THE SIMULATION OF GAS TURBINES BY A STATE OF THE ART ANALOG DEVICE Wi 1 1 iam M. Sheperd 7 MAR 1979 0»- SECURITY CLASSIFIC ATION OF THII FACE fWh»n f).r. Enfr»d) REPORT DOCUMENTATION PAGE i report numHA 3. GOVT ACCESSION HO. 4 TITLE f«nd Suftfff/.; THE SIMULATION OF GAS TURBINES BY A STATE OF THE ART ANALOG DEVICE 7 AUTHORf»J SHEPHERD, WILLIAM M. RRAD INSTRUCTIONS BRTCPE COMPLETING FOJ»M >• WCl»ICMT'lCAT»LO&Hui s. type of report ft Fimoo covered THESIS ft. FMFOWINS O»0. HI^OST NUMIf * ft. CONTRACT OH GRANT HbMieKC.j • PERFORMING ORGANIZATION NAME AND ADDRESS MASS. INST. OF TECHNOLOGY 16. PROGRAM ELEMENT. PROJECT TASK AREA ft WORK UNIT NUMBERS ' 1 I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA, 93940 mnE 03] MAY 78 IV NUMBER OF PAGES JIM. TT MONITORING AGENCY NAME ft ADDRESS/// ditlmrwnt Imi Conrrolllng Otllc*) IS. SECURITY CLASS, re/ (M. »«j>ort; UNCLaSS I la. OECL ASSlFl CATION/' DOWN G« AD, nG SChCOULE I*. DISTRIBUTION STATEMENT (ml (hit RftWPTlj APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED 17. DISTRIBUTION STATEMENT (at thm eft* tract M(M*« /n •laek 70, // dlllmrmnt Irmm Kmporl) IS SUPPLEMENTARY NOTES It. KEY WORDS fConffnu* on tmrmtmm tidm II nmcmmmmty and Identity my mloek niatwj GAS TURBINES; ANALOG DEVICE; SIMULATION OF GAS TURBINES 20 ABSTRACT (Cmntlnum on rmvmrmm •/*» H n»c»»»my and Ijmntttf my mlmmk nummt) SEE REVERSE. DD | jam 73 1473 EDITION OP « NOV •• II OBSOLETE (Page 1) S/R 0102-014- 4601 | UNCLASS SECURITY CLASSlFICATlOM O' Twit PA0« r***- Ox<» *•"»» -2- THE SIMULATION OF GAS TURBINES BY A STATE OF THE ART ANALOG DEVICE by William McMichael Shepherd Submitted to the Department of Ocean Engineering on 12 May 1978 in partial fulfillment of the requirements for the Degree of Ocean Engineer and the Degree of Master of Science in Mechanical Engineering. ABSTRACT A simulator was designed to model the operation of a single shaft gas turbine engine. The engine characteristics were represented by a third order non linear mathematical model implemented with analog computation. The use of "state of the art" integrated circuitry allowed for a considerable reduction in the space and power required for this device compared with conventional analog methods. A hard-wired desk-top machine was fabricated to simulate the real time dynamic behavior of a gas turbine prime mover. Thesis Supervisor: Henry M. Paynter Title: Professor of Mechanical Engineering Thesis Reader: A. Douglas Carmichael Title: Professor of Power Engineering T195835 THE SIMULATION OF GAS TURBINES BY A STATE OF THE ART ANALOG DEVICE by WILLIAM MCMICHAEL SHEPHERD 11 B.S.A.E., U.S. Naval Academy (1971) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF OCEAN ENGINEER AND THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1978 (c) William McMichael Shepherd -2- THE SIMULATION OF GAS TURBINES BY A STATE OF THE ART ANALOG DEVICE by William McMichael Shepherd Submitted to the Department of Ocean Engineering on 12 May 1978 in partial fulfillment of the requirements for the Degree of Ocean Engineer and the Degree of Master of Science in Mechanical Engineering. ABSTRACT A simulator was designed to model the operation of a single shaft gas turbine engine. The engine characteristics were represented by a third order non linear mathematical model implemented with analog computation. The use of "state of the art" integrated circuitry allowed for a considerable reduction in the space and power required for this device compared with conventional analog methods. A hard-wired desk-top machine was fabricated to simulate the real time dynamic behavior of a gas turbine prime mover. Thesis Supervisor: Title: Thesis Reader: Title: Henry M. Paynter Professor of Mechanical Engineering A. Douglas Carmichael Professor of Power Engineering -3- ACKNOWLEDGEMENTS The author wishes to acknowledge Professors H. M. Paynter and A. D. Carmichael for their assistance, advice, and guidance in the completion of this paper. Special thanks is also extended to Mr. Ken Busick of the Beede Instrument Company, and Mr. John Carter of the Henschel Corporation for providing much needed hardware . -4- TABLE