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Basic Concepts
1. Basic Concepts
1.1 The Operational Amplifier The operational amplifier is a versatile amplifying device, originally intended for use in analog computers to perform linear mathematical operations. Forty years of development of the operational amplifier's internal circuit design reflects, to a significant extent, the development of electronic components from vacuum tubes to monolithic integrated circuits. An increasing refinement of the operational amplifier's properties has shifted the emphasis of its ap plications from laboratories to industry. Due to its high performance, ver satility, and low price, the operational amplifier now dominates the field of analog electronic systems. We generally define the operational amplifier as a direct-coupled amplifier with a high gain and a low level of inherent noise, capable of stable operation in a closed-feedback loop. The exact meaning of these characteristics will be given in Chapters 2 and 11. It should be mentioned here that the term directcoupled does not imply an upper limitation of the amplifier's frequency re sponse but, on the contrary, an extension of the operating range to zero frequency, or infinitely long periods. The direction of signalflowfrom input to output in an operational amplifier is given by the triangular shape of its symbol in Figure 1-la. Three of the four illustrated terminals represent the three signal terminals of an actual operational amplifier. These are the inverting input, noninverting input, and output. The fourth signal terminal, the ground, may be either actual (Figure 1—lb) or only virtual (power supply common in Figure 1-lc). In either case, it represents symbolically a group of at least two terminals intended for the supply of energy. In addition to the above-mentioned signal terminals, the actual operational amplifier is, if necessary, fitted with further terminals for frequency compen sation, offset nulling, or supply-current setting. INVERTING INPUT
GROUND NON-INVERTING INPUT GROUND
T
SUPPLY LEADS
GROUND (a)
(C)
Figure 1-1. Symbol of an operational amplifier with signal terminals (a) and actual realization of the ground terminal (b, c).
Basic Concepts
(b)
(a)
Figure 1-2. Assignment and sign convention for input and output voltages and currents (a). In the simplified form, the ground terminal is omitted (b).
The ground signal terminal provides a reference point for the three others. The sign convention for the input voltages v~, v+, output voltage u(), input currents /~, / + , and output current /0, is shown in Figure l-2a. When there is no risk of confusion, the ground is usually omitted in the amplifier symbol and the terminal voltages are indicated merely by adding a letter (Figure 12b). The absolute values of signal voltages u~, v+, and v{) are normally limited by the supply voltages V$ and V$ . Unless there are special circumstances, V$ and V$ have nominal values of +15 V and - 1 5 V, respectively. The corresponding signal swings of both input voltages and of the output voltage are also symmetrical in both polarities, usually in the range of ±10 V. Many operational amplifiers can run on other supply voltages, both symmetrical and unsymmetrical (e.g., +5 V and 0 V), as well. The output current adapts to the load, which may be passive or active, with permissible operating points (y0, /0) in any of the four quadrants. An outstanding transfer property of an operational amplifier is its high sensitivity to the difference between the two input voltages and its insensitivity to their absolute value. This property leads to the introduction of the commonmode input voltage ucm for the superimposed common component of the input voltages, which will be rejected by the amplifier, and the differential input voltage ud, to which the amplifier does respond. Whereas the definition of the differential input voltage is obvious (Figure l-2a), v - vn the definition of the common-mode input voltage is rather arbitrary, ^d
*>cm = V+ +
(i.i)
KV6,
as it depends upon the value chosen for the constant K. Two choices used in practice are K = 1/2 and K = 0. The first choice preserves symmetry, vcm =
v~ + v+
but it leads to formal difficulties in the definition of operational amplifier parameters. Thus, the second choice is preferred, for which the commonmode input voltage vcm is identified with the noninverting input voltage v+, vcm = v\
(1.2)
This second choice is also justified by considering the function of the closedfeedback loop, where in most cases the noninverting input serves as a forced reference point with which the inverting input is compared. The difference
5
1.2 Operational Circuit
between the two definitions is practically negligible, since the differential input voltage is usually insignificant compared with the common-mode input voltage range.
