-2262 - 89S- UNTiVERSITY OF MICHIGAN ENTG.N:E'ERIiNG RESEARCH INSTITUh'E ELECTRONIC DEFENSE GROUP TECHNICAL ME3MORAINDWt NO. 22 SUBJECT: Klinkhamer's Method of Deternining Filter or Amplifier Transfer Functions BY: A. B. Macnee DATE: September 1955 I. INT'RODUCTION The process of network design breaks down logically into three parts: (1) specification of the network characteristics which woul.d be ideal for the job, (i) determination of a transfer function approximating these ideal characteristics and (3) synthesizing one or more filters having the transfer function determined by step (2). In filtering applications the engineering specifications can frequently be stated in iterms of the minimum permissible attenuation in one or more stopbands and the maximum permissible attenuation in one or more passbands. Such a specification is given graphically in Figure 1 where it is indicated that the attenuation a must satisfy the ATTENUATION| IN NEPERS a al I if a2 2Aaa, I O WF WQ2 ~ 3 I4 NN WS6 W7 FREQUENCY IN RADI PAN T / SECONDS FIC. I A TYPICAL FILTER SPECIFICATION

-2following requirements. c < 2A1 for c < ow2 Cx > c1l for (1) a < 2Zy2 for C C >92 for 7 w. The problem of finding a rational function f(p) t(p), Z2(p), etc. (2) of the complex frequency variable p = a + Jw which meets the requirements of Eq. 1 in an optimum manner experimentally, has been very neatly solved by Klinkhamner. Because this solution does not appear to have been used by many engineers in this country, who might profitably do so, this memorandum summarizes the approximation method and gives an example of its application to a particular problem. Klinkhamer's method of approximation enables one to determine experimentally, with the aid of an electrolytic tank or sheet of conducting paper, the poles and zeroes of a rational filter transfer- function which approximates the given attenuation specification in a Tschebycheff manner in each pass- and stop-band. The pole-zero locations are determined uniquely, without any trial and error process (to the precision of the experimental set-up employed), and the particular rational function is optimum in the sense that it employs the minimum number of poles necessary to achieve the given specification. This, in general, means that the network synthesized to have this transfer function will require the minimum possible number of elements. II. THE CHA\RACTERISTIC FUNCTION The problem is to find a rational function f(p)/g(p) having poles in the left half of the complex frequency p-plane and zeroes in this plane or on the axis of imaginaries (jw-axis), such that the attenuation J. F. Klinkhamer, "Emperical Determination of Wave-Filter Transfer Functions with Specified Properties," Philips Research Reports, 3, pp. 60-80 and pp. 378-400.

-3g(p) a = in f-(P) nepers (3) satisfies the following requirements: (a) in a finite number of pass-bands Pl P2, -—, Pn the attenuation lies between co and co + 2l1, oo + 2L2a,..., %o + 2Qbck (where the Lt's are less than 0.115 neper - 1 db) and (b) in a finite number of stop-bands S1, S2,..., Sm the attenuation lies between ao + Cl''Yo + 12' *'*' loo + am and cso + cl + 2s l' o + a2 +2 2, o + m + m (where +L} cvl,'''' cm are greater than 1.73 neper = 15 db, and lAr 1' y ha mmay range from 0 to G) It is expedient to define a new function 2 2 |3h(p) = I3 - e2,o (4) which will be called a characteristic function. It is clear that the poles of this characteristic function hjf are the same as the poles of the function gjf which one wishes to determine, but if the attenuation i.: oscillates in an equal ripple fashion between ao and c'o + 2Ln in the pass-bands, the zeroes of h/f 2 all lie on the jw-axis, and they are all double. If one considers initially that the nL m are all infinite, then h/,f is a function having all its zeroes and poles on the jwraxis which characterizes the proposed filter function just as completely as g/f does. To make use of this characteristic function one must translate -the given specifications on g/f, (a) and (b), to appropriate specifications on h/f. III. SPECIFICATIONS ON THE CHARACTERISTIC FUNCTION As long as the difference between the stop- and pass-band attenuations.:1, c2''' cm exceeds 15 db one may write in the stop-bands This restriction of all the points of infinite loss to the jP-axis can be removed later.