OF CONTENTS Page Title Page 1 Abstract 2 Acknowledgements 3 Table of Contents 4 I . Introduction 7 A. Background B. Design Concept 9 II. Modeling 11 A. General Approach 11 B. Steady-State and Dimensional Analysis .... H C . Frequency Analysis 16 D. Dynamic Equations 16 1. Assumptions 16 2 . Rotor Inertia 17 3 . Fluid Capacitance 17 E . State Equations 21 F. Engine Data, Assumptions, and Characteristics 2 4 1 . Introduction 2 4 2. Design Point Sepcif ications 25 3. Compressor and Turbine Characteristics 25 -5- Pa9e III. The Operational Amplifier 2S A. Introduction 29 B. Basic Operation 29 C. Inverting OP AMP 30 D. Inverting Summer 31 E . Integrator 32 F. Logarithmic Generator 33 G. Antilog Generator IV. Analog Computation 36 A. Introduction 36 B. The Practical Log Circuit 36 V. Preliminary Design 41 A. Introduction 41 B. Electronics 41 1. Logic Circuit 41 2. DYSYS Simulation and Scaling 42 C. Peripherals 4 5 VI. Implementation and Checkout 47 A. Procedure 47 B. Log Generation 47 C. Exponentiation 47 D. General Circuit Design 48 -6- Page VII. Conclusions and Recommendations 51 A. Feasibility of the Design 51 B. Design Refinements 51 References 53 Appendix I - Machine Characteristics 55 Appendix II - State Equations 62 Appendix III - Test Data 66 Appendix IV - DYSYS Simulation 93 Appendix V - Design Drawings and Sketches 97 Appendix VI - Specifications 1^ -7- I . INTRODUCTION A. BACKGROUND This paper describes the concept and construction of a hard wired analog device, pictured below, which simulates the operation of a gas turbine engine. Recent work on gas turbine simulation has focused largely on digital or hybrid computing. To accommodate the nonlinear elements of a gas turbine simulation in this fashion requires a perturbation or piecewise linear technique. A pure analog approach offers a continuous -8- solution to these nonlinear characteristics, which facili- tates a simulation in real time. At this time, analog computation of complex non- linear systems is not feasible on devices such as the EAI 680 computer currently installed in the M.I.T. Joint Computer Facility. With the miniaturization of both digital and linear functions into the now-common integrated circuit or IC chip, it becomes possible to design extremely compact analog devices. The IC component cost of such a simulation is rapidly approaching the cost of purely passive electronic elements. The modeling approach to the analog simulation, physical modeling, follows the work of Markunas closely. Physical modeling attempts to explain the engine char- acteristics within a framework of physical princples and laws which describe the functions occurring in the real machine. Simplicity is generally a strong point of the model. The model which has been implemented in the analog simulation represents a general machine "invented" by Markunas, who estimated the necessary characteristic relations between the independent variables. Markunas conducted digital and hybrid simulations of his "general" gas turbine, but a full analog implementation was not investigated. -9- This paper is essentially a design manual which traces the development of a hard-wired analog simulator. It is intended to give the reader a brief overview of the analog devices currently available, and their application to a computational circuit. To facilitate future work with the simulator, a detailed user's manual has been written. This document contains explicit details relating to the construction and operation of the simulator. The user's manual, the simulator, and associated hardware are under the supervisor of Professor Henry M. Paynter, Department of Mechanical Engineering. B. DESIGN CONCEPT The following characteristics were desired in the hard-wired simulator: 1. Real time, non- linear third order simulation of the dynamics of a single shaft gas turbine engine, from start-up to overload conditions. 2. Cost to be under $600. 3. Size of the device such as to be readily portable and convenient to set-up and operate. 4. Layout and operation of the simulator to be comparable with typical industrial -10- and military engine control consoles. 