+ Lý thuyết cơ bản về OPAMP và ứng dụng+ Đặc tính và sơ đồ hệ thống hồi tiếp+ Đáp ứng của hệ tuyến tính+ Hệ phi tuyến+Dao động cầu wein+ Mạch điều tần
OPERATIONAL AMPLIFIERS: Theory and Practice OPERATIONAL AMPLIFIERS Theory and Practice JAMES K ROBERGE Massachusetts Institute of Technology JOHN WILEY & SONS, Inc New York - Chichester - Brisbane - Toronto - Singapore To Nancy PREFACE The operational amplifier is responsible for a dramatic and continuing revolution in our approach to analog system design The availability of high performance, inexpensive devices influences the entire spectrum of circuits and systems, ranging from simple, mass-produced circuits to highly sophisticated equipment designed for complex data collection or processing operations At one end of this spectrum, modern operational amplifiers have lowered cost and improved performance; at the other end, they allow us to design and implement systems that were previously too complex for consideration An appreciation of the importance of this component, gained primarily through research rather than academic experience, prompted me in 1969 to start a course at M.I.T focusing on the operational amplifier Initially the course, structured as part of an elective sequence in active devices, concentrated on the circuit techniques needed to realize operational amplifiers and on the application of these versatile elements As the course evolved, it became apparent that the operational amplifier had a value beyond that of a circuit component; it was also an excellent instructional vehicle This device supplied a reason for studying a collection of analytic and design techniques that were necessary for a thorough understanding of operational amplifiers and were also important to the general area of active-circuit design For example, if we study direct-coupled amplifiers in detail, with proper attention given to transistor-parameter variation with temperature, to loading, and to passive-component peculiarities, we can improve our approach to the design of a large class of circuits dependent on these concepts and also better appreciate operational amplifiers Similarly, the use of an active load to increase dramatically the voltage gain of a stage is a design technique that has widespread applicability The vii viii Preface integrated-circuit fabrication and design methods responsible for the economical realization of modern operational amplifiers are the same as those used for other linear integrated circuits and also influence the design of many modern discrete-component circuits Chapters to 10 reflect the dual role of the operational-amplifier circuit The presentation is in greater detail than necessary if our only objective is to understand how an operational amplifier functions However, the depth of the presentation encourages the transfer of this information to other circuit-design problems A course based on circuit-design techniques and some applications material was taught for two years During this period, it became clear that in order to provide the background necessary for the optimum use of operational amplifiers in challenging applications, it was necessary to teach material on classical feedback concepts These concepts explain the evolution of the topology used for modern amplifiers, suggest configurations that should be used to obtain specific closed-loop transfer functions, and indicate the way to improve the dynamics of operational-amplifier connections The linear-system theory course that has become an important part of most engineering educational programs, while providing valuable background, usually does not develop the necessary facility with techniques for the analysis and synthesis of feedback systems When courses are offered in feedback, they normally use servomechanisms for their examples Although this material can be transferred to a circuits context, the initial assimilation of these ideas is simplified when instruction is specifically tailored to the intended field of application Chapters to and Chapter 13 present the techniques necessary to model, analyze, and design electronic feedback systems As with the circuitrelated material, the detail is greater than the minimum necessary for a background in the design of connections that use operational amplifiers This detail is justifiable because I use the operational amplifier as a vehicle for presenting concepts valuable for the general area of