Electronics - Operational Amplifiers

8/15/2019 Electronics - Operational Amplifiers

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Kennedy, E.J., Wait, J.V. “Operational Amplifiers”

The Electrical Engineering Handbook Ed. Richard C. Dorf

Boca Raton: CRC Press LLC, 2000


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© 2000 by CRC Press LLC

27Operational Amplifiers

27.1 Ideal and Practical ModelsThe Ideal Op Amp • Practical Op Amps • SPICE Computer Models

27.2 ApplicationsNoninverting Circuits

27.1 Ideal and Practical Models

E.J. Kennedy

The 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 mA709, designed by the late Bob Widlar, then with Fairchild Semiconductor), the availability of opamps 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 Amp

An 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 e,

between the two inputs, so that the output voltage as indicated in Fig. 27.1 is

E.J. KennedyUniversity of Tennessee

John V. WaitUniversity of Arizona (Retired)


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(27.1)

The general characteristics of an ideal op amp can be summarized as follows:

1. The open-loop gain A OL is infinite. Or, since the output signal v out is finite, then the differential input

signal v e must approach zero.

2. The input resistance R IN 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. 27.1 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 voltagechange due to a common input signal will be zero. This common signal is referred to as a common-

mode signal, and manufacturers 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 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. 27.2.

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 e must approach zero, or the input

current is

(27.2)

The feedback current i F must equal i I , and the output voltage must then be due to the voltage drop across R F , or

(27.3)

FIGURE 27.1 Configuration for an ideal op amp.

v A v v out OL B A = -( )

i v v R

v R

I I I = - = -e

1 1

0

v i R v i R R

R v o F F I F

F I = - + = - = -

æ è ç

ö ø ÷ e 1


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effects of the PSRR and CMRR are represented by the input series voltage sources of DV supply /PSRR andV C M /CMRR, where DV 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 as

(27.5)

where A OL is the finite dc open-loop gain, while poles are at frequencies w p 1, w p 2 , . . . and zeroes are at wZ 1,etc. The differential input resistance is Z IN , which is typically a resistance R IN in parallel with a capacitor C IN .

Similarly, the common-mode input impedance Z CM is established by placing an impedance 2Z CM in parallel

FIGURE 27.3 A model for a practical op amp illustrating nonideal effects. (Source: E.J. Kennedy, Operational Amplifier Circuits, Theory and Applications, New York: Holt, Rinehart and Winston, 1988, pp. 53, 126. With permission.)

A s

A s

s s

OL Z

p p

( )

( )

( )

=

+æ

è ç

ö

ø ÷ + × × ×

+æ

è çö

ø ÷ +

æ

è çö

ø ÷ + × × ×

1 1

1 1 1

1

1 2

w

w w


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with each input terminal. Normally, Z CM is best represented by a parallel resistance and capacitance of 2R CM (which is >> R IN ) and C C M /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, V OS , 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. 27.3 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

amp were shorted. In a similar 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.,

(27.6)

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 v I is shorted then the output voltage due to

the nonideal effects would be (using the model of Fig. 27.3)

(27.7)

provided that the loop gain (also called loop transmission in many texts) is related by the inequality

(27.8)

Inherent in Eq. (27.8) is the usual condition that R 1 >( )

I I I

I I I B B B

B B =+

= -+ -

+ -

2; offset

E E R

R

E E

R

R R R

R

R

F F

F F

F

out rms volts /Hz ENV

ENI ENI

2 212

1

2

2 222

1

2

2 2 222

1

2

1 1

( ) ( )

( ) ( )

=æ

è ç

ö

ø ÷ + + + ´

+æ

è çö

ø ÷ + + +

æ

è çö

ø ÷ - +


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(27.11)

where k is Boltzmann’s constant and T is absolute temperature (°Kelvin). To obtain the total output noise, onemust multiply the E 2out expression of Eq. (27.10) by the noise bandwidth of the circuit, which typically is equal

to p/2 times the –3 dB signal bandwidth, for a single-pole response system [Kennedy, 1988].

