Operational AmplifiersIdeal and Practical ModelsThe concept of the operational amplifier(usually referred to as an op amp ) originated at the beginning of the Second World War with the use of vacuum tubes in dc amplifier designs developed by the George A. Philbrick Co. [some of the early history of operational amplifiers is found in Williams, 1991]. The op amp was the basic building block for early electronic servomechanisms, for synthesizers, and in particular for analog computers used to solve differential equations. With the advent of the first monolithic integrated-circuit (IC) op amp in 1965 (the A709, designed by the late Bob Widlar, then with Fairchild Semiconductor), the availability of op amps was no longer a factor, while within a few years the cost of these devices (which had been as high as $200 each) rapidly plummeted to close to that of individual discrete transistors.Although the digital computer has now largely supplanted the analog computer in mathematically intensive applications, the use of inexpensive operational amplifiers in instrumentation applications, in pulse shaping, in filtering, and in signal processing applications in general has continued to grow. There are currently many commercial manufacturers whose main products are high-quality op amps. This competitiveness has ensured a marketplace featuring a wide range of relatively inexpensive devices suitable for use by electronic engineers, physicists, chemists, biologists, and almost any discipline that requires obtaining quantitative analog data from instrumented experiments.Most operational amplifier circuits can be analyzed, at least for first-order calculations, by considering the op amp to be an “ideal” device. For more quantitative information, however, and particularly when frequency response and dc offsets are important, one must refer to a more “practical” model that includes the internal limitations of the device. If the op amp is characterized by a really complete model, the resulting circuit may be quite complex, leading to rather laborious calculations. Fortunately, however, computer analysis using the program SPICE significantly reduces the problem to one of a simple input specification to the computer. Today, nearly all the op amp manufacturers provide SPICE models for their line of devices, with excellent correlation obtained between the computer simulation and the actual measured results.The Ideal Op AmpAn ideal operational amplifier is a dc-coupled amplifier having two inputs and normally one output (although in a few infrequent cases there may be a differential output). The inputs are designated as noninverting (designated + or NI) and inverting (designated – or Inv.). The amplified signal is the differential signal, v, between the two inputs, so that the output voltage as indicated in Fig. 1 isFIGURE 1 Configuration for an ideal op amp.The general characteristics of an ideal op amp can be summarized as follows:1. The open-loop gain AOL is infinite. Or, since the output signal vout is finite,then the differential input signal v must approach zero.2. The input resistance RIN is infinite, while the output resistance R O is zero.3. The amplifier has zero current at the input ( i A and i B in Fig. 1 are zero), but the op amp can either sink or source an infinite current at the output.4. The op amp is not sensitive to a common signal on both inputs (i.e., v A= v B); thus, the output voltage change due to a common input signal will be zero. This common signal is referred to as a commonmode signal, and manufa cturers specify this effect by an op amp’s common-mode rejection ratio (CMRR), which relates the ratio of the open-loop gain ( A OL) of the op amp to the common-mode gain (A CM ). Hence, for an ideal op amp CMRR =∞.5. A somewhat analogous specification to the CMRR is the power-supply rejection ratio (PSRR),which relates the ratio of a power supply voltage change to an equivalent input voltage change produced by the change in the power supply. Because an ideal op amp can operate with any power supply, without restriction, then for the ideal device PSRR =∞.6. The gain of the op amp is not a function of frequency. This implies an infinite bandwidth.Although the foregoing requirements for an ideal op amp appear to be impossible to achieve practically, modern devices can quite closely approximate many of these conditions. An op amp with a field-effect transistor (FET) on the input would certainly not have zero input current and infinite input resistance, but a current of <10 pA and an R IN= 10 12Ωi s obtainable and is a reasonable approximation to the ideal conditions. Further, although a CMRR and PSRR of infinity are not possible, there are several commercial op amps available with values of 140 dB (i.e., a ratio of 107). Open-loop gains of several precision op amps now have reached values of >107 , although certainly not infinity. The two most difficult ideal conditions to approach are the ability to handle large output currents and the requirement of a gain independence with frequency.Using the ideal model conditions it is quite simple to evaluate the two basic op amp circuit configurations, (1) the inverting amplifier and (2) the noninverting amplifier, as designated in Fig. 2.For the ideal inverting amplifier, since the open-loop gain is infinite and since the output voltage v o is finite, then the input differential voltage (often referred to as the error signal ) v must approach zero, or the input current isThe feedback current i F must equal i I, and the output voltage must then be due to the voltage drop across RF , orFIGURE 2 Illustration of (a) the inverting amplifier and (b) the noninverting amplifier.The inverting connection thus has a voltage gain vo/vi of-RF/RI, an input resistance seen by vI of R1 ohms [from Eq. (27.2)], and an output resistance of 0 . By a similar analysis for the noninverting circuit of Fig. 2(b), since v is zero, then signal vI must appear across resistor R1 , producing a current of vI/R1, which must flow through resistor RF . Hence the output voltage is the sum of the voltage drops acrossRF and R1 , or As opposed to the inverting connection, the input resistance seen by the source v I is now equal to an infinite resistance, since R IN for the ideal op amp is infinite.Practical Op AmpsA nonideal op amp is characterized not only by finite open-loop gain, input and outputresistance, finite currents, and frequency bandwidths, but also by various nonidealities due to the construction of the op amp circuit or