1.2 Operational Circuit The operational amplifier itself is only a part of the complete system, though it is often the most important one. The second, functionally determining part, is the external feedback network. Figure 1-3 shows a general configuration with one operational amplifier, signal source, and load (1). The feedback network, represented by the crosshatched area and consisting of passive and active electronic and electromechanical components, is terminated at nodes, which provide the connection to the signal terminals of the operational am plifier, signal source, and load. The whole configuration, consisting of an operational amplifier, feedback network, load, and signal source, constitutes the operational circuit. Its input quantity is the signal-source voltage vs or current /s and its output quantity is the voltage v0 or current current i0 at the load terminals. It should be noted that, in general, the operational circuit output need not be identical with the operational amplifier output, and the amplifier ground (the power supply common, according to Figure 1—lb and 1-lc) need not be directly connected to one of the terminals of the signal source or load, although this is usually the case. With the exception of operational circuits operating as reference sources, oscillators, or multivibrators, the output quantity (u0, /0) is related in a certain way to the input quantity (us, /s). This relationship, expressed mathematically, is the operational equation of the circuit. The most valuable property of circuits with operational amplifiers is the great insensitivity of the operational equation to the spread in the parameters of the operational amplifier itself, and to changes in the load and signal source (i.e., the changes in their load and internal resistances). The first fact leads to the definition of an ideal operational amplifier (Section 1.3), while the second leads to a reduced concept of the operational circuit as a mere con nection of the operational amplifier and feedback network. It is the insen sitivity of the operational equation to the properties of a naturally inconstant OPERATIONAL AMPLIFIER
FEEDBACK NETWORK
LOAD SIGNAL SOURCE OPERATIONAL CIRCUIT
Figure 1-3. An operational circuit with one operational amplifier, one signal source, and one load.
Basic Concepts
active component, the amplifier, that makes the behavior of the operational circuit mathematically predictable. The operational equation thus becomes essentially a characteristic of the feedback network alone. The operational circuit (Figure 1-3) may be extended by the addition of further signal sources, operational amplifiers, and loads. The realization of the feedback network need not be confined to electrical means. The feedback loop may also be closed by employing kinds of signal other than voltage or current, such as magnetic induction, Lorentz force, mechanical stress, strain and piezoelectric charge, heat, temperature and thermoelectric voltage, light and photoemission current, etc. The only but principal limiting condition is that the feedback stability of the resulting closed loop must be preserved.
1.3 Ideal Operational Amplifier and Ideal Operational Circuit The seemingly absurd aim of every designer of operational amplifiers is to design an amplifier that is functionally invisible, in the sense that it does not affect the operational equation of the circuit. This abstraction is an ideal operational amplifier, a useful conception that allows rapid analysis of the nominal (i.e., desired) behavior of the operational circuit, or allows the design of the operational circuit on the basis of a given mathematical or even func tional description, with results immediately and exactly applicable to an actual situation. Actual operational amplifiers, to a certain extent, approach their ideal. However, at a given state of technology, there is always a trade-off between quality, precision, and the higher complexity and price of the op erational amplifier. • The ideal operational amplifier is an operational amplifier with zero dif ferential input voltage and zero input currents for any output excitation and any common-mode input excitation, ud, I - , /+ = 0
for any
y0, /0, vcm.
(1.3)
• The ideal operational circuit is an operational circuit obtained by sub stituting an ideal operational amplifier for the actual one. • The ideal operational equation is the operational equation of an ideal operational circuit. The value of these concepts will be appreciated in Chapter 4. As may be seen from Eq. (1.3), the ultimate quality of the operational amplifier depends upon the quality of its input side (upon the deviation of its input voltage and input currents from zero). To put this into the language of Chapter 2, an ideal operational amplifier has, at all frequencies, infinite open-loop gain, infinite common-mode rejec tion ratio, infinite common-mode input impedances, and zero input error sources, as can also be deduced from Eq. (1.3). Because of the infinite gain, the magnitudes of the differential input impedance and output impedance are immaterial. However, since the real impedance values of an actual operational amplifier (with finite gain) make the dynamic errors of the operational circuit worse, one normally also associates the idea of an ideal operational amplifier with an infinite differential input impedance and a zero output impedance.
References
The data sheet of an ideal operational amplifier thus contains only zeros and infinites. The mathematical analysis of an operational circuit in a particular direction (e.g., calculation of noise, closed-loop gain) can be conveniently simplified, if one neglects at the very beginning what is unimportant, that is, if one idealizes some nonessential parameters of the operational amplifier. In this sense, an idealized operational amplifier is one in which some parameters have ideal values (zero or infinity).
1.4 Summary 1. The operational amplifier has four signal terminals, though often only three of them are drawn—the two inputs and the output. The fourth signal terminal is ground. 2. The common-mode input voltage vcm is identical to the voltage at the noninverting input v+. 3. The ideal operational amplifier has zero differential input voltage and zero input currents under any conditions.
References 1. Philbrick, G.A. (ed.) (1969) Operational Amplifiers (A Lightning Empiricist Literary Supplement), Part I, II, III, Philbrick/Nexus Research, Dedham, MA.