- n f 1.73 nepers, (5) or e 5.64 (6) Squaring both sides gives g -2.. > fl e ='O 31.8. (7) Therefore according to Eq. 4 II2 - I2 < Ig2 2 318 (8) From Eq. 8 one sees that in the stop-bands the attenuation obtained with h differs from that with, by at the most f 12 Q (In! 1). -0155 nepers =.135 db. 31.8 Therefore as long as the stop-band attenuation is at least 15 db greater than thle passband attenuation one can assuae that in the stopband. An Ii(P- J n |()| (9) L(f) f(P) In the passbands Qn (YO 6 nepers (10) which may vary between 0 and 2o'A9 nepers. T'.ak.ing the exoonential of both sides of this equation gives _2 I e - = 1 + 5 + 2.. (11) As long as 5 is small compared to unity, only the first; two terms in the series of Eq. 11 need be considered. r'hus f e g 1 - 26 (12)

and substituting from Eq. 4 gives Bh| e 0" 25 (13) which varies between zero and 4&Yv over the passband. Therefore in the passband the attenuation of the characteristic function In ao -Laos ~n(25) (14) which varies between - m and In2 + 1/2 RnCav. If the maximum passband attenuation 26av does not exceed 0.13. nepers (.87 decibels) the error introduced by the approximation of Eq. 12 on the specification of h will be at the most.0478 nepers (.41 db). In summary, if one wishes a transfer function g/f such that the attenuation given by Eq. 3 will satisfy conditions (a) and (b), it is necessary to choose the characteristic function h f such that: in the pass-bands Pv ~In r -Co must be n (2) + 1/2 v (15) in the stop-bands Sv, In - -c'o must be (cI (16) The approximation problem is thus reduced to finding a rational function h/f having all its zeroes and poles on the jw.ax:iws and sati sfy:ing the Eqs. 15 and 16. IV, HE ELEECTROLYTIC TAINK AIALOG FOR D]ETERMJNINJ G h/f The analog between the potentials and currents in a electrolytic tank (sheet of uni.form conducting paper) and the attenuation and phase of rational function is well known, 1'2,3 This is illustrated in Figure 2. A sheet of conducting paper is used as A. R. Boothroyd, E. C. Cherry, R. Maker, Journ. I.E.E., Part I, Vol. 96, pp. 163-177. 2R E. Scott, Proc. I.R.E., Vol, 40, pp. 970-973. 3J. F. Klinkhammer, Philips Res. Repts., Vol 3, pp. 60-80, pp. 378-400. lW. H. Huggins, Proc. I.R.E., Vol. 36, pp. 421-424.

