THE UNIVERSTITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR A LINEAR PHASE BAND-PASS FILTER Technical Memorandum No. 82 3697-1-T Cooley Electronics Laboratory Department of Electrical Engineering By: C. E. Lindahl Approved by: r H. A. Farris A CEL publication is given a memorandum designation due to reservations in one or more of the following respects: 1. The study reported was not exhaustive. 2. The results presented concern one phase of a continuing study. 3. The study reported was judged to have insufficient scope. Project 3697 CONTRACT NO. DA-18-119 sc-1357 U. S. ARMY SIGNAL PROCUREMENT AGENCY FT. GEORGE G. MEADE, MD. February 1961

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv 1. INTRODUCTION 1 2. LOW-PASS, ALL-POLE FUNCTION 1 3. DIRECTION FINDING CONSIDERATIONS 4 4. TRANSFORMATION OF A LOW-PASS FUNCTION TO A BAND-PASS FUNCTION 6 5. REALIZATION OF THE BAND-PASS FUNCTION 7 6. MEASUREMENTS AND CONCLUSIONS 13 APPENDIX A CALCULATION OF BAND-PASS POLE POSITIONS FROM LOW-PASS POLE POSITIONS 16 APPENDIX B DERIVATION OF ELEMENT VALUES OF PARALLEL RLC NETWORK WITH POLE-ZERO LOCATIONS AND THE INDUCTOR, L, SPECIFIED 20 APPENDIX C METHOD OF CALCULATION OF 3-db-DOWN FREQUENCIES FOR EACH TUNED CIRCUIT 22 LIST OF ILLUSTRATIONS Figure Page 1 Plot of numbers of degrees error from linearity vs. radian frequency. 2 2 Plot of filter attenuation vs. radian frequency. 3 3 ~Low-pass to band-pass transformation procedure. 7 4 Pentode plate circuit configuration. 8 5 Circuit diagram of linear phase band-pass filter. 10 6 Test setup for the alignment of each tuned plate. 12 7 Measured phase shift of linear phase filter. 14 8 Measured relative voltage gain of linear phase filter. 15 iii

ABSTRACT The design and realization of a linear phase band-pass filter for use as a predetection filter in a spinning goniometer radio direction finding system is described. The characteristics of the resulting unit are as follows: (1) center frequency, fo = 20,000 cps; (2) bandwidth at 3 db-down points, LAf = 500 cps; (3) attenuation for bandwidth of 1,000 cps is greater than 10 db; (4) phase deviation from linearity at the 3 db points is less than one degree. iv

A LINEAR PHASE BAND-PASS FIILTER 1. INTRODUCTION This report describes the design and realization of a linearphase band-pass filter. It is to be used as a predetection filter in a spinning-goniometer, radio direction-finding system. The filter has the following specifications: (1) f = center frequency = 20,000 cps Z2) Lf = bandwidth at 3-db-down points = 500 cps (3) Attenuation: greater than 10 db when the bandwidth is 1000 cps (4) Phase deviation from linearity:'e, S 10 at the 3-db-down points. 2. LOW-PASS, ALL-POLE FUNCTION The pole locations for an all-pole linear phase function are given in Table I for orders n = 1, 2, and 3. It is of interest to calculate two sets of curves as a function of the normalized radian frequency, W, and as a function of the order of the approximation, n, for these allTABLE I POLE LOCATIONS FOR AN ALL-POLE LINEAR PHASE FUNCTION Order Pole Location n = 1 -1.000000 n = 2 -1.500000 + jO.866025 f -2. 32185 n = 3 1-1.838907 + jl.754389 pole functions. One is a plot of the number of degrees error from linearity vs. radian frequency; the other, a plot of filter attenuation vs.

