s. I LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 510.84 izer mo. 131 -140 cop .3 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN NOV 1 «2 SFP 1 P id % L161— O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/phaseplanetheory137ribe 5/0.84 XJL&r DIGITAL COMPUTER LABORATORY n©. 1 37 UNIVERSITY OF ILLINOIS / URBANA, ILLINOIS REPORT NO. 137 PHASE PLANE THEORY OF TRANSISTOR BISTABLE CIRCUITS by Sergio Telles Ribeiro June G } 1963 fni S + r r ^i 1S being submitted in Partial fulfillment of the requirements 2 ^o Sgree ° f D ° Ct0r ° f Philos °P h Y ^ Electrical Engineering, S. ? ; ^ T S Su gP^ ted \ n P art b Y ^e Office of Naval Research under contract Nonr-l834(15 ). ) DIGITAL COMPUTER LABORATORY UNIVERSITY OF ILLINOIS URBANA, ILLINOIS REPORT NO. 137 PHASE PLANE THEORY OF TRANSISTOR BISTABLE CIRCUITS by Sergio Telles Ribeiro June G } 1963 (This work is being submitted in partial fulfillment of the requirements tor the Degree of Doctor of Philosophy in Electrical Engineering, May. 1963, and was supported in part by the Office of Naval Research under contract Nonr-l83^(l5 ) . ) 13? ACKNOWLEDGMENTS The author wishes to express his deep appreciation to his advisor,, Professor W. J. Poppelbaum, for his advice, encouragement, and valuable sug- gestions to improve the manuscript. Thanks are also due to his colleagues Bruce E. Briley for his invaluable help with proofreading and Gab or K. Ujhelyi for his cooperation in the preparation of Chapter 6; to Mrs. Phyllis Olson for her unusual skill in typing this text, and to Kenneth C. Law and his draftsmen for their great ability and good will in the execution of the figures. Finally, the author wishes to thank his wife, Laura Beatriz, for her unconditional support, patience and understanding. TABLE OF CONTENTS INTRODUCTION 3. STUDY OF THE FLIPFLOP EQUATION FOR THE CASE OF A RECTANGULAR TRIGGER 3.6 Under-Triggering and Back-Triggering 3.6.1 Under-Triggering 3.6.2 Back-Triggering 3.6.3 Discussion . , , 3»7 Summary ...,.,,, Page 2. THE FLIPFLOP DIFFERENTIAL EQUATION ... k 2.1 Introduction ........ k 2.2 The Transistor Pair Transfer Equation .!!!!!!!!)] 4 2.3 The Asymmetric Flipflop ............ ]_5 2.4 The Eccles -Jordan Flipflop ...... ' \ 2.5 Triggering . . .... . ... . . . , t \ \ \ \ \ 2.6 The Approximation Problem ......... 2.7 Summary ........ 19 24 31 38 42 50 58 70 3.1 Introduction ............. . i,p 3.2 Phase-Plane Analysis of the Basic Flipflop Equation ! ! ! 43 3.2.1 General Remarks .............. L-s 3.2.2 Existence of Singularities ......,.!."]"* 45 3.2.3 The Nature of the Singularities .....!]].' 3.2.4 Diagonalization of the Characteristic Matrix of the' System ............ . . . . . , , , # # 3.2.5 Comments on Figs. l4 .......... 3.3 Trajectory Equations ............ ° \ \ \ \ y 2 3.4 Separatrices ..... nn 3.5 Trajectories and the Action of the Trigger ....... 84 3.5.1 Turning the Trigger ON and Possibility of "Under-Triggering" ............... 84 3.5.2 Virtual Singularities and the Trajectory ...... 89 3.5.3 Turning the Trigger OFF ........... 3.5.4 Discussing Trigger Duration ...... 3.5.5 The Concept of "Optimum Trigger Duration" .!,'.'! 3.5.6 Possibility of "Back-Triggering" .......... 3 *5. 7 ^Trajectory After the Trigger is Turned OFF ...!.' 94 .......... 95 .......... 96 .......... 100 .......... 105 .......... 112 90 90 92 93 ANALYSIS AND DESIGN TECHNIQUES ................ 113 4.1 Introduction ............ . ti-o 4.2 Definitions of Time Intervals ... na 4.3 Calculation of a Time Interval Over a Trajectory by an Iterative Formula ................. 2.16 4.4 Graphical Constructions ...,...,,,[* ° [ 12 6 4.4.1 The @>,x) Plane Method . . . . .' . .'.'.' \ \ \ \ \ \ 12 6 4.4.2 A Simple Method on the (x,y) Plane ....... ,° 129 4.4.3 An Approximate Method on the (x,y) Plane ...... 133 4.5 Approximate Analysis of Waveforms ............. 135 4.5.1 Collector and Base Currents ........ .' \ \ 136 4.5.2 Collector Voltages ............. i ] | j 137 ■IV- TABLE OF CONTENTS (CONTINUED)- CONCLUDING REMARKS BIBLIOGRAPHY Page ANALYSIS OF DESIGN TECHNIQUES (CONTINUED) 4.6 The Influence of Parameters on Transition Times --Simplified Equations , -.kq 4.6.1 The Optimum Flipflop ............ 2_4l 4.6.2 The Total Charge Interchanged Between the Transistor Bases . „ . . „ . . . . 4.6.3 Collector Voltages --Maximum, Minimum and Settled Values ............... -tliq 4.6.4 Peak Values of Base Current .........[]. 150 4.7 The Problem of Circuit Optimization ...,..,',,'.,. 151 4.8 Summary ................ jco 148 154 154 5. EXTENSION OF THE THEORY .............. 5.1 Introduction ............. 5.2 Case When T is Negligible . [ . ', . . 154 5-3 Case of Negligible External Capacitances .....',['.. 155 5.4 Nonsymmetric Eccles -Jordan Flipflops .......'.'.'. 157 5.5 Other Types of Trigger ..*..,.....,,', l6l 5«5»1 Introduction ............... 2.61 5.5.2 Impulse Trigger ......,....,.., l6l 5-5 . 3 Exponential or Sinusoidal Triggers ......... 164 5.6 Use of Integral Transformations .........,..], l64 5.7 Summary ..................... 2.66 6. EXPERIMENTAL EXAMPLES ............... 2.68 6.1 Introduction ............ 6.2 Measurement of t, C. and C ......... 6.3 Equation Parameters 1 . ..?............ 2.73 6.4 An Illustrative Example ............. 2.74 6.4.1 Graphical Method A '. 183 6.4.2 Approximate Graphical Method B .......... 190 6.4.3 Iterative Numerical Method ............. 194 168 168 197 7 . 1 Summary ............... -1 q 7 7-2 Conclusions ............... 2.Q8 7.3 Further Investigations ....... 1 qA 200 -V- LIST OF TABLES Table 1.1 1.2 II III IV V VI VII VIII. 1 VIII. 2 IX X XI XII COEFFICIENTS OF (2.103) (SEE FIG. 9) . COEFFICIENTS OF (2.104) (SEE FIG. 9) . SINGULARITIES (SEE FIG. 9) SINGULARITIES (SEE FIG. 9) ...... . A SUMMARY OF THE NATURE OF THE SINGULARITIES IN ALL POSSIBLE SITUATIONS (SEE FIG. 9) ..... . IMPULSE VALUES FOR CHANGES IN V(i .......... , PARAMETERS OF THE SEPARATRIX EQUATION (3.64) . RESULTS OF EQUATIONS (3.62) FOR THE BRANCHES OF THE TRANSITION SEPARATRIX INSIDE REGION II PARAMETERS OF (3=83) AS FUNCTIONS OF THE PARAMETERS OF (3.79) . . . . ................. . PARAMETERS OF (3.9*0 AS FUNCTIONS OF THE PARAMETERS OF (3.90) ...................... DEFINITIONS OF TIME INTERVALS OVER A TRAJECTORY (SEE FIG. 21) .................... DEFINITION OF THE PARAMETERS OF EQUATION (4.3) . . 1 PARAMETERS FOR THE TWO EXPERIMENTAL FLIPFLOPS .... PARAMETERS AND CONSTANTS INVOLVED IN THE EQUATIONS REPRESENTING THE TWO EXPERIMENTAL FLIPFLOPS ..... DESCRIPTION OF APPROXIMATE TRAJECTORIES BY METHOD A . COMPARISON OF TIME INTERVALS OVER THE TRAJECTORIES OF TRANSITIONS BOTH CALCULATED BY METHOD A AND MEASURED . COMPARISON OF TIME INTERVALS OBTAINED BY METHOD B WITH EXPERIMENTAL RESULTS ................. COMPARISON OF TIME INTERVALS OBTAINED BY A VARIANT OF METHOD B WITH EXPERIMENTAL RESULTS .... PARAMETERS AND TRAJECTORY KEY ORDINATE FOR THE ITERATIVE NUMERICAL METHOD .............. COMPARISON OF TIME INTERVALS OBTAINED BY THE ITERATIVE NUMERICAL METHOD WITH EXPERIMENTAL RESULTS ..... Page 39 40 h9 50 55 lh 83 83 101 106 117 119 175 176 187 188 192 193 195 195 •vi- LIST OF ILLUSTRATIONS Figure 1 2 3 k 5 6 7 8 9 10 n 12 13 li+ 15 16 17 18 19 20 21 LOADED COMMON EMITTER COUPLED TRANSISTOR PAIR, UNLOADED TRANSISTOR PAIR, WITH JUNCTION CAPACITANCES NEGLECTED . . . . . . ... ASYMMETRIC FLIPFLOP . . , . . , . . TRIGGERING SOURCES . . . . . . . . . . THE GENERAL ECCLES-JORDAN FLIPFLOP . , . . . , TRIGGER WAVEFORMS . . . . . . . . . . . TRIGGER SOURCE . . . . . . , . . . . , , THE APPROXIMATION OF tanh xBY|5(x) . . . , . EFFECT OF VARIATIONS d n OF g(x) = d Vu - c Vll x, C THE POSITION OF SINGULARITIES, . ... 7 .... , ON CANONIC SYSTEM TRAJECTORIES WHEN THE SINGULARITY IS NODE* ......... CANONIC SYSTEM TRAJECTORIES WHEN THE SINGULARITY IS A SADDLE POINT . . . ........ TRAJECTORIES IN THE SYSTEM OF THE FIRST APPROXIMATION, CASE OF A STABLE NODE WITH \x | > \\ | - (CORRESPONDS TO FIGURE 10b) ,,„,,.„?,, A . , TRAJECTORIES IN THE SYSTEM OF THE FIRST APPROXIMATION, CASE OF g(x) = d CASE OF A SADDLE POINT, WITH X > 0, X < Ot. p "V(i V|j/ Vu c , AS IN TABLE I Vu SEPARATRLX FOR A SYSTEM WHERE d T . T > 0, c < 0, AND d TJ /c TT | < 1/7 ...... ?^ ... f 1 ^ EFFECT OF TRIGGER ON PHASE PLANE PORTRAIT. .... TRAJECTORIES AFTER TRIGGER TURN-OFF. ....... TRAJECTORIES AFTER TRIGGER TURN-OFF IN TIME DOMAIN, CORRESPONDING TO CASES ILLUSTRATED IN FIGURE 17 . . UNDER-TRIGGERING ................. BACK-TRIGGERING ................. DEFINITION OF TIME INTERVALS OVER A TRAJECTORY IN THE TIME DOMAIN, RELATED TO THE PHASE PLANE (SEE TABLE LX ) Page 5 8 12 17 20 25 27 33 51 6k 66 61 68 69 79 80 86 87 88 97 107 ■vu- Figure 22 LIST OF ILLUSTRATIONS (CONTINUED) Page NOTATION FOR A TRANSITION CALCULATION 125 23 APPROXIMATE GRAPHICAL METHOD A OF TRAJECTORY CALCULATION ON THE PHASE PLANE 2k PASSIVE NETWORK YIELDS THE EQUATION FOR THE OUTPUT VOLTAGE . . . . . . . . , , ILLUSTRATION OF THE HYBRID METHOD TO ANALYZE A GENERAL ECCLES-JORDAN FLIPFLOP ........... 26 BASE CURRENT DUE TO CHARGE STORAGE . . . 131 138 159 170 27 T, BASE VOLTAGE RISE ....... 171 28 COLLECTOR VOLTAGE RISE UNDER INJECTED CURRENT .... 29 TRANSITION CURVES FOR CASE 1 . . ... 172 ... 177 30 TRANSITION CURVE FOR CASE 2, WITH W = 0.kk5 I78 31 TRANSITION CURVE FOR CASE 2 } WITH W = O.667 32 CASE 2, W = O.667 - GRAPHICAL METHOD A . . . 33 CASE 2, W = O.667 - SIMPLIFIED GRAPHICAL METHOD A. . . l8l 3^ CASE 2, W = O.667 - APPROXIMATE METHOD B 182 179 180 -Vlll- ABSTRACT A.l Introduction It is our purpose to establish a theory describing the state transition of transistor flipflops making use mostly of phase plane techniques, but using also a time or even frequency domain point of view whenever helpful. Based on such a theory, we further wish to devise practical engineering methods of analysis, design, and optimization of transistor flipflops. We shall restrict ourselves to considering the asymmetric (Fig. A.l) and the Eccles-Jordan (Fig. A. 2) flipflops, with constant current T fed into the common emitters, and except for a short discussion, we shall consider only the case of a rectangular trigger. A ° 2 The Flipflop Differential Equation Based on the differential equations relating terminal voltages with the charges at the junctions and diffusion tails in a transistor we obtain the transistor pair' characteristics below. W k = 2 [1 " (-!) ktanh X J (A. la) o l - a Z k =W k + ~^~ w k ^. lb) where k - 1, 2, is the transistor index used in Figs. 1 and 2, and the symbol o _ dx x - t- ; besides, x = x 1 ~ * 2 = normalized base-to-base voltage Throughout this volume the expression "transistor pair" refers to the pair of identical transistors with the constant current I_ fed into the common emitter, as in Figs. 1 and 2. h -IX- -X- i r i a 2 Z 3 t 3 * -XI- qv k t X k ~ 2kT ~ normalized base voltages ' X Ck W k ~ al~ " normaliz ed collector current E X Bk k _ ^Y~ ~ normal ized base current S t - — - normalized time variable t' = real time variable t = collector time constant of the transistors Equations (A.l) therefore, relate the collector and base currents of the transistor pair to the base-to-base voltage. Analysis of the feedback networks yields a differential equation in the variable x fc for each value of k (k = 1, 2 for the Eccles -Jordan, but k = 1 for the asymmetric flipflop), respectively related to trigger plus base currents ^k + ^k^ and collec tor currents i „ } £ f k. We shall concentrate our attention on the asymmetric flipflop. The result, after normalization, is: T T T + T + T i o oo ioioo f T -o R — - x + x + x = 2 P Jb + w 2 + -i° (e x + § x ) + _£ (Qi + Zi )j (A. 2) where : f The k m the denominator is the Boltzmann constant. -Xll- T. = R.C. l 11 T. - R.C lO 1 O T . - R C. Ol O 1 A y -- time constants T = R C o o o J IJ + E ■D - B S C aLR ~ biasin S condition "TE o \ ~ £~T~ - normalized trigger current actually T! fed into the flipflop circuit Since G ± is the trigger current actually fed into the flipflop, it depends on the nature of the trigger circuit, and depends also on x itself. In general, if the trigger circuit of transistor T is represented by a current source of intensity i and shunt interval conductances G , we get K = s - G, x k k k (A.3) where, - tk -, • a S k " ccT ~ nomallzed trigger current source intensity. E Use of (A.l) with (A. 2) and (A.3) will result in a second order non- linear differential equation in x containing terms in tanh x along with its first and second derivatives, and under a forcing function s (t), and its first derivative s^t). This equation shall be called "the flipflop equation." -Xlll- A.3 Piecevise Linear Approximation A general solution to the flipflop equation is not known. Let us consider, however, instead of (A.l), the characteristics below: ^ k = |[1 - (-D%(x)] o 1 - a / . w k + — ^ — w t , (as before) a (A. 4a) (AAb) where, r -1, -XV- where o y = x The singular points (points of y = y = o) are given by the solutions of D - ex - ( A . 9 ) i.e., D c x = " (A. 10) and the nature of these singular points can be analyzed by studying the natural frequencies of the system (A.ll) below in a neighborhood around each singularity. o x = y ode b y = — x y a a a J J r (A.ii) This analysis shows that, if the trigger is OFF, and under proper biasing conditions, these singularities are two stable nodes, ^ one in each region I and III, and a saddle point'"'' in region II. The action of a trigger of amplitude W is essentially to shift the stable nodes by an amount Ax = ^ , and the saddle point by an amount Ax = - 22 u c (parameters always calculated in the correct regions!). In a neighborhood of a stable node both natural frequencies of the system are real and negative (by definition). In a neighborhood of a saddle point the natural frequencies of the system are real and have opposite signs (by definition). -XVI- When a singular point is shifted out of its proper region we say it has become "virtual" because its nature still determines the behavior of the system in that region, but it does not really exist. There is a value of W above which the saddle point and one of the stable nodes become virtual. Then the representative point P of the system will describe a trajectory towards the remaining stable node. When the trigger is turned OFF this remaining stable node (and also the other two singularities) returns to its resting position; P will now move towards this point, which is, in effect, one of the two stable states of the flipflop. Figures A. 3a, b, c, d illustrate a transition. Clearly, improper biasing may result in other singularity configura- tions which will not correspond to flipflop behavior. Study of some geometrical properties of the flipflop phase plane portrait, expecially the study of the separatrices and of certain properties of the trajectories can be made by analytical solutions to equation (A, 8). The usefulness of this approach is limited, however, by the complexity of the algebraic expressions involved. A ° 5 Engineerin g Methods for the Solution of Flipflop Problems These methods consist essentially of the drawing of figures like A. 3, i.e., phase plane portraits which are good approximations to the true phase plane portrait of a trajectory. The time durations of any portion of a trajectory, say, between two points P a and P is given by: [^ i X a But if y(x) is a straight line, -xvii- 10 z * ^ V E2 ' V E1 = V V C1 = E C1 + ^Cl " v i V C2 = E C2 + ^02 " V 2 V l " V 2 = V (2.6) (2.7) (2.8) (2.9) (2.10) (2.H) In the system above: a) The symbol x means |p- ; and t 1 is the time variable. € = electron charge b) r\ = JL f with < k = Boltzmann constant V T = absolute temperature c) R_ . R are the load resistances. ; HL1' L2 d) a , q , a„ are the charges stored in transistor T capacitances: ; q Ek> 4 Bk' 4 Ck K q at emitter diffusion tail and depletion layer of the emitter-base Ek junction. q at the base diffusion capacitance. Bk q at the depletion layer of the collector to base junction. UK. e) i , i are the saturation currents of transistor T , respectively of its ' J Ek' d Ck K emitter-base and collector-base junctions, measured with the opposite terminal short circuited to the base. f ) a , a the normal and inverse alphas of transistor T respectively from ' Ek' Ck emitter to collector and from collector to emitter. -7" Obs.: Notice the unusual convention of signs, which has been adopted in this report for convenience only. Comments: The system has 11 equations and 13 variables; usually variables v and v 2 would be independent, and then any other variable can be expressed as functions of them. However, this system of equations can be "considerably reduced by making the following simplifying assumptions (see Fig. 2). The transistors and operation points are such that the following relations hold, accordingly simplifying equations (2.l) through (2.6): J E1 (ex P T]v E1 - 1) » a cl J cl (expTiv cl - l) * -a^ ( 2 .12) J E2 (eXpllV E2 " 1} >y a C2J C 2 (eXpT1V C2 " l} * "^02 < 2 ' 13 ) a E1 J E1 (expriv E1 - 1) » j cl (expr)v ci - l) * -J^ (2,14) a E2 J E2 (expriv E2 - 1) » J C2 (expr]v C2 - l) « -J^ ( 2 .1 5 ) (1 " "ei^ei^^^ei " 1} y> (1 " a ci )j ci (expT]v ci ' 1} * " (1 " a ci )j ci (2.16) (1 - a E2 ) J E 2 (eXpT1V E2 -!)»(!- a C2 h C2 (eX ^ V C2 ' l} ~ "^ " a c2 )j C2 (2.17) For each transistor, the following relations hold, accordingly simplifying equations (2.l) through (2.6): q El < = T C2 1 G2 ( 2 -20) where T Q . is the "collector time constant" of transistor T.. The transistors are such that the following relations hold, accordingly modifying equations (2.1) through (2.6): T C1 = T C2 = T (2.21) °E1 = a E2 = a (2.22) J E1 = J E2 = J (2.23) T, a, j independent of any system variable. (2.24) Assumptions (2.12) through (2.17 ) divide the system (2.l) through (2.11) into two interdependent systems, namely (2.1) through (2.8) and (2.9) through (2.11). The first has eight equations and nine variables, so that all variables are determined if one of them is given; v is the "natural" independent variable. The second system has three equations and seven variables, so that four variables must be specified; in general, v ± and v g would be given (thus specifying v by ( 2 .ll)) and also i^ and i^ which are solutions of the first system of equations. These approximations restrict our analysis to unsaturating matched transistors. Constancy of parameters with respect to system variables as well as negligible junction capacitances are fairly strong assumptions, since they cannot occur in practice [l 5 ] ; however, we feel that the increased complexity involved in trying to take into account such things is too high a price to pay for the small increase in accuracy to be obtained. ■10- We obtain the system of equations below by substituting (2.12) through (2.2*0 into equations (2.l) through (2.8): o l . X E1 " T1 C1 + a a "Cl o 1 . X E2 = Tl C2 + a 1 C2 i cl = aj E expTlv E1 i c2 = aJ E expriv E2 1 - a . hi = T1 C1 + ~^T x ci 1 - a . 1 B2 = Tl C2 + a ' X C2 i El + X E2 -""E V E2 ' V E1 = V (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) Here we have neglected all terms j Q ^ a c ^ c ^ i 1 " a ck^Ck' in the same spirit we have used the approximation JEk' eXpT1V Ek* JEk feXpT1V Ek " X) (2 ' 33) which is not strictly valid. Actually, it is only valid as long as v E1 is positive, but not if ^ is negative. However, to use this approximation (or should we call it a substitution!) for all the range of v Ek is equivalent to connecting a current source of strength j £k in parallel with the emitter-base junction of T, , in a direction such as to yield input current zero when v Ek •11- This will make very little difference as far as results are concerned, but will considerably simplify the algebra involved. To be sure, it will tend to compensate for the fact that we have already neglected the collector-base junction saturation current, yielding even a better composite characteristic for the transistor pair. Equations (2,9) through (2.1l) are now irrelevant since we seek to express the currents as functions of v, which can be obtained just by solving system of equations (2.25) through (2.32). For convenience, we apply the following transformations of variables to this system: t= T (2-34) X Ck w, = k a V ^Bk z k - aI E > k = 1, 2 (2.35) k = 1, 2 (2,36) X = I TJV (2.37) R Q is the output resistor as shown in Fig. 3, From now on, the symbol x will be used for ^ , unless otherwise specified. We obtain: rO o a(w 1 + w 2 ) + (w x + w 2 ) - 1 (2.38) W 2 = w 1 exp2x (2.39) W 2 = ( W l + 2xw )exp2x (2.40) -12- FIGURE 3: ASYMMETRIC FLIPFLOP -13- o 1 - a o l - a Z 2 = V 2 + "IT" W 2 (2.^2) from which we get, using (2.38), (2.39) and (2.40): ^ + j- + x(l + tanh x)| w 1 = I (1 - tanh x) (2.43) This equation can be written in the form: Mdt + Ndw =0 (2.kk) If this is done, then we have: \5^ " 5t J = ja + x(l + tanh x )j " «(*) (2.45) where g(t) is only a function of t, i.e., it is independent of w ; here we assume that x ■■■■ x(t) is a given function of t, independent of w . Then u(t) = exp (2.