Transfer-Function Synthesis with Computer Amplifiers and Passive Networks

Transcription

1 955 WESTERN JOINT COMPUTER CONFERENCE 7 Transfer-Functin Synthesis with Cmputer Amplifiers Passive Netwrks M. v. MATHEWSt AND w. w. SEIFERTt Summary-The study f dynamic systems n an analg cmputer ften invlves the synthesis f cmplex transfer functins. Techniques frm the field f netwrk synthesis are cmbined with methds used in electrnic-differential-analyzer wrk t prvide effective means fr realizing these transfer functins with a minimum f cmputer equipment. The basic ideas f assciating high-gain amplifiers phase-inverting amplifiers with resistr-capacitr netwrks are applied in rder t btain three systematic methds fr synthesizing transfer functins f varius degrees f cmplexity. The methds described are illustrated by an example. INTRODUCTION M ANY analg-cmputer prblems require the synthesis f cmplex transfer functins. Realizatin f these transfer functins with a minimum f cmputing equipment is a practical bjective that can be achieved by the use f passive netwrks in cmbinatin with cmputer amplifiers. Prcedures fr realizing stable transfer functins with passive netwrks cntaining resistances, inductances, capacitances have been described by Darlingtn,l Guillemin,2 thers, but in many instances these methds have practical limitatins. A synthesis invlving inductrs frequently is cmplicated by inherent lsses distributed capacitances. Furthermre, inductrs, particularly when used at the lw frequencies encuntered in cmputers cntrl systems, tend t be large expensive. Highly versatile methds, which avid the limitatins inherent in passive-netwrk synthesis, may be derived by assciating active elements such as amplifiers with passive netwrks. The basic ideas f assciating highgain amplifiers phase-inverting amplifiers with passive netwrks were develped frm feedback-amplifier thery3 were discussed by Blackman, Bde, Shannn in a classified reprt in 946 by Belve 4 in 950. In the cmputer field, full advantage has nt been taken f the ptentialities f these methds because they have nt been develped int a systematic synthesis prcedure that uses cmputer amplifiers as the active elements. This paper describes three systematic synthesis methds which are particularly adaptable t cmputer t Massachusetts Institute f Technlgy, Cambridge, Mass. S. Darlingtn, "Synthesis f reactance 4-ples which prduce prescribed insertin lss characteristics," Jur. Math. Phys., vl. 8, pp ; September, E. A. Guillemin, "A Summary f Mdern Methds f Netwrk Synthesis," Advances in Electrnics, vl. III, Academic Press, Inc., New Yrk, pp ; H. W. Bde, "Netwrk Analysis Feedback Amplifier Design," D. Van Nstr C., Inc., New Yrk, N. Y., pp ; C. Belve, "Synthesis f Active Lw-Frequency Netwrks," Navrd Reprt N. 49, Bureau f Ordnance, Washingtn, D. c.; Nvember, 950. applicatins. First, the well-knwn methd that emplys ne amplifier a feedback netwrk is presented in its general frm. Next, an riginal 3-amplifier design that ffers distinct advantages in the synthesis f cmplicated functins is described. The third methd presen ted is a generalized versin f the basic technique used fr the slutin f differential equatins n a differential analyzer. In this last scheme, the active elements are integratrs rather than amplifiers. The cmputer amplifiers utilized in the synthesis prcedures are assumed t be ideal amplifiers which draw zer input current, have zer utput impedance, prvide any desired real gain. The assumptin f a perfect amplifier is valid in the frequency range frm direct current thrugh the audi spectrum because, fr these frequencies, cmputer amplifiers have been perfected t such a degree that their deviatins frm the ideal are as small as the parasitic errrs in resistrs capacitrs. SYNTHESIS USING ONE FEEDBACK AMPLIFIER In electrnic differential analyzers, integratin is perfrmed by assciating a simple rc netwrk with a high-gain amplifier, as shwn in Fig.. If an ideal am- Fig. -Blck diagram fr integratr used in electrnic differential analyzers. plifier with infinite gain is assumed, then analysis f the circuit yields fr the uput vltage e = - Gc f.' eidt.+ E), () where E is the value f the vltage acrss the capacitr at t = O. These same simplificatins give fr the transfer functin f this circuit e ---, RCs where s is the cmplex-frequency variable. Fig. 2 shws the simplest generalizatin f the basic integratr arrangement f Fig.. If the same simplifying assumptins used t derive () (2) are applied t (2)