radian frequency. These are plotted in Figs. 1 and 2, respectively. The error formula for this case is: -1an (_l)n () (1) The attenuation functions for each order are: n = 1: a(U) = lo gl0 (l + 2) (4) 2 4 n =:'x(w) = 10 log10(9 +3W + ) (3) 2 4 6 n =3: a(w) = 10 1og10(225 + 45W2 + +D4 + 6 By looking at the attenuation 45 and error curves simultaneously, one 40 - 35 L | / / I | can see that the function of order 35 n n2/ n3/ n = 3 will satisfy our requirements. J 30 2 / / / _From the attenuation plot (Fig. 2),, 25 W 20 W | / | / 1/ l the 3-db-down points for n = 3 occurs h 20 20 W o / /. / at w = 1.75 radians per second. Calz15 -lo/ / / 0 culations show that this frequency 0 = L /1 / l / l l is w = 1.75567 radians per second. I0 1.0 20l I3.0 4 From the phase error plot the error 0 1.0 2.0 3.0 4.0 FREQUENCY, (IN RADIANS/SEC) in degrees from linearity for n = 3 and w Z 1.75 radians per second is Fig. 1. Plot of numbers of degrees approximately one degree. error from linearity vs. radian frequency. In order to normalize the trans-. Balabanian, Network Synthesis (Eng Inc., 1958).

~ouanb aij uTspTJ'sA uoT' fenuoa axIT jo'oTd j''.' (33S/SNVtlV8 NI ) m'A:N3nO38-I 0*g O't 0' 0'~ 0'1 0 0 m z ~TTL t 0 01 Uj ~ r~~~~~~~~tpl ~l~ 91

fer function (n = 3) so that the 3-db-downpoint occurs at wA 1.0 radian per second, we can divide the pole positions by 1.75567. We then obtain as the pole positions for n = 3: p = - 1.32267, p = - 1.04741 + jO.99927. The transfer function to be synthesized is therefore F(p) =.7718 (5) p3 + 3.4175 p + 4.8664 p + ".7718 Before proceeding further it is of interest to examine some aspects of direction finding which are pertinent to this problem. 3. DIRECTION FINDING CONSIDERATIONS One of the important factors affecting the ability of the human operator in using a spinning-goniometer radio direction-finder with accuracy is the shape of the propeller pattern on the cathode ray tube. If the ends of the pattern are rounded off, it becomes increasingly difficult to make an accurate bearing determination. Such rounding off of the pattern is called "blur" and may be caused by a number of factors. Before second detection a quantitative measure for blur has been derived.1 The expression is Percent blur 100 (1 - (6) where: where: yr is the ratio of the magnitudes of the two sideband voltages which were originally of equal magnitude when obtained from the spinning goniometer. S. F. George, "Direction Finder Bandwidth Requirements," NRL Rpt. No. R-3182.

Let us now determine the blur that would occur if the signal were tuned to the 3-db-down point of the proposed linear phase filter. It will be assumed that the goniometer spin rate is 30 revolutions per second and the filter bandwidth is 500 cycles per second. This spin rate causes a separation of 60 cycles per second between the two sidebands. For n = 3 of the normalized curve of attenuation vs. frequency (Fig. 2), the 3-db-down point falls atw= 1.756 radians per second which corresponds to 250 cycles per second of the filter. Hence, a frequency separation of 60 cycles per second corresponds to 60/250 x 1.756 = 0.421 radians per second of the normalized frequency scale used. Assuming that the signal is tuned to the 3-drb-down point, it can be seen from the curve that the difference in sideband amplitudes originally equal in magnitude will now be approximately 1.55 db. The quantity 7r can be calculated as follows: {Difference in Sideband 1 = s z ='01 log10 r(7) Voltage Magnitudes (in dbJ r With tZs = 1.55 db, yr = 0.837. From Eq. (6) it is found that the percent blur is 8.87. When the signal is properly tuned it can be shown that the blur is less than 0.5%. Hence, it is rather easy to determine whether or not the signal is properly tuned in by watching the shape of the propeller pattern. As a consequence of the above the blur caused by the linear phase filter will not affect the bearing accuracy, when the signal is properly tuned. 5