VTVM 0 -,x__.3 (b) FIG. 2 THE POTENTIAL ANALOG OF A RATIONAL FUNCTION IN TERMS OF ITS POLES AND ZEROS IN THE COMPLEX FREQUENCY PLANE, an analog of the complex frequency p-plane. This sheet should be infinite in extent, but in practice a circular electrode a-t same distance from, and concentic with, the origin can be used to represent the point at infinity. It is also possible to use a conformal transformation to map the infinite p-plane into a finite area, The variable resistors in Figure 2(a) are adjusted. so that the current flow in each lead is the same. Positive unit currents are connected to flow into the paper at the points analogous to the pole locations of the

-7function being similated, and negative unit currents are withdrawn at the zero locations. It can then be shown that the potential along the line on the conducting paper corresponding to the jco-axis is proportional to the logarithm of the magnitude of the rational function whose poles and zeroes are located as indicated in Figure 2(b). This potential can be measured experimentally with a moving probe and a vacuum-tube voltmeter as indicated. It is also possible to show that the current flow at right angles to the j -axis in Figure 2(a) is proportional to the rate of change of the angle of the rational function having the poles and zeroes indicated in Figure 2(b). The general approach to determining h/f with the potential analog is: (1) Set-up a non-ansalytic function meeting the specifications (15) and (16) by introducing band electrodes into the electrolytic tank at the locations of the pass- and stop-bands and adjusting the potentials of these electrodes to meet the specifications; and (2) replace these band electrodes by a row of needle electrodes carrying the same currents as the bands. A band electrode carrying n unit currents into an electrolytic tank can be divided into n sections each carrying unit current as indicated in Figure 3(a). ZThe potential along 6i n2 ~~t ~ ~ ~ ~/ i /<4 \.... 1\I\\\ (\o II // 1 ( 1,"',, i, ~IlJ I ~7-. i /,~,' l'' I - (b)It I I II I I I I I I 1 FIG, 3 (a) A BAND ELECTRODEf (b) AN EYi V.ALN1T ARRAY OF NEEDLE SOURC~ES, A~ND (c).~ri~EN~ L~ AQr B~ AND ALO~NG THE~. SORESDKE (. TL-.(:..:LT

the line of such an electrode might be as indicated by the solid curve of Figure 3(c). If now this band electrode is replaced by a row of needle electrodes each carrying unit current and located at the midpoints of the equal current divisions of the band electrode as shown in Figure 3(b), it can be shown that the potential due to these needles will approximate that due to the band as indicated by the dashed curve in Figure 3(c).1 Along the line where the electrode was located at a potential VO the potential will now vary between - X and Vo + 2n2 as indicated. At other points in the electrolytic tank the approximation will, of course, be much better. Designating the non-analytic function setiup by the band electrodes as AnIF(p)!, the conditions that F(p) must satisfy in order that when the band electrodes are replaced by rows of needle electrodes the conditions of Eqs. 15 and 16 will be met are: in the passbands, Pv', n.jF-:i must be. In (~\;)3 (17) in the stopbands, Sv, ~nlFI-ao must be ov + n (2) (18) The steps of a feasible, although not t'he most direct, experimental procedure are indicated in Figures 4- and 5. Klinkhammer, loc, cit o, Sectio n 35 -.

~~~~jW~~j +2 P)j 2 +2 - -2 N A 4 N: FIG 4 POTENTIAL ANALOGS OF a) p AND () IN THE COMPLEX FRQUENCY PLANE \ / (l) Electrodes are introduced into the potential analog (electrolytic tank or conducting paper) at the locations where passe and stop-bands are desired, which might be as indicated in Figure 3(a): A sufficient number of double unit currents are then permitted to flow to and from these electrodes to —estab1ish their potentials as required by Eqs. 17 and 18, The potential as measured along the jc-axis might then appear as shown in Figure 5(a); it is hypothesized, for the purposes of this example, that in order to establish these potentials it is necessary to force four unit currents into each of the stop-band electrodes and to withdraw eight unit currents from the pass-band electrode. (2) The band electrodes are divided into elements each of which delivers or withdraws two unit currents from the potential tank or paper. In principle the distribualternact~e pIrocedbure will be developed_ shor~tly.,

inIFI ~n I I n f ).xn(2) - <p.!a, a,' WsI~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. Iw 0~~~~~~~~~~~~~I II 0 jan(2) It I —iII li (' 4(a, a b)~~~~~~~~~~~~~~~~~~~~~~~~~~~U FIG TON -N AN TE — A FORA I N ~~~~~~~~~~~~~.2a 2 ~~~~~~~ ~I N O I T, O O O 1 Ws w 2In(Ln(,) I~~~~~~~~~~~~I (a) (b) (c) FIG. 5. POTENTIAL OR ATTENUATION FUNCTIONS ALONG THE jw-'AXIS FOR A TYPICAL EXAMPLE OF:(a)THE NON-ANALYTIC FUNCTION F(p), (b) THE CHARACTERISTIC FUNCTION h (pl, AND (c) THE TRANSFER FUNCTION f