SPICE Computer Models

The 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

a 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. 27.4(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. 27.4(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 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. 27.3.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. Such a model is shown in Fig. 27.5. 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.

E kT R

E kT R

E kT R F F

12

1

22

2

2

4

4

4

=

=

=


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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].

FIGURE 27.4 Some simple SPICE macromodels. (a) A near ideal op amp. (b) A linear model for a 741 op amp. (c) The

Boyle macromodel.


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A comparison of the SPICE macromodels with actual manufacturer’s data for the case of an LM318 op amp

is demonstrated in Fig. 27.6, for the open-loop gain versus frequency specification.

Defining TermsBoyle macromodel: A SPICE computer model for an op amp. Developed by G.R. Boyle in 1974.

Equivalent noise current (ENI): A noise current source that is effectively in parallel with either the nonin-

verting input terminal (ENI+) or the inverting input terminal (ENI–) and represents the total noise

contributed by the op amp if either input terminal is open circuited.

Equivalent noise voltage (ENV): A noise voltage source that is effectively in series with either the inverting

or noninverting input terminal of the op amp and represents the total noise contributed by the op amp

if the inputs were shorted.

Ideal operational amplifier: An op amp having infinite gain from input to output, with infinite input

resistance and zero output resistance and insensitive to the frequency of the signal. An ideal op amp is

useful in first-order analysis of circuits.

Operational amplifier (op amp): A dc amplifier having both an inverting and noninverting input and

normally one output, with a very large gain from input to output.SPICE: A computer simulation program developed by the University of California, Berkeley, in 1975. Versions

are available from several companies. The program is particularly advantageous for electronic circuit

analysis, since dc, ac, transient, noise, and statistical analysis is possible.

Related Topic

13.1 Analog Circuit Simulation

FIGURE 27.5 A SPICE circuits-approach macromodel.

V2D6

D5

Isc+

Isc –

R01

R02

G0 = I/R02

D4D3

E2

VOUT

L0

V1 (7)

(18)(17)(14)(13)

(3)(+)

(–)

(Input)

(2)(1)

(4) (7)

(10)

(3)

(9)

(11)

(12)

(5)

(15)

(23)(16)

(6)(19)

(20)

(6)

(21)

(22)

RpsIps

VCC

Cp4

Rp4

Rp3

Rp2

VN VP

DN2 DP2

DN1

ECMRR EPSRR

CCMI

RCMI

CCM2 RCM2

DP1

RN1

CN2

RN2>>RN1

Rp1

Rslew

E1=1xV(15)

E2=1xV(6)

Cp3

Cp2

Cp1

Rz1G3G2G1 G4

D2

D1

VEE

VOS

Is –

Is+

VCCCIN RIN

-VEE

(4)

+ –

+ – + – + –

+ – + –

– ++

–

+

–

Islew+

Islew –


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References

G.R. Boyle et al., “Macromodeling of integrated circuit operational amplifiers,” IEEE J. S. S. Circuits,

pp. 353–363, 1974.

J.A. Connelly and P. Choi, Macromodeling with SPICE, Englewood Cliffs, N.J.: Prentice-Hall, 1992.

FIGURE 27.6 Comparison between manufacturer’s data and the SPICE macromodels.


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E.J. Kennedy, Operational Amplifier Circuits, Theory and Applications, New York: Holt, Rinehart and Winston,

1988.

L.W. Nagel, SPICE 2: A Computer Program to Simulate Semiconductor Circuits, ERL-M520, University of Cali-

fornia, Berkeley, 1975.

J. Williams (ed.), Analog Circuit Design, Boston: Butterworth-Heinemann, 1991.

27.2 Applications

John V. Wait

In microminiature form (epoxy or metal packages or as part of a VLSI mask layout) the operational amplifier

(op amp) is usually fabricated in integrated circuit (IC) form. The general environment is shown in Fig. 27.7.