electronic circuit and system design The material included here has been used as the basis for two rather different versions of the M.I.T course mentioned earlier One of these concentrates on circuits and applications, using material from Chapters to 10 Some application material is included in the examples in these chapters, and further applications from Chapters 11 and 12 are included as time permits Some of the elementary feedback concepts necessary to appreciate modern operational-amplifier topologies are also discussed in this version The second variation uses the feedback material in Chapters to and Chapter 13 as its central theme A brief discussion of the topology used Preface ix for modern operational amplifiers, such as that presented in portions of Chapters and 10, is included in this option The applications introduced as examples of feedback connections are augmented with topics selected from Chapters 11 and 12 A laboratory has been included as an integral part of both options In the circuits variation, students investigate specific circuits such as directcoupled amplifiers and high-gain stages, and conclude their laboratory experience by designing, building, and testing a simple operational amplifier In the feedback version, connections of operational amplifiers are used to verify the behavior of linear and nonlinear feedback systems, to compare time-domain and frequency-domain performance indices, and to investigate stability We have found it helpful to have ready access to some kind of computational facilities, particularly when teaching the feedback material The programs made available to the students reduce the manual effort required to draw the various plots and to factor polynomials when exact singularity locations are important Both versions of the course have been taught at least twice from notes essentially identical to the book The student population consisted primarily of juniors and seniors, with occasional graduate students The necessary background includes an appreciation of active-circuit concepts such as that provided in Electronic Principles by P E Gray and C L Searle (Wiley, New York, 1969), Chapters to 14 An abbreviated circuits preparation is acceptable for the feedback version of the course Although a detailed linear-systems background stressing formal operational calculus and related topics is not essential, familiarity with concepts such as polezero diagrams and elementary relationships between the time and the frequency domain is necessary Some of the more advanced applications in Chapters 11 and 12 have been included in a graduate course in analog and analog/digital instrumentation The success with this material suggests a third possible variation of the course that stresses applications, with feedback and circuit concepts added as necessary to clarify the applications I have not yet had the opportunity to structure an entire course in this way It is a pleasure to acknowledge several of the many individuals who contributed directly or indirectly to this book High on the list are three teachers and colleagues, Dr F Williams Sarles, Jr., Professor Campbell L Searle, and Professor Leonard A Gould, who are largely responsible for my own understanding and appreciation of the presented material Two students, Jeffrey T Millman and Samuel H Maslak, devoted substantial effort to reviewing and improving the book x Preface Most of the original manuscript and its many revisions were typed and illustrated by Mrs Janet Lague and Mrs Rosalind Wood Miss Susan Garland carefully proofread the final copy James K Roberge Cambridge, Massachusetts February, 1975 646 Compensation Revisited applications Quantities such as the upper limit to crossover frequency for reliably stable operation and the uncompensated open-loop transfer func tion are best determined experimentally Furthermore, many amplifiers have peculiarities that, once understood, can be exploited to enhance per formance The feedforward connection used with the LM101A is an ex ample Another example is that the performance of certain amplifiers is enhanced when the compensating network (or some portion of the com pensation) is connected to the output of the complete amplifier rather than to the output of the high-gain stage because effects of loading by the net work are reduced and because more of the amplifier is included inside the minor loop The time a system designer spends understanding the