2 8 955 WESTERN JOINT COMPUTER CONFERENCE the analysis f Fig. 2, the transfer functin f the circuit is fund t be (3) --=s (5) If Y A Y Bare 2-terminal rc netwrks, all their where ples zers must alternate alng the negative real axis f the cmplex-frequency plane the lwest C. [ N(S)] I = hm (6) 8-"0 critical frequencies must be zers. Cnsequently, the S ples zers f the transfer functin als must lie n The sum this axis, but tw ples r tw zers may ccur tgether, the lwest critical frequency may be a ple. Any transfer functins meeting these cnditins can be written in the frm (7) ----, D(s) where can be selected s that j D(s) j can be realized as 2-terminal rc netwrks. IMAGINARY COMPLEX PLANE (4) is btained by making a partial-fractin expansin f [N(S) _ CIS]. (8) s The resulting netwrk is shwn in Fig. 3, where the values f Ri C i are in hms farads. Substitutin f 3-terminal netwrks in place f the 2-terminal netwrks f Fig. 2 yields a useful generalizatin f this methd f synthesis. The resulting circuit, illustrated in Fig. 4, can be analyzed in terms f the REAL POSSIBLE POLE-ZERO LOCATIONS WHEN YA AND YB ARE 2-TERMINAL RC NETWORKS Fig. 2-Blck diagram fr -amplifier realizatin with 2-terminal netwrks. The synthesis f Y A can be carried ut several ways, ne f the simplest being t exp j in the frm C I Fig. 4-Blck diagram fr -amplifier realizatin with 3-terminal netwrks. input, utput, transfer admittances f the netwrk. These admittances are defined fr the A netwrk by the rela tinshi ps (9) (0) where the currents vltages are shwn in Fig. S. A similar definitin applies t the B netwrk. The vlt- Fig. 3-Frm f netwrk. Fig. 5-Definitin f admittances.

3 Mathews Seifert: Transfer-Functin Synthesis with Cmputer Amplifiers Passive Netwrks 9 ages in the circuit f Fig. 4 are related by the equatin Furthermre, e2(ya22 + YB22) = ei(ya2) + e(yb2). () (2) Slutin f () (2) yields fr the transfer functin eolei J..L (3) As J.t becmes infinite, eolei appraches the negative f the rati f the transfer admittances, (4) REALIZATION WITH THREE AMPLIFIERS Althugh few theretical limitatins are impsed n the type f transfer functin realizable with the singlefeedback-amplifier methd just described, the use f additinal amplifiers permits increased flexibility in the realizatin f the transfer functin. This flexibility can reduce the number f passive elements required t btain a given functin, decrease the spread f element values, simplify the synthesis calculatins. Such expedients are particularly imprtant when cmplicated functins with many ples must be realized. One synthesis methd using three amplifiers is develped t demnstrate that any transfer functin can be realized in this way. Once the particular methd is understd, many pssible variatins becme bvius. The circuit fr the 3-amplifier realizatin is shwn in Fig. 6. This circuit differs frm the l-amplifier realiza- The errr caused by a finite J.t can be evaluated frm (3), which is the exact expressin fr the realized transfer functin. The errrs can be determined either as the displacements f the ples f the realized transfer functin frm the desired ples r as the errr in the amplitude phase f the realized transfer functin at real frequencies. T keep the errr small, it is necessary that at all frequencies YA22 + YB22 Y B2» J..L (5) At high frequencies, either r bth f the utput admittances Y A22 YB22 may tend t becme infinite. If such is the case, the B netwrk shuld be s designed that Y B2 als ges t infinity at high frequencies. The transfer admittance f a 3-terminal netwrk frmed entirely f resistances capacitances can have nly simple ples 5 which must lie n the negative real axis f the cmplex-frequency plane but may have zers which lie anywhere in the cmplex-frequency plane except n the psitive real axis which need nt be simple. The ples f eolei in (4) fllw frm the ples in Y A2 r frm the zers f Y B2, while the zers f el ei fllw frm the zers f Y A2 r frm the ples f Y B2. Cnsequently, little theretical restrictin is placed n the type f transfer impedance that can be frmed by using a circuit f the type shwn in Fig. 4. Several general prcedures fr synthesizing 3-terminal rc netwrks have been given in the literature. 5,6 These prcedures are t lengthy t include here, but an example emplying ne f the methds 6 is given later. The principal restrictins impsed n this realizatin are the cmplexity f the synthesis calculatins, the large number f elements, large range f element values which may be required. 5 A. Fialkw. Gerst, "The transfer functin f general tw terminal-pair rc netwrks," Quart. Appl. Math., vl. 0, pp. 3-27; April, E. A. Guillemin, "Synthesis f rc-netwrks," Jur. Math. Phys., vl. 28, pp ; April, 949. Fig. 6-BIck diagram fr 3-amplifier realizatin. tin shwn in Fig. 2 nly by the additin f the C D netwrks the inverting amplifiers driving these netwrks. As will be brught ut in the discussin, 2- terminal netwrks are sufficient t realize any transfer functin; hence, this case is cnsidered. The vltages in the system bey the relatins e2(ya+yb+ Y c+y D ) =et(ya - Y c)+e(yb- Y D ), (6) e= -J..Le2. (7) Slutin f (6) (7) yields fr the transfer functin frm ei t e As J.t becmes large, (8) assumes the limiting frm (9) If the desired transfer functin is expressed as a rati f tw plynmials ID(s), the admittances must satisfy the relatin