A pair of + and – regulated power supplies (or batteries) may supply all of the op amp in a system, typically

with ±10 – ±15 V. The ground and power supply buses are usually assumed, and an individual op-amp symbolis shown in Fig. 27.8. Such amplifiers feature:

1. A high voltage gain, down to and including dc, and a dc open loop gain of perhaps 105 (100 dB) or more

2. An inverting (–) and noninverting (+) symbol

3. Minimized dc offsets, a high input impedance, and a low output impedance

4. An output stage able to deliver or absorb currents over a dynamic range approaching the power supply

voltagesIt is important never to use the op amp without feedback between the output and inverting terminals at all

frequencies. A simple inverting amplifier is shown in Fig. 27.9. Here the voltage gain is

V out /V in = –K = –R F /R 1

The circuit gain is determined essentially by the external resistances, within the bandwidth and output-driving

capabilities of the op amp (more later). If R F = R 1 = R , we have the simple unity gain inverter of Fig. 27.10.

Figure 27.11 shows a more flexible summer-inverter circuit with

v 0 = –(K 1v 1 + K 2 v 2 + . . . + K n v n )

where K i = R F /R i .

FIGURE 27.7 Typical operational amplifier environment.


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FIGURE 27.8 Conventional operational amplifier

symbol. Only active signal lines are shown, and all sig-

nals are referenced to ground. FIGURE 27.9 Simple resistive inverter-amplifier.

FIGURE 27.10 A simple unity gain inverter, showing (a) detailed circuit; (b) block-diagram symbol.

FIGURE 27.11 The summer-inverter circuit, showing (a) complete circuit; (b) block-diagram symbol.


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The summer-inverter is generally useful for precisely combining or mixing signals, e.g., summing and

inverting. The signal levels must be appropriately limited but may generally be bipolar (+/–).

The resistance values should be in a proper range since (a) too low resistance values draw excessive currentfrom the signal source, and (b) too high resistance values make the circuit performance too sensitive to stray

capacitances and dc offset effects.

Typical values are from 1 MW and 10 k W. The circuit of Fig. 27.12 shows a circuit to implement

v 0 = – 4v 1 – 2v 2

Noninverting Circuits

Figure 27.13(a) shows the useful noninverting amplifier circuit. It has a voltage gain

V 0 /V 1 = (R 2 + R 1)/R 1

= 1 + (R 2/R 1)

Figure 27.13(b) shows the important unity gain follower circuit, which has a very high input impedance, which

lightly loads the signal source but which can provide a reasonable amount of output current milliamps.

It is fairly easy to show that the inverting first-order low-pass filter of Fig. 27.14 has a dc gain or –R 2/R 1 and

a –3-dB frequency = 1/(2pR 2C ).Figure 27.15 shows a two-amplifier differentiator and high-pass filter circuit with a resistive input impedance

and a low-frequency cutoff determined by R 1 and C .

FIGURE 27.12 Simple summer-inverter.

FIGURE 27.13 Noninverting amplifier circuit with resistive elements. (a) General circuit; (b) simple unity gain follower.


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Op amps provide good differential amplifier circuits. Figure 27.16 is a single amplifier circuit with a differ-

ential gain

A d = R 0/R 1

Good resistance matching is required to have good common-mode rejection of unwanted common-mode

signals (static, 60-Hz hum, etc.). The one-amplifier circuit of Fig. 27.16 has a differential input impedance of 2R 1. R 1 may be chosen to provide a good load for a microphone, phono-pickup, etc.

The improved three-amplifier instrumentation amplifier circuit of Fig. 27.17, which several manufacturers

provide in a single module, provides

1. Very high voltage gain

2. Good common-mode rejection

3. A differential gain

4. High input impedance

FIGURE 27.14 First-order low-pass filter circuit.

FIGURE 27.15 A two-amplifier high-pass circuit.


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Operational amplifier circuits form the heart of many precision circuits, e.g., regulated power supplies,precision comparators, peak-detection circuits, and waveform generators [Wait et al., 1992]. Another important

area of application is active RC filters [Huelsman and Allen, 1980]. Microminiature electronic circuits seldom

use inductors. Through the use of op amps, resistors, and capacitors, one can implement precise filter circuits

(low-pass, high-pass, and bandpass). Figures 27.18 and 27.19 show second-order low-pass and bandpass filter

circuits that feature relatively low sensitivity of filter performance to component values. Details are provided

in Wait et al. [1992] and Huelsman and Allen [1980].