subtleties of a particular amplifier is well rewarded in terms of the performance that he can obtain from the device Important features of the various types of compensation discussed in this section are summarized in Table 13.1 This table indicates the open-loop transfer functions obtained with the different compensations The solid lines represent regions where the transfer function is controlled by the com pensating network, while dotted lines are used when uncompensated ampli fier characteristics dominate The minor-loop feedback networks used to obtain the various transfer functions from two-stage amplifiers are also shown Comments indicating relative advantages and disadvantages are included Table 13.1 Implementation and Effects of Various Types of Compensation One Pole Transfer Function Network Conservative, general-purpose compensation for systems with frequency-independent feedback and loading Changing capacitor value optimizes bandwidth as a function of attenuation provided by feedback network Slew rate inversely proportional to capacitor size Table 13.1-(Continued) Two Pole Transfer Function Network Improved desensitivity and lower error coefficients compared with one-pole systems with identical crossover frequencies Loop parameters must be selected to insure that crossover occurs in the 1/s region of the characteristics for adequate stability Instability generally results with capacitive loading or low-pass major-loop feedback networks Poor recovery from overload With Zero Transfer Function Network Zero is used to offset effects of pole associated with load or feedback network, and must be located as a function of this pole Major loop becomes unstable if pole is eliminated A small-value capacitor, indicated with dotted lines, improves minor-loop stability 647 648 Compensation Revisited Table 13.1-(Continued) Slow Rolloff Network Transfer Function Useful for systems with an additional loop-transmission pole at an uncertain location Adding more rungs to ladder network increases the range of frequencies over which the additional pole can be located and results in greater uniformity of phase-shift character istics Prolonged settling time compared to one-pole compensation when additional looptransmission pole not present Feedforward Typology Transfer Function input stages (noninverting) Output stage (inverting) Feedforward Capacitor Highest bandwidth Most useful when bandwidth of first stage or stages is less than that of rest of amplifier Can result in substantial slew-rate improvement Limits amplifier to use in inverting connections only Values and results critically dependent on specific details of amplifier performance Sensitive to capacitive loading or other sources of nega tive loop-transmission phase shift PROBLEMS P13.1 An operational amplifier is available with a fixed, unloaded open-loop transfer function a(s) 10 ~ 4s + Problems 649 This amplifier is to be used as a unity-gain inverter A load capacitor adds a pole at s = -106 sec-' to the unloaded open-loop transfer function Compensate this configuration with an input lead network so that its looptransmission magnitude is inversely proportional to frequency from low frequencies to a factor of five beyond the crossover frequency Choose element values to maximize crossover frequency subject to this constraint You may assume high input impedance for the amplifier P13.2 Design an input lag network and an input lead-lag network to compen sate the capacitively loaded inverter described in Problem P13.1 Maximize crossover frequency for your designs subject to the constraint that the loop transmission is inversely proportional to frequency over a frequency range that extends from a factor of five below to a factor of five above the cross over frequency P13.3 An operational amplifier is connected as shown in Fig 13.50 in an at tempt to obtain a closed-loop transfer function V0(s)_ -s(0.1s + 1) V =(s) Vi(s) Determine element values that yield an ideal closed-loop gain given by this expression Measurements indicate that the open-loop transfer function of the ampli fier is approximately single pole and that the transfer-function magnitude is 104 at w = 103 radians per second Needless to say, the configuration shown in Fig 13.50 is hopelessly unstable with this amplifier R R C2 vi Figure 13.50 r Double differentiator _- -0 V 650 Compensation Revisited Find appropriate topological modifications that will stabilize