(3) Needle electrodes are then located. at -1he centers of the elements determined in (2), and double unit currents are forced to flow into or out of these needles, This is illustrated in Figure 4(b) where four needle electrodes are indicated in the pass-band and two in each stop-band. The potential distribution along the jw-axis corresponding to this newst electrode configuration is shown in Figure 5(b). This func-.;ion is the potential -.,naPlog of the squared magnitude of the characteristic function h f; the zeroes of this function are double and are located wherever a needle is withdrawing a double current from the analog while the poles are located where currents flow into the analog, (4) The squared magnitude of transfer functions g is related to the charach 2% teristic function -by Eq. 4. Therefore adding the constant e to the function in Figure 4(b) gives the potential analog of I~i2 as shown in Figure 4(c). One can show that the only effect of adding a constant to i1! is to cause the zeroes to move along the streamlines from their original positions toward the poles. The zeroes of 1gj2 will lie where h(p) h( -p ) (19) Taking the logarithm of both rides gives h 2 en iT1 = -2o (20) which means that the zeroes of are located on tthe -2%' equipotential in the (p) f o f or the F(p) potential plane, In particular, if the minimum passband attenuation is zero nepers this is the zero equipotentiaL The potential along the jC-axis for the pole-zero configuration of Figure 4(c) is shown in Figure 5(c), Comparing Figures 4(a) and (c) one sees that the poles and zeroes of the transfer function | (p~j are located at the intersections of' certain equi.potentials and streamlines of the potential analog of the function F(p), If the minimum pass-band attenuation is zero, the zeroes (points of infinite gain) are located on the zero equipotent ial; and the poles (points of infinit;e loss) are located one n(2) equpoential on the i,-axis. If one sets up the potential analog of F(p), it is a straight forward process

-2to experimentall determine the location of these equipotentials., To complete the locations of the poles and zeroes, however, the stream lines in this potential analog must also be located. For the function ~n F(p) = An IF(p)i + j arg F(p) (21) illustrated in Figure 4(a) the real and imaginary part would vary along the jc-axis as indicated in Figure 6, In our potential analog the potential along the j-saxis is proportional to in IFJ, and the current flow into the tank is proportional to arg F. Thus -V w FIG.6 REAL AND IMAGINARY PARTS OF THE LOGARITHM OF THE NON-ANALYTIC FUNCTIONS F (jw). between the equipotential bands, there is no current fglow into the tank, and the arg F is necessarily a constant. A method of measuring arg F(p) directly is to set ulp the conjugate potential analog of F(p). Tn this conjugate configuration the equipotentials are the stream lines of the original configuration, rWe know that the stream function is a constant along the jw-axis in every transition region between a pass-btand and a stop-band. The stream function will change by an amount proportional to the current flow from the band

-13 electrodes as one traverses a band between two transition regions. Experimentally then, one can set up the conjugate potential map by withdrawing the original set of band electrodes, corresponding to the pass- and stop-bands of the proposed filter design, and inserting a new set of band electrodes located in all the transition regions of the filter specifications. The potentials between these new electrodes should be made proportional to the number of double unit currents flowing to the band electrode which was originally located between them. This is illustrated in Figure 7. With the electrode configuration shown in Figure 7(a) it is established that four unit currents must flow to each stop-band electrode and eight unit currents be withdrawn from the passband. electrode to satisfy the specifications of Eqs. 17 and 18. The equipotential V=O can then be traced as indicated. Next the conjugate electrode configuration is set up as indicated in Figure 7(b). There is only a-single pair of transition bands in the example illustrated. If the potentials of these V^ V electrodes are made + - and - respectively, then V0 is proportional to the four 2 2 double unit currents flowing from the pass-band electrode in Figure 7(a). The equipotentials in 7(b) corresponding to the streamlines which pass through the centers of the section of the pass band electrode in 7(ae) carrying double unit currents are at V0 (2k-l) V= - + - -2 Vo k = 1 2,.., n (22) where n is the number of double unit currents, In the example illustrated n = 4. Now the poles of g/f lie at the intersections of the "streamline" equipotentials in Figure 7(b) with the &c1 + ln(2) equipotentials in 7(a) and the zeroes lie at the intersection of these same stre'amlines with the V - 0 equipotential. in 7(a). For the example in Figure 7 the pole locations are indicated by crosses and the zero locations by circles.

