in the lowpass lattice prototypes8may be produced. However, to save space, only a singletable is given. Table 1 shows the element values for n = 4and passband ripple 0-5 dB with various selectivity factors.</p><p>4 Synthesis procedure</p><p>Recently, the author3'4 has discussed a method by whichthe lattice lowpass prototype may be transformed into acascade bandpass network containing double-resonatormonolithic crystal filter elements with bridging and seriescapacitors, which will not be repeated here.</p><p>However, the synthesis procedure can be outlined andconsists of the following steps:</p><p>(a) Approximation step: From Fig. 1, one may find anelliptic function filter that meets the requirements.</p><p>(b) Lowpass lattice prototype: Obtain the elementvalues directly from table, e.g. Table 1, or from comput-ation using eqns. 1 17.</p><p>(c) Simplifying the lowpass prototype: By using theproperty that the admittance level of the network betweenany pair of inverters may be scaled by a real positive factorj32 without affecting the response, if both inverters areadmittance scaled by the factor 0, one may</p><p>(i) equate all but the last inverter so as to permit com-plete adsorption of the series inductive arms of the inverterto the adjacent lattices</p><p>(ii) interchange the lattice arms(iii) enable equimotional inductances of two m.c.f.</p><p>elements to result. (In practice, however, the m.c.f. ele-ments may be designed to have different motional induct-ances or electrode areas to assist spurious suppression in thestopband).</p><p>144 ELECTRONIC CIRCUITS AND SYSTEMS, SEPTEMBER 1978, Vol. 2, No. 5</p></li><li><p>Table 1: Element values for n = 4 and passband ripple 0-5 dB</p><p>196A1</p><p>Bl</p><p>20</p><p>22</p><p>24</p><p>26</p><p>28</p><p>30</p><p>32</p><p>34</p><p>36</p><p>38</p><p>40</p><p>dB2-924</p><p>2-670</p><p>2-459</p><p>2-281</p><p>2-130</p><p>2 000</p><p>1-887</p><p>1-788</p><p>1-701</p><p>1-624</p><p>1-556</p><p>dB63-20</p><p>59-81</p><p>56-70</p><p>53-83</p><p>51-15</p><p>48-64</p><p>46-28</p><p>4404</p><p>41-92</p><p>39-89</p><p>37-95</p><p>1212121212121212121212</p><p>1-68102-39751 -68422-40391 -68812-41091 -69282-41841 -69852-42641 -70532-43481-71342-44361-72322-45281-73482-46221 -74872-47181 -76522-4816</p><p>1 -22791 -69921 -23811 -70401 -24971 -70931 -26271-71511-27731-72131 -29371 -72801-31201-73521 -33261 -74301-35571-75121-38161 -75991-41081-7691</p><p>- 00463- 0 0054- 00565- 00065- 00681- 00078- 0 0809- 00092- 00950-00107-0-1107-00123-0-1280-00140-0-1470-00159-0-1678-00179-0-1907- 00201-0-2158- 00225</p><p>0-86060-59390-86420-59340-86830-59280-87280.59220-87800-59160-88370-59100-89020-59030-89740-58970-90540-58900-91450-58840-92460-5877</p><p>3-15077-44242-87406-75712-64426-18512-45075-70032-28565-28392-14334-9223201954-60511-91094-32441-8152407411-73013-84931 -65433-6463</p><p>(d) Bandpass filter: Finally, apply the narrowband low-pass to bandpass transformation and the network trans-formation suggested by the author3 to obtain the requiredcascade network containing double-resonator monolithiccrystal filter elements with bridging and series capacitors.</p><p>5 Example</p><p>For the fourth-order Cauer-parameter filter, having select-ivity factor 1-624 or 6 = 38 and ultimate attenuation inthe stopband 40 dB, we obtain the lowpass lattice proto-type whose element values are</p><p>Z-! = 1-7487</p><p>Xx = 1-3816</p><p>Bl = -01907</p><p>At = 0-9145</p><p>and</p><p>L2 = 2-4718</p><p>X2 = 1-7599</p><p>B2 = - 0 0 2 0 1</p><p>A2 = 0-5884Since there are only two sections, and if the admittancescaling property is used to anticipate equimotional in-ductances for the final m.c.f. elements, then the scalingfactor 0 in this example is given by</p><p>where</p><p>k = = 0-8411After having adsorbed the series elements