the system (without changing the amplifier) and will result in a closed-loop transfer function that approximates the desired one at frequencies below 100 radians per second P13.4 A sample-and-hold circuit is constructed as shown in Fig 13.51 The unloaded open-loop transfer function of the amplifier is a(s) -10 (0.ls + 1)(10 s + 1) The sum of the open-loop output resistance of the amplifier and the on resistance of the switch is 100 Q (a) With R = 0, is this circuit stable in the sample mode (switch closed)? (b) Determine a value for R that results in approximately 450 of phase margin in the sample mode (c) Estimate the time required for vo(t) to reach % of final value following initiation of sampling when the value of R determined in part b is used You may assume that the capacitor is initially discharged, that vr is time invariant, and that the circuit remains linear during the transient P13.5 An externally compensated operational amplifier that uses minor-loop feedback to generate an approximate open-loop transfer function a(s) X 10-4 Ye(s) is available The amplifier is connected as a unity-gain voltage follower The spectral content of anticipated input signals is such that a closed-loop V, R pP 1_ Figure 13.51 Sample-and-hold circuit vF o- Problems 651 bandwidth in excess of 106 radians per second degrades the noise perform ance of the connection Determine a compensating element that will result in a closed-loop transfer function A(s) 10- 6s + for the voltage-follower connection P13.6 An operational amplifier of the type described in Problem P13.5 is con nected in the log circuit shown in Fig 13.9a Experimental evaluation shows that this connection will be acceptably stable if the loop crossover frequency is limited to MHz Determine a compensating element that insures sta bility for any input-signal level between and +10 volts Estimate the time required for the incremental output signal of the circuit to settle to 1% of final value when a small step change in input voltage is applied at an operating point Vr = 0.1 volt P13.7 A two-stage operational amplifier has a d-c open-loop gain of 106 and is acceptably stable in connections involving frequency-independent feed back provided that compensation is selected which limits the crossover frequency to MHz This amplifier is used as a unity-gain inverter to amplify 10-kHz sinusoids, and a major design objective is to have the input and the output signals of the inverter exactly 180* out of phase Discuss the relative merits of one- and two-pole compensation in this application Also indicate the effect that the two types of compensation have on the magnitude of the closed-loop transfer function at 10 kHz P13.8 The uncompensated, open-loop transfer function of a two-stage amplifier is a(s) 10, (10- s + 1)(10-5s + 1)(5 X 10- 8s + 1)2 The two lowest-frequency poles result from dynamics that can be modified by compensation, while the location of the higher-frequency pole pair is independent of the compensation that is used The amplifier is compensated and connected for a noninverting gain of 10 You may assume that the compensation used does not cause significant loading of the minor loop This closed-loop connection is excited with an input ramp having a slope of 104 volts per second The differential input signal applied to the amplifier is observed, and it is found that after a starting transient, the steady-state value of the signal is 10 mV 652 Compensation Revisited (a) Determine a single-pole approximation to the amplifier open-loop transfer function (b) Refine your estimate of part a, taking advantage of all the information you have available about the amplifier (c) Assuming that this amplifier described is an LM301A, what compen sating element is used? (d) Suggest alternate compensation that results in the same crossover fre quency as obtained with the compensation described, a phase margin in excess of 60*, and essentially zero steady-state ramp error Deter mine element values that implement the required compensation for an LM301A P13.9 The material discussed in connection with Fig 13.24 indicated that the steady-state error of a closed-loop operational-amplifier connection in re sponse to a ramp can be reduced to insignificant levels by using two-pole compensation An extension of this line of reasoning implies that if threepole compensation is used, the steady-state error will be nearly zero for parabolic excitation