5. Accuracy of the steady-state simulation to be considered of secondary importance compared with the representation of large scale transient effects. 6. The ability to investigate various fuel control schemes in a closed loop control, and the ability to operate with different torque-speed load characteristics. 7. Utilization of the general purpose, low cost IC operational amplifier in a hard wired configuration to accomplish all mathematical operations. -11- II. MODELING A. GENERAL APPROACH The gas turbine model selected by Markunas will be briefly explained. The modeling approach is broken down into: steady state and dimensional considerations, frequency analysis, and dynamic equations. Markunas1 estimated performance parameters and the dynamic equations are combined in state-variable format. These equations appear in Appendix II and form the basis for the analog signal flow used in the design. B. STEADY STATE AND DIMENSIONAL ANALYSIS 3 Horlock establishes non-dimensional functions in describing the performance of compressors and turbines. A set of relations are developed to describe the performance of a turbomachine in its simplest terms. In the subsequent development, gas turbine state points are used, numbered in the following convention, with total values indicated by a zero in front of the state point: -12- £>o-v/\%. ,^k = W-£L > In order to present meaningful performance relations, compressor data should be corrected following the above parameters. 4 . Identically for a turbine, it can be shown that the same relations hold and that the parameters of interest in turbine performance are {V\ y s^l . 1 "Z. )&>\&- All inlet conditions assume values at state point 4, and all exit values are at state point 5. Torque across the turbine is i*z . As shown before, The characteristics for a given turbomachine can now be specified as functions of dimensionless or pseudo- dimensionless groups: -15- Compressor: Turbine : Po4 £ (Ik* * \ p«z Mj^-2 / Fc* ~ +( pc* > 'iTVA -16- C. FREQUENCY ANALYSIS Modeling of the processes which take place in a gas turbine can generally be classified as mechanical, fluid, or thermal in nature. To reduce the scope of the problem to a more manageable field, only the most significant energy domains and dynamics were considered. Markunas defines three dynamic regimes for the gas turbine: 1. Low frequency, (f -s^ .1 hz) where rotary inertias and thermal capacitances dominate the dynamics. 2. Medium frequency (f ^ 1 hz) where rotary inertias and fluid capacitances dominate. 3. High frequency (f^t" 100 hz) where the fluid dynamics are dominant. Markunas established that rotor dynamics and fluid capacitances were the primary influences on gas turbine dynamics. Thermal capacitances and fluid inertias were neglected. D. DYNAMIC EQUATIONS 1. Assumptions The gas turbine model is assumed to operate with a perfect gas in one-dimensional flow. Components are treated in lumped parameter fashion. -17- 2. Rotor Inertia For the rotor, a summation of torques give the following: tt T4* where «k S^Cr = angular acceleration, rpm/sec 2 XLc^, = rotary inertia, f t-lbf-sec "\" = load, compressor, or turbine torque, ft-lbf 3. Fluid Capacitance For the one-dimensional lumped parameter fluid, the following symbology is used: £> = fluid density, lbm/ft3 y{ = axial distance along duct, ft S/y = mean axial velocity of fluid, ft/sec *£$ = ratio of specific heats = Cp/Cv Cp = specific heat at constant pressure Cv/ = specific heat at constant volume \**\ = Mach number = V* / ^< Cv = Local sonic velocity = >/ J*- ^ ^ = enthalpy of the fluid in ft-lbf/lbm V^L = lumped fluid volume -18- A general mass conservation equation for the one dimensional fluid stream may be written: Integrating over a constant area duct, with fluid pro- perties constant at a given axial distance or dimension -^ — ■ «E~ Ci by definition. Integrating over the length of the duct ^5r JfMy * ^^Cf^V^cU so ^.