iw jw v - a +2n (2) +4 v=a+ n(2) 8 / (a) (b) FIG. 7 (a) AN ELECTRODE CONFlGURATION ESTABLISHING THE FUNCTION BnF(p), AND (b) THE CONJUGATE ELECTRODE CONFIGURATION V. S'JlY\~Y OF THEE EiERx [,m ~,U PROCED7OE The experimental procedeure nEessary to determfrni the poles and zeroes of the transfer function g(P) such that the attenuation,. Eq 3 satisfies the specifications f(p) that, in the passbands, PF < n 4 - C. v, (23) and in the stopband, SE n can beesumv m ari e ano ~ s ( 24o ) can be sunmarized as Follows:

(1) Find the proportionality cnstant C of the electrolytic tank or conducting paper. To do this one introduces two needle electrodes into'4he electrolytic tank, one carrying a positive unit current and the other a negative unit current,located at let us say, p = a as indicated in Figure 8, Then the potential difference VO between the I o Vo.-C O -a _a +a_ +0 2 2 FIG. 8 SET-UP FOR CALIBRATION OF THE POTENTIAL ANALOG OF THE COMPLEX FREQUENCY PLANE, + a points i s given by a.a. o = C n(- + a) - C in (-.a) - n~ + a) +-C in ( - a) = -C In (9) (24) and the proportionality constant of the tank is V c (25) (2) Put band electrodes in-to the tank (or pal..>.1. band electrodes on the conducting paper) along those parts of tLhe jw-axis where passe and stop- bands are desired. Then feed an integral number of unit currents to each band until. (a) the potential oP each passband electroe where an attenuation ripple of + ovl is desired becomes Vp i ln Aa (26) and (b) the potenia of eac stop bnd electrode where a minimum attenuation of,v is required satisfies the relation v Us (Or C Lr ( )1~ (27)

If the attenuation in the stopbands is further restricted to vary between (Yv A.-v and - + C., the stopband electrode potentials should be sV S C ['v 2 i n (eLcv 1).. In (2) + AxY] (28) which simplifies to v 2 n l'v ] (29) ijf A) - 0.1. (3) Check whether there are portions of any o' the stopband electrodes at a potential below the electrolyte potenti-al in the iLrinmed.-atie vi ci.nity of the electrodes. If this occurs, shorten these electrodes until this is no longer true. Make the same check for the passband electrodes. (4) Check the band potentials twith these shortened electrodes and determnine whether the potentials required in (2) can be achieved wilth fewer unit currents. (D) Plot the equipotential and if A.- J is not infinite, the equipotential = v (31) (6) Remove the band electrodes inserted in (2) and replace them by the conjugate set. There will now be band electrodes in the tank wherever transition regions in the specified attenuation characteristic occur. Adjust the currents flowing to these new electrodes so as to makle the potential difference be;tween any adjacent electrodes equal to N times the current taken by the intervening electr;xode in the original electrode configuration (N is an arb.itrary constant). ('7) Designating the lowest potential in this conjugate configuration as the zero reference potential, plot out the equipotentials Vm = N(2 + m) (32) where m = 0, 1,..: until the highest potential in the tanka is exceeded.