^3) [8]. We have then: g(t)dt is an integrating factor for equation u(t) - exp -g + 1(1 + tanh x)xdtf (2.46) But j(l + tanh x)xdt = /(l + tanh x )dx = x + SmJ (cosh x) + const. (2A7) Hence, ignoring the constant, t X Q! 1 - tanh x , exp — u(t) = exp(- + x) • cosh x - ~L — " (2.48) -Ik- The differential equation can now "be easily solved, Let F(-w ,t) = "be the solution; then: dF 3w\ = u(t) (since M = l) and £-*<*> SI (2.U9) We get from (2.46) F(w^t) = w x • u(t) + f(t) = (2.50) and from (2.^9) and (2.50) N|i(t) = w g • ji(t) + ?(t) (2.51) Therefore f(t) = - | (1 - tanh x)u(t) (2.52) So f(t) = - | /(l - tanh x) ' u(t)dt (2.53) From (2.U8) into (2.53) f(t) = - (§ exp - + const.) 2 c ^ a (2.510 From (2.50), explicitating w , we finally get w = I (1 - tanh x)[l + Cexp(- -)] (2.55) -15- where C is an arbitrary constant determined by some initial charge stored in the bases prior to the moment t = t Q when the circuit takes on the configuration described by (2. 1+3). Therefore, for all practical purpose Cexp(- -) is zero, since we can assume this configuration to be in existence for an arbitrarily long time; we finally have, using (2.56), (2.39), (2.4l) and (2A2): The "transfer functions": w = — (l - tanh x) 2 = - (l + tanh x) and the "input functions": z. = 1 1 - a 1 2 a (l - tanh x) - x • sech x 2 2 a For completion, we have, with k = 1, 2 o f N k 1 Z k = ( ~^ 2 1 Jl - a /, . , x o ,2 (l + tanh x) + x • sech^ 1 - a o 00 ^/0\2 , x + x - 2(x) tanh x a ,2 sech x (2.56) (2=57) (2.58) (2.59) (2.60) which are the characteristic functions for a transistor pair connected as in Fig. 2. As a corollary of this solution, we notice that, at all times, the following relations hold: -16- W l + W 2 = 1 => ^1 + ^2 = aI E ^^2-^r 1 => H^h. = ^--Kj (2.61) which check with (2.7): i El + i E2 ~ ^^ 2„3 The Asymmetric Flipflop The asymmetric flipflop shown in Fig. 3 has a feedback network whose configuration depends on the trigger circuit, as shown in Figs, ha and kh. However, we can obtain the equilibrium equation for this network assuming an arbitrary trigger current i, which can be replaced by its value when a trigger is specified (see section 2.5). Analysis of the circuit of Fig. ^a yields: Vo° v °i + (T i + T oi + T o)\ + v i = R ^ + V„ + E c + Vio (? + °bi' ^ "s- ' -a /O O / . . \ . . (i + i„. ) + R (i + i p1 ; o C2 B s C o io v (2.62) where a) T - R.C. b ) T o = R o C o c) T. = R.C ' io i o d) T . = R C. ' oi oi e) R = R. + R ' s 1 o (2.63a) (2.63b) (2.63c) (2.63d) (2.63e) -IT- 'S it ST *o to Ci '■R 61 %t rfn m? /w /777 a) FEEDBACK NETWORK AND EQUIVALENT CURRENT SOURCE FEEDING TRIGGER CURRENT i t . •to \® V- it^vGi VG rfn /m b) THE CURRENT SOURCE WITH AN INTERNAL CONDUCTANCE 6 IS REPLACED 8Y AN IDEAL VOLTAGE SOURCE WHOSE OUTPUT IS r. FIGURE 4- TRIGGER SOURCES -18- Notice that in this asymmetric flipflop (Fig. 3) ^ a v. Let us apply to (2.62) the same transformations of variables used in section 2.2, i.e., equations (2.3*0 through (2.37), and also (2.6U) below (a similar transformation for the trigger current): (2.61+) a h dx The result, with x again standing for — , will be T T T + T . + T i o 00 i 01 o o x + x + x = p { 2 B + 2w 2 + 2 ^ (8 + § x ) + 2 ^ (0 + Z] _)} 2 T (2.65) where a) B = T B R s + E C (2.66a) E o h ) = 1 ~ a -i (not explicit above) (2.66b) «0 p-^wft (2 - 66c) If we expand (2.65) by replacing z according to (2.58) we get: T T T + T . + T i OOO 1 01 O O — x+ x + x ?) T T R "] - p {(1 + 2B + p) + (1 - P)tanh x + 2 -^ (0 + 8^ + 2 ^ (S - z Q )j where •19- z = - x sech x (2.68) The condition given by 1 + 2B + p = (2.69) is called state-symmetry. 11 In practice the effect of the base current on biasing, represented by the term pp, ' is usually small (see (2.66b)), so that the state-symmetry condition can be approximated by setting 1 + 2B = 0. Of course the larger Ot is (closer to l), the better this approximation will be. Equation (2.65), whether in this form or its more explicit form (2.67), is the general asymmetric transistor flipflop equation. 2.4. The Eccles -Jordan Flipflop Analysis of the two feed-back circuits of the flipflop of Fig. 5 leads to the following pair of differential equations : T..T T- + T + T ik ok 00 1k 01k ok o O X 1 + Z X T + X T ,2 k t k k (2.70) = P k ^2B k + 2w, + 2 !^(§ k + § k ) +2 ^( +z ) ok rith k = 1, 2; I = 1, 2; I £ k; t as in (lU.l) and (ik.k) and: a) T ik - R ik c ik < 2 - 71a) Notice that this term implies that the base-to-base voltage i,v (in this case identical to v ) will have a constant component equal to 1(1 - a)]L,R . 1 P J±i s -20- 0- Q -J U. 5 Q ay UJ -J o o in -J < (Z UJ Z UJ UJ I I- to UJ cc ID u. -21- b) T , = R , C . (2.71b) ok ok ok c) t = R C , (2.71c) iok lk ok d) T .. = R.. C, (2.71d) ' oik ok lk e > R sk " R lk + R ok < 2 - 71e > I_, n R , + E_ f ) B=-^ ^ (2.71f) 2 aI E R ok ) \ = | T)v k (2.71g) h) p k = J iaI E R , k (2. 7 lh) i) 0, = -4- (2.711) E j) w and z are given by equations (2.56) through (2.59) with x replaced by (x - x ). (2.71j) Notice that i is the trigger current actually fed into the flipflop; i is the trigger current put out by the pure current sources (see Figs. K). Executing (2,71j)j w and z turn out to be: it K. V k = I jl + ("l) k tanh( X;L - x 2 )j" (2.72) 1 \l_ Zm a \ 2 1 a j ~ a a [1 + (-1) tanh(x 1 - x 2 )] + (-l) (x^ - x 2 )sech (x 1 - x g )| (2.73) -22- 4 Taking (2.72) and (2,73) into (2.70) we get: + x k = ?k {& + 2B k + P k } TT T + T ., + T , ik okoo ik oik ok o X + X, k " - (2.7U) - (-l) k (l - P k )tan h (x 1 - x e ) + 2 ^ (g + § ) + 2 !i* (^ + (-1)%) ^ ok where, with k = 1, 2: a ) p = I - a - ^ (2.75a) a; p k a R . ok b) z Q - i (% - S 2 )seeh 2 ( Xl - x 2 ) (2.75b) ; Equations (2.7*0 describe a general Eccles -Jordan transistor flipflop. The asymmetric flipflop equation (2.67) is a special case of (2.7*0, and so is the symmetric flipflop, represented by equation (2.77), as follows: Let us consider the case of a symmetric (but possibly unsymmetrically biased) flipflop. Subtraction of (2,70, k = 2) from (2.70, k = l) yields: T T T. + T . + T iooo 1 Ol OO X + X + x = p J2(B 1 - B 2 ) + 2(w 1 - w 2 ) T 2 T T (2.76) + 2 ^ [(§- - e) + (§ x - § 2 )] + 2 ^ [(e x - e 2 ) + (z, - Z g)]} This furnishes (2.76) below, which can also be obtained directly by subtraction of (2.7*+, k = 2) from (2.7*+, k - l): Notice that (-l) = -(-l) • ■23- T .T T . + T . + T 1 O OO 1 Ol O O — — x + X + X T T T. o R 2(B 1 - B 2 ) + 2(1 - p)tanh x + 2 -^ [0 + z] + 2 ^ [© - 2z Q ] (2.77) where the indices 1, 2 have been dropped from all parameters P whenever P = P ,• in the case of the variables we have as before: a) x = x., - x. (2.78a) b) e = (2.78b) c ) z Z l - Z 2 = tanh x + 2z~ a j from (2.73) and (2.78a) (2.78c) d j z = — x sech x (2.78d) Inspection of (2,7^-) and (2.75) shows that the symmetric flipflop is formally equivalent to the asymmetric' one. Furthermore, if it is biased in such a way that B = B (not necessarily symmetric biasing), then it is formally equivalent to a state-symmetric asymmetric flipflop. We should stress the importance of this conclusion, since it actually doubles the effectiveness of the theory! But better yet, it allows us to compare the performance of any symmetric flipflop with its asymmetric counter- part, for we will actually be comparing formally equivalent things. In this report, the word "asymmetric" is reserved for the flipflop of Fig. 3» The Eccles-Jordan with different parameters on either side will be called "nonsymmetric . " -2k- We can anticipate here that (as will be seen in the following) that all other things being equal, the asymmetric flipflop besides being simpler, performs better than the symmetric flipflop with the same parameter values. 2.5 Triggering The equations we have found so far take the trigger into account in a rather general form. Among the infinite possible trigger waveforms, however, we shall discuss only those which are the most important in practice, and study them in some detail. First we consider the type of trigger circuit. It can be a: a) Voltage source with negligible internal resistance. b) Current source with negligible internal conductance. c) Voltage source with considerable internal resistance. d) Voltage source with considerable internal conductance. So for the trigger current waveform, the following are three important types (see Fig. 6): a) Rectangular. b) Impulse. c) Pair of rising and decaying exponentials. d) Sinusoidal wave plus constant, simulating usual trigger. We will study type a) in detail, and discuss types b), c) and d) (see section 5-5 )• The asymmetric and symmetric flipflops will be treated together (since they are formally equivalent), and an indication will be given as to how a basically similar method applies to the more general nonsymmetric Eccles- Jordan flipflop. This is justified because of the much greater practical importance of the asymmetric and symmetric case, compared to the more academic importance of the nonsymmetric case. a, •25- 0=0 to+T a) RECTANGULAR TRIGGER. A e> t O t b) IMPULSE TRIGGER. 0=0 0=6t>(l-exp-f7) 9=Bo e*p^n o t c) PAIR OF RISING AND DECAYING EXPONENTIALS. 0*0 0«0 o (, " s,n ^¥ L) & s ° €> t d) SINUSOIDAL WAVE PLUS CONSTANT, SIMULATING USUAL TRIGGER. FIGURE 6' TRIGGER W ?m:z\:.> -26- In the treatments of both voltage and current triggering sources, we will assume that the trigger has a definite duration, outside of which current sources have zero output current, and voltage sources are isolated from the input of the flipflop (say, by means of an open diode gate). One way of simulating this condition is to require that, outside the duration of the trigger, the voltage or current of the triggering sources are such that the normalized trigger currents 9^ be zero, for any k. Let us consider the various types of trigger: type a) is a trivial case, since the variable x is specified; case b) is both theoretically and formally a special case of d) and case c) is shown to be formally equivalent to d), which is not surprising since one case can be converted into the other by Thevenin or Norton transformations. We will establish expressions for 9 in each of the special cases mentioned above except case a). Case b) : See case c). Case c): Let the trigger circuit be as shown in Fig. 7a. We get, with k = 1, 2 : G,R . k ok tk " 2p. X k _ _ G, R , o o k ok o 9 - 9 - —r x, k tk 2p, k (2.79) (2.80) where ) = -^ is the normalized trigger current put out by (2.8la) E the ideal current source. Q = — is the normalized trigger current actually fed (2.8lb) k aI E into the flipflop terminals. ■27- »tk FIGURE 7a: TRIGGER CURRENT i k PRODUCED BY CURRENT SOURCE CLEARLY, ifc— *tk~" G k v k- FIGURE 7b: TRIGGER CURRENT i k PRODUCED BY VOLTAGE SOURCE. CLEARLY, l k"TT^tk" v K , "Hk'" G k v k •Pifk'GkVtk AND G k = ^ FIGURE 7: TRIGGER SOURCE -28- G is the trigger circuit internal conductance, (2.8lc) k Nov case b) is obtained by letting G k - in equations (2.79) and (2.80). Case d): Let the trigger circuit be as shown in Fig. fb . We get, with k = 1, 2: R R , ok „ ok _ „ (2.82) x . , - ~ — =- x, k 2p A tk 2p A k g v - _i_ 8 - _i- £ (2.83) where X tk 2 nV tk - nv is the normalized source voltage. (2.84) He nee, independently of the nature of the trigger circuit the form of 6 is the same. In other words, the voltage source x^ with internal resistance R can be converted into a current source of intensity 6^ and internal conductance G, given by: P. ok (2.85) x, tk 2p k R k tk r * _ _L (2.86) k From now on s k will be used meaning either 9^ or e* k , whatever the case may be. ■29- Furthermore, we will require that whatever the waveform of s , it must be nonzero only for a time interval (t , t ), being zero at any other aK DlC time. From now on we will use only s, and G, to describe the trigger. For the asymmetric flipflop, we will have only s, and G, ,' for the symmetric Eccles-Jordan s - s p = s, and we require G, = G p = G. For the non- symmetric Eccles -Jordan we will keep s and G,, k = 1, 2, in their respective independent equations . We can also account for the possibility that G has one value when the trigger is ON and another when it is OFF by using an index \x to indicate which case is being considered: Then u- = or 1 will indicate respectively trigger OFF or ON, and G, will assume the appropriate one of its two possible KU values, G, ~ or G, , . ' kO kl With this notation, the general flipflop equations are: For the asymmetric case : T.T T. + T . + T + T. R G, 1 O OO 1 01 o io o tu O /_, _ _ % — r- x + = x + (1 + R G )x T 2 T . v s \1 J J r-i rs-n \ /-, w i io oo f{± - a) io s \ pj(l + 2B+p)+(l-p )tanh x - — x + ( A — ^ — ~ ~ + R~ ) „ io ,0\2, , 2 — — - (x) tanh x sech 2 x + 20Hg + f7) } (2.87) For the symmetric Eccles-Jordan case (not necessarily symmetric trigger ) : -30- TT T +T.+T +T.RG, i QQ9 , i 01 2 1Q ° ^ i + (1 + R G )x 2 x + T s u = p 2(B - B ) +2(1 - p)tanh x - 2 I — T io oo / "(l -a) io s\o -IT- — + rJ io /0\2, , — (x) tanh x /T. R N 2 r - / io o s sech x + 2( — s + p- s [2.88) where x = x - x 2 s — S, - Sp G - G, - G u l|i 2p (2.89a) (2.89b) (2.89c) All other coefficients being the same for both (2.89d) values of circuit index k, the index has been dropped. For the general nonsymmetric Eccles -Jordan flipflop: T..T ik'ok oo T ik + T oik + T ok + T iok ok tk)i o , ± + R Q v — 2 — x k + T k v sk k|i p k |(1 + 2B k + P k ) - (-l) k (l - P k ) tanh x ("D J " T iokoo / (l - a) liok ^sk\ o _ lipk ( o } 2 tanh . ,2 sech x /\ok o sk k = 1, 2 (2.90) -31- 2.6 The Approximation Problem We have thus established the nature of flipflop behavior through equations (2.65), (2.70), (2.74) or their respectively equivalent forms. It is clear that the chances of success of an attempt to solve such equations exactly are very slim indeed [18]. Therefore, in order to get useful results, we must bow and try to find approximate solutions to these equations. There are two possible approaches to this problem. One is to find an approximate model to the circuit. This model must be described by solvable equations. We then consider such solutions as approximations to the exact solution of the original problem. The other approch is to approximate the original equations directly, rather than the model. The latter approach would probably allow more accurate results to be obtained, since an error introduced in approximating, say, i Ck (v) would not necessarily propagate through i pk (v) to i Bk (v), and then to ^(v). That is to say that each function would be calculated exactly for our original model and then each one of them would be approximated as well as possible. However, as we will show, this question does not necessarily affect the nature or even the complexity of the equation or system of equations we must analyze. For in fact, since an exact solution is not to be found but instead we must be content with an approximation, we might as well linearize the problem. This would involve dividing the x-axis into regions where the function of x appearing on the right-hand side of the equation can be approxi- mated by a linear function of x. Let N be the number of such regions. Then we would reduce our problem to that of solving N linear second degree differential equations valid in their respective regions and match the solutions -32- at the "border between every pair of adjacent regions. On both the phase plane and time domain, these regions are N strips separated by (N - l) lines of x = constant. One of the great advantages of this type of approximation is the relative simplicity of its applications, whether in the problem of analyzing the properties of the circuits involves, or in the actual computation of transition times and waveforms. If this direction is chosen, then we have decided to pay a price in exactness for the advantages of simplicity and usefulness. Then it is a question of inspection to see that the most promising way to arrive at con- sistent results is to approximate the transistor pair model, i.e., equations (2.56) through (2.59) must be the basis for the definition of a "reasonable" piecewise linear model [26]. Of course the problem centers on how to linearize tanh x. There is a good degree of arbitrariness here, but we will select the following three region approximation (see Fig. 8). tanh x w 7 - 0.721 This criterion, of course, is quite arbitrary, and there is nothing to prove that it is the best; however, it also reduces the maximum absolute difference between the two functions to about 0.120 at the worst point. This is not a minimum; if 7 is selected to minimize this maximum absolute error rather than the integral, then it would be 7 ~ 0.71^ and the maximum absolute error would be about 0.115. So, for simplicity, we could for example, take 7 = 0.7, or alter- nately (and perhaps better) take 7 = 7. Whatever the criterion may be, the intention of using such a factor 7 is to reduce somewhat the error by which the solution to the differential equation is affected due to the piecewise linearization process. This would be the ideal criterion if it were not impractical. A five region approximation could also be used with perhaps better accuracy, but increased labor involved both in computation and analysis. In general the accuracy can be improved by increasing the number of regions, which results also in increased labor. Since the technique does not change in essence, we shall use the three region approximation in this dissertation. •35- Equations (2.56) through (2.59) become; w. = |{i+ (-i)V*)} (2.93) z, = k 2 a o / xk 1 z k = ( " 1} 2 j^- 2 [1 + (-l)S(x)] + (-l) k Sep' (x)} (2.9^) a a o 00 X + X cp'(x) + (S)V(x) (2.95) with k = 1, 2. The prime indicates differentiation with respect to x. Also, 1 o , / \ z Q = - xcp'(x) (2.96) And obvious ly, 0, x < - cp'(x) = < y t -i + T < x ' 7 (2.97) Differentiation of cp'(x) will clearly consist of two impulse functions, since cp f (x) is constant everywhere except for two discontinuities. We get: (x) is an impulse function of strength 7 occurring at x - 0. The flipflop equations (2.87), (2.88) and (2.90) will be approximated by the following equations, respectively: For the asymmetric case, (2.87) is approximated by: -36- TT T+T.+T+T. RG i °9? , _i 2i 2 1Q ° ^ $ + (1 + R G )x 2 1 s M- = P (1 + 2B + p) + (1 - p)cp(x) - — x + T /, T. R io oo f 1 - a. io s \ o T + R- ,X o cp'(x) ^(8)VW + 2^^ (2.99) For the symmetric Eccles-Jordan case (also not necessarily symmetric trigger), (2.88) is approximated by: T.T T-+ T.+T + T. P G L2 9° + -i 2i 2 !2_°_fc!: £ + (l + R G ) s |_l p { 2(Bl - B 2 ) + 2(1 - pMx) - 2 [>°x° + (i^2 !±2 + Qx}p'(x) 2 ^2 ( g)V(x) ♦ 2(^2 8 + ^ .) (2.100) with X — x_ - x_ s = S l " S 2 G u " G lu = °2u (2. 101a) (2.101b) (2.101c) All other coefficients being the same for both values (2.101d) of circuit index k, the index has been dropped. -37- For the general nonsymmetric Eccles-Jordan flipflop, (2.90) is approximated by: ik ok op ik + oik + T ok + T iok R ok G tku. o , , = P k {(l + 2B k + p k ) - (-l) k (l - p k ) H H Cfl H H H|<^ co co fl + X CD O H|<^ V V P CD 1 X H| <-. ?H V V + 43 •H -H X H|f> i H X OJ r* P ft OJ H H H H cti OJ ^ ox II T3 w o ft OJ OJ. H H OJ. H H O •H P S CU I ft OX

, O 3 o 0) a ■H CO o d cu ft CU Cti o ■H P co O -ko- o\ H CO O H OJ o CO u H o OJ M < E-i tQ 4 c5 X O H O tQ K O X O J*i •H o 4 £ o •H OJ + o X co ^4 ■H K H « O G X m OJ ft to -P 0) •H O •H '"■H tH CD o o of ft > X ft > X X X > > ^ rd H H CD H I a •H TS a •H Ch OJ Ti CO ■H O CO PS Pn ft OJ M ft OJ ox t>s OJ. M o •H p X J*i ft H I ox M X i a co CD > H QJ CO 6 CD d ■p co 0) o +3 •H o eS ft a3 o a> ,cl -p 0) CO ! 3 cd cu rQ > fl o >d a cu ft CD tj a. > M o •\ i fn > O A a > 3 M a -p co P a C O cd •H •H bO O (1) •H h