^.V^ *. ^/W* ^f A,ds is approximated by an average density, f • Lumped fluid volume (V^L.) since -19- where the subscipts 1 and 2 identify inlet and outlet conditions across the engine, respectively. Writing a similar equation for the conservation of momentum neglecting viscous shear and body forces, -£*+- Integrating over area, ^s Integrating over length of the duct 4 (4-0 - *<*)- <5r VyAv r— -a 4- PA, PN~1 _ Q -=- o I L- *3<- J-z- -20- ^5T x^)=[^Xp^*^^J H-~ Since total pressure . *, p^^^rf)— Assuming V = 1.4, and M £ .5, the momentum equation can be written: 4-X^)~ *£fa.-*^ Similarly for energy conservation with no external work or heat addition, 4- ^L. Integrating over area and length, U =. o -21- -L ( voo 4 3t- W VT\ YAqs where Since K.= + ""2#C- Substituting , "2. ■*- ^Z •=. cvT (j_ * *(V~i) ^ ^ -2 Tc^ 15*. To-s TS'Z. Tc*4 P« Po* Substituting notation for the first and third differential equations, -24- T^ = (f^^^^T^^^V^T^ Introducing the torque equation as the third system equation , ^fr* -it -fCI where ^LUV = fuel lower heating value, Btu/lbm W\p = fuel flow, lbm/sec 'Tt- * ^ = load torque, ft-lbf r^a^ TW^ v--,— ' ) where the thermistor provides temperature compensation. One of the drawbacks to this scheme is that the number of components necessary to generate the log function has gone from 3 to at least 9, with proportional space and cost increases. This configuration was not evaluated. The third and most expensive choice is to utilize a dedicated logarithmic device. The particular unit tested was a Texas Instrument log amplifier. This is a dual log generator on a single DIP package, at a cost of -38- approximately $2.50. With appropriate circuit manipulation, the desired log/anti-log operations can be performed. The TI log amplifier was evaluated in depth. Results are documented in Appendix III. The log amplifier was the foundation of the simulator design. Other dedicated log/ antilog modules exist which offer increased dynamic range, temperature stability, and accuracy, but alternative devices were an order of magnitude more costly than the TI log amplifier. C. EXPONENTIATION - ALL CASES In order to achieve either fixed or variable power exponentiation, the use of logarithms is basic. The signal flow for a fixed power exponent (power function generator or PFG) circuit is diagrammed below: yvww LjCBj (Z^ -WWW - - * i. --A/VWV By a judicious selection of values for \£L and R^ current can be multiplied or divided using an operational amplifier in the ratio: -39- l To be able to exponentiate to a variable power requires a cascaded log/antilog arrangement. The use of a segment of the state equations is made to demonstrate the concept. It is desired to generate the following: where K. represents the scaling of several fixed parameters, and X^^PRX } 8^ and ^ are all problem variables. The circuit selected must process: wc As a result of thermal instabilities, the interior section of the signal flow was replaced with a four-quadrant multiplier. The multiplier essentially compresses the separate functions into a single chip. By circuit manipulation, -40- a multiplier can also divide, square, and take square roots. A range of multiplier devices currently exist, with the price of the least expensive about $3. Test data on the multiplier used in the simulator was collected in Appendix III. To allow as compact a circuit as possible, the multiplier was frequently utilized in its dividing and exponentiating modes. Complete device specifications are included in Appendix VI. -41- V. PRELIMINARY DESIGN A. INTRODUCTION The simulator design proceeded in a parallel fashion. In an iterative process, estimations were generated for the electronic layout, which allowed the cabinet and control panel to be sized and designed. The simulator is configured to resemble a real-world engine control panel. Additional concepts for the per- ipheral design which were not incorporated were: 1. Proportional fuel control schemes with variable gains. 2. Torque-speed loads characteristic of pumps, propellers, generators, and other real world devices. 