4 0 955 WESTERN JOINT COMPUTER CONFERENCE Y A - Y e A majr advantage f the 3-amplifier synthesis ----=--. (20) prcedure is the simplicity f the calculatins required \. t btain the element values. The spread f element val In rder t realize the admittances as rc netwrks, (20) ues is determined by the spread f the terms in the is separated t give expansin f j given in (5) the crrespnd- ing expansin f D(s)j. The arbitrary zers f Y A - Ye =- (2) can be chsen by a trial--errr apprach t cntrl this spread. D(s) Y B - YD = --, (22) where is an arbitrary plynmial which des nt alter the realized transfer functin. The realizatin fllws the methd used t btain Y A Y B in Fig. 2. The fractin j is exped in the series given by (5), the terms in the resulting expansin are divided between the A C netwrks s all the elements have psitive values. The additinal freedm gained frm allwing negative terms in the expansin makes pssible the realizatin f any j with 2-terminal rc netwrks, prvided that the tw fllwing cnditi"ns are met: ) The zers f lie n the negative real axis; 2) The rati j ges t infinity n faster than s as s becmes infinite. An identical prcedure is used t realized D(s)j. The errr intrduced by a finite gain j.j. can be evaluated frm (8), which is the exact expressin fr the realized transfer functin. The methd is the same as the methd already described fr evaluating the errrs in the I-amplifier realizatin. Hwever, in (8) the pssibility exists f changing the synthesis prcedure slightly t realize exactly the desired transfer functin with a finite JJ-. Relatin (8) may be rewritten in the frm Y A - Ye +: (23) YB(H :) -YD(l- :) (YA+Yc) If the desired transfer functin is again designated j D (s ), it can be realized by making ( Y A - Y e =- ) ( ) D(s) VB +- -Y D -- =---- (YA+Ye). G( (24) (25) If JJ->, the A, B, C, D netwrks can always be realized as 2-terminal rc netwrks by using expansins f the frm given in (5). The exact realizatin, btained at the expense f including additinal elements in the B D netwrks, is justified nly in special instances because the errrs caused by a finite JJ- in the apprximate realizatin are usually less than the errrs due t parasitic behavir f the elements. SYNTHESIS WITH INTEGRATORS Anther synthesis apprach, different frm the tw already described, cnsists f cmbining a number f simple transfer functins t prduce a desired functin. A transfer functin f the frm. e ams m + am_ls m a F(s) = - =, (26) ei sm + bm_s m b where the a's b's are any real cnstants, can be instrumented as shwn in Fig. 7. This methd, ften Fig. 7-Integratr realizatin. applied in analg-cmputer wrk, is particularly suitable fr realizing transfer functins having a small number f ples, particularly if the cefficients in the functin require frequent change. Since the cefficients appear directly as the gains f amplifiers, almst n calculatins are required in the synthesis. The principal errrs generated by this realizatin stem frm the limited frequency range ver which the physical integratrs apprximate true integratrs. Althugh in this methd the departure frm ideality restricts the frequency range ver which a transfer functin may be realized, such a limitatin presents n serius drawback in many cmputer applicatins. A mre imprtant cnsideratin is that if high-rder functins are t be synthesized, an excessively large number f active units is required, the equipment reductin effected by using the methds discussed in the previus sectins assumes practical significance. ILLUSTRATIVE EXAMPLES The fllwing equatin typifies the cmplexity f the transfer functins frequently encuntered in the analy-