so P(x,y) = P(x,y) Q(x,y) - Q(x,y) J (3.2] b ) taking A ±1 x + A lg y A 21 X + A 22 y > (3.3) where the A are the coefficients of the terms of first order of P and Q, in the order indicated. The solution of (3.3) in parametric form with parameter t will con- tain exponentials of i^t and \ t, where \ Q and \ p are the two characteristic frequencies of the system, i.e., the two eigenvalues of the matrix (\^)t gi ven by the solutions of A u -X 21 A 12 A^-X = (3.M By definition, the singularity P (x ,y ) of the original system is called: (i) Stable node if \ and \ a are real and negative, i.e., x(t) and y(t) contain only damped exponentials . (ii) Unstable node if \ and \ are real and positive, i.e., x(t) and y(t) contain only growing exponentials . Ciii) Saddle point if \ and \_ are real and have opposite signs, i.e., both x(t) and y(t) include one growing exponential. -45- (iv) Stable focus if \ q and ^ are complex with negative real part, i.e., x(t) and y(t) undergo damped oscillatory motions, (v) Unstable focus if ^ and ^ are complex with positive real part, i.e., x(t) and y(t) undergo growing oscillatory motions. The study of singularities is important mostly because if a system maintains the same qualitative properties in a relatively large neighborhood of a singularity, then the nature of the singularity will give us considerable information about its behavior in this neighborhood. 3-2.2 Existence of Singularities Taking into account equations (2.91) and the group (2.93) through (2.96), equation (2.103) was obtained from (2.6 5 ) and (2. 77), with parameters as described in Table I; for convenience, we repeat (2.103): V + V + %u X = d V + f ^)t8(x + i) - 5 (x - £)] + rav s° + ns (2.IO3) where V = I, II, Illf is the reglon lndeXo We define a new variable y = % f and express (2.103) as a system of two first order equations: x = y D y = _ V Vu - — ^ X a V ^ b — — t- v a J V J > (3.5) where D = % + t(i)[ b ( X + i) - 6 ( x _i)] +mv g + ns -46- Notice that d is a constant in each region, and that the impulses occur in the borders between regions I and II and between regions II and III (this is a direct consequence of the way p7*, h) Only one (any of the three) singularities is real, the other two being virtual. Figure 9 illustrates these conclusions. We will now prove that whenever: i) three singularities are real (case g-i), the extremes are stable, the median is unstable; ii) only one singularity is real (cases g-ii or h), it is stable. 3.2.3 The Nature of the Singularities We have discussed the existence of singular points, and established the importance of several relationships among parameters upon the position of JS REAL STABLE NODE >JS VIRTUAL STABLE NODE MS REAL SADDLE PT >JS VIRTUAL SADDLE PT. )ERLINE SITUATION EUR ow we should Lons (310) can Ox (3.14) point P and Q by ns about (3.15) (3.16) (3.17) FIGURE 9: EFFECT OF VARIATIONS d^ , OF g(x)= dw^-c^x, ON THE POSITION (AND EXISTENCE !) OF SINGULARITIES. SEE TABLES 1,111,11. UNIVERSITY Of ILLINOIS LIBRARY CURVE POSITION t in* m SYMBOLS o MEANS REAL STABLE NOOE ® MEANS VIRTUAL STABLE NODE X MEANS REAL SADDLE PT X MEANS VIRTUAL SADDLE PT • BORDERLINE SITUATION o« FIGURE 9 EFFECT OF VARIATIONS d^ , OF g(x)-d„ M -c^«,ON THE POSITION (AND EXISTENCE !) OF SINGULARITIES SEE TABLES H.TJI.IS. UNIVERSITY Of- ILLINOIS LIBRARY -52- singular points, and therefore upon their real or virtual nature. Now we should like to study the nature of these singularities. The system of equations (3I0) be written in the form of (3.1 ): can x = P(x,y) y - Q(x,y) (3.14) The "system of the first approximation about the singular point ^ X Vu' y vu)" has been defined as being the system formed by replacing P and Q by their respective first terms of corresponding Taylor series expansions about vu Let us define the new variables : X = x - x„. ,• Y=y-v The system of equations (3.14) becomes (3.15) * m hi****,,** + K^r^y* y" vu^vu' Comparison of (3.1*0 with (3.10) shows that: P(x,y) = y Q(x,y) -554 -S± x _S± a v a v y (3.16) (3.17) Therefore -53- P x ( V y Vn } = °> VV y vn' = x c "b V|i'-Vn' a v ' V VH ,3 V " " a ScKu'^J (3.18) Let D - - -»» aM = _ _Vu. a.. n (3.19) Then (3.l6) becomes X = • X + 1 • Y Y = D • X + E • Y (3o20) The characteristic equation of this system is ■\ 1 D E - X = 0, i.e., \ - EX. - D = (3.21) So ^p = l{ E ±^ 2 + ^'} (3.22) i.e. = \ IE - 7e 2 + kB | a V^ E + f Je 2 + k~D (3.23) -5*- and we have the following rules : fi) \ and \_ real and negative => stable node. v ' a p (ii) \ and \„ real and positive => unstable node. v ' a P (iii) \ and \„ real and opposite signs => saddle point. v ' a p (iv) \ and X, complex conjugates LX P with negative real part r> stable focus . (V) A. and \ n complex conjugates v ' a P with positive real part => unstable focus. Application of these rules to equation (3-22) or (3.23) is straight- forward; replacement of D and E by their expressions in terms of parameters (through equations (3-19) and Table i) will produce the results summarized in Table IV, as can be shown by the following analysis: Theorem 1 . If x or x exist they are stable nodes . Proof : From definitions of E and D, and from Table I for regions I and III we have V a v T +T.+ T+T. RG _ _ _i_oi_o_io_oja < Q (3o2l+) (t.t /t) V=I,III V 1 O D a v V=I,III T i T o/ x2 1 + R G -^<0 (3.25) Therefore, |Ve 2 + kl) | cannot be greater than |e|; we will show next that E 2 + i+D > 0: in fact, from the expressions above for E and D, we have f ,2 (T + T . + T + T. R G ) - k(l + R G )T T E 2 + l^D = x Q1 ° 1Q ° V a (3.26) (t.t hy 1 o Consider the numerator M of the fraction above: ■55- TABLE IV. A SUMMARY OF THE NATURE OF THE SINGULARITIES IN ALL POSSIBLE SITUATIONS (SEE FIG. 9) Possible Situations Exists 'iu 'IIu 'IIIu 'Iu 'IIu "III|a 'iu 'IIu 'IIIU 'in 'IIu 'IIIu •J Does Not Exist V V n/ V V V sj V s/ ^ Nature Parameter Condition Stable Node c > and -^ < - I llfi c TT 7 IIu or Stable Node C IIu < ° and IIu 7 c nn > ° and IIu IIu 7 C IIu : and c^ : ' + 7 Stable Node IIu or Stable Node Saddle Point c < and -S± 1J -n c IIu <-i c TT < IIu and Stable Node IIu IIu < -56- M = (T + T . + T + T. R G f - Ml + R G )T T v i oi o 10 o n B u I < ■ T i + 2 Vo + T o + 2(T i + T o )T oi + T oi + 2(T i + T oi + 'o^ + T 2 R 2 G 2 - Mr .T - i+T .T R G o i(i i o i o s u. ■ A - 2T i T o + T o + 2(T i - T o )T oi + T oi + U Voi - 2 < T 1 + T oi - T o )T o R l^ 2 2 2 O 1 |i 2 2 = (\ t * \ - \f - SEiVo^ol + T i - T o) + \\% + T oi + Vol So M = [T . + T. - T (1 + T R.G )] 2 + T [T + k T ] > (3-2?) oi i o o l u oi oi o This implies that E 2 + kV > 0, and so \ Q and \ are real and negative in both regions I and III, which means that x and Xj^j if they exist, are stable nodes, which was to be shown. Notice that even if one of them (or both) is virtual, its action upon its corresponding region will be that of a stable node. This follows from the proof of Theorem 1 and from the nature of the coefficients of equation (2.103) The orem 2 . If p7>l> > I and if x exists, then it is a saddle point. Parameters as defined in Table II; this is a necessary condition for bistable behavior (but not sufficient), for otherwise, either x or x^ will exist, but not both! This is stated in an equivalent way in conclusion h, page W. -57- Proof : From definitions of E and D, and from Table I for region II we have; E - I in L II T . + T 1 O . + T + T R G + p7T 1 o lO O n *' 1 - a a T. R 10 s "T" + R~ o- 1 T.T 1 O T *' io < (3.28) D = : IIp: . T ^u - P7 ^ a II T .T 1 O — + P 7T io > (3.29) Therefore \Je 2 + 4d' | > Ie^ \ q and \ p are then real with opposite signs, and consequently x is a saddle point as was to be shown. The0rem 3 ° If ^ < V th ^n the existing x^ will be a stable node . Proof : Using the calculations made for the proof of Theorems 1 and 2 we get, if region II is considered: E + k-D = TN "T .T 1 O . T 10 (3.30) where N - M + 2(t + t + T + t. R G )p7 1 01 o 10 o n -^ 1 - a 10 s L a t + Fj + ipy 1 - a 10 s a t + r J (3.31) /T.T /T T N' = M' - Ipyr. +1 ( -i-2 + pn (-T IO U V T T P/ iq (3.32) 1 No bistable behavior is possible! -58- where M' stands for the sum of the first three terms of N, and p7ty < I was used. Then T .T N' = M« + I -~^ (3-33) U. T since M > 0, M* > and N 1 > 0. Therefore N > 0; as a consequence, \^ and \ are real and negative and if x exists it is a stable node. Of course the proof of Theorem 1 is independent of EZ* , and therefore it is valid under the present hypothesis of p7^ < I . Thus the proof of this Theorem 3 is complete. 3 . 2 . k Diagonalization of the Characteristic Matrix of the System The eigenvalues and corresponding eigenvectors of the characteristic matrix are given by: (3-34) Therefore, I. = \.k ± Z I = OL, p. Let k. = 1 arbitrarily, and I = \^. Column i can be normalized by multiplying it by a factor / -_ SIX. - but it is more convenient not to normalize it. The polar matrix of r is 1 > \7 + i i (3-35) Its inverse is -59- •1 1 ^ a (3.36) The diagonal matrix of r is \ ° K and the diagonal form of r is [28] (3.37) r = r • r, • r~- p d p (3.38) Therefore : 1 D E (3.39) The system of equations (3.20) can be expressed in matrix form X o 1 D E, (3.40) whose solution is: = e 1 D E r with T = t - t, X = x(o) Y(0) (3.41) -6o- where, by definition, given a diagonalizable matrix A, with polar matrix A^ and corresponding diagonal matrix A^, A A d . A -l e = A ' e "A P P (3^2) and for any diagonal matrix Q q x Q 02 if Q *2 (3.^3) Therefore, from (3-39), (3-M), (3-^), ($M), we get: 'X (3M) and this turns out to be % \- x a a v . P V v We ' e " e \ T \ T Y Q l-e a + eP \ T \ 6 T- WV + V (3^5) In another form -61- x o X 3 - Y o V -Vt, + Y o V - - Vg ' Y , V 'Va + Y . V (3.46) and we recall the definitions of X and Y: X = x - x VU> Y=y ' = v. since y =0 V|_t ■ /J ' J V\i (3.15) Equations (3.^6) furnish the integration constants A and B of the differential equation as functions of natural frequencies and initial values Substitution of (3-15) into (3. 46) yields the general equation for the flipflop trans ition, if we keep in mind that: (i) Changing from one region into another changes all parameters; therefore one must be careful in calculating new initial conditions, new singularity position (which can be virtual!) and new natural frequencies. (ii) The same thing occurs if there is a change in trigger level, in particular when it is turned ON or OFF. (iii) Impulses produce discontinuities in y (and therefore in Y), but everything else remains unchanged. Remember that the effects of such impulses depend on a } which changes from one region to another; the impulses due to effects of base current occur when a is changing from a to a and also when it is changing from a to -Li JJ_ a III' in each case we w iH take the average of the two adjacent values of a . -62- The results are presented in equation (3-56), section 3.3. However equation (3.K6) is also a very practical form; discontinuities of X and Y can be calculated from equations (3-15) whenever they occur. Some more information can be obtained from the "canonical system of the first approximation"; this is defined as: (3.47) where p a (3-W) Of course, system (3.V7) is equivalent to system (3.UO) under the transformation of variables (3.^8). Such a system clearly has a unique singularity at the origin. Solution of (3-^7) is: (3A9) i.e. A $ = $ e X T a > X = X Q e V (3.50) from which we find -6 3 - t - t, 1 $ - — Smi x, — = ii,I \ a p (3.51) and therefore, (3.52) which is the equation for the phase plane trajectory in this canonic system. If the singularity is a stable node, then \ < and \ < 0, and we assume either |\J |\ R | or |\J |\ | (which is irrelevant, since the P a' ordering of X and \ is arbitrary!), From ( 3 o 52 ) we get : (Xp/\^) froA ) *. P a $ (3.53) and, by differentiation, ax _ V d$ \ a (Xp/A^) $ $ v P' cr (3.54) Therefore, (3.53) defines a family of curves of the parabolic type in the plane ($,X), for x = f(*) [1]. From (3-51) it is seen that the direction of movement of a repre- sentative point P over any of these trajectories is clearly towards the origin. Figures 10 show the two types of parabolic curve according to whether (a) |X a |<|X0|, 04> (b) IXalHXpl FIGURE 10: CANONIC SYSTEM TRAJECTORIES WHEN THE SINGULARITY IS A NODE. IF IT IS A STABLE NODE, i.e., IF X a 8 X£ ARE NEGATIVE, THEN THE REPRESENTATIVE POINT P MOVES TOWARDS THE ORIGIN (SINGULARITY) WITH INCREASING TIME. -65- |\ I < |\ R | or |\ J > |\ I . If the singularity is a saddle point, then we take X . < < X (again no loss of generality) and inspection of (3.53) and (3.54) shows that (3-53) represents a family of hyperbolic curves with the coordinate axes as asymptotes [l]. From (3.5l) it is seen that the direction of movement of the representative point P along any of these trajectories is found to be towards smaller values of |$| and larger values of |x|. See Figs. 11. We wish to know how these curves transform into the (X, Y) plane, i.e., the system of the first approximation, which, although not canonic^ also has a unique singularity at the origin. The best way to see this is to find the lines in the (X, Y) plane that correspond to the Oandx axes. From (3.48) (definition of $ and X.) we get: $ axis :x=0 => Y = X X a ► (3-55) X axis : = => Y = X X P J We realize that in the case of a node with |\ I > \X n \ , i.e., with 1 a' ' p 1 ' ' X < Xg < 0, the trajectories have the <£ axis as the direction of their axes of symmetry, and that they are tangent to the X axis at the singularity point (origin), as shown in Fig. 9h. In the case of a saddle point, the curves are asymptotic to both axes. The linear transformation conserve all these properties, and also the direction of motion of the trajectories with increasing times. Therefore we can get a fairly good picture of the family of trajec- tories in the (X, Y) plane if we know them in the ($,x) plane. The results are qualitatively illustrated in Figs. 12 and 13. Figure 14 qualitatively shows some portraits of the original system (x ; y) from what we have found so far [l, 22]. -66- i>* (o) \ a >0, \0 (b) Xa<0, X^O COMMENT : THE REPRESENTATIVE POINT P MOVES IN THE DIRECTION OF HEAVY ARROWS WITH INCREASING TIME. FIGURE I!' CANONIC SYSTEM TRAJECTORIES WHEN THE SINGULARITY ISA SADDLE POINT. ■67- COMMENT: SLOPE OF a a' = X, SLOPE OF bb' = X SLOPE OF mm' x„x a A /3 x a +x /3 OX FIGURE 12: TRAJECTORIES IN THE SYSTEM OF THE FIRST APPROXIMATION, CASE OF A STABLE NODE WITH |xJ>|Xo| -(CORRESPONDS TO FIGURE 10b). -68- COMMENTS : SLOPE OF aa' = X a SLOPE OF bt> = Xg SLOPE OF mm= X a Xft THE CURVES ARE HYPERBOLIC, ALL ASYMPTOTIC TO LINES aa abb', WITH MAXIMA AND MINIMA ON LINE mm'; ALL CURVES CROSS THE X-AXIS PERPENDICULARLY. HEAVY ARROWS INDICATE DIRECTION OF MOVEMENT OF P WITH INCREASING TIME. FIGURE 13: TRAJECTORIES IN THE SYSTEM OF THE FIRST APPROXIMATION, CASE OF A SADDLE POINT, WITH fx a > V mm~^ — ■>* Ox (c) NEGATIVE BIAS ON gU)= dj. <0 but d i / x/ c ff M ond or to a her from dg dx (f) STABLE SYSTEM: c v : >0: FIGURE 14: g(x)= d Wft -c^x; d V/i a c u ^ AS IN TABLE I refore, (o) BALLANCED g(x) dj =0; but c^ <0 Ox Ox O « O « (b) POSITIVE BIAS ON g(x): d^ >0 but |d K /Cjj| <"'r. o- +l/y (e) MONOSTABLE SYSTEM IN E: d n M / c II/i <_l/y FIGURE 14 glx)= d^-c^x, d„ M 8 c u ^ AS IN TABLE I -70- It is worth reminding that, due to the piecevise linear character of our complete system in (x,y), in each region the solution of the corresponding system of the first approximation is exactly equal to the solution of the original system. 3-2.5 Comments on Figs . l^t- a) Balanced g(x): d = 0; this situation would usually correspond to a state symmetric (d^ = 0) untriggered (a = ) system. However, we could have this same situation if d^ / 0, nW = -d^, and u = 1, since d iiu = d no + MnW > in this case d m = °- b) and c) Nonzero bias on g(x): a / 0; this situation would correspond either to a state asymmetric (d^ = ) untriggered (u = ) system, or to a partially triggered (no matter what kind of state symmetry) system (d in = d no + nW * °)- d) and e) Monostable systems in either region I or III; lip > — ; either from v • Cl1 ^ a large bias in an untriggered system, or from an adequate trigger in any potentially bistable system. ' f ) Stable system: c > 0; the system is not a flipflop. General Comments 1) Notice the relationship between d (bias or triggering or both, d IIl = d II0 + nW ^ and the P osi tion of the singularities: the value of ^ dx at a singular point and its nature: n Potentially bistable system is any system which would exhibit bistable behavior if adequately biased, i.e., any system for which c TT „ < 0: therefore, the system of Fig. 13f is not potentially bistable. II0 -71- dg dx > -> stable singularity (in our case, stable node) dg -^ < => unstable singularity (in our case, saddle point) Of course, the exact nature of the singularity also depends on a and b V VJ! 2) The position of the singularity, if it is real, must be inside its corre- sponding region; otherwise, it is virtual. 3) The singularity nature and position is just a translation of the coefficient of the differential equation; this is one way to interpret the "action" of a singularity over its corresponding region no matter where it happens to be and justifies calling it "virtual" if it happens to be outside its region of influence: it does not really exist, but it would exist if the parameters C V|_i aM d vu. Were the same throughout the phase plane as they are inside the proper region. In this sense, this would-be singularity adequately trans- lates the coefficients of the differential equation inside its proper region. h) We have used a coded tag to describe the singularities in this figure: SN = Stable Node SP = Saddle Point R = Real V = Virtual The last symbol of the tags on the singularities is one of I, II or III, and designates the proper (corresponding) region of the singularity; so, if the position of the singularity agrees with this symbol, it is real; otherwise it is virtual. The symbol R or V is therefore redundant, but has been used for clarity. -72- 3„3 Trajectory Equations Solution to equation (2.103) is o v = x J VU V|i (3.56) where a) V refers to the region: V = I, II, III u refers to the trigger: u == 0, 1=> trigger OFF, ON h ) A A c are parameters corresponding to a given value of the pair J avu' pvu' vu * V|a. c ) \ \ are the natural frequencies corresponding to a value of the pair V\x, ' avp/ pv^i d) t is the instant of time when the pair of indices assumes a new value. ; Vu e) The most common sequence of index pair values is V^ = 10, II, III, III1, IIIO. f ) Let us use, in general, a prime to mean: a 1 , V, c', d ! , differential equation parameter values in the previous V\i condition solution parameter values in the previous V|i condition natural frequencies in the previous Vu condition values of t, x, y at the end of previous Vu condition A*, A' C ! or p t', x', y' Actually c = x is the abscissa of singular point X --and here X^ has nothing to V ^do wife X = X - x used in the previous section. yyx ■73- Unprimed symbols will refer to parameters and variables throughout the present V|i conditions, which, whenever necessary, will be indicated. t Q = f x Q = x' U y o = y + a with U = magnitude of any impulse occurring during the transition from the previous value of vn to the present value; it is zero if no impulse occurs at this point. It may be a function of y Q or y\ as in Table V. t 0> x (y y = values of t, x, y at the beginning of present V|a condition. h) A _ + ( x o - x *\ - y o (x n - X*)\ - V, A _ __0 J a J o " \_ - X P a C - £ = x- c where -d (x*,0) - (-, 0) = location of singularity corresponding to present V(i conditions. -TR- IABLE V. IMPULSE VALUES FOR CHANGES IN Vu V\x Condition Change d o ■H -P •H w s a) U M H CI O •H -P •H W d EH H O P From 10 II III III1 IIIO III1 III II To II III nil IIIO nil in n 10 Magnitude of Impulse U 10 +2p — |W -Hp7 'io 2 T J +Hp7 -=- y* •2P ^ |„, -2P -f |W| +Hp7 'io 2 y^ T J T -Hp7 — y 1 + 2p-^ |V| Comments W > y is initial value or y in region II y' is final value of y in region II W > W < y^ is the initial value of y in region II y' is the final value of y in region II W < H is the symmetry factor; H = 1 for asymmetric flipflop. 2 for symmetric flipflop. -75- i) x .1 a 2a -b + k f b - ^~^\ j ) And for complet eness we repeat equation (2.IO3) in the new notati on 00 _o ax + hx + ex = d. for a given Vfi Now equation (3.56) becomes x = A e a ° a 3 J + A e + x >s a > y = A X e a ° + a A e 3 ° a a 3 ^ In the phase plane, from (k) and (h) ; (3.57) 1 (x - X*).\ - y ^ T^ - x*)\ y- a ^0 (3.58) °r, in another form: e Vp (t - t o ) . /> - x *)\, y 1 a x ~ x **a - y l (x - x *)\ p - y/p (* - x*;x p - y c (3.59) Notice that t and X are normalized and have no dimensions! Obs.: Equations (3.58) and (3.59) are phase plane trajectory equations, and the calculation of transition time is hased on them. Also based on them is the phase Plane graphical construction which simplifies not only transition time calcula- tions but also the analysis of waveforms and trigger. and design optimization of both circuit -76- In this analysis the trigger is assumed rectangular, so the term ns = unW is imbedded into d,* the impulse terms ms - (jmW[s(t - t Q ) _ 6 ( t _ t _ T )] and also f(x)[g(x + h - o(x - ^)] are both considered in g), with their magnitudes represented by U. This is summarized in Table V. Therefore, expansion of g) yields for this special case: Discontinuity y = y 1 - h Q y = y ? + V y n = y* + h c c y = i 1 - h ,2 Transition Through Line I -* II x = 1 7 II -* III x = 1 + 7 III -* II X = 1 + 7 II -> I X = 1 " 7 (3.60a) (3.60b) (3.60c) (3-60d) where .) I - ^II 7 2a II b) m and a as in Table I c) H is called the "symmetry factor" H = 1 for the asymmetric flipflop 2 for the symmetric flipflop We can write equations (3.60) in more explicit forms, and also obtain the inverse functions : ■77- Discontinuity M y = ^{^i + hh>' - 1} > (ii) y' = (1 + iyj 0^0 (i) 7 n - (1 + iy') y (ii) y' = ^{wrr^ - i}J (i) = ~H- >/i - W 1 ' + y o ~ 2£ r + 1 > (ii) y' = (1 - i^)^ (i) y Q = (1 - iy«)y« (ii) y« -^{-N^n^r +1 j J Transition Through Lines I -» II II -* III III - II II - I x = - i (3.6la) 1 x = + — 7 1 x = + - 7 1 x = - — 7 (3.6lb) (3.61c ) (3.6ld) Notice that in cases (a) and (b) we have the I to III transition, i.e., 7 > through the transition, whereas in cases (c) and (d) we have the III to I transition, i.e., y < through the transition. The signs of the square roots are selected from this physical consideration. 3«^ Separatrices We are using the word separatrix in a somewhat looser sense than its strictly mathematical meaning [17]. We shall call a "separatrix" any phase Plane trajectory which divides the phase plane portrait of the system into qualitatively distinct families of trajectories. The concept of "qualitatively distinct" is purposely vague; this means bat a separatrix will be so with respect to some stated qualitative distinction -78- between the two families of trajectories in which it divides the phase plane portrait of the system. In order to define a separatrix, all we need to do is to find one of its points Q, let Q = (x ,y ), and enter x Q and y Q instead of x Q and y Q into equation (3-58) or (3. 