4. TRANSFORMATION OF A LOW-PASS FUNCTION TO A BAND-PASS FUNCTION In the preceding section a low-pass function was discussed; however, a bandpass function is desired. The usual transformation used for this purpose is obtained by replacing the complex variable p of the trans2 2 P + w fer function by -Ap., where w~ is the midband radian frequency of the filter and Awis its radian bandwidth. If one knows each pole position (as we do in this case), he can transform each pole individually; however, in each case a quadratic equation must be solved. Examples of this are shown in Appendix A. It is shown there that if the ratio of the bandwidth to the center frequency is small enough, a simplification can be made. If the required conditions are met (see Appendix A), the following approximations are valid for transforming low-pass pole positions to the band-pass case: Low-pass Pole Positions Band-pass Pole Positions (Approximate) p = -a p (2 )a + (8) p = -a + jb p = -A(a + jb) j (9) P = -a - Jb - p = -Aw( )+ jw (10) 2 o For each pole transformed to the band-pass case a zero must be added at the origin of the complex plane. The process can be divided into four steps, which are illustrated in Fig. 3. Formulas (8), (9), and (10) are implemented as shown above in order to calculate the pole positions of the band-pass filter. However, a word of caution is in order. Had the ratio of ~-been greater than co approximately 0.05, the effects of the complex conjugate poles and the 6

ji ij jw p - plane p - plane p - plane X X1 X p = - 1.32267 p = 0.661335 (c) Multiplying the p = - 1.04741+j0.99927 p = 0.5237050JO.499635 results of (b) by Aw (a) Low-pass pple positions (b) Multiplying low-pass pole positions by one-half jW X p - plane p = - 2077.65~j125,664 +jW0X pII+ = - 1645. 27~J124, 094 p = - 1645.27~j125,234 X -iWO X (d) Adding +jw0 and -JW0 to (e) Resulting band-pass the results of (c) and adding pole positions for the linear phase filter (n=3) three zeros at the origin Fig. 3. Low-pass to band-pass transformation procedure. three zeros at the origin would have caused the phase characteristics to be adversely affected. In other words, the phase characteristics of the low-pass function are not preserved under the given low-pass to band-pass transformation; however, in the narrowband case they are sufficiently close to the desired characteristics that the transformation can be applied. 5. REALIZATION OF THE BAND-PASS FUNCTION For realization of the transfer function it was decided to realize each complex conjugate pole pair along with an associated zero at the origin of the complex plane as a separately tuned interstage between pentode tubes acting as current sources. As is well known, the overall transfer function for such a combination is the product of the transfer functions of each stage. The circuit used for each plate circuit is 7

shown in Fig. 4(a). The transfer function, (e/i), for this circuit is: R e (P + )(11) C[p2 + ( + + - P It is possible to derive expressions for the element values in jw a L b RP T RS b Aa (a) (b) Fig. 4. Pentode plate circuit configuration. terms of the pole-zero positions shown in Fig. 4(b) and the inductance, L t(ee Appendix B). Certain realizability conditions have to be observed: (1) ra > c' (2) a + b > c'(2a - c') If the inequalities of Eqs. 12 are satisfied, then the desired relations are: R = c' L (13) 1 (14) L(a2 + b + c'" - ac') Rp =. 2a -- c' 8~~~~~~~~(5

It must be noted here that Rs is not merely an ohmic resistance, but it includes any loss component which the inductor, L, itself may have. Hence, it is possible that the zero location indicated in Eq. 11 cannot be realized because the Q of the inductor is not high enough, since the closer the zero is to the origin the higher the Q required. It was for this reason that in the design equations (13), (14), and (15) emphasis was placed upon picking L first instead of, say, R or C. The proper choice of zero locations, p = -c', in each circuit can compensate for the adverse effect on the phase of the complex conjugate poles clustered around p = - Jwo' The desired phase for the filter (n = 3) can be calculated from e = -Kw, where K = 1; at the 3-db-down point, e = -(1)(1.75567) = -1.75567 radians = -100.59250. From the polezero plot, the phase-shift of the function was calculated for 19,750, 20,000, and 20,250 cycles per second and was found to be 100.6880, 1.2240, and -98.231~, respectively. Hence, the error, ee, in degrees from the ideal was 0.096~, 1.224~, and 2.2618~, respectively. Moving the zero location of each plate circuit to the left by the proper amount, exactly compensates for the error, ee, at the midband frequency. We shall let the zero location of each circuit be the same. It turns out that for exact compensation at 20,000 cycles per second, c' = 894.71. The calculated error at 19,750 and 20,250 cycles per second is now 1.140 and 1.150, respectively. With this value of c' the Q of the inductance must be 140.45* The circuit used is shown in Fig. 5. The first stage is a converter (6BE6) which changes the input frequency from 455 kc to 20 kc. The plate circuit of the 6BE6 is tuned to 20,000 cps. The voltage divider following the tuned circuit was inserted to prevent the overall gain from 9