Linear-system considerations show that stability is possible if two zeros are combined with a three-pole rolloff For example, a loop transmission L(s) = 10"(10- s + 1)2 S has approximately 80* of phase margin at its crossover frequency Find a compensating-network topology that can be used in conjunction with minor-loop compensated amplifiers to provide this general type of open-loop transfer function Discuss practical difficulties you anticipate with this form of transfer function P13.10 A two-stage operational amplifier that uses minor-loop compensation is loaded with a capacitor that adds a pole at s = - 106 sec- to the unloaded open-loop transfer function of the amplifier The desired open-loop trans fer function including loading effects is a(s) ~ X 1011(5 X 10- 6s + 1) Find a compensating-network topology that can be used to effect this form of compensation Determine appropriate element values assuming that the effective input-stage transconductance of the operational amplifier used is X 10-4 mho Problems 653 P13.11 A two-stage operational amplifier is connected as an inverting differen tiator with a feedback resistor of 100 ko and an input capacitor of AF What type of minor-loop compensating network should be used to stabilize this configuration? Determine element values that result in a predicted crossover frequency of 104 radians per second with a value of X 10-4 mho for input-stage transconductance When this type of compensation is tried using an LM301A operational amplifier, minor loop stability is unacceptable, and it is necessary to shunt the compensation terminals with a 3-pF capacitor in addition to the network developed above for satisfactory performance Describe the effect of this modification on closed-loop pe-formance P13.12 A certain application necessitates an operational amplifier with an ap proximate open-loop transfer function a(s) 104 2/3 Find a compensating network that can be used in conjunction with an LM301A to approximate this transfer function The phase shift of the approximating transfer function should be - 60* i 50 over a frequency range from radian per second to 106 radians per second INDEX Absolute-value circuit, 458 Active filters, 525 All-pass transfer function, 530, 536 Amplitude scaling, 513 Amplitude stabilization, 487 Analog computation, 502 Analog multiplier, 501 Analog-signal switching, 471 Antilog circuit, 462 Automatic gain control, 488 Auxiliary equation, 115 Band-gap voltage, 252 Band-gap voltage reference, 288 Bandpass amplifier, 135 Bandwidth, 94 Base-to-emitter voltage temperature coefficient, 252 Base-width modulation, 310 Base-width modulation factor, 313 Block diagram, 22 Bode plot, 85 Bose oscillator, 502 Butterworth filter, 508 Butterworth transfer function, 80 frequency response of, 89 step response of, 83 Bypass capacitors, 343 CA3039 integrated circuit, 544 Capacitive load, 169, 564, 597 Capacitor selection, 448 Cascode amplifier, current-source loaded, 320 Characteristic equation, 44, 112 Chopper stabilization, 522 Closed-loop gain, Closed-loop zeros, 133 Cofactor, 44 Collector FET, 390 Common-emitter amplifier, 33 current-cource loaded, 315 Common-mode input, 256 Common-mode rejection ratio, 259, 434 Compensating capacitor, 199 Compensation, 165, 557 nonlinear, 240 series, 165 Compensation that includes zero, 597 Complementary Darlington con nection, 292, 397 Complementary emitter follower, 328 Conditional stability, 234 655 656 Index Convolution, 68 Crossover distortion, 329 Crossover frequency, 148 Current repeater, 392 Current source, 322, 452 Darlington connection, 275 Deadzone, 26 Decibel, 84 Decoupling capacitor, 171 Demodulator, 474 Describing function, 217 table of, 223 Desensitivity, 25 Determinant, 44, 46 Differential amplifier, 254, 397, 449 Differential input, 256 Differential output, 255 Differentiator, 561 Diode-connected transistor, 390 Direct-coupled amplifiers, 249 Divider, 211, 244 Dominant pole, 78, 168 Double integrator, 452 Drift referred to the input, 250 from resistor mismatches, 266 from transistor mismatches, 262, 268 Duty-cycle modulation, 500 Electronic switch, 518 Emitter follower, 326, 327 Error coefficients, 97 Estimating open-loop gain, 65 Exponentiating circuit, 547 Feedback compensation, 196 Feedback-network compensation, 563 Feedforward, 304, 421, 630 FET preamplifier, 424 Field-effect transistor, 323, 471, 491 Final-value theorem, 69 First-order system, 78 frequency response of, 