59). Also x* must be known (x* as defined in (3.56I1)). We have a special interest in defining two types of separatrix over the whole portrait : (i) "Transition Separatrix" which divides the portrait into two families of trajectories: those that cross through region II and those that do not. This separatrix, as illustrated in Figs. 15 and l6, is composed of four branches: A, B; C, D; A, B; C, D. These lines will be given special names : a) ABCD: (I to III) transition separatrix. b) ABCD: (ill to l) transition separatrix. c) ABBA: end-point separatrix. d) DCCD: initial-point separatrix. Since a) ABCD separates the lines which are trajectories from region I to region III from the lines which are not so. b) ABCD separates the lines which are trajectories from region III to region I from the lines which are not so. c) ABBA divides the portrait into two sets of trajectories: those that terminate in x and those that terminate in X TII • d) DCCD divides the portrait into two sets of trajectories: those that originate in region I and those that originate in region III. (ii) "Critical Separatrix," whose importance will be later explained, divides the portrait into two families of trajectories: those that lie partly -79- v 51 M A o z < o V 1=1 o A =t M UJ where x is in region II (x*,0) is the singularity corresponding to region II \^ and *£ are the natural frequencies inside region II We find r A» y D> y A' y D respectively from T V V ^B' ^C through equation (3.59): (a,ii), (b,i), (c,ii), (d,i), after the replacements; y A"* y ' y jD-v ^A^y'' ^-y y B -.v y c " y '' ^v yc^ y (3.62) -82- We get the following equations - (1 + h B )y B (3.63a) y n = (1 + h n h r (3 = 63b) D v J C ,,7 C y- A = (l + £y g )y s (3-630 y B = (1 + ^c)yc (3o63d) b ) The Critical Separatrix Determining Points : Of course those trajectories which cross the x axis at points x = - j and x = ~ are the two branches of the critical separatrix. c ) The Separatrices : The two separatrices, as illustrated in Fig. 15, are given by the general trajectory equation (3.58) or (3-59) (they are entirely equivalent). Since here we are more concerned with geometrical properties, we prefer the latter, which we repeat here as equation (3-6U), leaving out the exponential of time, and replacing (x Q ,y ) by (x Q ,y Q ). So the separatrices are given by: f (x-x«V-yfo f (x-x*)^ -y| X P } t(* Q - **>\rTJ - lTx Q - x*)\ p - yj and Tables VI and VII furnish the values of the parameters. Besides the separatrices, three more lines at each stable node are important in the qualitative study of trajectories in the phase plane. These lines are: (by definition) assuming \ Q < \^ (see Fig. 12): a) The "tangent line": y - \ p (x - x*), which is a tangent to all trajectories at point (x*,0). -83- TABLE VI. PARAMETERS OF THE SEPARATRIX EQUATION (3.64) Calculate by -- Equation (3.63) Table II Equation (3„56i) and Table I Parameter X Q (depend on fi) X* X. 1 i = a, p Branch Value Transition Separatrix Branch A 1 7 y A in iIji Branch D 1 + — 7 y D X IIIH N.IIIH Branch A 1 7 y A X III(i x inm Branch D 1 7 y D X T N.in Critical Separatrix Branch E 1 7 I|I ii|j. Branch E + — i 7 X III^ ^illlH — Notice that the index (i merely indicates if the trigger is ON or OFF; the actual direction and effects of the trigger (if ON) must be computed through Tables I and II. B TABLE VII. RESULTS OF EQUATIONS (3.62) FOR THE BRANCHES OF THE TRANSITION SEPARATRIX INSIDE REGION II Branch Equations (3.62) Furnish Transition Separatrix Branch B_, B y crV (x " x n } y B = y B = y a (+ £) Branch C., C y p = y(x - x n ) y C = V + 7) Tq = y (--) y^\ 7 ) Notice that both y.'s and X.'s, I = a, &, depend on ,1, so all y's depend on p, -Qk- b) The "direction line": y = \ (x - x*), which does not cross any trajectory except at point (x*,0), where it crosses all trajectories. c) The "max-min line": v = . p . (x - x*) which crosses all trajectories at a p (x*,0) and also every trajectory at its point of maximum or minimum value of y(x). d) The line described in (t ) also applies to the case of saddle points, except that, at the singularity (x*,0) it crosses only the asymptotes, since the other trajectories do not pass through the singularity. 3.5 Trajectories and the Action of zhe Trigger The siinplest trigger is the rectangular current trigger. In fact the action of a trigger of a different waveform will, in general, differ in detail, hut not in principle from the action of the rectangular 'rigger. For this reason we have considered it important enough to he the oasis of this work, and, in this section, we will discuss its action in a qualitative manner. 3.5.1 Turning the Trigger ON and Possibility of "Under-Triggering " ..gures 16 show various possibilities of trigger action upon the phase plane portrait—singularities and separatrices--and also the corresponding initial value of y. Let us assume that the flipflop is in stable state I, i.e., the representative point P is at stable node I, when a rectangular trigger of amplitude W is applied (a positive trigger). The immediate effect of the trigger upon the phase plane portrait to shift the stable nodes to points respectively £k x and ^ IT1 "to the right, and to shift the saddle point by £x to the left, where Ac y = x yl - x vQ , as given by Table II. -85- At the same time P is shifted upwards to a point at (x ,y ), i.e., the x coordinate does not change, but y goes to an initial value y = ^ W (3-65) Suppose that the singularities have not been shifted out of their proper regions (Fig. l6b ) . The transition separatrix in region I approaches the critical separatrix (whose shape changes!), thus reducing the minimum initial Value y Qmin ° f y necessar y fo ? a complete transition to occur. Let X^ denote the singularity (x ,0), and let y^^ be the least initial ordinate at x IQ , for which a I to III transition occurs. There will be two possibilities: a ^ y < y Omin' then P wil1 follow some parabolic trajectory and tend towards x j]_' Here there are yet three possibilities. lo y < : \/ Sc I° P wil1 not cross the x axis, moving towards X , in an overdamped manner. 2 ° y = ^a^l 1 P w111 not cross the x axis, following a straight path towards X , approaching it in a critically damped way. 3» y Q > ^ a ^Xj : P will cross the x axis, moving towards X in an underdamped way. In case 3 we might still distinguish the two possibilities of P going through region II (entering it under point y ) or not. Si y > y Qmin' then P wil1 als0 follow a parabolic path towards X in the X[J. underdamped way, as in (a. 3) above, but it enters region II above point y before reaching the x axis. Then a transition occurs. This effect will be called "under-triggering" (Fig, 19 ). -86- x A oN I* a i z »- a: UJ cr ■87- -88- Ni MK ,- -« r -89- One might then expect that the minimum value W . of W necessary to mm J cause a transition would be slightly less than the value W of W necessarv crit J to shift X^ and X^ out of their proper regions (and into virtual existence!). But this is usually not so, as it will he proved at the end of this section that, under certain (usual) conditions, of all possibilities mentioned above only (a.l) occurs, all others being impossible for the type of circuit under consideration. Therefore, under-triggering usually does not occur for this type of circuit. However, it is a possibility, especially in a general equation, whose coefficients were related in some other manner. We will later study this effect in some detail. 3-5.2 Virtual Singularities and the Trajectory We establish that W^ = W^^ = W Q ; suppose that W > W Q ; then the portrait becomes as in Fig. l6d. There is only one real singularity, and this is the stable node X im ; the end-point separatrix vanishes* (since all lines must now terminate at X mi ); the critical separatrix and part of the initial-point separatrix also vanish, and the remaining part of the initial-point separatrix loses its meaning. From its initial position at (x J0 ,y ), P "sees" only the virtual stable node X n (somewhere in region II or III) and starts to move on a parabolic path towards it. However, before reaching x__, P crosses the line x = - - where 11 7 y suffers a discontinuity (-^y), ' and enters region II where now it "sees" only the virtual saddle point X m (somewhere in region i) and changes its trajectory into a hyperbolic path asymptotic to the line y = \ in «(x - Xj ) (a remaining part of the initial-point separatrix, and here we see why it is meaningless); t o See footnote on page 90. -90- f inally, P crosses the line x = + - , where y suffers a discontinuity (+\j), and enters region III. Once in region III, P will "see" only the real stable node x m , towards which it will start to move in a parabolic path (in an overdamped manner, as will be shown). 3.5.3 Turning the Trigger OFF The next event with the trajectory of P is the turning OFF of the trigger. Rigorously, the trigger may be turned OFF as soon as P has progressed far enough into region II, i.e., to a point where, after the negative jump of y caused by the trigger turning OFF, P finds itself at side III of the end-point separatrix (of the [X = system, of course). On the other extreme, we could leave the trigger ON for an indefinite amount of time. We are interested in establishing a criterion with which to judge the adequacy of the trigger duration. 3.5.U Discussing Trigger Duration Assuming the trigger is sufficiently long to produce a transition, we recognize five possibilities for the trigger (see Figs. 17 and 18): (i) Too short: if it is turned OFF while P is still in region II. (ii) Short: if it is turned OFF long before P reaches the line x = x , but after P is in region III (Figs. 17a, b and l8a, b ) . P Discontinuities caused by impulses + Ay £ — respectively. dx -91- (iii) Fair: if it is turned OFF approximately as P crosses the line x = x . (iv) Long: if it is turned OFF long after P crosses the line x = x III0 , but before it approaches X (Figs. 17c, d and l8c, d). (v) Too long: if it is turned OFF after P is already close to X im . Of course, these definitions can be formalized and made exact: so, a sufficient rectangular trigger starting at t Q = and having duration T is said to be : (i) Too short: if x(t) < - - (ii) Short: if + - < x(t) < x - e 7 v J IIIO 0- (iii) Fair: if ^ - e Q _ < x( T )< x^ + e Q+ (iv) Long: if x^, + 6 Q+ <_x( T ) < x JnjL - ^ (v) Too long: if x - e < x(t) where e Q _, e Q+ , ^ are positive numbers such that all the inequalities above can be satisfied. No matter which case occurs, y will suffer a discontinuity equal to (-y Q ); from the point (x(t), y(T)), P will jump to (x(t), y(T) - y Q ), and then it will move towards X III0 by some parabolic path, thus completing the transition. NOTE: Of course, P never reaches * 1II0> but given an arbitrary neighborhood e ino of x ino^ there is a time T nio (e iiio ) such that for an y t:> T ino (e iiio^ P is in € III0 . We can therefore, arbitrarily select a neighborhood N of X III0 , and ' by definl "tion, say that a transition is complete after P enters N for the last time. -92- 3.5.5 The Concept of "Optimum Trigger Duration " Call: and P T _ the point (x(t), y(T>) > P T+ the point (x(t), y(T) •■ y n ) (3.66) etc., be vectors with the same coordinates as the points designated by the symbols under the arrows. By definition, let and let P T _, P T+ , R III0 (W,T) = |P T+ - X mo i (3.67) Given a rectangular trigger of amplitude W, we define "optimum duration T*" as that value of T for which R iiio (t) is a minimum. That is: R mo (W,T*) = min R IIIQ (W,T) (3.68) Since for each value of W there is a corresponding value T*, we con- clude that, for a given circuit, T* is a function of W: T* = e(w) (3.69) This function 9(w) is a characteristic of the complete circuit, i.e., flipflop and triggering circuit together. Notice however that the definition of T* is somewhat arbitrary, and probably there is no ideal criterion on which to base a definition of optimum duration. It certainly depends under what criterion we would like to have the transition optimized. -93- A more practical definition is as follows: given a rectangular trigger of amplitude W, we define "optimum duration T* M as the value of T for which x(T) = x . That is X(T * } = X IIIO (3.70) The discussion presented in this subsection can be applied to a III to I transition, if we make the necessary (and obvious) changes. The criterion for the definition of T* expressed by equation (3.70) is the most useful, and will be used throughout this report. So, unless other- wise specified, the expression of "optimum duration" or the symbol T* will imply "as defined by equation (3.70)." 3-5.6 Possibility of "Back-Triggering " We have said that after the trigger is turned OFF, if it is suffi- ciently long , P (whose y coordinate has suffered a negative discontinuity equal to (-y )) "sees" only X mo towards which it moves by some parabolic path. However, we must ask ourselves if this is always true. There seems to be nothing in the nature of the equation to warrant this assumption. The objection is: "The position (x(T-), y(T-)) of P at the moment of turning the trigger OFF might be such that the new position of P, (x(T+), y(T+)) (where y(t+) = y(T-) - y Q ) would be under the branch A of the transition separatrix, and therefore P would return to X^, rather than going to X » " This effect will be called "back- triggering. " In fact, the possibility of back-triggering is small, unless the coef- ficients of the differential equation were not related by the circuit parameters (representing some other analogous type of bistable device). -9k- We will prove that, under certain (usual) conditions, hack-triggering is not possible for the circuits under consideration. The conditions that make back-triggering impossible are the same that make under-triggering impossible. Actually, these two characteristics are closely related. We will presently discuss these effects in some detail, explain their interrelation, and find the conditions that make them impossible to occur. 3.5.7 Trajectory After the Trigger is Turned OFF Let us assume the trigger duration is sufficient and that no back- triggering occurs. Figures 17 and 18 illustrate the four cases as considered below: (i) x(T) < x III() , y(T+) > (ii) x(T) < x III0 , y(T+) < (iii) x(T) > x IIIQ , y(T+) > (iv) x(T) < x III0 , y(T+) < Figure 19 illustrates the three possibilities in the case of optimum triggering: x(T) = x III( y (i) y(T+) > (ii) y(T+) = (iii) y(T+) < We should point out that x(t) is quite arbitrary, since we have absolute control of the trigger duration T; but, for a given circuit, y(T+) is ' We point out again that back-triggering refers only to the case of sufficient trigger, i.e., there must be an interval of time T min < T < T max for which a normal transition would occur. T < T^ means insufficient trigger duration; T would be imposed by back-triggering. m a v max -95- a function of x(t) given by the phase plane portraits of the differential equations (with trigger ON and OFF). This means that, for a given differential equation, selection of T = T* will lead to a certain value of y(T*+) for which y(T*+) < 0; the equation and also the trigger amplitude W will determine which relationship holds . It would he interesting to know how the coefficients of the equation and the circuit parameters, as well as the trigger amplitude affect the curve y(T) versus x(t). It would also be useful to know how T* and y(T*+) depend on W for a given equation. In the following chapters these questions will be considered. 3.6 Under-Triggering and Back-Triggering Let us analyze the possibilities of under-triggering and back- triggering for (i) a given differential equation of type (2.103) with coefficients defined in the three regions (ii) ignoring, in this section, the relationships established in Table I, but still assuming (iii) a rectangular trigger (i.e., (3*6) and (3.7) hold) and that, as before (iv) the function f(x) = f(y), the magnitude of the impulses occurring at x = + - , are: - 9 6- r e-O , at x = + i f(y) = ^ undefined for any x f + - (3.71) with m. m.. I _ _ r 11 or — , according to whether the flipflop is II 2a- [I a^ asymmetric or symmetric. 3.6.1 Under-Triggering Suppose a trigger of amplitude W < W Q is applied to the system; assume P was at X = (x I0 ,0) before the trigger was turned ON. P jumps to a point P Q = (x 1Q ,y Q )> i* ? ls above branch A of the transi " tion separatrix of the triggered system, then a transition will occur, with P going to X m , rather than to X^ (under-triggering). Fig. 19 illustrates this effect. Therefore a transition will occur if and only if y^ > y Q 10 where v is the ordinate of branch A of the transition separatrix of the J al0 triggered system (u = l) at x = x^. The problem is to find the value W min of W that, for a given system, will cause the point (x I0> y ) to be on branch A of the transition separatrix of the triggered system (i.e., u = l). We could keep the expression f(y) = -&y at any other point x { + -, taking I = Z„ , Vu indicating region and state of trigger; but this would be irrel- evant, since the function is multiplied by zero at these points anyway. •97- -98- As could be expected, we will see that it is not possible to solve this problem analytically, but only by a graphical or iterative numerical procedure. In fact, we have a set of formulae, repeated here for convenience, that can be used to solve this problem: From (3.13) and the Obs . at the end of section 3.2.2, d d v ^ c vu c vu V (3.72) from which we get "10 c 10 I n w 11 ""' C I1 " C I1 ■II "110 c 1 10 d-r-T x = + ■ w III c lzl c lzl > (3*73) < w <: w„ => x™ < x Tn < i < w < w Q => > x in > - - '0 10 II 7 From (3.62), using the notation \ and X as before pv^x ■ 1 y Bl =: " X alll" ( 7" fx ni ) - W < W Q => y B1 > (3-7*0 y Al = y Bl^ 1 + ^ y Bl^ Co. -x*jr j A1 - j b1 ilearly y A1 > y T (3.75) with 2 a n ' H = 1 for the asymmetric flipflop 2 for the symmetric flipflop Also, from equation (2.103) itself (see section 3.5.1): -99- m I (3.76) And finally, the general trajectory equation, expressed by (3.59) be rewritten as : can (x i - X *K - yj- fo . f(\ - **fr p - y . A (x. - x*)\ - y (x j " x *^ - 57- (3.77) where (x^y^ and (x^y^) are any two points over the same trajectory, in the same region, and x* is the abcissa of the singularity and \ and \ are the OL (3 natural frequencies corresponding to that region; so, in (3.77) let: (x.,y.) = (x ,y ■ ); x* = x "> II UyyP = (- py A b \ a and ^ - ^ and ^ ► (3.78) J Considering (3-73), (3-7*0, (3-75), (3-76) and (3.78), (3.77) yields m T /d + nW d N a i aI1 V c n c ia >, all d II + nWN X alll (7 + — c I III , d + nW x, d + nW N + \ ,„ f - + all \ 7 c II y = < m /d + nW d N alii \y 1 __ d n + nW 111 311 £\ 1 + d n + nW \ in y alii V 7 c /, d + nW^ ^ (3-79) which can be numerically solved for W. -100- If a real and positive root W x can be found for (3-79)j such that W < W , then we have under-triggering whenever the trigger duration W satisfies rl 0' w „ < w < w A (3.8o) rl In this case, W . = W T (3.8l) mm rl If no such W exists, then no under-triggering can occur, and we rl have : w . = w n (3-82) mm Expansion of the numerator and denominator in the fractions appearing in (3=79) furnish: K W + V I " 11 . f K Bl' W + V I PI1 ( 3„ 83 ) (w 2 + p w + ^ kv 2 + P pl w + Q pI al , H+> , v m P Q ¥ M . P 9 Q~ T as in Table VIII. 1. with K aI , M aI , P ar H ar ^ pI , iip r rp T , *p X An entirely analogous analysis could be made for the case of a III to I transitions reverse the signs of all coordinates and trigger, and change subscripts I to III, A to A, B to B. 3„6.2 Back-Triggering We will consider only the possibility of back-triggering in the case of too-long trigger, i.e., P is assumed to be at X im = (x nil ,0) when the trigger is turned OFF. -101- TABLE VIII. 1. PARAMETERS OF (3.83) AS FUNCTIONS OF THE PARAMETERS OF (3.79) K al M a I "al ) aII1 v W - 1 ♦»m(l-£ ii. -102- P jumps to a point P Q = (x III:L * -7 Q )f if p is under branch A of the transition separatrix of the untriggered system (u. = 0), a transition will occur, with P returning to X Q , rather than going to X III0 (hack-triggering) . Figure 20 illustrates this effect. Therefore, a transition will occur if and only if : -y l y a01 l {3 ' Qk) where v is the ordinate of branch A of the transition separatrix of the 'aOl untriggered system (u = 0) at x = Xj-q^- The problem is to find the value W of W that, for a given system, will cause the point (x mi > -y Q ) to be on branch A of the transition separatrix of the untriggered system (i.e., u. = 0). As for the case of under-triggering, we will see that this problem cannot be solved analytically either, but only by a graphical or numerical iterative procedure . A set of formulas similar to the one used in the case of under- triggering can be used to solve this problem. We had obtained (3.72) which we repeat here for convenience: x = _YH = _v_ + n-iL-w (3.72) v ^ c vu c vu V We get . -103- 110 c II 110 k III0 c " III IIIO x TT1 = -£- + -2- w 111 C II1 C II1 < W < W_ => > x T „ > - - III 7 III n TT 1111 c iin c iin < W < W„ => x (3^85) < (2 " 7 )x mo J Again from (3.62 ), retaining the notation \ and \ avu pvu ' y B0 X aII0'^7 +x ii0^ (3*86) y A0 = yBO'^-^Io) (3.87) From equation (2.103) itself: m ■y^ = in a w III (3.88) (we have implicitly taken true ) . L III " a i and m Hl = m i> but the y ma y be not strictly And, for convenience, we repeat the general trajectory equation (3.59), in the form (3.77); (x i - **K -_iA a r (x i x * )x e ■■ y il^ T^^TX-Ty/ " (X J - **h - y J (3.77) with (x i ,y ± ), (x ,y ), x*, \ and \ as before J J Now let a 3 -104- ■> (x^y.) = (x im ,-y ): (x^y^) = (+ -, y^ )i? X* = X IIIO (3.89) x a and \ = X aliio and Vno J Considering (3.85), (3-86), (3.87), (3.88) and (3.89), (3.77) yields: r m J III_ 01110 V c no/l , d^^ + nW d TTT III tt ( m m — W + X TTT . — - a allxOV c. "s 'III1 'iiia alllO iX i + d ^ alio V 7 c IICL A d m ^ aino^ c moy / V v. m -i III a HI W + X ' d III + nW d III PIIIOV c c ina X pino > A i d II + "alio \7 c IIC ix fl , !ll^ . ! bII ° V " c na + x A III PIIIO \7 c ina (3.90) which can be numerically solved for W. If a real and positive root W n can he found for (3-90), then we have hack-triggering whenever the trigger duration W satisfies W TTT < W rill (3.91) In this case, W = W TTT max rill (3.92) and in order to obtain a permanent transition, we must have W < W max (3.93) If no such W exists, then no value of W will cause back-triggering, rill -105- Expansion of the numerator and denominator in the fractions appearing in (3.90) furnish: L(W ' + p ain w + %aiJ ' V + Vn W - Q PII / (3 ' 94) with K a in' M aiir p aiir %air K pin> M pm> p pnr Q pm as in Table VIII ° 2 ° An entirely analogous analysis could be made for the case of a III to I transition: reverse the signs of all coordinates and trigger, and change subscripts III to I, I to A, B to B. 3.6-3 Discussion It is clear from inspection of either Fig. 20 or equation (3.79) and of either Fig. 21 or equation (3.90) that the necessary and sufficient conditions for the impossibility of occurrence of under-triggering : m_ 'W ' IVll > a~ W ^ °< W < W (3.95) back-triggering s X HI 1 ' ' Villi > a— : W ' 0X _-,.