5 Mathews Seifert: Transfer-Functin Synthesis with Cmputer Amplifiers Passive Netwrks sis f systems where an analg-cmputer representatin may be desired: (s+ ) C s + ) (27) (4s + ) ( s + ) ( + s + ) T illustrate the l-amplifier 3-amplifier synthesis prcedures, this functin is realized by each f the methds. Synthesis with integratrs is s direct that it will nt be illustrated. The functin was selected nt t represent any particular type f system but rather t give an example which is neither trivial nr impracticable t realize by either f the methds. Fr l-amplifier synthesis, the functin f (27) is put int the frm e (s + :) (s + 6) -8 ei S2 + 2s + 6 (28) The A netwrk then is synthesized by the methd f Guillemin 6 t have a transfer admittance Y Al2 = (s +!) (s + 6) by realizing a driving-pint admittance (s + ) (s + 8) YAll = -r (s + :) (s + 6) (29) (30) in such a frm that bth zers f Y Al2 ccur at infinity. The zers f Y All are arbitrary except that they must be selected s that Y All can be realized as the input admittance f an rc netwrk. The B netwrk is synthesized t have a transfer admittance S2 + 2s + 6 Y Bl2 = (3) by cnnecting in parallel three netwrks with the shrtcircuit transfer admittances 6 Y BI2 (O) = s Y BI2 () = S2 Y (2) _ B2 - (s+)(s+8) (32) (33) (34) The driving-pint admittances f the netwrks are selected as Y Bll (0) = (s+ f) (s+ 4) s(s + 4) YEll (I) = Y Bll (2) = (35) (36) the realizatins are carried ut t cntrl the zers f the transfer admittances. The resulting realizatin is shwn in Fig. 8, where the impedance f the netwrks I T '--...,l.-- Fig. 8-Example f -amplifier realizatin. Element values in meghms micrfarads. has been increased by a factr f 0 6, the realized transfer functin differs frm the specified transfer functin by a cnstant gain factr f i. The number f required passive elements is 8; the range in capacitr values is 48 t, the range in resistr values is 6 t. The 3-amplifier synthesis can be carried ut effectively by selecting = (s + 6). (37) Substitutin f this particular G{s) alng with D(s) frm (27) int (2) (22) gives, except fr a cnstant multiplying factr, YA-Yc =-- s + 6 (s + :) (S2 + 2s + 6) YB-Y D =. (38) (39) The right-h sides f (38) (39) nw may be exped in partial-fractin frms as s Y A - Yc = s + 6 (40)

6 2 955 WESTERN JOINT COMPUTER CONFERENCE.6s 8.86s yb-yd=s , (4) 2 s+l s+8 which may be realized as shwn in Fig. 9. The imped Fig. 9-Example f 3-amplifier realizatin. Element values in meghms micrfarads. ances f the A C netwrks have been increased by 5.6 X 0 3, the impedances f the B D netwrks by 0 6 The realized transfer functin differs frm the specified transfer functin by a cnstant gain -6-2 g-8 fa -24 w :::> -30 :E «-36 I I'- - '\\, -- THEORETICAL RESPONSE, I-AMPUFIER REALIZATION --<> AMPLIFIER REALIZATION PHASE\ \ \ \ \', \ \ \ \ vr' I I I I I I --, FREQUENCY IN RAD/SEC Fig. IO-Respnse data. " \ ""... r ;>--.-::_- \ AMPLITUDE r-t C/l W w -60$ w z -90- w!fl I factr f 8. The number f required passive elements is 9. The range in capacitr values is 6.43 t, the range in resistr values is 7.7 t. The range f element values fr this example is s small that n practical difficulty arises in building the netwrks; cnsequently, n necessity exists fr reducing the spread by adjusting. T test the synthesis prcedures experimentally, the realiza tins shwn in Figs. 8 9 were cnstructed frm paper capacitrs carbn resistrs. Element sizes were adjusted t be within per cent f the required values. The measured amplitude phase characteristics f the realizatins tgether with the characteristics cmputed frm (27) are shwn in Fig. 0. Agreement between the measured curves the desired characteristics demnstrates the sundness f the synthesis. At high frequencies the deviatins in the phase curves fall within the limits f errr f the experimental prcedure emplyed. CONCLUSIONS The example shws that either f the tw synthesis prcedures illustrated in this paper is practical fr realizing a 4-ple transfer functin. The l-amplifier methd utilizes a minimum f active equipment but requires the mre cmplex synthesis calculatins a greater number f passive elements. If mre cmplex transfer functins are t be realized, the passive netwrks assciated with the lamplifier realizatin becme impractical because f the number f elements the spread f element values. Therefre, this methd is mst suitable fr realizing transfer functins n mre cmplicated than (27). The 3-amplifier realizatin requires mre active equipment but simplifies the synthesis calculatins reduces the cmplexity f the passive netwrks. As the cmplexity f the transfer functin increases, the advantages f the methd rapidly becme mre apparent. If sufficient active equipment is available, the third methd, integratr realizatin, may be used. The calculatins fr this synthesis are the simplest f the three methds, the cefficients f the transfer functin may be changed easily because they appear directly as amplifier gains. The three methds present a systematic apprach t the realizatin f a wide variety f transfer functins which cannt be btained practicably by using nly passive elements.