FIGURE 27.16 Single-output differential-input amplifier circuit.

FIGURE 27.17 A three-amplifier differential-input instrumentation amplifier featuring high input impedance and easily

adjustable gain.

A R

R

R

R V V d = - +

æ è ç

ö ø ÷

-( )02

12 11

2


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Of course, the op amp does not have infinite bandwidth and gain. An important op-amp parameter is the

unity-gain frequency, f u . For example, it is fairly easy to show the actual bandwidth of a constant gain amplifier

of nominal gain G is approximately

f –3 dB = f u /G

Thus, an op amp with f u = 1 MHz will provide an amplifier gain of 20 up to about 50 kHz.

FIGURE 27.18 Sallen and Key low-pass filter.

FIGURE 27.19 State-variable filter.


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When a circuit designer needs to accurately explore the performance of an op-amp circuit design, modern circuit

simulation programs (SPICE, PSPICE, and MICRO-CAP) permit a thorough study of circuit design, as related to

op-amp performance parameters. We have not here treated nonlinear op-amp performance limitations such as slew

rate, full-power bandwidth, and rated output. Surely, the op-amp circuit designer must be careful not to exceed the

output rating of the op amp, as related to maximum output voltage and current and output rate-of-change.

Nevertheless, op-amp circuits provide the circuit designer with a handy and straightforward way to complete

electronic system designs with the use of only a few basic circuit components plus, of course, the operational

amplifier.

Defining Terms

Active RC filter: An electronic circuit made up of resistors, capacitors, and operational amplifiers that provide

well-controlled linear frequency-dependent functions, e.g., low-, high-, and bandpass filters.

Analog-to-digital converter (ADC): An electronic circuit that receives a magnitude-scaled analog voltage and

generates a binary-coded number proportional to the analog input, which is delivered to an interface

subsystem to a digital computer.

Digital-to-analog converter (DAC): An electronic circuit that receives an n -bit digital word from an interface

circuit and generates an analog voltage proportional to it.

Electronic switch: An electronic circuit that controls analog signals with digital (binary) signals.

Interface: A collection of electronic modules that provide data transfer between analog and digital systems.

Operational amplifier: A small (usually integrated circuit) electronic module with a bipolar (+/–) outputterminal and a pair of differential input terminals. It is provided with power and external components,

e.g., resistors, capacitors, and semiconductors, to make amplifiers, filters, and wave-shaping circuits with

well-controlled performance characteristics, relatively immune to environmental effects.

Related Topic

29.1 Synthesis of Low-Pass Forms

References

Electronic Design, Hasbrook Heights, N.J.: Hayden Publishing Co.; a biweekly journal for electronics engineers.

(In particular, see the articles in the Technology section.)Electronics, New York: McGraw-Hill; a biweekly journal for electronic engineers. (In particular, see the circuit

design features.)

J.G. Graeme, Applications of Operational Amplifiers, New York: McGraw-Hill, 1973.

L.P. Huelsman, and P.E. Allen, Introduction to the Theory and Design of Active Filters. New York: McGraw-Hill,

1980.

J. Till, “Flexible Op-Amp Model Improves SPICE,” Electronic Design, June 22, 1989.

G.E. Tobey, J.G. Graeme, and L.P. Huelsman, Operational Amplifiers, New York: McGraw-Hill, 1971.

J.V. Wait, L.P. Huelsman, and G.A. Korn, Introduction to Operational Amplifier Theory and Applications, 2nd ed.,

New York: McGraw-Hill, 1992.

Further Information

For further information see J.V. Wait, L.P. Huelsman, and G.A. Korn, Introduction to OperationalAmplifier

Theory and Applications, 2nd ed., New York: McGraw-Hill, 1992, a general textbook on the design of operational

amplifier circuits, including the SPICE model of operational amplifiers; and L.P. Huelsman and P.E. Allen,

Introduction to the Theory and Design of Active Filters, New York: McGraw-Hill, 1980, a general textbook of

design considerations and configurations of active RC filters.

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