external connections. A complete model for a practical op amp is illustrated in Fig. 3.FIGURE 3 A model for a practical op amp illustrating nonideal effects.The nonideal effects of the PSRR and CMRR are represented by the input series voltage sources of V supply/PSRR and V CM/CMRR, where V supply would be any total change of the two power supply voltages, V +dc and V–dc , from their nominal values, while V CM is the voltage common to both inputs of the op amp. The open-loop gain of the op amp is no longer infinite but is modeled by a network of the output impedance Z out (which may be merely a resistor but could also be a series R-L network) in series with a source A ( s ), which includes all the open-loop poles and zeroes of the op amp aswhere A OL is the finite dc open-loop gain, while poles are at frequencies p1, p2, . . . and zeroes are at Z1, etc. The differential input resistance is Z IN , which is typically a resistance RIN parallel with a capacitor C IN. Similarly, the common-mode input impedance Z CM isestablished by placing an impedance 2Z CM in parallel with each input terminal. Normally, Z CMis best represented by a parallel resistance and capacitance of 2R CM (which is >> RIN) and C CM/2. The dc bias currents at the input are represented by I B+ and I B–current sources that would equal the input base currents if a differential bipolar transistor were used as the input stage of the op amp, or the input gate currents if FETs were used. The fact that the two transistors of the input stage of the op amp may not be perfectly balanced is represented by an equivalent input offset voltage source, VOS , in series with the input.The smallest signal that can be amplified is always limited by the inherent random noise internal to the op amp itself. In Fig. 3 the noise effects are represented by an equivalent input voltage source (ENV), which when multiplied by the gain of the op amp would equal the total output noise present if the inputs to the op ilar fashion, if the inputs to the op amp were open circuited, the total output noise would equal the sum of the noise due to the equivalent input current sources (ENI+ and ENI–), each multiplied by their respective current gain to the output. Because noise is a random variable, this summation must be accomplished in a squared fashion, i.e.,Typically, the correlation (C) between the ENV and ENI sources is low, so the assumption of C 0 can be made. For the basic circuits of Fig. 27.2(a) or (b), if the signal source vI is shorted then the output voltage due to the nonideal effects would be (using the model of Fig. 3)provided that the loop gain (also called loop transmission in many texts) is related by the inequalityInherent in Eq. (27.8) is the usual condition that R1 << Z IN and Z CM. If a resistor R2 were in series with the noninverting input terminal, then a corresponding term must be added to the right hand side of Eq. (27.7) of value –I B+R2 (R1 + R F)/R1.On manufacturers’ data sheets the individual values of I B+ and I B– are not stated; instead the average input bias current and offset current are specified asThe output noise effects can be obtained using the model of Fig. 3 along with the circuits of Fig.2 aswhere it is assumed that a resistor R2 is also in series with the noninverting input of either Fig. 2(a) or (b). The thermal noise (often called Johnson or Nyquist noise) due to the resistors R1 , R2 , and RF is given by (in rms volt2/Hz)where k is Boltzmann’s constant and T is absolute temperature ( Kelvin). To obtain the total output noise, one must multiply the E2 out expression of Eq. (27.10) by the noise bandwidth of the circuit, which typically is equal to /2 times the –3 dB signal bandwidth, for a single-pole response system [Kennedy, 1988].SPICE Computer ModelsThe use of op amps can be considerably simplified by computer-aided analysis using the program SPICE. SPICE originated with the University of California, Berkeley, in 1975 [Nagel, 1975], although more recent user-friendly commercial versions are now available such as HSPICE, HPSPICE, IS-SPICE, PSPICE, and ZSPICE, to mention few of those most widely used. A simple macromodel for a near-ideal op amp could be simply stated with the SPICE subcircuit file (* indicates a comment that is not processed by the file).SUBCKT IDEALOA 1 2 3*A near-ideal op amp: (1) is noninv, (2) is inv, and (3) is output.RIN 1 2 1E12E1 (3, 0) (1, 2) 1E8.ENDS IDEALOA (27.12)The circuit model for IDEALOA would appear as in Fig. 4(a). A more complete model, but not including nonideal offset effects, could be constructed for the 741 op amp as the subcircuit file OA741, shown in Fig. 4(b)..SUBCKT OA741 1 2 6*A linear model for the 741 op amp: (1) is noninv, (2) is inv, and*(6) is output. RIN = 2MEG, AOL = 200,000, ROUT = 75 ohm,*Dominant open - loop pole at 5 Hz, gain - bandwidth product*is 1 MHz.RIN 1 2 2MEGE1 (3, 0) (1, 2) 2E5R1 3 4 100KC1 4 0 0.318UF ; R1 2 C1 = 5HZPOLEE2 (5, 0) (4, 0) 1.0ROUT 5 6 75.ENDS OA741 (27.13)The most widely used op amp macromodel that includes dc offset effects is the Boyle model [Boyle et al., 1974]. Most op amp manufacturers use this model, usually with additions to add more poles (and perhaps zeroes). The various resistor and capacitor values, as well as transistor, and current and voltage generator, values are intimately related to the specifications of the op amp, as shown earlier in the nonideal model of Fig. 3. The appropriate equations are too involved to list here; instead, the interested reader is referred to the article by Boyle in the listed references. The Boyle model does not accurately model noise effects, nor does it fully model PSRR and CMRR effects.A more circuits-oriented approach to modeling op amps can be obtained if the input transistors are removed and a model formed by using passive components along with both fixed and dependent voltage and current sources.This model not only includes all the basic nonideal effects of the op amp, allowing for multiple poles and zeroes, but can also accurately include ENV and ENI noise effects.The circuits-approach macromodel can also be easily adapted to current-feedback op amp designs, whose input impedance at the noninverting input is much greater than that at the inverting input [see Williams, 1991]. The interested reader is referred to the text edited by J. Williams, listed in the references, as well as the SPICE modeling book by Connelly and Choi [1992].运算放大器理想及实际模型运算放大器（通常称之为op amp）的概念起源于第二次世界大战开始时，乔治•a •菲尔公司对真空管直流放大器应用的发展〔一些运放的早期历史于1991年在威廉姆斯被发现〕。