7 955 WESTERN JOINT COMPUTER CONFERENCE 3 Simulatin by Mdeling N. L. IRVINEt AND L. DAVIst INTRODUCTION S IMULATION by mdeling may be classified int three related methds.. We may use analg mdels which bey the same laws as the phenmena we wish t study. Instruments such as netwrk analyzers, slide rules, electrlytic tanks are examples f devices that are used t make analg mdels. 2. We may use mathematical mdels t describe phenmena we wish t study. Quite ften we resrt t high-speed cmputing machines r differential analyzers t slve specific prblems frm the equatins derived in ur ma thema tical mdels. 3. We may subject scaled mdels f equipment t actual r simulated envirnments. Wind-tunnel testing f airfils is a ntable example. Actual equipment may als be subjected t simulated expected r knwn envirnments, fr example, experimentatin with persnnel equipment in high-altitude chambers. Many f ur cmplex phenmena may be studied effectively by simulatin. The design f multidimensin filters, smetimes called space filters, may be studied in this manner.l We are all familiar with the prblem f detecting a message frm an electrical signal which cntains the message nise. Suppse we wanted t detect a particular gemetric shape in a field which cntained a large number f cnfiguratins as well as the desired frm. Let us chse the letter Z as ur gemetric shape, surrunded by a number f ther cnfigura tins; we culd detect ur letter Z by surveying the entire field with a filter which had the letter Z detailed t give full transmissin while the respnse f the remainder f the filter gave small transmissin. Obviusly, when the cnfiguratin f the filter matches the desired infrmatin, a high degree f transmissin is btained, cmpared with that btained frm a nnmatching cnditin. If we were t reverse the Z t OS we wuld nt be able t find ur message. 2 A special case f this type f filtering is that in which a small aperture is used t scan the field. This technique may in general be treated as a unidimensinal prcess, it has been rather thrughly explred fr many scanning devices, such as equipment fr the facsimile prcess fr televisin. In general, space filtering may be extended t n dimensins, where clr, intensity, lcatin as well as gemetric shape may be intrduced. The analgy t Aerjet General Crp., Azusa, Calif. P. Elias D. S. Grey, "Furier treatment f ptical prcesses," Jur. Opt. Sc. Amer., vl. 42, p. 27; February, R. C. Jnes, "Detectin f Targets Against the Sky Backgrund," Rep. 535, Plarid Crp., Cambridge, Mass.; Apri2, 954. between the unidimensinal electrnic-filter prblem the multidimensinal filter prblem is quite cmplete. The methds f analysis f space filtering are an extensin f the usual techniques fr dealing with filters in electrnic circuits. In bth cases, the utput frm the filter is a linear weighted average f the input infrmatin; thus, the weighting functin f a filter is determined. In bth cases, the prblems are as fllws:. T design a methd fr searching fr a given type f signal. 2. T reprduce a signal, with discriminatin In favr f a desired signal against thers. 3. T imprve a signal, i.e., t remve distrtin. DISCUSSION In electrnic circuitry, the input utput messages are described in terms f frequency respnse frequency cntent; thus, the independent parameter is time. With certain restrictins, any single-valued functin f time has a Furier transfrm as a functin f frequency. Similarly, any general field distributin in space has a Furier transfrm, but since the space distributin is a functin f lengths, the transfrmed distributin is a functin f reciprcal lengths r wave numbers. Rapid changes f the functin in space will give relatively large high-wave-number cmpnents in the spectrum, just as rapid changes f amplitudes f electrical signals give large high-frequency cmpnents. A unidimensinal field distributin will have a unidimensinal Furier transfrm. In the case f a planar field distributin, the transfrmed functin will be f tw wave-number variables where the tw-cmpnent psitin vectr frms a tw-cmpnent wave-number vectr. Let us lk mre clsely at the similarity between the transfrms used in electrnic filter design the n dimensinal transfrms. The transfrm f f(x) is as fllws: under certain cnditins f f(x) = - F(k)e-ikxdk. 27" -00 Fr the n-dimensinal field, we define the field psitin vectr R, it can be shwn that under suitable cnditins the functin fer) has a Furier transfrm () (2) (3)