being too high and causing saturation effects in the final stages. 6BE6 20,000 KC 0.1 6AU6 19,750 KC 0.1 6C4 547oK 2L: L 2TP2 0 71 -: s IOmh R c L C TPI 0 0IK 4 T7K 220K JA ~C 010 1 -.T..0.5 lo 1 5 10 - ITK 0.5 1 o 6AU6 20.250 KC 0.1 6C4,. Rs o' --- o1 R, 0.5 0.1 Rs3 220 K 2W 0.5L IK IK T (C~) @ ~~~~~~ TVV 0 250V iw Fig. 5. Circuit diagram of linear phase band-pass filter. TABLE II SUMMARY OF ELEMENT VALUES FOR EACH TUNED PLATE CIRCUIT f 20,000 cps f = 19,750 cps f3 = 20,250 cps L1= l10mh ~ Q = 140.45 L2 = lOmh @ Q = 140.45 L3 = 10mh @ Q = 140.45 C1 = 0.0063320,f C2 = 0.0064936 4f C = 0.0061770 lf R = 48,436 Q R = 64,278 Q R = 67,572 Q Theiplate circuit w asfolowed yacathoe scon ge2 3 to This circuit was followed by a cathode follower (6C4) in order to mini10

mize the capacitance effects of the potentiometer. If the potentiometer had been placed directly across the tank circuit, its resonant frequency would have been changed from approximately 0 to 25 cps, depending upon the position of the potentiometer. The fourth stage (6AU6) was tuned to 20,250 cps. Isolation between the output and the last tuned circuit was provided by a cathode follower (6C4). Greater than normal decouplingwas used in the plate voltage supply line to prevent regeneration effects. A word is now in order concerning alignment procedures. Each of the inductors obtained had a Q > 140.45; hence, it was necessary to lower the Q of each to the proper value by the addition of a small series resistance, R. This resistance consisted of a few turns of No. 30 B. & S., constantan wire wound on a 3/16" diameter phenolic coil form. Through the use of a Boonton Type 260-A Q-meter each inductor was individually adjusted to the proper Q at its operating frequency. The next step was to adjust each tuned circuit to the proper center frequency and bandwidth. Throughout the alignment procedure an HP-225B electronic counter was used to measure frequency. The test setup is shown in Fig. 6. With the converter operating, a test signal of 20,000 cps was applied to the signal grid of the 6BE6. The tuned circuits of the second (6AU6) and third tuned stages were shunted by 560-ohm, one-watt resistors. An oscilloscope was connected to the filter output for monitoring purposes. The capacitor C1 was placed across L1 and adjusted until a maximum deflection was obtained on the oscilloscope at exactly 20,000 kc. Then R was adjusted to give the proper bandwidth. This was accomplished in the following way. First, the 3-db-down frequencies of each tuned circuit were calculated. These are given in Table III for convenience. 11