86 step response of, 79 Fourier series, 218 Frequency modulation, 502 Frequency response, 81 Function generator, 234, 428, 497, 540 Gain adjustment, 165 Gain margin, 147 Gain-phase plot, 88 Gated operational amplifier, 472 Gaussian pulse, 104 Goldberg amplifier, 522 Grounding problems, 446 Gyrator, 456 High-gain stages, 309 Howland current source, 454, 549 Hybrid-pi model, 310 Hysteresis, 220, 234 Impedance scaling, 46 Initial-value theorem, 68 Inner loop, 197 Input bias current, 434 Input common-mode range, 434 Input compensation, 558 Input current, 269 Input-current cancellation, 271 Input-current measurement, 438 Input differential range, 434 Input offset current, 434 Input offset voltage, 434 Integrator, 11 Inverting amplifier, Inverted-transistor connection, 472 Jump resonance, 642 Index Lag network, 173, 179 Lag transfer function, 558 Laplace transforms, 67 properties of, 68 table of, 70 Lateral-PNP transistor, 386 Lead network, 172, 178 Lead transfer function, 558 Leakage current, 268 Limit cycle, 217, 510 Limiter, 232 Linearization, 209 LM 101 operational amplifier, 401 LM101 A operational amplifier, 63, 657 Minor-loop compensation, 573 Minor-loop instability, 600 Modulator, 474 MOS capacitor, 391 Multiplexer, 473 Multiplier, 211, 468 Negative feedback, 24 Negative impedance converter, 455 Nichols chart, 150 Node equations, 33 Noninverting amplifier, Nonlinear oscillators, 496 Nyquist criterion, 139 206, 406 LM108 operational amplifier, 415 LM1 10 operational amplifier, 416 LM 118 operational amplifier, 421 LM121, 425 Load capacitor, 169, 564, 597 Load regulation, 169 Log circuit, 12, 18, 462, 568 Loop, 44 Loop transmission, 6, 24 Low-current operation, 270 Offset voltage, 251, 472 Offset-voltage measurement, 438 One-pole compensation, 575 One-stage amplifier, 417 Open-loop gain, measurement of, 440 Open-loop transfer function, estimation of, 65 Operational-amplifier specifications, Magnetic suspension, 214 Major loop, 197 MC1533 operational amplifier, 374 Output Output Output Output MC1538R, 425 MC1539 operational amplifier, 375 yA702 operational amplifier, 421 pA715 operational amplifier, 421 yA726, 422 yA727, 422 pA733, 374 pA740 operational amplifier, 424 yA741 operational amplifier, 375 yA776 operational amplifier, 410 Miller effect, 300, 354 Minor loop, 197 433 Output amplifiers, 327 impedance, 36 resistance, 49, 169 stages, 425 voltage range, 434 Pade approximate, 530 Path, 44 Peak detector, 459 Phase detector, 537 Phase margin, 147 Phase plane, 11 Phase-shift oscillator, 116, 231 Phase shifter, 536 Piecewise-linear circuit, 461 Piecewise-linear network, 540 658 Index Pinched resistor, 389 Positive feedback, 9, 497 Power amplifier, 202 Power ground, 447 Power-supply decoupling, 444 Precision rectifier, 457 Pulse signal, 71, 72 Quadrature osdillator, 486, 519 Quarter-square multiplier, 468 Slew rate, 364, 371, 435, 633 Slew-rate measurement, 440 Slow-rolloff compensation, 604 Soft saturation, 224 Speed regulator, 203 Split collector transistor, 387, 393 Square-rooting circuit, 244 Stable-amplitude oscillation, 231 Stability, defined, 109 Ramp error, 101 Rejection amplifier, 134 Resistor selection, 447 Step response, 187 Substrate PNP transistor, 387 Summing amplifier, 10 Summation point, 22 Super-3 transistor, 385, 415 Resolver, 243, 537 Superdiode, 457 Right-half-plane singularities, 130, Supply-voltage-rejection ratio, 434 measurement of, 439 Supply-voltage sensitivity, 434 143 Rise time, 92 Root contours, 136 Root-locus construction rules, 121 Root-locus diagram, 119 Routh criterion, 112 Sallen and Key circuit, 525 Sample-and-hold circuit, 103, 475, 519, 650 Saturating nonlinearity, 219 Schmitt trigger, 234, 497 Second-order system, 79, 119 frequency response of, 87 step response of, 81 Second-stage drift contributions, 279 Settling time, 94 with lag compensation, 193 709 operational amplifier, 305 Signal-flow graph, 44 Signal ground, 447 Tangent approximation, 210 Taylor's series, 210 Thermal protection, 422 Thermal runaway, 330, 339 Three-stage amplifier, 296 Three-mode integrator, 516 Time delay, 192, 530 Time-division multiplier, 501 Time scaling, 513 Tracking filter, 538 Transconductance multiplier, 468 Transient response, 76 Two-pole compensation, 586 Two-port network, 359 Two-stage amplifier, 198, 305 compensation of, 356 Unity-gain frequency, 148 Sine-wave shaping circuit, 540 Single-ended output, 255 Sinusoidal oscillators, 485 Six-mask process, 383 Van der Pol's equation, 510 Vertical-PNP transistor, 387 Voltage reference, 286, 519, 553 Index Voltage regulator, 169, 292 Wien-bridge oscillator, 485 Zener diode, 227 659 MIT OpenCourseWare http://ocw.mit.edu RES.6-010 Electronic Feedback Systems Spring 2013 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms [...]