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1.1 The Operational Amplifier The operational amplifier is a versatile amplifying device, originally intended for use in analog computers to perform linear mathematical operations. Forty years of development of the operational amplifier's internal circuit design reflects, to a significant extent, the development of electronic components from vacuum tubes to monolithic integrated circuits. An increasing refinement of the operational amplifier's properties has shifted the emphasis of its ap plications from laboratories to industry. Due to its high performance, ver satility, and low price, the operational amplifier now dominates the field of analog electronic systems. We generally define the operational amplifier as a direct-coupled amplifier with a high gain and a low level of inherent noise, capable of stable operation in a closed-feedback loop. The exact meaning of these characteristics will be given in Chapters 2 and 11. It should be mentioned here that the term directcoupled does not imply an upper limitation of the amplifier's frequency re sponse but, on the contrary, an extension of the operating range to zero frequency, or infinitely long periods. The direction of signalflowfrom input to output in an operational amplifier is given by the triangular shape of its symbol in Figure 1-la. Three of the four illustrated terminals represent the three signal terminals of an actual operational amplifier. These are the inverting input, noninverting input, and output. The fourth signal terminal, the ground, may be either actual (Figure 1—lb) or only virtual (power supply common in Figure 1-lc). In either case, it represents symbolically a group of at least two terminals intended for the supply of energy. In addition to the above-mentioned signal terminals, the actual operational amplifier is, if necessary, fitted with further terminals for frequency compen sation, offset nulling, or supply-current setting. INVERTING INPUT
GROUND NON-INVERTING INPUT GROUND
T
SUPPLY LEADS
GROUND (a)
(C)
Figure 1-1. Symbol of an operational amplifier with signal terminals (a) and actual realization of the ground terminal (b, c).
Basic Concepts
(b)
(a)
Figure 1-2. Assignment and sign convention for input and output voltages and currents (a). In the simplified form, the ground terminal is omitted (b).
The ground signal terminal provides a reference point for the three others. The sign convention for the input voltages v~, v+, output voltage u(), input currents /~, / + , and output current /0, is shown in Figure l-2a. When there is no risk of confusion, the ground is usually omitted in the amplifier symbol and the terminal voltages are indicated merely by adding a letter (Figure 12b). The absolute values of signal voltages u~, v+, and v{) are normally limited by the supply voltages V$ and V$ . Unless there are special circumstances, V$ and V$ have nominal values of +15 V and - 1 5 V, respectively. The corresponding signal swings of both input voltages and of the output voltage are also symmetrical in both polarities, usually in the range of ±10 V. Many operational amplifiers can run on other supply voltages, both symmetrical and unsymmetrical (e.g., +5 V and 0 V), as well. The output current adapts to the load, which may be passive or active, with permissible operating points (y0, /0) in any of the four quadrants. An outstanding transfer property of an operational amplifier is its high sensitivity to the difference between the two input voltages and its insensitivity to their absolute value. This property leads to the introduction of the commonmode input voltage ucm for the superimposed common component of the input voltages, which will be rejected by the amplifier, and the differential input voltage ud, to which the amplifier does respond. Whereas the definition of the differential input voltage is obvious (Figure l-2a), v - vn the definition of the common-mode input voltage is rather arbitrary, ^d
*>cm = V+ +
(i.i)
KV6,
as it depends upon the value chosen for the constant K. Two choices used in practice are K = 1/2 and K = 0. The first choice preserves symmetry, vcm =
v~ + v+
but it leads to formal difficulties in the definition of operational amplifier parameters. Thus, the second choice is preferred, for which the commonmode input voltage vcm is identified with the noninverting input voltage v+, vcm = v\
(1.2)
This second choice is also justified by considering the function of the closedfeedback loop, where in most cases the noninverting input serves as a forced reference point with which the inverting input is compared. The difference
5
1.2 Operational Circuit
between the two definitions is practically negligible, since the differential input voltage is usually insignificant compared with the common-mode input voltage range.