3. Overspeed alarms and stall audio noise circuits. B. ELECTRONICS 1. Logic Circuit The logic processing of the electronic circuitry was broken down into 3 separate grid structures, which appear as design drawings in Appendix V. Use was made of bread-board panels to avoid the inflexibility of other techniques. Concurrent with testing various mathematical -42- operations, the circuit design was modified to balance the competing parameters of simplicity, size, accuracy, and cost. The appropriate equation variables were normalized about their design point values, and then scaled to provide a circuit signal that was approximately 75% of the maximum rates input for the device. The steady state design point values for all variables were established by a DYSYS program and used for the normalization. Additional notes on the wiring techniques and breadboard layout are included in Appendix VI. 2. DYSYS Simulation and Scaling A DYSYS simulation of the system equations was run as it appears in Appendix IV. Significant comments on the program are : 1. The conditional value of FC was changed to a fixed value of 3.0 to facilitate the circuit logic. 2. Markunas estimated a parabolic "pump-type" load torque, which appears as .0000977 (3) **2. 3. The modeling parameter DT is called TT in the program to avoid confusion with the DYSYS time-step variables, DT. -43- The scaling procedure started with the following listings : a. Constants Tb-z. STi-V (Z b. Machine Characteristics (fe) ^l^ (m-^sj i"*- ) _L.i~sa f^-2 OfeS.U«*4 ^oS^ 'Oe:»iCr«a -44- c. Input Variables at Design Point d. State Variables at Design Point Tc*4 «s lT7'^r 6L The DYSYS simulation was run to verify Markunas's simulations. Use was made of the program to assist in adding voltage and current values to the circuit design, and to evaluate the effect of fixing the discontinuous constant, FC. FC was conditionally assigned the value of 2 or 4 in Markunas work. The steady state effect of assigning FC as 3 was investigated and adopted. The DYSYS simulation was also run off - design to generate a test case for the logic circuit. The simulation was run at approximately 50% fuel flow and steady-state values were generated for all elements of the equations. -45- Some difficulty was encountered in determining the correct value for the lumped parameter fluid volume. The value which appeared to be consistent with Markunas1 data was 144 ft . An approximation based on real world data 3 gave 15 ft as a more likely figure. Since the volume represents a time constant scaling for the fluid capacitance, it has no significance for the DYSYS steady state values. Changes in the volume assumed can be readily investigated in the simulator dynamic response. The scaled voltage values and the system variables they represent are drawn on the 3 circuit plans in Appendix V. C. PERIPHERALS The main elements of the remaining design work included the chasis, power supply and distribution, instrumentation and control hardware, and breadboard architecture. The design drawings and sketches included in Appendix V illustrate the final peripheral design. The photograph below shows the internal layout of the simulator. -46- 4 <£ Appendix VI, specifications, contains additional data on the peripheral equipment, its wiring, and operating parameters. -47- VI. IMPLEMENTATION AND CHECKOUT A. PROCEDURE To establish the characteristics of the various devices and to investigate the methodology of the design, 10 tests were conducted. The significant results are discussed here in the context of the design process. All test data is included in Appendix III. B. LOG GENERATION Tests 1 and 2 determined the voltage - current relations for the transdiode - connected OP amp, using both 2N2222 and 2N2904 transistors. It became apparent that the allowable voltage values of the log circuit were so narrow that only current variations could be used for the signal, over about a 1 1/2 decade range of values. C. POWER EXPONENTIATION Tests 3, 4, 5, and 6 were designed to varify the fixed power exponentiation circuitry. A workable configuration was wired in test 6, using a current - dividing resistive network. Tests 7, 8, 9, and 10 were conducted to validate the variable - power exponentiation procedure. The inability of the transdiode - connected OP amp to provide a stable -48- cascaded log circuit led to the investigation of a 4 quadrant multiplier in Test 8. The Exar 2208 multiplier characteristics were investigated to generate hookup diagrams, trimming values, and allowable voltage values. With Test 9, a simple log/multiplier/simple antilog circuit was wired and observed. Stability was significantly improved, but was still considered below the accuracy level desired. In Test 10, a Texas Instrument 441 log amplifier was tested to determine its log/antilog characteristics. The log- amp was incorporated into the variable expon- entiation circuit and the stability evaluated. The steady state accuracy of the log amplifier in the variable exponentiation circuit was somewhat better than the simple log configuration. Since the dedicated log amplifier offered better compensation characteristics than the simple log configuration, it was substituted at all points in the circuit design. D. GENERAL CIRCUIT DESIGN Concurrent with the test sequence, the overall circuit design was changing as new test data was generated. The more significant problems which caused iteration of the circuit are briefly discussed. One immediate limitation -49- exists using the TI log amplifier module. Looking at "TL 441 Transfer Characteristics" included in Appendix III, the log amplifier only has a useable output range of 0 - 450 millivolts. This can be doubled by wiring an OP amp to the input side of the log amplifier to use the dual input capability of the device. Still, the output accuracy is limited, since various offsets for multipliers and OP amps can easily reach 10 millivolts. As a general rule, it was decided to trin the circuit at a limited number of points, rather than dealing with the offsets required for each component. It was intended in the final circuit design to operate all components at one dual-polarity supply voltage. This choice nominally would be + 8 volts, the maximum rating of the most sensitive device, the TI log amp. Test 10 demonstrated that significant nonlinearities were intro- duced into the multiplier characteristics if the supply voltage were reduced. Three solutions were postulated: 1. Run the multipliers at + 15 volts, and the OP and log amps at + 8 volts. 2 . Compensate the multiplier to have linear characteristics at + 8 volts supply. 3. Replace the multiplier modules with log amplifiers to process the same function. -50- The first choice was selected in an effort to keep the circuit design as simple as possible, in as much as the single - voltage supply was more of a packaging consideration than an electronic one. -51- VII. CONCLUSIONS AND RECOMMENDATIONS A. FEASIBILITY OF THE DESIGN From the test data collected, a real time simulation can be conducted with the logic circuits wired as designed. The techniques developed for the implementation of non- linear differential equations should be applicable to a wide range of system models. Markunas defined his engine in essentially non dimensional or pseudo-dimensionless groups. It was not possible to investigate the ability of the simulator to represent different geometric machines, however, it is felt that future work will verify the utility of the simulator and the nonlinear model. B. DESIGN REFINEMENTS The simulator as designed could be significantly enhanced with more general torque and fuel control schemes. If the simulator could demonstate sufficient generality in modeling the behavior of real world gas turbines, an autonomous "observer" circuit could be configured for a wide range of engines. This observer device, using the logic developed in the simulator, could be an extremely compact unit. It could provide improved control of fuel, speed, temperature, and pressure dynamics. Such a circuit could be incorporated in one -52- or two monolithic LSI chips, to be externally trimmed for a particular engine or operating condition. -53- REFERENCES 1. A. L. Markunas, "Modeling, Simulation, and Control of Gas Turbines", MIT Thesis, M.S., 1972. 