(8)''The inttersect:ions of the oequ:.potent:ials of Eq. 32 wi th that of i'f:. 30 13cate'the zeroes of'. 1..Tl Yetc while the intersectios with the equi. otential of Eq 31 locate the poles. V;..,N:.X. 0i A'IIO IT.,w.XP].:RIDM!INTAL PhROCT;*',:URT. pan papl.ying K:Linkhammer s met;hod it, i.s corenvi ent to use Teledel.tos conductiang paper for the potent~ia.l an.ralog. E;ectrodes are painted on thuis paper with an air-drying Sjje. condiuct:ing si:lver pai.t A t i,..ypical o)ole-zero configuration for a network function.is show.rn:in F.igure 9(a), ) By' syimmetry one sees that the current flow across the 1.ine corres.tnd:.ing to3'the'T-axis wil X. alwttays be zero. Lt is therefore always possible:or ex:per:menrtal ournroses to r e-, lace thie oenniure p-pilane analog by a semi -cxircular anaJ].oe;:.orrespondi.ng to the uIper or lioe? half of the p-plane only'.:if in this'process any current; sources on the a-axis are hal.ved, no chane:i.n the potential or current flow, across -the jo-axi.s i s introduced, Thi.s -orocess casm be carried one stepo furthe hen one recal3s that adding ro:les and zeroes in Lhe r;.ght half plane at the m:irror images of those in the left half i;).lane has the effec' of doubling the attenuation (analog: potential) along the j.) a..is and reducing the phase shif-t to zero, This same effect can be obta:ined by cutting our half-p.lane Qpoten-tial analog in half again leaving the quarter plane analog as illustrated:I n Figulre 9(c), If the current sources corresponding to poles and zeroes withi.n the pl.re are hnajved anld those on the axes are reduced to one quarter of their values in Fi.lmre 9(a). the poteni. a1s nme-arnsured in the F'igure 9(c) analog ll.3. be exactly the sar:e as those JIn 9(a), rTeled-eltos?ir is manufactured bv the,Wrestern Union Telegra:h Co., 60 Hudson't7r eet, N.T.'Ior 1 Y. The paper used. type L-39, yhere has a res:i.stance of 1500 C..ohms/ i ossiqare.. suiLtable pa:in t is GCeneral Cement Mf;lg Cg'o's';. S. Iver Pr'.nt''No. 2.-2

-18jw I jw (a*( X 0 0' x 0' (b) (C) FIG 9. REDUCTION OF THE POTENTIAL ANALOG TO AN EQUIVALENT QUARTER PLANE ANALOG FOR THE MAGNITUDE OF A NETWORK FUNCTION, The reason for wishing to reduce the-potential analog to a quarter plane lies in the desirability of reducing errors due to the finite size of the analog plane. Normally in the potential analog the point at infinity is replaced by a circular electrode at a radius large compared to the distance from the origin to the region where measurements are to be made. Experimentally it has been determined that if the radius of the tank is ten times the radius of that portion used for measurements the attenuation errors introduced do not exceed 0.4 db. For a given maximum dimension of the electrolytic tank or conducting paper use of the quarter plane allows one to double the frequency scale (inches/rad/sec.) employed. For example, teledeltes paper is available in rolls 31 inches wide. With a radius of 31 inches for the quarter plane analog measurements can be made from the origin cut to a circle 3.1 inches from the origin. If a larger useful region is desired for the analog, some scheme for overcoming the error due to finite tank size must be employed. One approach is to employ a conformal

transformation which maps the entire p-plane into a rectangle.1'2'3 In this case infinity maps into a single point at a finite distance from the origin; the drawback of this approach is the necessity of transforming the locations of the poles and zeroes in the rectangular w-plane back into the p-plane through the transformation p = j sn(w,k) (33) When w is complex, evaluation of p from Eq. 33 requires making six entries into elliptic function tables in which interpolation in two directions is required. Another approach, is to employ a double-layer construction of the potential tank. The lower layer is connected to the upper along the circular boundary and represents an inversion of that part of the p-plane lying outside the boundary. The points corresponding to p=O and pao are located at the centers of the upper and lower layers of the tank as indicated in Figure 10(a). This same idea might be employed with two sheets of conducting paper insulated everywhere except at the boundary as indicated in Figure 1O(b). ORIGIN INFIN TY (a) I (b) INFINITY FIG. 10 TECHNIQUE FOR SIMULATING THE INFINITE P-PLANE WITH A FINITE (a) ELECTROLYTIC TANK OR (b) CONDUCTING SHEET. 1 Fano, R. M. "A Note on the Solution of Certain Approximation Problems in Network Synthesis", Jan. Frankl. Inst., Vol. 249, No. 3, pp. 189-205, March, 1950. 2 G. L. Matthaei, "Conformal Mappings for Filter Transfer Function Synthesis", Proc. IRE, Vol. 41, pp. 1658-1664. A. R. Boothroyd, E. C. Cherry, and R. Makar, "An Electrolytic for the Measurement of Steady-state Response, Transient Response, and Allied Properties of Network," Proc. IEE, Part 1, Vol. 196, p. 163, May 1949.