IODS VARI I. uDB OSCILLATOR ATTENUATION TO INPUT OF STAGE LPAD NETWORK PADit J t BEING ALIGNED (5) ELEC T RON I C COUNTER (2) Fig. 6. Test setup for the alignment of each tuned plate. The method of calculation of these frequencies is shown in Appendix C. After introducing 3 db of attenuuation into the input signal channel by means of the variable attenuator, the oscillator is tuned to the resonant frequency of the circuit and the deflection on the oscilloscope noted; then the 3 db of attenuation is removed and the oscillator frequency is changed until the same oscilloscope deflection as noted before is obtained. TABLE III TABULATION OF 3-db-DOWI/N FREQUENCIES, FOR EACH TUNED CIRCUIT f = 20,000 cps f = 19,750 cps f = 20,250 cps 20,333 cps 20,014 cps 20,514 cps 19,672 cps 19,490 cps 19,990 cps The frequencies at which this occurs are the 3 db down frequencies. The shunt resistor, Rp, is changed until the frequencies listed in Table III are obtained. The first circuit is then properly tuned. The next step is to remove the 560-ohm shunting resistor from the second tuned stage, 12

shunt the first tuned circuit with this same resistor, and proceed in the same manner as described. Finally, we align the last stage. By carefully making these measurements and adjustments, we obtain an overall transfer function which will be correct without any further adjustment. 6. MEASUREMETS AND CONCLUSIONS The phase shift of the filter was measured and the results are shown in Fig. 7. A Hewlett-Packard Model 524C electronic counter with a plug-in Model 526B time-interval unit was used to measure this phase shift. By measuring the time interval between the positive-going input voltage and the positive-going output voltage and by knowing the frequency of the input wave, we can calculate the phase shift of the filter from: e = 36OTf; where e is the phase shift in degrees,T the measured time interval in seconds, and f the frequency in cycles per second. With the electronic counter an accuracy of + 1 degree can be obtained. The measured curve clearly shows that the original phase specification has been met. Figure 8 shows the measured relative voltage gain of the filter. An interesting observation can be made regarding whether or not the direction-finding receiver has been properly tuned to the desired signal. If the output of the filter is observed on an oscilloscope, the null of the double-sideband signal is distinct when the signal is properly tuned; it is not distinct if the receiver has been mistuned. In conclusion it can be stated that a linear-phase band-pass filter meeting the desired specifications can be designed and constructed using the methods contained in this report. 13

+100 + 80 +60 +40O O3 +20 U3 0 ~J3 LL I. 2-20 -40 -60 -80 -100 f- 300 f- 200 f- 100 f o fo+ 100 fo+ 200 fo'300 FREQUENCY (IN CYCLES/SEC.) Fig. 7- Me~asured phase shift of linear phase filter. 14

f I i FT~~i~i-F~T-TT~r-ilF I-CiT tj ~ t ElE-~~~~~~~~~~~~~~~~~~~~Fig.~_$-lt-iI~ 8. Meaure reatv voltg gan fliner phs filter. f~5 l ti I I ~ - I I i. T1 t 1 1i 1. i j t IjI I fi I~~t -i-e _ t I t jt I fi i:t~l i~t-i. li-T. i # I,; -t~l iff~~~~~~~~~~~i T + I~ ~~~~~~~~ ti j +-~iijt +~~+ 4- 44 + -~~~~~~~~~i- ii j.i 600 f + 400 f + 200 f + 200 f +400 f + 60 0 0 0 0 0 0- l-t It~ i1If t- i FREQUENCY (IN CYCLES SEC.)-. Fig 8.Meaure reatie vltae gin f lnea, pbase filter.

APPENDIX A CALCULATION OF BAND-PASS POLE POSITIONS FROM LOW-PASS POLE POSITIONS Two cases are to be considered: (1) real low-pass poles; and (2) pairs of low-pass, complex conjugate poles. In order to make the 2 2 P +w transformation Amp is substituted for p in the low-pass transfer. function. This will be carried out below for cases (1) and (2), and an approximate transformation will be derived. Case 1: Transfer function of the form F(p) = (real pole). Substi2 2 p + P +__ 1 tuting ()p... for p, we obtain: tuting (awKp F(p) 2 2 2 ( 2 (A.1) p +w p + + (Aw)ap+ wo () + a The solution for the pole positions is obtained by setting the denominator equal to zero. Therefore, 2 2 p + (w)ap + w 0 +(O-w)-w 2 - -+ IPf 2 + ()-n o 2 JW (A.2) If now ( < < 1 then 2w ote the zero intransfoduced rmat the origin of the cplex plane by the 16