2. H. M. Paynter, "First Interim Technical Report - Computer Simulation of the Power Conversion System for Nike-X Power Plants, DA-49-129-ENG-542, Arthur D. Little, Inc. 3. J. H. Horlock, Axial Flow Compressors, Butterworths Scientific Publications, London, 1958. 4. J. H. Horlock, Axial Flow Turbines, Butterworths Scientific Publications, London, 1966. 5. D. G. Shepherd, Principles of Turbomachinery , MacMillan Co., New York, 1956. 6. D. R. Ahlbeck, "Simulating a Jet Gas Turbine with an Analog Computer", Simulation, September 1966. 7. J. F. Sellers and C. J. Daniele, "Dyngen - A Program for Calculating Steady-State and Transient Performance of Turbojet and Turbofan Engines" , NASA TN D-7901, 1975. 8. J. R. Szuch and W. M. Burton, "Real Time Simulation of the TF-30-P-3 Turbofan Engine Using a Hybrid Computer", NASA TM x 3106, 1974. 9. J. R. Szuch and K. Seldner, "Real Time Simulation of F-100-PW 100 Turbofan Engine Using the Hybrid Computer", NASA TM x 3261, August 1975. 10. Nishio and Sugiyama, "Real Time Simulation of Jet Engines with Digital Computers" , ASME Publication 74-GT-19. 11. F. D. Jordan, M. R. Hum, and A. N. Carras, "An Analog Computer Simulation of a Closed Brayton Cycle System", ASME Paper 69-GT-50. -54- 12. B. D. Maclssac and H. I. H. Saravanamuttoo, "An Investigation into the Dynamic Performance of a Variable Pitch Turbofan Using a Hybrid Computer " , ASME Paper 76-GT-31. 13. B. D. Maclssac and H. I. H. Saravanamuttoo, "A Comparison of Analog, Digital, and Hybrid Computing Technique: for Simulation of Gas Turbine Performance", ASME Paper No. 74-GT-127. 14. J. K. Roberge, Operational Amplifiers: Theory and Practice, Wiley and Sons, New York, 1975. 15. W. C. Jung, IC OP-AMP Cookbook, H. W. Sams and Co. , Indianapolis, 1977. 16. J. Carr, OP-AMP Circuit Design and Application, Tab Books , Blue Ridge Summit, PA, 1976. 17. D. H. Sheingold, NonLinear Circuits Handbook, Analog Devices, Norwood, MA, 1974. 18. K. Tracton, Integrated Circuits Guidebook, Tab Books, Blue Ridge Summit, PA, 1975. 19. RCA Transistor Thyristor and Diode Manual, Harrison, NJ, 1969. 20. The Linear Control Circuits Data Book for Engineers, Texas Instruments, Dallas, 1975. -55- APPENDIX I MACHINE CHARACTERISTICS -56- -57- -58- A N 1 o . 00 sQ P 0 -3 o Q <8 -59- »j**.«-^*»— **-»> «M> - ^ J N j K1 V> vtf . rr 00 0 q /O p 8 -60- - A . N . r - v o 8? . h . «? ft. 0 -I -61- N \s -62- APPENDIX II STATE EQUATIONS -63- ? y ^— > c I c 1 -I 8 ? rv1 I I* * i be A v5 t 1-^ J ri 02 ii a? H -64- o j S r \S r fa He M H c a. It 8 5 I' -65- N i it J ii * 1 y Y v5o i -4 n 8r C y i i i ~> * f Vy -66- APPENDIX III Test Data A. TEST 1 AND 2 19 From reference data, the transdiode voltage current characteristics was estimated for various values of saturation current. The log circuits diagrammed subsequently for the 2N2222 and 2N2904 transistors demonstrated low base-emitter saturation currents, thus making the exponential characteristic curve more abrupt. 0^e.€KTT D^cftf^N^ ^~ Bl^i W^M-lAQ^f. -67- Test Data 1 = *. (EQ/26Mv) e EQ(Mv) I (MA) 470 .004 548 .015 581 .045 609 .121 616 .178 Estimated IS (PA) 5.04 x 10 1.05 x 10 8.87 x 10 8.13 x 10 9.14 x 10 -11 -11 -12 -12 -12 Test Circuit -68- TEST 2 CIRCUIT DIAGRAMS AND DATA :»4 ~2F\o4 -4. ^"2 - -*.TS" 0\A^ i_ v^ \ K *s^° .1/i.V -;h.i--> .v- <,-- .*/*?> -res" ,ca"7<1 ■srs^ . Cs^c ~3%4 A-"2^> A4S 4^T 41^ 4ST7 ^"5>? -S"c>1 i i i — N N( y o 7- ^ \ ( I O C \j 1 1 li. 0 d -74- TEST 4 CIRCUIT DIAGRAMS AND DATA >^<1 r* -75- TEST 5 CIRCUIT DIAGRAMS AND DATA .• ^ -76- TEST 6 CIRCUIT DIAGRAMS AND DATA O > r * 1 — V#/V — C4 yi r « -4 o 8 1 ■z u -77- C. TEST 7 It is desired to validate the circuit segment for the operation P£ 3C The operational sequence is as follows: $■-*& '< The circuit described in the test data was set up to evaluate the feasibility of the operation. The desired operating characteristic of the cascaded logger - antilogger is picutred as follows: rAk -1 M- ii. Methodology The functional method used to exponentiate to a variable power was to regulate the multiplier output current in the ratio called for by the exponentiation. For example : IF CC i_ ^ ex . oih >- X .Ln\k /v/; «S n. -81- For multiplier scaling the integral buffer OP amp in the 2208 was used for the current division with a resistive bridge as follows, Muir>£Ue<2 i- T^ov^-fciZ o^ r-A/WW-i such that i-, = ±2 and the output voltage of the multiplier is thus controlling the current input to the antilog amplifier transdiode. The multiplier output is nominally: i-O V- 4l ^.