„ alio \7 c no. + \ p iiio v" c inoy •107- y 'O . BOTH ARE DEFINED ONLY J IF t *t(x |0 »' FIGURE 21: DEFINITION OF TIME INTERVALS OVER A TRAJECTORY IN THF flMfi DOMAIN, RELATED TO THF PHASE PLANE. (8EeTaBLE». n w/ v - d v n w - , Si " c vo Si Si So Si v SoSi •108- Ax (3-98) Notice that if either of conditions (3-95) and (3-96) is not met, then there will exist real positive solutions W^ or W rm to (3-79) and (3-90) respectively. We clearly see that conditions (3-95) and (3-96) are formally the same . a) If we make the further assumption that the coefficients of equation (2.103) except d are the same in regions I and III (a realistic assumption!), then the only difference between them is the effect of different values of the coefficients with the index u (neglecting variation of capacitances). b) If we make the further assumption that the coefficients of equation (2.103) are invariant with u (i.e., trigger circuit is fixed!), then the two conditions are identical. This shows a close relationship between these two effects, i.e., they have the same intrinsic nature and origin from the circuit parameter point of view, and if any distinction exists between them, it is due to the fact that the circuit itself is not the same in each case (unless the two conditions above hold ! ) . One further assumption leads us to an interesting point: assume that G = 0, u = 0, 1 (3-99) i.e., consider the case of a trigger circuit that closely approximates a true current source. Then we have the Theorem k . If equation (2.103 ) describes either an asymmetric or a symmetric flipflop (coefficients as in Table i), and if the trigger circuit satisfies (3.99) above, then no under-triggering or back- triggering can occur. -109- Proof. Conditions a and b can be expressed as a single condition: m V ' n> i (3.100) since c vu = 1 for v = ^ XII > n = o, 1 and with K a - -(b ♦ Vb 2 - Ua' ) i T.T 1 O a = T 2 T. + T . + T b = 01 T L = = 2p T . io T n = 2p R s R o as in Table I, for regions I and III. Therefore, (3. 100) becomes, after expansion and simplification: 2T. < 10 { (T 1 + T oi + V + ^ T i +T oi +T o )2 - UT i T o'} T (3.101) This becomes 1 < 2 ^ + 7) \(° + cr + r ) + ^( c + cr + r) 2 - kcr [ = f(c,r) (3.102) -llO- where C R „ c = — , and r- T o i Now, let us make a change of variable, as follows r = r c = Sr so that f(c,r) becomes g(&,r): g(&,r) = \ (1 + \) |(6r + 5r 2 + r) + V( 5 r + &r 2 + r) 2 - 4sr 2 ' j or g (6,r) - | (r + l) {(1 + 5 + Br) + J(l + 6 + Br) 2 - ^'} 1 (3.103) Condition (3.102) has become: 1 < g(B,r) (3.1CA) Let us find the partial derivative of g(8,r) with respect to r: k^ a i("( 1+6+5r ) W(l +6+ Br) 2 -k b + (rflhlu f, 2{l + \P T \} &r 2 l_ v I V(i+ 6 + Br) - h& J (3.105) Then ' The discriminant always positive, since (l + 6 + &r) - *+& - (l - B + Br) + U& 2 r > so g(6,r) is a real positive number. -Ill- %^>o, for all 6 > 0, r > (3.106) Therefore lim g(6,r) < g(s,r) r-O r r > s-*o 8 > (3.107) But lim g(8,r) r^O 8-^0 = 1 (3.108) Sub stitution of (3.108) in (3.IO7) yields 1 < g(8,r) r > 8 > (3.109) and therefore condition (3.100) holds, and the theorem is proved. Comments : 1) Notice that the theorem is very strong, in the sense that condition (3.109) is strong; in fact, 1 is not only a lower bound for g(5,r), but it is an infimum of g(s,r), i.e., its greatest lower bound! So, anything at all upsetting the assumptions made, may invalidate the theorem, and in this case either under-triggering or back-triggering, or both, at least in principle, could occur! 2) Notice also that (3.IO6) is weak. In fact, dg(6,r) or > lim hi^il 5>0 8^ (3.110) -112- So, in fairness, we should point out that the possibility of occurrence of under-triggering or back-triggering in practice is not so great, since any "reasonable" value at all of c and r ought to satisfy condition (3.100) with a good margin. 3.7 Summary In this chapter we have analyzed the phase plane characteristics of the basic flipflop equation in the case of a rectangular trigger. Conditions related to the existence and nature of singularities were discussed and three theorems were proved with respect to this point. Some properties of the system were also established by diagonalization of the characteristic matrix of the system (in effect, considering two possi- bilities, respectively for the two possible types of singularity). A general trajectory equation was established, and the geometry of the separatrices was discussed, as well as the action of a trigger upon the system phase plane portrait, with special attention to the effects of turning the trigger ON and OFF. Here the possibilities of under-triggering and back- triggering were discussed, and a theorem on the conditions for such a possibility to exist was proved for an important special case. h. ANALYSIS AND DESIGN TECHNIQUES U.l Introduction At this point we would like to utilize this information we have about the bistable system represented by equation (2.103) to the purpose of developing some analysis and design techniques . Specifically, we wo ild like to find effective methods to solve the following problem: given a flipflop, its loading and triggering circuits, find the transition wave forms of: (i) base currents and voltages (ii) collector currents and voltages We would then be able to find the optimum trigger duration . Further- more, knowledge of the base and collector currents and voltages as functions of time would help to improve the design of the overall system [l, k] . And finally, if the influence of the circuit parameters upon wave form characteristics is known, we would have a means of optimizing the design of the system towards approaching some transition requirements [7], Lastly, if the transistors are given (t is given) and also the trigger, but if the circuit (resistors and capacitors) is abritrary, then the lower limit in transition time can be calculated, and a convenient figure of merit for transistors describing their performance in switching circuits can be defined, and would certainly be useful in the selection of transistors for switching applications (see Chapter 5) [10, 20] . ^.2 Definitions of Time Intervals We have divided the range of the variable x into three parts, which were called regions I, II and III, Remember that x is a normalized form of By effective we mean: "a sensible compromise between accuracy and ease of application." ■113- -Uk- the base-to-base voltage. Let us consider the variables w fc , which are normalized forms of the collector currents (of transistors 1 and 2 f or k = 1, 2, respec- tively), as given by equation (2.93). It is clear that, as long as x is in region I, w 1 = and w g = 1; whenever x is in region III, this situation is reversed, with v^ ■ 1 and w g = 0; in both these cases one of the transistors is cutt off, and the other is con- ducting a fixed current, i.e., the transistors are inactive; they are active only when x is in region II. Def . k.l. So, in a I to III transition, from the point of view of collector currents, the time during which x is crossing region I, from x(t Q ) towards - -, is really a delay. It will be called "delay time" and designated by 1^. If the circuit were settled, x(t Q ) = x^ otherwise, it may happen that x(t Q ) / x^. 1 , 1 . Def. 4.2. The time interval when x is in region II, going from - - to + - , is characterized by activity of the transistors, and variation of collector cur- rents. It will be called "active time" and designated by 1^. Def. 4.3. And finally, for the time interval when x is already in region III, from + - until final settling in a neighborhood N m0 of * 1110 , the collector currents are constant (having reached their final values) and the transistors are again inactive. It will be called "complementary time" and designated by T . C However, many things can happen while x is in region III. Def. k.k . The time interval in which x goes from + - to x III0 , with trigger ON, is called "balanced time" and designated T g . De f. k.5 . The time interval between the moment the trigger is turned OFF and the moment x settles inside N m0 is called "settling time" and designated T g . This time interval is often called "discrimination time." -115- Def ' k ° 6 ° The time lnter val between the moment x reaches x^ (or wo ad reach X IIIO lf the trigger were kept ON) and the moment the trigger is turned OFF is called "trigger excess overtime" and designated T^. We will make the convention of using negative values of T^ if the trigger is turned OFF before x reaches x^, i.e., given the function t(x), then T^ = T Q - t(x mo ) where t(x lII0 ) is calculated assuming a sufficiently long trigger (or measured!) and T is the trigger duration. Def ° h ° 7 ° The time interval between the instant x reaches + i and the instant the trigger is turned OFF is called "trigger overtime" or simply "overtime/ 1 and designated T^ i.e., T QV = T Q - t(+ i). With our criterion for optimum trigger duration T* (see equation (3.70)) we will have: T e = t ^ x nio ) (^.ia) and if T Q = T|, then from Def. k,6, T E0 = ° C^.lb) And we define "optimum overtime" T* : T ov = T ? " *C + 7) = T R (^ic) - ef ° k ° Q ° In the case of too long a trigger, we define the "long settling time, ' T Lg , as the time interval between the moment x reaches + - and the moment it settles inside a given neighborhood N of x III1 1111° Def\_ih9* Finally, we call "l-III transition time," T TR , the total time interval between the moment the trigger is turned ON and the moment x crosses the line X IIIO° -116- The following relations are obvious from the definitions: T TR = T D + T A + V " X(t ) = X I0 (U - 2a) T c - T QV + T s (U.2b) T 0V " T B + T E0 (4 - 20) T EQ = implies T QV - T* y = Tg (U.2d) Naturally, all the above definitions apply equally well to a III- I transition by replacing III by I and I by III everywhere. The symbol T^ denotes the "ill- I transition time." Table IX contains these definitions of time intervals, which are also illustrated in Fig. 21. U„3 Calculation of a Time Interval Over a Trajectory by an Iterative Formula It is not possible to explicit y in (3.59). Therefore, given an initial point P^x^r and the abcissa x f of another point P f , in order to find the other coordinate y of P f such that P Q and P f are on the same trajectory, we must apply an iterative procedure to (3-59); the time intervals can be found from (3. 58). Equation (3.59) can be expressed as ■M yi a vr=fv^ Read: P n "whose coordinate are" (x Q ,y ). -117- TABLE IX. DEFINITIONS OF TIME INTERVALS OVER A TRAJECTORY (SEE FIG. 21 ) Symbol Name T D B EO "OV TR I>* OV Delay Time Definition t(-y) - t. Comments Active Time Complementary Time t(+£) -t(-i) - t(- i) 7 If circuit was settled at X I1 at V ^ x iq\* V otherwise t(x ) f- t . t is the instant the trigger is turned ON. Balance Time t(x III0 , u = 1) - t(- i) Settling Time Trigger Excess Overtime Trigger Overtime t - t, s t t g is the instant when P enters N for the last time. t(x- = l) is the 'IIIO' M ' instant P crosses x= x assuming the trigger stays ON (|i = 1) all the time ' t(x III() , u = l) Transition Time Trigger Duration Optimum Trigger Duration and Overtime respectively *fl " ^ 7) s tg is the instant the trigger is turned OFF. t, Defined only if *o = t(x io^ to letting t.Q - t(x ), equivalent Notice that it is measured since t (not t(x m )!) until t . 1U s Of course, only approxi' mately realizable. J Q t(x ni0 , ix = 1), -118- for region V and trigger condition |i. Parameters M Q , M , Q q , Q^ are gi Table X. We obtain either one of two formulae lven in y n = M - Q i ^n+l a U a {h.ka) or a K £rv rM -y " a J n n+1 P V (l*.lrt>) The only difference between them is a question of convergence. In fact, given two implicit functions f and g of y, the equation f (y) = g(y) (^5) can be solved by an iterative procedure by means of a formula such as f(y n+1 ) - s(y n ) ^° 6 ) i.e, n+1 = f _1 [g(yj] (+.7) Let y„ be the solution of equation (U.5) The iterative formula (U.6) will converge, i.e., lim y n = y f , if and n->oo only if there is a number e > such that, If |y Q - y f | < 6, there is a positive number K such that : No loss of generality, since the symbols f and g can be interchanged. -119- TABLE X. DEFINITION OF THE PARAMETERS OF EQUATION (4.3) Parameters of (4.3) M a %c K Q, P Expressed as Functions of the Parameters of (3.59) (x - x*)\ a (x A - x*)\ - v 'a J (x - x*)k (3 (x " x*)^ - y Q Comments: a) Region V^ trigger condition u b) x* = x Vu C) V = W V W d ) ( x o' y o^ = any S lven Point on the branch of trajectory under conditions Vu. e) x = some abcissa such that there is an ordinate y satisfying the condition: "(x,y) is on the same trajectory vu branch as (x ,y )." f ) y is to be found. -120- *V, n+1 % < K, K < 1 (fc.a) n However, it is clear that ^ lim n+1 :'(y f ) ST ^ (4.9) where the prime means "derivative with respect to y_ . Now (4.8) and (4.9) imply that (4.6) converges if and only if g'(y f ) f 1 !^ < 1 (4.10) Therefore, one of formulae (4.4a, b) converges and the other diverges. There is no way to know a priori which one will converge, since we would need the solu- tion y of (4.3) to answer this question. However, assuming we start from a good initial guess y , if we calculate y g and y^ from both formulae (4.4a, b), the initial tendency should be clear. Another way would be to differentiate both sides of (4.3) with respect to y, and compare the two results for the initial guess y^ hoping that compar- ison at y would yield the same qualitative result. Call f(y) and g(y), respectively, the sides of (4.3) with the larger and smaller absolute value at point y , and take that of formulae (4.4a, b) which conforms to (4.7). If eventually, the selected formula diverges, then we should try to improve the initial guess y 1 and repeat the procedure outlined in the previous paragraph. A method of extrapolation usually allows improvement of any trial y n using the previous results for y^ and y^ 2 : from (4.7) we write (4.1l) below: -121- Ay n - f'^gty J] - y (4.11) So, to y Q _ 2 and y^ there correspond respectively the variations Ay n _ 2 and Ay^; the tvo points (y^, Ay^) and (y^Ay^) define a line (y ,Ay ):' w n' ^ n ■r- /X-l \ / t%„ 'n-2 y V Vl n - 2 ^n-2, ^n ' 'nl 5~ " V + (Vl " ^-2 5^ ) C^J Let * - then: v - y n-2^n-l - Vl ^ n -2 „ % y n Ay" T^ (4.13) J n-1 jr n-2 Try substituting y n into (4.1l). Stop when Ay n is small enough. This method, even though more involved, would speed up the convergence, it is more tolerant with respect to initial guesses, and stabilizes the method to the point of usually producing a convergent sequence of numbers y -* v even n ''s in a case for which, if directly applied, equation (4,7) would diverge. If the problem consists in finding time intervals only, and we are not concerned with other characteristics of the trajectory, then the trajectory equations (3-57) in the time domain could be used directly. They can be written as ^n is the ex "trapolated, or expected, value of Ay . -122- Xf . A> T ° f ♦ A p > T ° f + x* (U.lUa) y . A X e V ° f + A ft X R e^ 0f (U.lto) J f a a p P where T = t - t n , and x* is the abcissa of the singular point corresponding to the region; clearly (x Q ,y ) is the initial point. So: x n = Ae^f A.^ + x* (4.15) n+1 a P and, by the use of a method like the one expressed by equation (4.13), xAT , + (x T . - x ,T ) m "-1 n n " 1 n - 1 n (4.l6) n+1 X n " X n-1 Now, T , would be used in (4.15). The numbers x and x are ' n+1 n_i - n respectively the results of (4.15) when fed with T^ and T n . Clearly AT = T - T n (4.17) n n n-1 When AT is small enough, the process is stopped and T corresponding n wx to x could be fed into (4.l4b) to find the corresponding value of y. There remains the problem of how to find a fairly good initial value for the iterative procedure. One good way is to assume that, in a crude approximation, in regions I and III, P moves in a straight line towards the singularity (virtual or real), and that in region II it moves parallel to the asymptote of positive slope. Therefore : In regions I and III, with x* standing for the corresponding singularity: -123- (x * - x f )y y i = " x* - x Q (^.18) In region II, with \ being the positive natural frequency: y i = V x f - V + y (^.19) Both in (^18) and (k.± 9 ), (x ,y Q ) is the initial position, and we wish to find the first approximation y x to the ordinate y f corresponding to the abcissa x . To find a first approximation T Q1 to the time interval for P to go from (x Q ,y ) to (x f ,y f ), consider the first approximation to the trajectory as being a straight line from U Q ,y Q ) to (x^). We know that, whatever the trajectory y(x) may be, T 0f = J Vtt ** (^20) x o Therefore, if y( x ) is a straight line with slope y', going through (x ,y Q ) and (x f ,y 1 ), then P crosses the line x after a time interval: T 01 = ?-^ (^.21) and naturally, y' = -± 2 , k 22) x f - x ^.^; NOTE: In a general form, if a trajectory is a straight line between points fa : T ab = t b -\ ±s -12*4— T 1'jfoh (4.23) ab y* y a and v . = \U± (4.24) J x_ - x b a Whenever x and x are in different regions, we have to proceed by steps. Assume a I to III transition with x Q in region I and x f in region III; y is, of course, given, and y is to be found. Then (with the notation shown in Fig. 22): (i) With (x ,y ) as initial point on the equation, find y . at x =--l , by iteration. J a } a 7 _ (ii) Use equation (3.6la.i) to find y fe , at x^ = - -j . (iii) With (x^y, ) as initial values on the equations, find V , at x = + — J c' c 7 , by iteration. 