of the invertersto their respective lattices with excess susceptances addedto the source and load, and applying the narrowbandlowpass-to-bandpass transformation to scale to centre fre-quency of 10-7 MHz with passband ripple bandwidth of10 kHz and motional inductance of 30 mH the final filter</p><p>Rs</p><p>1 "</p><p>r "i</p><p>II-</p><p>= L cjc L ;p Tip r*</p><p>c,1-</p><p>- ~i</p><p>-</p><p>:Csi</p><p>r11</p><p>Co</p><p>1- -1</p><p>_J. J jcs2</p><p>L n</p><p> =A i kBi</p><p>= 0-7343</p><p>?.. m.c.f. 1 m.c.f. 2</p><p>Fig. 3 Fourth-order Cauer-parameter bandpass monolithic crystalfilterElement values:</p><p>C, = T359pF</p><p>C2 = 6-140 pF</p><p>C3 = 3-590pF</p><p>Rs = 2472 ft</p><p>RL - 2414ft</p><p>m.c.f. 1</p><p>L = 3 0 0 mH</p><p>C = 7-3731119X 10"3pF</p><p>Cm = 6-940 pF</p><p>Cbl = 1-601 pF</p><p>Cb2 = 0-297 pF</p><p>CSi = 13-83 pF</p><p>CS2 = 81-02 pF</p><p>Co = 2-5 pF</p><p>m.c.f. 2</p><p>L = 3 0 0 mH</p><p>C = 7-3741934X 10"3pF</p><p>C m = 1005 pF</p><p>ELECTRONIC CIRCUITS AND SYSTEMS, SEPTEMBER 1978, Vol. 2, No. 5 145</p></li><li><p>shown in Fig. 3 results. Fig. 4 illustrates the correlationbetween a measured filter response and the computed re-sponses with infinite crystal Q and a typical realisable Qof 20000.</p><p>6 Conclusion</p><p>The design of monolithic crystal filter using the originaleven-degree elliptic function was discussed, and illustrated</p><p>-40 -30 -20 -10 0 10 20frequency, kHz from 107MHz</p><p>30 40</p><p>Fig. 4 Amplitude response of the filter in Fig. 3computedmeasured</p><p>with an example. A comparison between the predicted andmeasured responses was given. The application of theeven-degree elliptic function may also be extended to thedesign of monolithic crystal filters having single-sidebandand bandstop characteristics.</p><p>7 Acknowledgment</p><p>The author wishes to thank Hy-Q International (Singapore)for its facillity in making the monolithic crystal filter.</p><p>8 References</p><p>1 SAAL, R., and ULBRICH, E.: 'On the design of filters bysynthesis',IRE Trans., 1958,CT-5,pp. 284-327</p><p>2 HOLT, A.C.J., and GRAY, R.L.: 'Bandpass crystal filters bytransformation of a low-pass ladder', IEEE Trans., 1968, CT-15,pp. 492-494</p><p>3 LU, S.K.S.: 'Cascade synthesis of monolithic crystal filterswith transmission zeros at finite frequencies', Electron. Lett.,1978, 14, pp. 45-46</p><p>4 LU, S.K.S.: 'Cascade synthesis of single-sideband monolithiccrystal filters', IEEE Trans. Circuits and Systems, (submitted)</p><p>5 BAUM, R.F.: 'A modification of Brune's method for narrow-band filters', IRE Trans. 1958, CT-5, pp. 264-267</p><p>6 SCANLAN, J.O., and RHODES, J.D.: 'Unified theory of cascadesynthesis', Proc. IEE, 1970,117, (4), pp. 665-670</p><p>7 DILLON, C.R., and LIND, L.F.: 'Cascade synthesis of polylithiccrystal filters containing double-resonator monolithic crystalfilter (MCF) elements', IEEE Trans., 1976, CAS-23, pp. 146-154</p><p>8 Internal report, available on request from the author9 CHEN, W.K.: 'Theory and design of broadband matching net-</p><p>works', (Pergamon Press, 1976)10 RHODES, J.D.: 'Theory of electrical filters' (Wiley, 1976)11 BYRD, P.F., and FRIEDMAN, M.D.: 'Handbook of elliptic</p><p>integrals for engineers and physicists' (Springer-Verlag, 1954)</p><p>S. K. S. Lu received the B.E. fromMelbourne University in 1967 andthe M.Eng.Sc. and M.E. from theUniversity of New South Wales in1969 and 1972, respectively. Heworked for the Research Laboratory ofthe PMG Department in 1968, andduring 1970-1972 he was with theengineering products division ofAmalgamated Wireless Australasia inSydney. Since 1972 he has been with</p><p>the University of Malaya. He was on study leave at theUniveristy of Leeds in 1976.</p><p>146 ELECTRONIC CIRCUITS AND SYSTEMS, SEPTEMBER 1978, Vol. 2, No. 5</p></li></ul>