... summation and integration Because of the performance and economic advantages of available units, present applications extend far beyond the original ones, and modern operational amplifiers are used as general purpose analog data-processing elements High-quality operational amplifiers' were available in the early 1950s These amplifiers were generally committed to use with analog computers and were not... outlined in this section The Closed-Loop Gain of an Operational Amplifier 3 Vb Figure 1.1 1.2.1 Symbol for an operational amplifier Closed-Loop Gain Calculation The symbol used to designate an operational amplifier is shown in Fig 1.1 The amplifier shown has a differential input and a single output The input terminals marked - and + are called the inverting and the noninverting input terminals respectively... the designer tailors the circuit 14 Background and Objectives he uses to his own specific, detailed requirements, and to the particular operational amplifier he chooses A balanced presentation that combines practical circuit and system design concepts with applicable theory is essential background for the type of creative approach that results in optimum operational- amplifier systems The following chapters... spectrum of circuit and system design problems, and the material is structured to encourage this type of transfer Feedback is central to virtually all operational- amplifier applications, and a thorough understanding of this important topic is necessary in any challenging design situation Chapters 2 through 6 are devoted to feedback concepts, with emphasis placed on examples drawn from operationalamplifier... circuit and system design Several interesting and widely applicable circuit-design techniques are used to realize operational amplifiers The design of operational- amplifier circuits is complicated by the requirement of obtaining gain at zero fre quency with low drift and input current Chapter 7 discusses the design of the necessary d-c amplifiers The implications of topology on the dy namics of operational- amplifier... Fig 1.2 Note that the topology shown is common to both the inverting and the noninverting connection when input points are grounded It is important to emphasize the difference between the loop transmission, which is dependent on properties of both the feedback elements and the operational amplifier, and the open-loop gain of the operational amplifier itself 1.2.2 The Ideal Closed-Loop Gain Detailed... effectively applying operational amplifiers, and often represents the differ ence between inadequate and superlative performance Several examples of the way in which compensation influences the performance of a repre sentative integrated-circuit operational amplifier are used to reinforce the theoretical discussion included in this chapter PROBLEMS P1.1 Design a circuit using a single operational amplifier... also necessary to provide operating power to the operational ampli fier via power-supply terminals Many operational amplifiers use balanced (equal positive and negative) supply voltages The various signals are usually referenced to the common ground connection of these power sup 2 The notation used to designate system variables consists of a symbol and a subscript This combination serves not only... INTEGRATED-CIRCUIT FIERS OPERATIONAL AMPLI381 10.1 Introduction 381 10.2 Fabrication 382 Contents xv Page 10.2.1 10.2.2 10.2.3 10.3 10.3.1 10.3.2 10.4 10.4.1 10.4.2 10.4.3 10.4.4 10.4.5 10.5 XI NPN Transistors PNP Transistors Other Components Integrated-Circuit Design Techniques Current Repeaters Other Connections Representative Integrated-Circuit Operational Amplifiers The LM1O and LM1O1A Operational Amplifiers... V,, 0 (1.11) Kirchhoff's current law combined with condition 2 shows that I + Ib ~ 0 (1.12) With Eqn 1.11 satisfied, the currents I, and I are readily determined in terms of the input and output voltages Vai Z1 b Va Z2 (1.13) (1.14) Combining Eqns 1.12, 1.13, and 1.14 and solving for the ratio of V, to Vi yields the ideal closed-loop gain V.V- Z2 Vi Z1 (1.15) The technique used to determine the ideal