-20As an:i1llustration of thi.s awppLroximation method, consider a filter specified as fol.lows: (C.. < L o no speci"fication. T<'.i. 6 0., 0,-,:.< O.5 db i. 54-6 < 1 7 a.28. no specification i 728 c* 1'.91 20 db <, -191 < c no specifica'tion Converting the specific-ation to nepers gives AT].= -.0289 nepers for 1' (S_ 1546 radians sec., and.:'> 2.31 nepers for 1.7238 W 1.91 rad. sec. Using the teledeltors pamper type -39 aZnd a 22 1/2 volt battery in series with a 34, 000 ohm resistor as a unit current source, the constant of the tank was found to be -0.91 units on the particular vacuum tube voltmeter employed. Then.according to Eq. 26 the potent:ial of the passband electrode sho'uld be > -o. 9i Vp in(.0o289) - 1.61, 2 and from Eq. 27 the potential of the stopband electrode should be -0. 91(2.31 +.693) -2.76 Since, for fixed cu.rrents, using'the quarter tank analog doubles the potentials everywhere, these requirements become V = 3.22 V <- -5.52 5

-21or that the difference in potential between the pass- and stop-band electrodes must exceed 8.74. Electrodes were painted on a piece of teledeltos paper as indicated in Figure 11(a); the radius of the quarter sheet employed was 18 inches and the frequency scale was 2 inches per radian per second - S 4- p (a~) (b) FIG.-II (a)QUARTER PLANE ANALOG OF EXAMPLE WITH PASS- AND STOP-BAND ELECTRODES AND (b) THE CONJUGATE ELECTRODE CONFIGURATION. The painted areas ari-.'. cru Connecting a single positive unit current lead to the passband electrode (p) and a single negative unit current to the stop electrode (s), the measured potential was Vp - Vs =7.0. This is less than the required value of 8.74, but when two unit currents are used to each electrode Vp - V5 = 14. At this stage one would normally trace out the equipotential V = Vp - 3.22 on which the poles should be located in accordance with Eq. 30. Since, however, the observed potential difference far exceeds the minimum required by the specification, some modifications of the design is possible. Since there is no pass- or stop-band electrode at the origin, odd numbers of poles were not considered worthwhile. One could, arbitrarily locate a pole on the negative real axis and a zero at infinity, but for the configuration being considered one would not expect such a pole-zero pair to effect the potential between the pass- and stop-band electrodes appreciably. It was therefore decided to take advantage

-22of the observed excess uipotential to exceed the given specifications The eqipotential V=Vp -5 was traced out as indicated by the solid line in Figure 12. This means the ripple in the pass-band should be reduced to 2Z;1 = 2 Exp [=-.0081 =.07 db, l- in(2 ) = 4.26 nepers = 37 db C The conjugate electrode configuration was then painted on a second sheet of teledeltos paper as indicated in Figure 10(b). Current was injected into the transition band electrode (between s and p in Figure 10(b)) and withdrawn from the electrode along the rest of the jw-axis. The magnitude of this current was adjusted until the transition band electrode was at a potential of 4.0 relative to the electrode along the balance of the axis. The equipotentials V = Vt - 1 and Vt - 3 were then traced. These are the dashed lines in Figure 12. The intersections of the dashed and solid lines in Figure 12 locate the poles of the transfer function f(p)/g(p) at P = -.55 + J 1.57, p = -.085 t j 1.583; and the zeroes of the function should lie at the intersection of the dashed lines with the jc-axis: p = t j 1.745 p =- j 1.88