1 (iv) Use equation (3.6lb.i) to find y d at x d = + y| . (v) With (x,,y,) as initial values, find the point y f> at x = x , by iteration, assuming that P crosses x = x f while the trigger is ON. (vi) Calculate all time intervals by (3-58). Then i 0f Oa be df (vii) Consider the trigger duration T^, and suppose that P crosses x = x after the trigger is turned OFF, either for the first or second time. (viii) With (x,,y,) as initial values of the coordinates, find (x ,y ), by iteration, where, after a time interval e e T = T - (T + T ), trigger turn-off occurs. ■125- -126- (ix) Find y e - y e - y Q . (x) With (x ,y ) as initial values, find y f , at x = x f , by iteration; there may be none, one, or two values, (xi) The time intervals can be calculated by (3-58), and, in this case, where P crosses x = x after trigger t turn-off, T = T + T X 0f W ef Of course, this algorithm, with slight modifications of detail, can be used to find optimum values for T„, instead of having T y as part of the data. This same algorithm can be applied to a III to I transition, after the obvious interchange of reference to regions I and III. See illustrative examples in Chapter 6. k.k Graphical Constructions k.k.l The ^,X) Plane Method Equations (3,15) and (3.U8) define the transformation of variables from (x,y) to ($,x), and can be written equivalently in a single pair of expressions : > <& = . - 1 . [+v(x-x*) - y] p a > (^.25) NOTE: Considering that y* = 0, ' Of course, if P crosses x = x twice after trigger turn-off, we must know up to which passage of P by x = x we wish to calculate T gf . ■127- Here equation (3-59) and (3.5l) are repeated for convenience, with (t - t ) replaced by T: VT $ = $ e a > X = X n e p J (4.26) X a °o X P *o (4.27) We see that it is possible to define a new pair of variables, say $ and %, as follows : $ = X = "\ > (4.28) where » K pV|a cuvu ^ (4.30) Also (4.27) becomes: £w VQ ^avu °vuO X VuO (4.31) And, of course, (4.28) becomes <& Vu VuO v "VU X vuO , (4.32) And finally, from (4.29) JV>u_ _ *JVu_ V ^ W W (4.33) Now , for each value of Vp. that occurs in the problem (this must be known ) : -129- a) Draw, on a linear graph paper, with $ and x axes marked on it, the VfJ. V|_l lines corresponding to the following phase plane lines: (i) The coordinate x axis with a scale on it. (ii) The direction of the vertical lines, b) On a log-log graph paper, mark the coordinated and x axes, and the direction of the trajectory lines, as well as a time scale, and also the scaled curve corresponding to the x axis, c ) Given a trigger amplitude W, draw the corresponding lines x = + - in the - 7 linear graphs where u = 1, and from these graphs, draw the corresponding curves in the log-log graph, by means of a point -by-point transportation. d) The log-log graphs representing the various (ff x" v ) planes provide a means for very fast calculation of times over the trajectory corresponding to the given trigger amplitude. e) The set of log-log papers plus the linear graphs allow a fast calculation of trajectory times for any trigger amplitude (within the bounds of the graph papers, of course). It is clear that this method is advantageous mostly in the case where several calculations must be performed for the same system. ^.2 A Simple Method on the (x.y) Plane This is a less accurate, but faster method, and more versatile in solving problems for several different trigger amplitudes. It will be called "the phase plane method A," It consists in approximating the system phase plane portrait with four straight line segments, under the following assumptions: -130- a) We assume that, in regions I and III, P moves in a straight line towards the corresponding singularity x , V = I, III,' (1 = 0, 1; and that in V(J. region II it moves parallel to and in the same direction of the asymptote nearest to it. b) The discontinuities of y at x = - - and at x = '+. - can be calculated by equations (3.6la.i) and (3.6lb.i) respectively, c) As a very fast method at the cost of obtaining a somewhat poorer accuracy, these discontinuities of y at the boundaries of region II (i.e., at x = + — ) can be completely ignored. d) So, as illustrated in Fig. 23, we take a linear graph paper, mark on it the scaled x and y axes, vertical lines x = + -, and singular points X vQ , and also the direction of the asymptotes related to the saddle point of region II. Then, given a trigger amplitude W, we mark the points X vl and also P . Suppose a I to III transition; then P Q = P:(x IQ , y Q ). (i) Draw the segment of the line PqX-^ contained in region I. a (ii) The intersection of this line with x = - - is y (iii) Find y by (3.6la.i). (iv) From y draw, inside region II, a line segment with slope \__ T (the positive asymptote slope); its pli intersection with x = + — is y . (v) Find y d by (3.6lb.i). (vi) Consider P d = (+ r, y d ); draw the line P^j-^. Let \, be the slope of this line, and call it X line i ' In the descriptions of these graphical methods, references to a line shall, in general, be made using the symbol for its slope. -131- W"< 1 **< -132- (vii) Consider P„:(x ,y ) be the intersection of lines • f .v- f *j f P d X IIIl aM X = X f ' (viii) Calculate Tj f - T Qa + T bc + T df by (U.2l), supposing that P crosses x = x while the trigger is ON. X (ix) Consider the trigger duration T , and suppose that P crosses x = x for the second time' after the trigger is turned OFF. (x) Find W^Oa^ 1 y = y-,e J e d and mark P : (x ,y ), the point immediately before e e e trigger turn-off occurs, (xi) Find y g = y g - y Q , and mark p e -( x e ^ e )^ the P oint immediately after trigger turn-off. (xii) Draw the line P~X~ III( y and mark P oint p f :(x f >y f ) of intersection of the lines P JLjjjq and x = x f (xiii) Find T _ by (U.2l), and then in this case where P crosses x = x after trigger turn-off, T 0f = T W + T ef Of course, after the obvious changes, the same algorithm can be applied to a III to I transition. This method does not apply with acceptable accuracy if P crosses x - x f for the first time after trigger turn-off. -133- 4„^„3 An Approximate Method on the (x,y) Plane This is a slightly more sophisticated method then method A; it will be called "the phase plane method B." We will use the phase plane equation (3.1l) which we repeat here for convenience : , d , - c x b dy Vu Vu Vu a . , ta = ' a v y ^ " if' % - d v + ^ nW (3.11) To plot a I- III transition in the phase plane, we do as follows (i) Find y f and x , for all Vu conditions, (ii) Draw the lines y = \ (x - x*) for x^, * 1TL > x III0 -> and x , and call them, respectively, the (3 __, p m> p iiio> and P III1 lines ° (iii) By (3.11), we find ^ at ? Q :{* 1Q ,y Q ) t and call \ Q this derivative, (iv) Draw a line with slope \ through P , and call this the \ line. (v) Check if the \ line intersects the 3 line inside region I. If this is so, call the intersection P : (x , y ) and call P : (- — a v 7 P Tn and x = - — . II 7 , y ) the intersection of the lines Otherwise, ignore the intersection of lines X n and B__. and call P :(- — II' a v 7 lines p and x = - — . , y ) the intersection of (vi) Find y b (x = - i ) by (3.6la.i) ■13^- (vii) Find y (x = + — ) by the iterative numerical procedure c* 7 using, for example, one of equations (k.k), or, instead, assume the trajectory in region II is a straight line of slope X RTT ° (The choice depends entirely on a compromise between accuracy and com- putation time.) (viii) Find y n (x - + — ) by (3.6ld.i). (ix) By (3.1l), calculate the slope — - at P -,:( + ~ + > y d } (for region III), and call 3 this derivative. (x) Draw a line with slope X through point P, and call it the A., line, d (xi) Find the intersection P : (x , y ) of lines X and III1' The trajectory in region III with a very long trigger is taken as segments P^P^ and P X of lines A,, and 3-r-r-r-i • Call this the line P,X TTT „. d IIIl d III! (xii ) Do as directed in (vi ) to (xiii) of method A, but modify instruction (x ) of that method to: (x) Find P :(x ,y ), the point immediately before trigger turn-off occurs by: y el = y a e W T Oa + V ] < y e2 = y el e VlIl [ V( T Oa +T bc + V r — y e = y el> if y el > y 3 y e - y e2> if y el < y ; -135- Again, after the obvious modifications, this applies to a III to I transition. An illustrative example is presented in Chapter 6. Obs.: Notice that this Method B can be a hybrid numerical and graphical method. Various such combinations can be made and we feel that, some of these combinations may be good compromises between speed and accuracy. 4.5 Approximate Analysis of Waveforms One of the most important aspects of transition waveforms is the time duration of the various phases of a transition as defined in section 4.2. The exact shapes of a particular variable (voltage or current) as a function of time is less important than its general characteristics, such as delay time, rise time, average form in each region, minimum and maximum values, etc. The exact shape is important insofar as it influences the calculations of these characteristics, especially the various time intervals elapsed between the definite changes in character of the curve, generally described by changes in the values of the pair of indices Vu. Furthermore, even if we do have the exact (analytic) solution of (2.103), it will not do us much good. We can solve the problem for the waveforms of all variables based on the solution of (2.103). But then--besides the fact that (2.103), and therefore any solution based on it, is already an approximation to the real problem--the important general characteristics of the waveforms are hidden in a fairly cumbersome analytical formula, which would take a considerable time of tedious labor to plot . Our aims consist mostly in analyzing and evaluating an existing flip- flop or improving its design, selecting a better trigger or loading circuits, -136- determining optimum trigger duration and better waveforms, and better understanding the operation of bistable circuits. For these purposes, an approximate plot of the several variables which could be obtained in a reasonably short time would be far more useful. In this section we will suggest some methods by which such graphs can be obtained. U.5.1 Collector and Base Currents Assume that an approximation to y(x) (phase plane) has been obtained,, consisting of four line segments,, one for each value of Vu (three line seg- ments in the case of optimum trigger duration). This can be obtained either by the second graphical method described in section k.k.2, or, if time durations are extremely important, by calculating time durations with one of the iterative techniques described in section k.3, and then using (4.21 ) and (4.22) to determine the position of line segments which would result in the same trajectory time intervals. a) With (4.21), several points can be marked over this approximate trajectory constituting indeed a (nonlinear) time scale. b) Or else, considering that x(t) has the form x = A + Be (4.3*) A, B and \ can be found for each value of Vu. By a) or b) above, or any equivalent method, plot x(t) and y(t). (i) Collector' Currents If (2.93) is assumed, an approximate graph of the collector current variables w, k - 1, 2, is immediate for they will be constants outside region II -137- (either or 1, whatever the case may he) and will be linear functions of x inside region II, so that, by just assigning new scales, the two curves w (t) can be obtained. More accurate curves can be obtained by using equations (2,56) and (2.57), which will yield results in closer approximation to the real transistor currents, than the model represented by equation (2.IO3). Use of a graph of the tanh x would allow a completely graphical procedure . (ii) Base Currents Here use of equation (2.9k) yields graphs of z , k = 1, 2, with almost equal ease. The first term is directly proportional to the corresponding collector current w , and the second term is proportional to y(t) in every region, since cp'(x) is a constant in every region. In this case, use of (2.58) and (2.59) to improve accuracy would hardly be justified. J+.5-2 Collector Voltages By inspection of Fig„ 2k we get immediately: k = 1, 2 dv k v oi = \ + R ik c ik IF " R ik^k + Sk + V^ y-i>e fr-36) I t k where t' is the nonnormalized time variable. Normalizing as before and setting u i = \ »»▼ v J = I* 2 (4.37) J 2 oj- we get -138- I 'ce i bk Vol it C ± C n * R< rfn FIGURE 24: rrn PASSIVE NETWORK YIELDS THE EQUATION FOR THE OUTPUT VOLTAGE. -139- VVTV 2 P k F^ K + \) - ^ k f# (M8) ok ok ,T E with k = 1, 2} I = 1, 2} $> f k. As before, we can interpret in terms of an equivalent current source of strength s and a parallel conductance G , k k according to (2.79), repeated here for convenience . Assuming a rectangular trigger, and making use of the index |i = 0, 1; •*- **-¥*"* k = 1 ' 2 (4 - 39) k Substituting (4,39) into (4,38), and using y = x , we get: u * = ^ + R iAA + ^ y k " 2 ? k FT < s k + z k > " s P k f# ^°) ok ok E with k = 1, 2; i = 1, 2; i / k. This equation is very general, and allows one to find both collector voltages of a general Eccles -Jordan flipflop (symmetrical or nonsymmetrical) if x 1 and x g are known and also holds for the asymmetrical flipflop, by dropping the indices k and &. For the moment we shall focus our attention on the asymmetric flip- flop and on the symmetric Eccles -Jordan. In the first case (asymmetric flipflop) we get: T R R T u = (1 + R.G )x + ^ y - 2p ^ (s + z) - 2p -*JL (kM) o o E And in the latter case (symmetric flipflop) we get by subtraction, and setting u - «g - ^ (^2) u - (1 + R.G^x + ^ y - 2P J z - 2P J ,W + fe^) (^3) with M 0, if trigger is OFF 1, if trigger is ON These equations immediately suggest the procedure for obtaining the collector voltage variable u(t) from x(t), y(t), z(t) and Wj it is clearly a very easy graph to obtain from the preceding ones, since, in each region, it consists of a constant plus a linear combination of the previous curves, with only the constant and possibly the coefficient of x(t) having different values for the two distinct trigger states. k.6 The Influence of Parameters on Transition Times - -Simplified Equ ations We would like to have some qualitative notion about the effects of the various parameters upon the overall transition time. We are also interested in learning something about the total charge fed into and removed from transistor bases and capacitors, and their relation, if any, with transition times. Besides that, some characteristics of waveforms, such as maximum, minimum and settled levels of collector voltages and peak base currents also interest us. At this point we must stress that we are searching for more qualitative criteria, i.e., first order approximation formulae which could help considerably t Notice that x = x ± - x g , i.e., the order of the indices is reversed in the two definitions. -141- in the evaluation and understanding of flipflops, and not for exact (or good) engineering design formulae . In this respect the character of this section is entirely different from the general character of this dissertation. 4.6.1 The Optimum Flipflop Let us assume (since it is possible in principle) that a flipflop has been constructed such that, if the trigger duration is optimum., i.e., if X e = X IIIO> then y e = °> where x e and y e are the coordinates of P , the position of P immediately after trigger turn off. For this flipflop, we can say that, in a first order approximation, the transition time is given by: y n x Tn - x x ^ y^ X„,., - x. 'IIIl 1 ■II 10 (4.44) ■mi ino Let us assume that x im - x III0 = x^ - * = Ac. Then TR y (Ac) 2 &M) From Table II: Ac = 1 + R5, s 1 / 1 + R G,\ (4.46) ■II IIIl Vl + R G, s 1. ) 2 [(n'¥ + J) 2 - (Htf] (^7) This comes from assuming a straight line approximation to the trajectory, neglecting the active time, and using equations (4.23) and (4.24) in regions I and III. Notice that state symmetry is not required for this approximate equation to be valid. So, -1*4-2- where : n f W + 1 1 + R s G l 1 + R sV l(J + Hty) "|2 (n'W + of - (Ht) 2 (k.kQ) ,) n" = 2 R c b) cp k = [1 + 2B k + p] o d) J = % if it is an asymmetric flipflop Ap = cp - cp ,, if it is a symmetric flipflop ) H = 1, if it is an asymmetric flipflop 2. 3 if it is a symmetric flipflop Assume: a = a xxi = a ^ and m I = m III ~ m ° m_ y = W = - W J a a (^9) where a and m are given in Table I; a) m = 2p T. 10 b) a = T.T 1 O And therefore -143- y = 2P1 w = 2pr_ ^0 T_. W R C. Oi o 1 (^.50) So y Td ♦ r sGi ) Lr I - 1 + R s gJ 2W . (U.51) It is clear that f ~ 1, if (1 - a)R « aR s o so that Hty *» Ho And assumption of state symmetry eliminates biasing from the formula: J = From (4„48) and (4.51) we get: R C. o 1 "TR T(l+R g ) si rR / u o \ 1 + R G n \ _ „. s 1 \ J + Hlr 1 + R G n s 0. 2W W #n/ R / 1 + R GJN _ " __s A s_l \ G + H R V ' 1 + R G n / 2W ■ o \ s oy v4 W + J ) 2 v2 (^52) And,, if t " 1 and J = 0, we get -lM- R C. o 1 TR T(l + R G, ) v s 1 ^s HK R + 2W • o • fou \ hk R + 2W o w k 2W (*.53) where K = 1 1 + R G n s 1 1 + RG, s As a final simplif ication, if G o = G i (equivalently, K = 0) we get: T, R C. s 1 1 T(l + R G.) s 1' bn> 1 - IR \2 o \ 2R ¥, •v s / {h.5k) Let us suppose that we have a fairly large trigger, and that p is i also sufficiently large 1 so that « 1 (h.55) Then taking the first two terms of the series expansion of the argu- ment of the natural logarithm, and. then taking the first term of the series expansion of the logarithm itself, we obtain: f These conditions are not a property of f lipf lops . In fact HR Q /2R W close to 1 is practical, since W « 10" 1 is practical. However, we assume that W has been chosen large to speed up the transition. -145- T~~ - -*-2i- ( ^ 56) s S X Equation (4.56) should be an acceptable first order approximation for the transition time whenever the flipflop and trigger satisfy all of the assumptions leading to it. Observe that there is a hierarchy of equations with more and more restrictive assumptions; all of them assume the optimum flipflop described before. Then: Equation (4.52) is very general; the only assumption is that the flipflop is either symmetric or asymmetric; Equation (4.53) assumes, further, that ty = 1, and J = 0; Equation (4.54) assumes, still further, that G = G , i.e., that K = 0; Equation (4.56), besides the above, assumes (4.55) to be valid. Also remember that all times are normalized with respect to T, so that nonnormalized times would not include T in the denominator. For example, (4.56) would read : H 2 T 2 Mi + r o. )vr l i oi-' Remembering that W = — — , we get : » in 4R C.(l + R Gjr s i v s 1' t T r = ; ; 72 (^-58) Another expression for T TR can be obtained as follows: Consider that the charge variation A}, of C. between the two stable states is given by (see O.69)): *k " ^IIIO - V ' C i = I^ I , (1 + H o ) (1 + W 2 ' 2 s (^.59) Therefore, /»AA 2 (1 + R S G Q ) 2 R C. s 1 (U.60) If G = G = (i.e., the trigger circuits are perfect current sources), and if ty ^ 1, R C. s 1 (4.61) This equation should give us a somewhat crude but satisfactory first approximation to the transition time. We insist that use of (4.59) should always be cautious, since some of the assumptions made in its derivation are somewhat vague, and others, if ligitimate, will seldom be fulfilled. So, (4.59) is usable for estimating results, i.e., as a kind of figure of meritj it is definitely not a design formula. t This approximation is made under our assumptions? ? - 0,. ♦ = 1, G Q == G ± , s< there is a cancelling out in the second term of (h.bO). At first glance it seems strange that the transition time does not depend on T in a first approximation. In other words , it seems strange that T is not a factor of prime importance in the transition time. The following discussion should account for this observation „ First of all, since the beginning, we have completely ignored the active region, and second, we have assumed that y,, the ordinate of P when entering region III, would be such that y =0, Of course, if T has some influence in the active region, it will determine the value of y } in assuming y d to have a convenient value, we have ignored the effects of the transistors, or in a better way, we have assumed that there is a relationship between the transistors and the passive network such that the optimum flipflop assumption is verified. In this sense, T should be related to T ., and therefore 01' (especially if this relation were found to be linear) T . could be replaced by its expression in terms of T „ Then T would be the prime factor in all those equations, and T would not appear at all. We could also have a linear com- bination of both parameters. That we have started using T . was a question of 01 * convenience; the assumptions made establish a certain relationship between T oi and T . We conclude that, after all, T is a very important factor in the transition time. Even more important than the approximation of T mo furnished by these TR formulae in the case of an optimum flipflop, is the following consideration: (i) Even if the flipflop is not optimum, the transition time should not be substantially different from the results obtained by the use of these formulae. They would be, at any rate, a first order approximation to the transition time. -11+8- (ii) They would certainly "be true in a qualitative sense, i.e., as indications of the relative effects of the various parameters, as well as the order of magnitudes and directions of change. k.6.2 The Total Charge Interchanged Between the Transistor Bases These were established in Chapter 2, equations (2.20) as Therefore, the total charge variation is ^B = m(X h Some relations can be established here, such as m h - ~f = — & + K s G 01 But let G = G Q ; then, from (k.6o) *U " ^ ' * • C 1 + R , G n) Also, TR (^•63) 1 (1 + R G n ) (h.6k) T TR " 1^2 I t / R^T (1 + W) i 2I.T J (1 + R s Wi Since the charge that enters one base is equal to the charge that leaves the other, we can talk about the charge transferred between the bases, though this transference is really only a mathematical cancellation, not physical transference. ■149- If G 1 = 0, T ™ =1 HA} _T A 2 B o i N 21 T Li R C. s 1 (^.67) Also, it is clear that, if G = 0, 1 Ol ^1 (4.68) which is very illustrative of the type of condition relating the transistor parameter, the passive network parameters, and the two stable states. Notice that for the symmetric flipflop, q B = q Bl " q B2 q i = q il - q i2 % = **B1 " ^B2 .''