1.2 Operational Circuit The operational amplifier itself is only a part of the complete system, though it is often the most important one. The second, functionally determining part, is the external feedback network. Figure 1-3 shows a general configuration with one operational amplifier, signal source, and load (1). The feedback network, represented by the crosshatched area and consisting of passive and active electronic and electromechanical components, is terminated at nodes, which provide the connection to the signal terminals of the operational am plifier, signal source, and load. The whole configuration, consisting of an operational amplifier, feedback network, load, and signal source, constitutes the operational circuit. Its input quantity is the signal-source voltage vs or current /s and its output quantity is the voltage v0 or current current i0 at the load terminals. It should be noted that, in general, the operational circuit output need not be identical with the operational amplifier output, and the amplifier ground (the power supply common, according to Figure 1—lb and 1-lc) need not be directly connected to one of the terminals of the signal source or load, although this is usually the case. With the exception of operational circuits operating as reference sources, oscillators, or multivibrators, the output quantity (u0, /0) is related in a certain way to the input quantity (us, /s). This relationship, expressed mathematically, is the operational equation of the circuit. The most valuable property of circuits with operational amplifiers is the great insensitivity of the operational equation to the spread in the parameters of the operational amplifier itself, and to changes in the load and signal source (i.e., the changes in their load and internal resistances). The first fact leads to the definition of an ideal operational amplifier (Section 1.3), while the second leads to a reduced concept of the operational circuit as a mere con nection of the operational amplifier and feedback network. It is the insen sitivity of the operational equation to the properties of a naturally inconstant OPERATIONAL AMPLIFIER
FEEDBACK NETWORK
LOAD SIGNAL SOURCE OPERATIONAL CIRCUIT
Figure 1-3. An operational circuit with one operational amplifier, one signal source, and one load.
Basic Concepts
active component, the amplifier, that makes the behavior of the operational circuit mathematically predictable. The operational equation thus becomes essentially a characteristic of the feedback network alone. The operational circuit (Figure 1-3) may be extended by the addition of further signal sources, operational amplifiers, and loads. The realization of the feedback network need not be confined to electrical means. The feedback loop may also be closed by employing kinds of signal other than voltage or current, such as magnetic induction, Lorentz force, mechanical stress, strain and piezoelectric charge, heat, temperature and thermoelectric voltage, light and photoemission current, etc. The only but principal limiting condition is that the feedback stability of the resulting closed loop must be preserved.
1.3 Ideal Operational Amplifier and Ideal Operational Circuit The seemingly absurd aim of every designer of operational amplifiers is to design an amplifier that is functionally invisible, in the sense that it does not affect the operational equation of the circuit. This abstraction is an ideal operational amplifier, a useful conception that allows rapid analysis of the nominal (i.e., desired) behavior of the operational circuit, or allows the design of the operational circuit on the basis of a given mathematical or even func tional description, with results immediately and exactly applicable to an actual situation. Actual operational amplifiers, to a certain extent, approach their ideal. However, at a given state of technology, there is always a trade-off between quality, precision, and the higher complexity and price of the op erational amplifier. • The ideal operational amplifier is an operational amplifier with zero dif ferential input voltage and zero input currents for any output excitation and any common-mode input excitation, ud, I - , /+ = 0
for any
y0, /0, vcm.
(1.3)
• The ideal operational circuit is an operational circuit obtained by sub stituting an ideal operational amplifier for the actual one. • The ideal operational equation is the operational equation of an ideal operational circuit. The value of these concepts will be appreciated in Chapter 4. As may be seen from Eq. (1.3), the ultimate quality of the operational amplifier depends upon the quality of its input side (upon the deviation of its input voltage and input currents from zero). To put this into the language of Chapter 2, an ideal operational amplifier has, at all frequencies, infinite open-loop gain, infinite common-mode rejec tion ratio, infinite common-mode input impedances, and zero input error sources, as can also be deduced from Eq. (1.3). Because of the infinite gain, the magnitudes of the differential input impedance and output impedance are immaterial. However, since the real impedance values of an actual operational amplifier (with finite gain) make the dynamic errors of the operational circuit worse, one normally also associates the idea of an ideal operational amplifier with an infinite differential input impedance and a zero output impedance.
References
The data sheet of an ideal operational amplifier thus contains only zeros and infinites. The mathematical analysis of an operational circuit in a particular direction (e.g., calculation of noise, closed-loop gain) can be conveniently simplified, if one neglects at the very beginning what is unimportant, that is, if one idealizes some nonessential parameters of the operational amplifier. In this sense, an idealized operational amplifier is one in which some parameters have ideal values (zero or infinity).
1.4 Summary 1. The operational amplifier has four signal terminals, though often only three of them are drawn—the two inputs and the output. The fourth signal terminal is ground. 2. The common-mode input voltage vcm is identical to the voltage at the noninverting input v+. 3. The ideal operational amplifier has zero differential input voltage and zero input currents under any conditions.
References 1. Philbrick, G.A. (ed.) (1969) Operational Amplifiers (A Lightning Empiricist Literary Supplement), Part I, II, III, Philbrick/Nexus Research, Dedham, MA.
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