jw j2.0 i.0 -3.0 -2.0 -10 ao FIG. 12. POTENTIAL PLOTS FOR POLE-ZERO DETERMINATIONS. r~~~~~~ g~ ~ ~~~

-24The complete transfer function can be written as f(p) K (p2 + 3.045)(p2 + 3.534) 9-(P) = K (p2 + l.llp + 2.773)(p2 +.17p + 2.533) where the constant K determines the reference gain. Choosing K = 0.621 gives a reference gain of zero decibels; for this case the calculated attenuation is plotted in Figure 13. One, finds, from this plot, that the attenuation does not meet the specification (dotted lines) at the high frequency end of the pass-band. From the position of the poles it is apparent that the poles at p = -.085 + jl.583 have the greatest influence on the attenuation at the edge of the passband. Moving this pair of poles to p = -.077 t j 1.583 and choosing a new multiplying constant K = 0.595 yields the attenuation characteristic plotted in Figure 14. VII. CONCLUSIONS Klinkhamer has developed a practical experimental procedure for determining the poles and zeroes of a rational network function satisfying a specification of constant attenuation levels in a number of pass- and stop-bands. This procedure, although experimental in nature, is not a cut and try process; the poles and zeroes are located at the intersection of certain well defined equipotential lines. These equipotentials are conveniently located with the aid of an electrolytic tank or a sheet of conducting paper. An important feature of the network functions obtained by this method is that they are optimum in that they meet the specifications with the minimum number of poles and zeroes, which in turn means that the networks synthesized to have these characteristics will have a minimum number of elements. If the specifications of the example of Section 6 were met with a bandpass filter derivable from a low-pass filter, for example, Klinkhamer, loc. cit. p. 395.

40 - -- - _ — f(p) 061 (p2 + 3.045)(p2+3.534) g(p) (p2 + 1.1lp+2.773)(p2+.17p + 2.533) 30 LAJI~~~~~~~~~~~~~~~~ I i 0.. _. -------. z 0 1.2 1.4 1.6 1.8 2.0 FREQUENCY IN RADIANS PER SECOND FIG. 13. RESULTS OF EXAMPLE

IN3WLSnfrl0V3 310d 831_jIV 31dVlVX3'tl'91 ON033S 83d SNVIOVl NI A3N3n0O38..I ~~~ 1iv ~~~~~~I,._ — O~i~~~~~~z',I I 1 I I 0 Z \~ ~~~(0g | t, + I d)(2Sb~+ zd) (d)I | O 2, 2262 A - G I - 229 BM 15 NOV55

-27a 6 pole function would. be required. Such a filter would have a stop-band s from o = 1.73 rad/sec to infinity and from zero to 0.895 rad/sec. If on the other hand the specification is met with a low pass design having a passband from zero to 1.55 rad'sec and a stop band from 1.73 rad,/sec to infinity, four poles are sufficient. The function obtained here, however, provides much more attenuation in the specified band from 1.73 to 1.91 rad/sec with the same number of poles.

DISTRIBUTION LIST 15 Copies Transportation Officer, SCEL Evans Signal Laboratory Building No. 42 Belmar, New Jersey FOR-SCEL Accountable Officer Inspect at Destination File No. 22824-PH-54-91(1701.) 1 Copy H. W. We-l —,'1ch, Jr. Engineering Research Institute University of Michigan Ann Arbor, Mi.chigan -opy J. A. Boyd Engineering Research Institute University of Michigan Ann Arbor, Michigan 1 Copy Document Room Willow Run Research Center University of Michigan Ypsi lanti, Michigan 17 Copies Electronic Defense Group Project File University of Michigan Ann Arbor, Michigan 1 Copy Engineering Research Institute Project File University of Michigan Ann Arbor, Michigan 28