. Aq. = Aq - £q l ^ll ^i2 ^ r (^.69) j W - W 1 - W £ but the triggers W^ and W 2 are assumed to occur simultaneously ^.6-3 Collector Voltages --Maximum, Minimum and Settled Values (i) Maximum: v^^ = V^ + R^ + q^) (ii) Minimum: v . . = v + R (l. fl + (l - a)L kmin Ck o v tl/, J & (iii) Settled: v„, w -•■ V„ n + R aL > C1I0 CI o (iv) (v) (vi) 1III0 = V 01 + R o (l - a K V C2I0 = V C2 + V 1 - ^h V C2III0 = V C2 + R o aI E y (^.70) j -150- where: k = 1, 2; I = 1, 2; I t k v = collector voltage of transistor T, when x = x n CkVO K vu V = collector supply voltage of transistor T k In case of the asymmetric flipflop, drop the indices k and k, and v = V . ClvO CI k.d.k Peak Values of Base Current Considering the optimum flipflop, and the approximate model whose equation is (2.103), it is clear that the peak base current would be given by: (Ml) z* = peak 7y d d == y X III1 Ax z* f if the flipflop is asymmetric with z* = i * - z*, if the flipflop is symmetric z 1 2 and the symbol "*" means the component of the current corresponding to base charge variation y The approximate form of ~, from, say, (^.5*0, and the approximate form of x , yield t(1 + R G, ) nW ,. . s 1 nw (1+ 72) y d : R_C, ' ' (1 + \\) s 1 so that Notice that n = pn' . -151- T «*, . „ T Z peak - 2 — 7W ° r Speak = 2 ~ 7 \ ^) 01 01 We stress that this value of z refers to, not only the approximate model of equation (2. 103), but to this model with all the restrictions implicitly imposed in the evaluation of y . . Therefore, such an expression is specially meant to give us an acceptably close idea of the values of the base current in any given case, when just a fast estimate is required „ ^-1 The Problem of Circuit Optimization Whenever one tries to state a problem of optimization, besides a clear statement of what is to be optimized, two basic questions must be answered. First: "Under what criterion?" Second: "What are the constraints?" The amount of material written on these optimization questions is very large . We shall not try to find complete answers here, but rather, to open the discussion by some pertinent approximations. The first question is what characteristics we could wish to optimize: trigger duration and amplitude (if not its waveform!), circuit parameters, or the transistor characteristics. This first question being decided, we could go on to the second question, and try to be specific about stating an optimization criterion , i.e., an interpretation of the word "improvement!" Of a host of possibilities, we can state the following three as examples : -152- a) The time interval between the moment when the trigger is turned ON and the moment the collector voltage is settled (under what criterion to decide this?) is to be minimized. Call this time the "collector voltage switching time., " T . b) The time interval between the moment when the trigger is turned ON and the moment the base (or base-to-base) voltage is settled, that is, the "base voltage switching time/' T , is to be minimized. c) Instead of minimizing switching times, one might wish to have a given delay time and a minimum active time, or a minimum switching time with a given delay. And so on I The above illustrates the point. We have already attempted to approach question number one, in a very tentative way, with respect to the variable x (see 3°5d and e) in defining, for a special purpose, a concept of "optimum trigger duration, " which was related to the minimization of a defined "transition time" T TR , for the variable x. The difficulties were apparent and that discussion stands as a good example of the issues involved. The second question is usually easier to settle, since constraints are naturally stated either as inequalities or as relations between the variables, or some other mathematical statement. To incorporate constraints in an optimization algorithm is still another thing; but it has been done success- fully for several problems, and, once stated, there is no a priori reason to expect the problem to be intractable. The theory presented so far suggests a number of techniques to approach optimization problems, once they are stated in a mathematical form. As a last observation, it is worth reminding ourselves that problems of optimization tend to raise questions of existence of solutions (realizability) -153- and that nothing has been said, for example, about the realizability of our hypothetical "optimum flipflop" so liberally used (as an approximation device) throughout this Chapter 4. k.Q Summary After defining a nomenclature for time intervals over a phase plane trajectory, we have presented some methods for the calculation of points and time intervals for a given trajectory., An iterative numeric procedure allows the exact calculation of y (x ) and t, (x ) if y (x ) and t, (x, ) are known, for any given pair of abcissae x and x., . a b Similarly, a fairly sophisticated graphical construction using two variable transformations, (x,y) -* ($,x) "* ($jX)> was also presented, and shown to yield accurate results, given the limitations of a graphical construction „ A more naive construction on the phase plane was described, which yields somewhat less accurate results,, but is extremely simple to apply. It was also suggested that some hybrid constructions graphical and numeric,, might be ideal for accuracy and practicability of use„ A graphical procedure to obtain fairly good plots of collector and base voltage and current waveforms was described „ Engineering interest in simple-minded formulae which can work as rules of thumb for the rapid evaluation of circuit characteristics has led us to discuss,, by means of an ultra- simplified model, a set of such relationships „ Finally,, the optimization problem was proposed in a first approach discussion,, 5. EXTENSION OF THE THEORY 5.1 Introduction We have, so far, confined ourselves to the asymmetric and the symmetric flipflops subjected to a rectangular trigger, and also we have implicitly assumed that neither C ± , C q or T is zero. In this chapter we shall discuss the problems involved in applying this theory to other situations, and indicate the methods and modifications involved. 5.2 Case When T Is Negligible This is a very unlikely possibility, but it may happen. In case it does, we can take T = as a good approximation. Then, the coefficients of the equilibrium differential equations apparently are meaningless! However, looking back to how these equations were established, we will see that T was used only as a convenient time normalization constant. Of course, if it is too small (or too large, as we shall seel) it ceases to be convenient, and some other time interval T (such as T Q± , for example) could be used as a time normalization constant. In performing this renormalization of time, we replace T with T in equation (2.3*0 and on all related equations from then on. By letting T = in equations (2.29) and (2.30), l Q± and l^ disappear from the expressions for i^ and i B2 . The result of this is that the charge storage in the base along with its related current will be negligible, and only the recombination component of the base current needs to be considered. Then, the equilibrium equations will not contain terms like z Q and § Q . Except for this, the theory is exactly the same, and applies exactly in the same way. -15^- -155- 5.3 Case of Negligible External Capacitances Again we have a possible, although unlikely situation, which becomes important especially because it can be solved in a special way, i.e., not just an extension of the general theory. By making t ±t 1 ±q} t q± and T q all zero in equations (2.87), (2.88) and (2.90), we would get, respectively: For the asymmetric flipflop, from (2.87) y = r 2 j (! - PHanhx - - (1 + R s G (i )x + (l + 2B + p) + 2 -2. s I cosh 2 x s o -^ (5.1) For the symmetric Eccles -Jordan flipflop, we get from (2.88) .2 cosh x of 1 R 1 y = r" {(! " PHanhx - — (1 + R G )x + (^ - B Q ) + ^ sf s J (5.2) For the nonsymmetric Eccles-Jordan flipflop, i.e., the most general case, directly from (2.90): (1 + R sk G ku )x k = P k {C 1 + 2B k + P k ) - C" 1 ^^ " ^k )tanh X (5-3) ( i > k R sk ,2 R sk + (-1; - — y sech x + 2 - — 3. ok ok so that, since x = x - x , we get: -156- P f TOT = (i + r 1 G ) { (1 + 2B i + p i } + (1 ' p i^ tanhx •' -r. ysech x + :! f: bl bl -Trrra{ (1 + aB a + p 2 ) - (1 - s2 2u R s2 2 R s2 p )tanhx + ^ — ysech x + 2 - — s^ o2 o2 (5A) w ith the result that: .2 cosh x y ■*! Si P 2 R s2 "t 1 + R sl G m^oT + (1 + R s2 G 2^ p 1 (l - P x ) p 2 (l - P 2 )" 1 + R _G n + 1 + R G si lu s2 2|-i J tanhx - x Pl (l + 2B X + P x ) p 2 (l + 2B 2 + P 2 ) 1 + R ,G n si lu 1 + R G s2 2u + 2 Pl R sl S l P 2 R s2 S 2 L(l + R sl G^)R 01 ^ 1+R s2S^ R 02- (5»5) So, in every case we have y = f (x),° of course we are assuming that s (t) is a rectangular function. Therefore, we can find x(t), or better t(x), by the formula: t - t, V e5 St, where v (|) = f (!) (5.6) and we mean that, if u changes, at a certain point, we must find its abcissa x , and continue the integration after x with the new function of x. a a It is easy to see from equations (2 087) and (2.88) that these cases are still exactly solvable even if only C Qk = and C ik / 0, in the same way as when C = C = 0. The only difference is that, in (5.1) and (5-2), instead ik ok R P of _2 cosh x as a factor on the right-hand side, we shall have the following R s modifications s -157- For the asymmetric case, from (2.87), replace, R in (5.1), ~ cosh x by L _ ( 5o? ) s sech x i R — « + — s R T o For the symmetric case, from (2.88), replace, R o 2 1 in (5.2), — cosh x by ± — (5.8) s sech x i R s —R + 2T o Note in (2.90) that, if C Qk = 0, even the nonsymmetric case is con- siderably simplified, since it will be reduced to a second order case, i.e., two first order equations. Then, if R = R the system can be exactly solved, S-L Sd * just like the symmetric case. Otherwise it would be approximately solvable, like the nondegenerate symmetric system. 5 ok Nonsymmetric Eccles-Jordan Flipflops The difficulty in the case of the nonsymmetric Eccles-Jordan flipflop is that there is no way (except for some extremely fortunate coincidence) to reduce the two equations (2.7*0 in x ± and x £ into a single equation. The fact is that this circuit has one more degree of freedom and there is no possible reduction to the previous cases. Nevertheless, we can do something about solving the system. Suppose that we carry out an approximation of equations (2.7*4-), taking cp(x) instead of tanh x just as we have done to obtain equation (2.IO3). The result will be the pair of equations expressed by (2.104), whose coefficients are shown in Table 1.2, and which is repeated below for convenience: -158- DO O i OO ., , O , a kv x k + b k x + c k x = a kv x + \v x + c kv x + d kv + f k (x)[6(x +i) - 6(x -i)] + V k + Vk (2.104) with k = 1, 2 These two equations are coupled only in the active region II (here the three regions are still defined in terms of the base-to-base voltage variable x = x - x ). Except for region II, each equation is of the same form as (2.103)1 Thus, we can define another plane where x and x^ are represented independently but on the same horizontal axis. Call it the x^. axis. In this plane, y and y would also be represented independently but on the same vertical axis. Call it the y fc axis. We will still divide this plane into three regions, but the region boundaries will be determined on the (x,y) plane, rather than on the (\?Y^) plane . That is to say: If x is in region I or HI of the (x,y) plane then x is in its region I or 113^ and x^ is in its region I g or Illg of the (* k> y k ) Plane. If x is in region II of the (x,y) plane, then both x ± and x^ will be in their respective regions II and II 2 of the ( x k ^y k ) pl ane ° Therefore, region II, which is nothing but the representation of the active region of both planes, in the case of the (x fe ,y k ) plane, will correspond to two regions, one for x ± and another for x^ These regions are determined by the values of x ± and x^ when \x ± - x g | = - (see Fig. 25 ). -159- t>~ C 3 s St CD Lfc tt < _1 -i6o- In turn, these values depend only on how they start, i.e., the relative values of their respective initial ordinates y and y -, and which one starts first (receives a trigger first). So, the regions for x 1 may not coincide at all with the regions for x , and besides they have a certain con- figuration only for a given transition: i.e., in the ( x k >y k ) plane the region configuration is a function of the system and of the triggers. Another plane is very helpful, and can be used. It is the (x ,x g ) plane, in which the active region is a strip of parallel lines going through the origin, intersecting the coordinate axes at points (+ —, + -) thus bisecting the first and third quadrants. The representative point Q of the system is the point of coordinates (x ,x ), and it is a simple matter to go from the time scaled trajectories of the two points P and P g in the (* k >y k ) plane to the trajectory of Q in the (x^x^) plane. Use of the (x ,x ) plane makes it easier for us to find the points (x la ,y la ) and (x 2 -, y g -) where (x 1& - x^-) = + | (the sign + according to the direction of the transition), i.e., the points where x enters or leaves the active region. Now, inside the active region, equations (2,10*0 form a system of two linear second order differential equations in x (t), k = 1, 2 . We can easily solve this system of equations for x (t ) and x g (t), y^t) and y g (t), and so, x(t) = x - x and y(t) = y^ - y 2 can be found, and from these, the points (x lc ,y lc ), (x 2 ^,y 2 ^) where x comes out of the active region. From then on, the equations (2.10*0 are again independent, and the remaining trajectories y^x ) and y 2 (x g ) can be found. Figure 25 illustrates this discussion. The case where both C ik and C Qk or just C Qk are negligible has already deserved special mention in the previous section, for it is exactly solved by equations (5»l) to (5.8). -l6l- 5.5 Other Types of Trigger 5.5«1 Introduction We have concentrated our efforts on a theory using a rectangular trigger for two main reasons: the wave form can often be approximated t>y a rectangular form, and a rectangular trigger lends itself easily to a phase plane treatment. We feel, however, that some comments are necessary on the most common nonrectangular trigger waveforms, such as those mentioned in 2.k. 5°5°2 Impulse Trigger A trigger can be considered as an impulse if the two approximate conditions hold: (i) W » W . av mm (ii) q^ « Aq i + £q f (5.9) where (i) W = average trigger current variable cLV W min = minimum rectangular trigger amplitude necessary for a transition (iii) q. = W ' T , is the charge transported by the trigger (iv) T, = trigger duration, assumed here to be well defined ( v ) &l ± , &l B , as defined in (4.6,2), are the total variations of charge between the two states, of, respectively, the input capacitors C, and the IK base storages <> -162- (vi) To simplify matters, we shall reason only with the asymmetric or symmetric flipflops in this section. If conditions (5.9) are met, there are two ways to compute transition times (in the case of an impulse trigger the transition waveforms are meaning- less); we will assume that T = T TR . As the crudest possible transition time evaluation, assume a rectangular trigger of amplitude W av and duration T t = T TR . Then 2 w = §PI W (5.10) J a av T . av oi r X IH0 ^ _ a(x IIIQ - x IQ ) TR "./ V~ mW ±n d J av X I0 (5.11) We find T' from its normal form T TR : HT . T . = 2i_ if T = T mp (5-12) TR (1 + RGJ ' t TR s av As a less crude method, assume the transition is complete when the charge fed by the trigger into the input capacitances and the bases is equal to the total charge variation between the stable states: ^ = Aq B + Aq 1 (5.13) We are implicitly assuming that the charge lost through both recom- bination inside the bases and the input resistances during the transition is small compared to the variation of stored charge. The trigger duration is again assumed to be optimum. -163- From (k.6k), Aq B = HTaI E (5.14) ^i = ^V 1 + R s G } (5.15) q t =aI E W av T ; =aI E W av T iR <5.1*) From (5.^3), after denormalizing T into T' , TR TR T m " vT" [T + V 1 + R .°o> ] ' lf T t = t tr ( 5 -"' av where H = 1, 2, is the symmetry factor. Further simplification in (5-12) and (5-17) is possible if G = 0. We get from (5.12) HT . *J R - -^ (5-18) av and from (5.17) T i R " v 5 " [T + T ol ] (5.19) av And we see that assumption of a constant value y of y(x) is equiva- lent to neglecting the transistor's collector time constant T with respect to T , which may be warranted or not. From (5.19), we conclude that TV = f- aL (5.20) mm IE \s * is the minimum transition time that can be obtained from the given transistors and trigger (by making C. = 0). As a result, we can use -164- q B - TaL (5.2D as a figure of merit of a transistor for use in switching circuits . And for the flipflop and trigger we have: IT'. = Hq^ (5-22) t mm B 5.5.3 Exponential or Sinusoidal Triggers Here the trigger waveforms are continuously changing functions of time, and therefore phase plane treatment is not indicated, since the time variable appears explicitly in the differential equation(s) and cannot he eliminated. We have to work either in the time domain or in the frequency domain by means of integral transforms. It is, in general, easy to solve, directly or, for instance, by Laplace transforms, the three region second order linear differential equation under an exponential or sinusoidal forcing function, so that, in any given problem, a numerical solution can always be found for wave- forms, transition times, etc. A theory covering these and other time- varying trigger waveforms, i.e., finding analytical expressions, relationships, approximate formulae and methods for the fast calculation of transition times, waveforms, etc., would be an entirely new proposition altogether, and clearly outside the scope of a phase plane theory of flipflops such as the present work proposes to be. 5 .6 Use of Integral Transformations In any of the three regions, (2.103) is a linear second order dif- ferential equation, and (2.10U) is a system of two linear second order differental equations. -165- Therefore integral transform- -or operational methods --except for other reasons --can be used, with whatever advantage one might have from them. In particular, Laplace transform methods could be used. The main advantage of these transform methods is that they simplify the solution of the differential equation under an arbitrary transformable forcing function, in our case, the arbitrary trigger „ One added advantage of these methods is that they make it easy to solve (2.104) for x, which is the system's state variable. The results are presented below, for completeness : From (2.103) one gets: W t (a) -*[s(t)] with J x(o) «*[x(t)] a = a + j a, is a complex variable X(a) - m y a + n a + d v *(a v a 2 + \a + c ) • W t ( a ) X a + y + — (5=23) where P :(x Q ,y ) is the initial point under each Vu-condition, From (2.104) one gets: with K* (a) =*[s.(t)] X k (a) =^[x k (t)] And also: X = X l " X 2 X '~~~ X 10 " X 20 ; y =: y l0 ' y 20 x = x ± - x 2 Parameters are as given in Table 1.2. -166- (m lv a 2 + n 1 a + d lv )(a 2 g 2 + b 2u a + c^)W tl ( a) - (^a^n^^K^+^a+c^)^ a {(a.a 2 + b. a + c )(a CT 2 +b a + c 2 ) - [ (*^-*^)o 2 + (^ v "^V )a + (c lv- C 2V )]} l l u ™lu u ^lu'^2 u '~2u u '~2u 2 a where P : (x ,y n ) is the initial point under each Vu condition. This makes it obvious why the approximate and graphical methods are important! We can also find X-^a) and X 2 ( a ), hy: K° 2 + Kv a + c kv } ' (a2x(a) ' x o a - y o } + a ' Kv a2 + \ a+ ^v h \i a) X v (o) = 2~ 2 k' a (m kv a + V + d^) (5-25) X kO J + y k0 2 a where P • (x ,y n ) is the initial point under each Vu condition. ™ kO v kO'^kCr Since X(a) would have to be found first, this makes it doubly obvious why the approximate and graphical methods are important. 5.7 Summary In this chapter we have shown how two degenerate cases (t = and C = C =0) relate to the theory presented so far. The case of 1 = was ik ok shown to be essentially included in the theory, since T has been used as a normalization constant for no other reason than that of convenience. The other (, flqp f r = C = 0. or C =0, have been shown to be exactly solvable, the first even for the nonsymmetric flipflop, and the second for at least the -167- asymmetric and symmetric flipflops, and possibly (if R = R ) for the si s2 nonsymmetric case which, at any rate, is reduced to a system of the second order. A phase plane method suitable for the general nonsymmetric Eccles- Jordan flipflop was given, and a discussion showed that there are areas where the nonsymmetric flipflop is equiA^alent to two independent asymmetric flipflops (regions I and III); the trajectories in the active region (region III) must be found by solution of the system in the time domain. Finally, we have discussed other types of trigger waveform. Besides the almost trivial impulse trigger, for which some relationships have been established, the other cases, such as exponential or sinusoid, cannot be treated by a phase plane theory. They can be treated analytically or numerically; however no general results are available. The equations must be solved in each specific case. Approximations (2.103) and (2.104) are also important in allowing a phase plane treatment of the most important cases (second order), besides allowing treatment in the time domain (directly) or in the frequency domain (integral transforms) for any case. Finally, we have briefly discussed the application of Laplace trans- form methods to (2.103 ) and (2.104), and presented special formula (5.23) and entirely general formula (3.2k), thus covering all possibilities. 