P =- (a % + j (Approximation) (A. 3) For this approximation to be true within 1%: 1.00 1 a2 0.99 w 2 0 oo0 > 1 _(2 (2()2 > 0.980 < =2 2 0.0199 )(2) < 0. 1411 or A 1 for 1% accuracy of 3-.544a the approximation (A.4) Case 2: Transfer function of the form, F(p) = (p- P)( - pi) where: P1 = -a + Jb, and P- = -a - jb. 2 2 P +w Substitution of (-)p for p gives: Again note the zero introduced at the origin of the complex plane for each of the poles by the transformation. 17

F(p) = 2 2 p +w [p +W 0 - ] 0 - -- (zw) p P (Aw)p rL (A 5) (aw)2 2 K 2 2 2 2 [P -(Aw)p1 + w I [p -(Aw)l + w ] Finding the zeros of each factor of the denominator, the following is obtained: P = jb(2 ) + jWO /1.]2 (A.6) P ~ 2 = _Wo(a 2 jb +-J% S-['-2..32 (A.6) W ~a - jb) a - jb 2: -w jb)+j% 1-[+ — (A.7) 2 o W 2 If2.. jb. 2 1<( << 1, then (A.6) and (A.7) become: 0 p = D( a+j ) + i., p = _W(a jb)+ (A.8), 2 - o Approximations (A.9) For these approximations to be true within 1%: >oo w a[ ~ jb2 0.99',00 >i1 -I > 0.9_ -2 0 a ja a- jb_2 Since j 1( _(m2 (a _b_ 2 o O the above inequalities will be true if 18

i -(w )2 a - - > 0.9801 1 2 1 b 0.9801 (~oL,2 la + jbl2 > 0.0199 0(O) I 2 1 < 0.1411 or _W1 for 1% accuracy of w < / 2./2 the approximation 0~ 3.544 Ja2 + b1 19

APPENDIX B DERIVATION OF ELEMENT VALUES OF PARALLEL RLI NEWORK WITH POLE-ZERO LOCATIONS AND THE INDUCTOR, L, SPECIFID P-PLANE x b Z(P)- -,.R-0 L M s C[p2 = 1 R 1 R(B.1) P P This equation can be put into trhe form: + Z(p) = 2 2 (B.2) C[p + 2cx p + w ] where: R R P P This equation can be put into the fom: z(p) = + c' ( 2)d 20o R C +' L PL

Solving for the pole positions, one obtains: +2 2 p = -c + j 0 o+ = - + jb (B.4) Therefore R 2 a 2 R C +-; b R= - ( +1) 2 (B.5) P p Since R C = -% by definition, R L(B.6) If Eqs. B.5 and B.6 are used now, R and C can be solved for in terms p of a, b, c', and L: 2 2 R 1-1 a +b (- + l) = c' ( ) + L = c' (2a - c') + P P 1 a2 +b + c2 _ 2a c' C (B.7) L[a + b + c2 - 2ac'] 2a = R C +' R C p C[2a - c' ] | =L[a +b +c'-ac'] |(B.8) p 2a -c' 21

APPENDIX C METHOD OF CALCUIATION OF 3-db-DOWN FREQUENCIES FOR EACH TUNED CIRCUIT It is found that R Z(P) = C[p2 + ( 1+ RSp + s + 1)] RC LC R Z(p) =+ c' C[p +2ca p + ] where: R R 1 R c= L 2RC L] and Wo LC [R P P o- lop, IF t For the purposes of this problem L i it will be assumed that Z(p) - C'Rp R R Rs- T _ < 1 andL < < w 1 5 sT C Fig. C.1. Tuned circuit. Therefore Z(p) 2 (C.1) C[p + 2cp + 3r At resonance ( = w): jr Iz(Wr)l - C[axs2t t 5 2l(C.2) 22

UNIVERSITY OF MICHIGAN 3 9015 03483 149811111111 3 0503483 1498