^^ v~yW &j£S6»; (s«^c It was determined that a resistive bridge must be used for the multiplier input from logging OA, sketched -82- as follows: ^vwvL - |MOL.v»fLvt"3>o . fc>*3^ ■ <5^>"S .^4^ . fo"2ST . <^rr 1 - fc€T7 . ^0 , &S \ .•€^4. .G**\l .<^L . — X • VD ^« t- • ■*-^ C; r- • E IT. r- * T— * 1 CJ ^. * * U 1 ^s •i *—■■» + 0^ c • + u u vt. *^v • E *»* (\ o cj r- • a *~\ w tr CN • E' r- O <.' V£ • rz * 1 * CJ V. v • ^J- C »— o rsi »— • CO X in CJ * Ct' <— ^v * * •^ * %w f\ ON * + * a CJ v> Cj • + * * E- * U. c C r- * T C_' > »■ X • ct «- D 1 fr E-- H c E- f % E-* * s- u l ♦ T- 1 CJ St-' z iT fl C. P. c t c. 3 + + K • • 1 CJ Ui 03 C 1 CN • * * 1 • 1 • > c\ es • Eh r- u 1 v^ rn PL. x^ * "V • »— b T— i— H v-^ ^v o • • > s^- C >H 1" r~\ CN c- Ci • • E • r (N en + ^-^ 1 CJ VC \** + fi. 1 * r 1 o m u o T— o M. + * ZS U c o u + * to BR • r- p. T— >— £n >' Cj- C3 ST. m rr u: V "n. N fN P- r* w >^ Sr O vC S-- * 4 c + *^ + S./ + :«.: + 3E % • 90 if • <* .^ -^ no rr c< V *N, V v. * ^ oc o ■f « u CJ + — ex, r- SE a?- r- • O a^ » • *tr" »— • • r- a 1 *_' '-x t- c p J s- =* r* ra >' >• >< n II II • c • • pn T— II T— f'i • • • |i ^ r- e C P' II II CT If II II II n II T II ^-N II ►X * d s- P" II II n ^"N .--N /~*v ii II II II 11 Ii CJ II II ii 1! f- ^^ Ii tf SF « >— U E < ^^ LP vc r* cj Eh CJ E-< CJ Cm CJ II E-- Cj E- E n C W u u". u c<< r»" a cr ^-* >w- ^^ rr Ct a? V ( 1 u o o u u n;< U, D E E- E- Eh f ' n 0. p Cr. C 1 r>: CJ uT n U) u i-i P c_ -'. >H >• >■ fi. c, sv >■ ■< 0Q U Q tx, UJ X ^3 Eh rC a CJ Eh u >- u a, E-» rr -95- EJ fc < E-- E- * E- U « E cc fc Eh ■X. * U 0 E- k (J t*- t«- U; *— V ^ ^v *-» u CM • » w cn X >h * % ♦ * o E- ^ u* 2t on v Eh 1 v- u O ^ >h o Eh 3- * •. V *~ r u o >•> &\ O u o * c 5? V E ( • o v. (jj CV c k ID O *^~ SE CL t* CN O • V C fc Eh CM • *~ a. *— » £-- l; «. C; 1 .-J ro [x «r £-■ P- U E- ip E- c K 2* + o w I l U E- V <_> E- E- C — ft- CJ *-- E- 1 U t- £• *-* >» E- E< + t- c x^ >-.- • V ^ » u c o Vt * * r- U C 1 ."« E ' 0 V CJ f i Br '.* U "—^ «w f rr 7< T' u X * r*- * + ♦ f- * * ? * F- *— h; r- o a % >H (V) II • o y O r. V r* lT> O ii II m Eh •« O u; m • o *- II (N II <— ■ r- f-. r' to u »- •. K- *" >< • i-^ M rt Ul E- " II v:- Ui • CJ #-'' vr E- 5-'- * O a • tN fi- * o CJ Cu w «— tH II CO II » >-« Ul 00 to sr: *~* * H «- rs tO Eh ^^ * >H CO >H »- to II >H Eh Ci Br a; u. -96- a CX vf > o P" O r- o o »— o O o o • o ■=T o o O • OC. • • c. ^- • o «— r- t- • in II II II II II o CN Li H V C; E-' 3f E X CN m. X r< M Cr O O • «*; X T~ • • ^~ o CN 1 c. M •a CN T~ IP a UP vr r*: ct O, sc CN CJ O t- Csi c C\ cr. c h- cr o a. cr vr on r- *3 ! C r- CN • ID cc T- T" C or • r o Ch CN r- T- > • r- r- r- • • (N C C cr CN c m a a • T— c> c o o o o t— • *— r~ cr 1— «~ • • c o o • • T— T— • a cr o i—' II II It II 1! II II II II n w CN CN M X to y 3 u E-n U c CJ O r~ C> C 2*- E p. o p u (X E- L: • o >": >*■ > U Eh E P c H W i~" r™ !*■ - M Li «a. H V." • • r" o LP ■3 cn C ^ ■ cn a o ^ — CN c l Eh • r* rs • • c • w E- T— O o r c C O ro vr cr m o c • o t/J O f s IT O • r a CN o r~ o o c- «- CC C m o r- en o • -3" o Ck, O vo fr, oo CN o Ci P~, en O ro o O • • • a cn • • cr oo o =r • *~ lT O tT' «■' • • t— T— IT) £3 II II li II Eh II II II II II CJ II II •a: > - r-g m a in vT. E- U4 a Uj E- O 10 s c fc u. u a: L> uH M C ) C • O » >- r>- >-« X >H >i X u e t» E~ C- r% W Eh :a Eh • o * X -97- APPENDIX V DESIGN DRAWINGS AND SKETCHES &B'P -L H(— I ®VWty- 6««i ®Wl/»V ij ®Vv £> ^Vvw CI ] i- ' 8WlW»- -WWwJjv> -AWW-i .1-(«^-."0 v*»*o CtRIP Z e^ v 1/WVl' 0 — WW^ L r W. * u. Til C«H-,> ©ww^ ^Ffc Ep> £^T^ OV5) «^»-tW) u aafcu i a. -101- h3 y a 3 Vl u 0 9 IV > c V 7 2 o u -102- APPENDIX VI SPECIFICATIONS A. ELECTRONIC COMPONENTS 1. Operational Amplifiers a. Fairchild a\k 741 Maximum ratings supply voltage + 18v Power dissipation 500 MW Supply current 2 . 8 MA Typical input offset voltage 1 mv b. Fairchild A