6. EXPERIMENTAL EXAMPLES 6.1 Introduction The present chapter has a double purpose. We wish to illustrate the application of some of the described procedures, and also to test the accuracy of the theoretical results as compared to experimental fact. No extensive program of experimentation is intended; only a few examples were treated which should suffice to provide seme feeling for the quality of the theory. The experiments we have carried out consist in triggering a flipflop with a rectangular current trigger from what was practically a current source, i.e., the collector of a transistor. The trigger had a reasonably good waveform but we did not attempt to obtain an exceptionally good rectangular shape. As for the flipflops themselves, we took two classes: one was a slowed-down flipflop where relatively large capacitors were paralleled with the T base-to-ground and T collector-to-ground terminals; the other had just parasitic capacitances, which were carefully measured. Only the asymmetric structure was used. In each case transition times and waveforms were measured and recorded for different values of trigger amplitude and various values of R ± and R . o Corresponding calculated values were found and comparisons between theoretical and experimental values are presented in the tables. The transistors used were the same for all flipflops, 2N1309's. 6.2 Measurement of T, C. and C The collector time constant T determines the influence of the base current terms upon the solution of the flipflop equation. Whenever T is negligible compared to the other time constants of the system, it becomes irrelevant and the base current terms may be ignored, as ■168- -169- in Chapter 5. But if T is comparable to the other system time constants, it becomes critical and must be carefully measured. The system used here was as follows: a) The transistor pair whose collector time constants (assumed equal) are to be measured were assembled into a switching amplifier, with no collector lead, and a current of 1 ma fed into the parallel emitters . b) A (periodically repeated) step voltage with amplitude just enough to switch the current from one transistor to another was applied to the base of T , and the base current of T g was recorded and integrated with respect to time (see Fig. 25). c) Since there is only a negligible voltage variation at the base of T (grounded lead) the parasitic capacitances have only a negligible effect on the measurement. Recombination current can also be neglected in compar- ison to the storage current. From Fig. 25 we obtain by integration T = 15.1 nsec (6.1) The parsitic capacitances have to be measured in situ. This can be done by measuring the time constants of voltage curves under applied step currents. So, Figs. 27 and 28 yield C. and C in all regions .' C. is found to vary slightly from one region to another (Figs. 6.5a and 6.5b) but C remains essentially the same in all regions. In calculating C it is necessary to sub- tract the injected current time constant (t = 15 nsec) from the total collector voltage time constant (t = 6l nsec) in order to obtain the true collector circuit time constant (t = R C = k6 nsec). 000 ' C^ is the base-to-ground capacitance of T^; C is the collector-to-ground capacitance of T (see Fig. 3 and also Fig. 5 for comparison). ■170- • — o 2 fi O (0 o o ^ d £ £ ii it •I CO 1- A ' u 1— • cy • « ^ ^ >1. . o CM o o 8i y E 8~ 9 8 o o C\J o o UJ O O C0 HI O < X o p IxJ O z UJ cc o: z> o UJ c/> < CD to CM UJ IT 3 O C o 4. ■171- — m wmmWffi ■im Miami ■rant :llil||llll ■:''":•■■ -^-L-';:^l § o EH M H CO H « CO < H O H -P O CD CO o o OJ co > •H o p CO > « 0) > 5 CO M K o > CO •H CD TJ CO LT\ O • O O 0\ ., d o CD i— -1 II co w o n CO H H O « * t— P H ta C5 • II <; CO EH > •H t^» o p £> CQ > > « w CO •• < CQ £ H EH 8 -172- FIGURE 28: COLLECTOR VOLTAGE RISE UNDER INJECTED CURRENT Curve : v vs . t Scales : Vert . : 90 mv/div Horiz.: 20 nsec/div Total time constant: ^^ = 6l nsec Time constant of the injected current: 1 Tj = 15 nsec Time constant of the collector circuit: r q = t =46 nsec o o o Since R Q = 1 KB, C q = h6 pf .1 Not shovn here. -173- This is not done in the measurement of C, since in normal operation the trigger circuit does contribute to C, whereas it does not contribute to C . o We get the values presented in (6.4). By way of approximation we have used as a value of C. in region II the average of its values in regions I and III, although in reality it is a continuously changing value. 6.3 Equation Parameters We have considered two possibilities: 1. A flipflop loaded with relatively large capacitors 2. A flipflop with only parsitic capacitances. For case 1 we used and had C. =0.02 uf and C =0.01 uf (6.2) R. = 2 kft and R = 1 kft (6.3) 1 o \ -»/ In case 2 there are only parasitic capacitances c ± -< 56.7 in region I 6l.7 in region II and C = 46.0 pf (6.4) 66.7 in region III The two resistors were chosen to be R. = 2 kfi and R = 1 kfi (6.5) 1 o Besides this we had Case 1: I g = 3 ma; 1^ = 1 ma, p = 10 (6.6) Case 2: I = 5 ma; L, = 0.9 ma, p - 10 with the values of p calculated on the assumption that — = h0 v In both cases the value of E was adjusted to make the system state c symmetric, i.e., to have the stable values of the base voltage of T symmetric with respect to ground. Table XI presents these parameter values in a convenient form. The equation coefficients are obtained from Tables I and II, and presented in Table XII. Notice that all three cases are normalized with respect to T. 6.k An Illustrative Example In order to illustrate some of the techniques described in the previous chapters, we shall consider case 2, under a rectangular current trigger of amplitude W = 0.667, i.e., I = 0.6 ma and we will calculate the delay, active, balance and transition times by three different methods, and compare the theoretical results with the experi- mental ones . a) Graphical Method A: see Figs. 32 and 33 b) Approximate Method B: see Fig. 3^ c) Iterative Numeric Method. •175- TABLE XI. PARAMETERS FOR THE TWO EXPERIMENTAL FLIPFLOPS Parameter Case 1 Case 2 C. 1 0.02 uf 61.7 + 5 --1- L+i. pf c o 0.01 (if 46.0 pf R. 1 2 kfi 2 kfi R o 1 kft 1 kft h 3 ma 5 ma h 1 ma 0.9 ma E c adjusted for state symmetry T. 1 40 (isec 123.4 + 10 --1- u+lJ nsec T. io 20 usee 61.7 + 5 -+1- nsec T . 01 20 usee 92.0 nsec T O 10 usee 46.0 nsec T n T T T 15.1 nsec 7 0.7 0.7 P 10 10 1 - a 0,007 t («) 0.979 .0-979 cp (») ■ -176- TABLE XII. PARAMETERS AND CONSTANTS INVOLVED IN THE EQUATIONS REPRESENTING THE TWO EXPERIMENTAL FLIPFLOPS Coefficient a^, vn 'vn a. f(S) \o *, III A *EI \ °al ^1 \ all "bit A "alii Bill Case 1 6 4 1.755 • 10 + 10 o- 1 •0- 4.63 • 10 3 + 86 rO- 1 -0 1 -6 i J [1 L+10-J 26.5 ' 10 3 60 ■9.275 ■ 10 3 15.1 10 3 w 5.26 • 10" 3 10 -1 L+1J 60 • w -10 ■ w -3 -2.40 • 10 -0.236 • 10" 3 -3.611 • 10 +0.913 ' 10" -3 -2. 40 10 -3 -0.236 • 10" 3 Case 2 r-22.9' 29.0 L 26.9- 1-16.6 38.3 L l8.o-J -10 +10J 75.2 81.8 488. 4 J 6 60 -28.65 ■ y 3.28 ■ W O.988 10 1 L+iJ 60 • W -10 • W -0.658 -O.O67 -1.46 +0 . iks. -0.608 -0.0610 •177- a) Upper curve: v. vs. t Lower curve : v vs . t o Scales Vert. : 0.25 v/div Horiz.: 10 p.sec/div b ) v . vs . t v . vs . t i Long trigger Optimum trigger v vs . t o v vs . t o Long trigger Optimum trigger Scales: ( X ert ' : W^/ [ Horiz. : 0.5 msec/div c) v. vs. v. (approximate) J Upper curve: long trigger [Lower curve : optimum trigger Scales: j^ert.: 12-5 (v/msec)/div ' Horiz.: 0.25 v/div FIGURE 29: TRANSITION CURVES FOR CASE 1 Comments : Part a) illustrates the relationship between the transition curves for T base voltage v. and T collector voltage v . Delay, active and balance times are apparent. The time duration of the transmission phases can be measured from it. Part b) illustrates the effects of trigger duration upon the waveforms of both v. and v 1 o. Part c) is an approximate phase plane portrait of the transition, both for the long trigger (upper curve, showing the return of P from X to X , after trigger turn-off) and optimum trigger. Both curves are slightly tilted to the right due to imperfect differentiation of v.. -178- FIGURE 30: TRANSITION CURVE FOR CASE 2, WITH W == 0.^5; v. vs. t Scales Vert. : 0.22? v/div Horiz. : 20 nsec/div -179- FIGURE 31: TRANSITION CURVE FOR CASE 2, WITH W = 0.667; v. vs. t Scales Vert. : Horiz. 0.225 v/div 20 nsec/div o o X I- UJ S -I < o X a. < o I 8 d UJ CO < o CM UJ o -181- < 4 o X Q. < EC CD a UJ u. CO I'- ll 1 CM o R M. 182- S * S tf> N * oj c\i <\i c\i c\i c\i CD a o x H UJ 2 UJ < X o oc 0. QL < I CD b CM UJ < o CD -183- 6.4.1 Graphical Method A We calculate ^ x^, Ax^, Ax^, x^, x^ and y Q by the formulae on Table II: x io = -vf~-v = -10 Ax_ = nW = 40.0 X I1 = X I0 + ^1 = 3 °"° X III0 = +P f^ * +P = +10 Ax = nW = 40.0 X III1 = X III0 + ^III = 50 -° m i y =-W=2.1 9 We also need I and ^ STI ; we find from the note on equations (3.60). m J = -S- = O.988 da and, obviously, - b n + ^11 - Si c n ni)c Vl = 2^ = °' lk2 Then (see Fig. 34) -184- (i) We draw the line P X x , v ' o II (ii) Intersection of P X T , with x = - - is P v ' o II y a 1 X H + 7 .. , no (iii) Using equation (3.6la.i) we find y^ from y g We have: \ - vt* and: I = 0.988 so: y b = 0.92 (iv) Draw, through P , the line of slope kmjj* whose intersection with x = + — is P . Since / c x all " °- lte through Ax = 2.86 causes Ay = 0.1+06 -185 so that y c = i-33 (v) P is found from P by (3„6lb.i) c y d - (i + % c )y c We have y c = 1.33 and: I = O.988 so: y a - 3.oo (vi) (Simplified in our case) we draw the line P n X^^ Tn d III1 and its intersection with x = X TTT „ is P . IIIO e y e = y d • — = 2 o5 2 X III1 " 7 The time intervals over the trajectorycan be calculated by using equation (4„2l), which we repeat here, for convenience, in a slightly different, but equivalent form; b a Ay y v • ■ / cl -186- Applied to region I we get: AX T y a T = — * fa -2 = 1+.31 d y y 1 y c A Vl y d Aacj.^. Therefore^ -iS ^ -S = 3.05 y e ^d T T = T D + T A + T B = 9.91 rp> = t • T = 65.1 nsec D D mt _ m • t = 38. 4 nsec A A T i _ T . 1 - 46,0 nsec B B T» = t t = 1^9.5 nsec Similarly we can find the trajectory for the other cases and fill out Table XIII. 1. Experimental values for the time durations over the trajectories are taken from Figs. 29 and 30, which are reproductions of pictures of oscilloscope images of the actual transitions. Observe on Table XIV. 1 the excellent agreement between experimental and calculated values of total transition times (T T ). Notice also that in case 1, the agreement for the partial time durations (T D , T a , T g ) is also excellent. But in case 2 the agreement for the partial time durations, although -187- PQ co O Eh O w ft o o CO H H X 3 OJ H H H X pq < En H X PQ < EH F- on OJ oo LT\ VO H c— H t- VO o a • o nJ c— o d CM H >i OJ :>> H o VO II ii n II ii II II II VO d vo 1 ^* o aJ ^ o t3 0) o 5 !>> >> >S >5 k 3 >5 OJ M -d- a !>> H « fl OJ co CO O Eh LTN -3 VO On On On no VO o OJ J" ■> 1^ o as o H o i>5 H >•» H a t- LTN > l>> >s >> >> >> OJ CO O >> VO VO d ii IS On H OJ II o r>s OJ t- H II OJ ON ■ o II OO oo H II O VO O oO II OJ LTN OJ II CD o3 a VO d O d t- VO vo' OJ -H- H O LTN J" 3- o ii IS VO H II o OJ ON ON o II VO OJ VO d n OO o H II O >> LTN o OJ II >> UA H II n3 a o t- vo° OJ LTN 1 H CD CO o3 o r-> oo d it 5e OO 1 o oo LTN II >? oo i O H CO OO OJ II ctf >> o3 >> II OO 1 o H On On II o >> II oo I O H CO oo OO II CD a OO d CO H o oo 1 oo 1 O H oo H ON • O o H 1 1 + OO H 1 OO -4- H + + OO -d- H + O H + -p H H H II H < H H CO. ^s oo -d- H i o a •H T) <1) U o p co cu bO Im 03 ■s -p O ■P tj 0) I o cu •rH bO •H H bO a O ■P to CD bO CD CO ct3 rQ CM o CO -p o CD «H =H CD CD EH -188- K O CO > Q K O P 2d H 9 E r T "| >H a cu H Q ft | o 2 a R Q fcj u3 4 S] O 2 W s H p S • X OJ X! CO H s a o pq H < 11 EH CO a i EH N O CO H « o g w 5 p§ Q Eh W S CO E-t < o II bO a •H M C. EH CM 0) en CCJ o vo VO d ii u ft LA CM CM -Hr VO H o VO 00 VO H ,3 VO 0O -5 LA OJ 1) co EH H O H co 6 a! & o LA & OJ -H/ CVI CO -H/ CO OO OV O ^f 1 OJ d Ij 6 o O -H/ -H/ a o o £ i fc- en H H 1 o LA I s - no OJ PO £ ft OJ en & H d s o ^ Eh O > EH O £ Eh co EH co •h cti •H -H •h aj E! 5J +3 bO H bD -P bD H CtJ H •rH 0) O

y =2,066 El => y d = 1.214 => y e = 2,294 TABLE XVIII, COMPARISON OF TIME INTERVALS OBTAINED BY THE ITERATIVE NUMERICAL METHOD WITH EXPERIMENTAL RESULTS Time Interval (in nsec) Region Theoretical Experimental 58,4 52 I 31 = 3 22 II 61,6 64 III 151 = 3 138 Total -196- a) to the piecewise linear approximation use for the most exact nonlinear differential equation . t>) to assumptions of lumped parameters, constant in each region. c) to the measurement of circuit parameters and transistor constants . d) possible departure of the transistor characteristics from the ideal one we have assumed. e) the imperfection of the rectangular trigger used in the experiments. Under the above considerations the error of about ten per cent in total transition times, along with the good agreement in waveform (in the case of method B, for example) is a satisfactory result. 7° CONCLUDING REMARKS 7.1 Summary The purpose of this investigation was to describe in detail the operation of flipflops from a mathematical point of view, and to devise, based on this mathematical description, practical methods of analysis, design and optimization of both flipflop and triggering circuits. The mathematical description has been accomplished with the establish- ment of equations (2 087), (2.88) and (2 .90) in Chapter 2, and with those qualitative aspects of their piecewise linear approximations --equations (2.99), (2.100), (2 „ 102) --which clearly apply to the original system- Methods of analysis and design were devised by means of a detailed study on the phase plane of the piecewise linear equations, taken as approxima- tions to the original nonlinear equations. The singularities of the system, the conditions for their existence and the dependence of their nature upon the system parameters, have been thoroughly described. The phase plane portrait of the system was described with some emphasis on separatrices, trajectories, and the influence of the singularity corresponding to a given region, whether this singularity exists in its proper region or has a virtual image in another region. Based on this study some engineering methods of analysis and design have been described in Chapter 4, and some simplified formulae for the rapid estimate of flipflop behavior have been presented in Chapter 5. The experimental example presented in Chapter 6 illustrates the use of some of these methods, and also, by comparing theoretical with practical results, some feeling is obtained for the adequacy of the various methods and for the type of approximations (piecewise linear at the model level) used in the theory. -197- -198- 7 ,2 Conclusions It is apparent that we have obtained a useful and,, for most practical purposes , adequate theory . We feel, however, that there are some questions to which we do not have even unsatisfactory answers. A first question is: why is it that the test of all methods when applied to the active region yields a path which obviously differs considerably from the true path? Even the crude device of assuming the path, in region II, to be a constant equal to y & would produce a result closer to the true one in that region ! Another question is : why is it that results are worse if the impulses (second derivatives of cp(x)) are considered than the results we get when they are ignored? We feel that the answer lies in a more detailed study of the relation- ship between a nonlinear differential equation (especially of the second order! ) and another equation which formally is a piecewise linear approximation to the nonlinear one„ Specifically, what are the effects of a) the break points (error in derivatives!) b) the error itself c ) the constancy of coefficients on the solution of the approximate equation with respect to the original one? The present investigation gives the impression that this type of approximation should be studied in detail and formalized . 7„3 Further Investigations There are three directions for further investigation: a) The study of approximate solutions to nonlinear differential equations by use of solvable formally approximate equations to the original one, such as -199- piecewise linear equations or other standard types of equations with known solutions o b) The polishing of the present theory by considering other types of approxi- mation, such as,, for example, approximation at the equation level. c) Application of the ideas we have described to more complex situations, for example, (i) considering the nonlinearity of parasitic capacitances, (ii) taking the collector-base junction capacitances into account, (iii) considering inductances in the passive circuit, (iv) considering the distributed nature of some of the parasitic capacitances. Advances in one, some, or all of these directions would certainly improve the present-day techniques of switching circuit design for digital computers. BIBLIOGRAPHY 1. Andronow, A. A. and C. E. Chaikin: Theory of Oscillations . Princeton University Press, 19^9° 2. Beaufoy, R. and To T. Sparkes : "A Study of the Charge Control Parameters of Transistors/' Proc. IRE (October i960) pp. I696-I705 3. Bashkow, T. R. : "Stability Analysis of a Basic Transistor Switching Circuit/' Proc, N.E.C. , 195^ 4. Cunningham, Wo J.: Introduction to Nonlinear Analysis . McGraw Hill Book Company, Inc . , 1958 5o Greiner, R. A.: Semiconductor Devices and Applications . New York: McGraw Hill, 1961 6. Hedvig, Thomas Ivan: "The Determination of Optimum Paths in the Phase Plane for Dynamical Systems Using Calculus of Variations Techniques." Electrical Engineering Doctoral Thesis, University of Illinois, 1961 7. Hughes, William Lewis: Nonlinear Electrical Networks . New York: Ronald Press, Co., i960 8. Ince, E. L.: Ordinary Differential Equations . New York: Dover Publica- tions, 19 9. Kryloff, N. M. and N. Bogoliuboff: Introduction to Nonlinear Mechanics . (Annals of Mathematics Studies, No. IT) Princeton University Press, 19^3 10. Lebow, L., R. H. Baker and R. E. McMahon: "The Transient Response of Transistor Switching Circuits," Tech. Report 27, Mass. Inst, of Technology, 1953 11. Lefschetz, S. : Lectures on Differential Equations . (Annals of Mathematics Studies, No. Ik) Princeton University Press, I9U6 12. Liapounov, A. M. : Probleme General de la Stabilite du Mouvement . (Annals of Mathematics Studies, No. 17) Princeton University Press, 19^7 13. Linvill, J. G. : "Nonsaturating Pulse Circuits Using Two Junction Transistors," Proc. IRE , vol. ^3 (July 1955) p. 826 lU. Liu, Ruey-Wen: "Two Dimensional Autonomous Oscillatory System Solution without Approximations by Analogy to Classical Dynamics," Electrical Engineering Doctoral Thesis, University of Illinois, i960 15. Middlebrook, R. D. : An Introduction to Junction Transistor Theory . New York: Wiley, 196cT 16. MilLman, Jacob and Herbert Taub: Pulse and Digital Circuits . McGraw Hill Book Company, Inc . , 1956 17. Minorsky, N. : Introduction to Nonlinear Mechanics . Ann Arbor: Edward Bros., 19^7 -200- -201- 18. Murphy, George M. : Ordinary Differential Equations and Their Solutions , Princeton (N. J.); Van Nostrand, i960 19. Poppelbaum, W. J. and N. E. Wiseman: "Circuit Design for the New Illinois Computer/' University of Illinois, Digital Computer Laboratory Report No. 90, August 20, 1959 20. Pressman, Abraham I.: Design of Transistorized Circuits for Digital Computers . New York: Rider, i960 21. Shockley, William: Electrons and Holes in Semiconductors , Princeton (N. J.): D. Van Nostrand Company, Inc., 1950 22. Stoker, J. J.: Nonlinear Vibrations in Mechanical and Electrical Systems . New York: Interscience Publishers, Inc., 1950 23. Tillman, J. R. : "Transition of an Eccles-Jordan Circuit," Wireless Eng . (1951) pp. 101-110 24. Valdes, Leopoldo B. : The Physical Theory of Transistors . New York: McGraw Hill, 1961 25. Vallese, L. M. : "On the Synthesis of Nonlinear Systems," Proc. Symp . Nonlinear Circuit Analysis , Polytechnic Inst. Brooklyn (1953) 26. Vallese, L. M. : "A Note on the Analysis of Flip-flops," Symposium on Nonlinear Circuit Analysis , Polytechnic Inst. Brooklyn (April 25-27, 1956) 27. Vallese, L. M. : "Transient Analysis of Second Order Flip-flops," Trans. AIEE, Communications and Electronics (1957) 28. Wilf, Herbert S. : Mathematics for the Physical Sciences . New York: John Wiley, 1962