Download Classification of Operational Amplifiers

Operational amplifier
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Various op-amp ICs in eight-pin dual in-line packages ("DIPs")
An operational amplifier, which is often called an op-amp, is a DC-coupledhigh-gain electronic
voltage amplifier with differential inputs and, usually, a single output.[1]. Typically the op-amp's very large
gain is controlled bynegative feedback, which largely determines the magnitude of its output ("closedloop") voltage gain in amplifier applications, or the transfer functionrequired (in analog computers).
Without negative feedback, and perhaps withpositive feedback for regeneration, an op-amp essentially
acts as acomparator. High input impedance at the input terminals (ideally infinite) and low output
impedance at the output terminal(s) (ideally zero) are important typical characteristics.
Op-amps are among the most widely used electronic devices today, being used in a vast array of
consumer, industrial, and scientific devices. Many standard IC op-amps cost only a few cents in moderate
production volume; however some integrated or hybrid operational amplifiers with special performance
specifications may cost over $100 US in small quantities. Op-amps sometimes come in the form of
macroscopic components, (see photo) or as integrated circuit cells; patterns that can be reprinted several
times on one chip as part of a more complex device.
Modern designs are electronically more rugged than earlier implementations and some can sustain
direct short circuits on their outputs without damage.
The op-amp is one type of differential amplifier. Other types of differential amplifier include the fully
differential amplifier (similar to the op-amp, but with two outputs), the instrumentation amplifier (usually
built from three op-amps), the isolation amplifier (similar to the instrumentation amplifier, but which works
fine with common-mode voltages that would destroy an ordinary op-amp), and negative feedback
amplifier (usually built from one or more op-amps and a resistive feedback network).
Contents
[hide]

1 Circuit notation

2 Operation
o
2.1 Golden rules of op-amp negative feedback
o
2.2 Real and Ideal op-amps

3 History
o
3.1 1941: First (vacuum tube) op-amp
o
3.2 1947: First op-amp with an explicit non-inverting input
o
3.3 1948: First chopper-stabilized op-amp
o
3.4 1961: First discrete IC op-amps
o
3.5 1962: First op-amps in potted modules
o
3.6 1963: First monolithic IC op-amp
o
3.7 1968: Release of the μA741 – would be seen as a nearly ubiquitous chip
o
3.8 1966: First varactor bridge op-amps
o
3.9 1970: First high-speed, low-input current FET design
o
3.10 1972: Single sided supply op-amps being produced
o
3.11 Recent trends

4 Classification of Operational Amplifiers

5 Applications
o
5.1 Use in electronics system design
o
5.2 Positive feedback configurations
o
5.3 Basic single stage amplifiers

5.3.1 Non-inverting amplifier

5.3.2 Inverting amplifier
o
5.4 Other applications

6 Limitations of real op-amps
o
6.1 DC imperfections
o
6.2 AC imperfections
o
6.3 Nonlinear imperfections
o
6.4 Power considerations

7 Internal circuitry of 741 type op-amp
o
7.1 Input stage

7.1.1 Constant-current stabilization system

7.1.2 Differential amplifier
o
7.2 Class A gain stage
o
7.3 Output bias circuitry
o
7.4 Output stage

8 See also

9 Notes

10 References

11 External links
[edit]Circuit
notation
Circuit diagram symbol for an op-amp
The circuit symbol for an op-amp is shown to the right, where:

: non-inverting input

: inverting input

: output

: positive power supply

: negative power supply
The power supply pins (
and
) can be labeled in different ways (See IC power supply pins).
Despite different labeling, the function remains the same — to provide additional power for amplification
of signal. Often these pins are left out of the diagram for clarity, and the power configuration is described
or assumed from the circuit.
[edit]Operation
The amplifier's differential inputs consist of a
input and a
input, and ideally the op-amp amplifies
only the difference in voltage between the two, which is called the differential input voltage. The output
voltage of the op-amp is given by the equation,
where
is the voltage at the non-inverting terminal,
is the voltage at the inverting terminal
and Gopen-loop is the open-loop gain of the amplifier. (The term open-loop refers to the absence of a
feedback loop from the output to the input.)
With no positive feedback, the op-amp acts as a switch. The inverting input is held at ground (0 V) by the resistor, so if
the Vin applied to the non-inverting input is positive, the output will be maximum positive, and if V in is negative, the
output will be maximum negative. Since there is no feedback from the output to either input, this is an open
loop circuit. The circuit's gain is just the Gopen-loop of the op-amp.
Adding negative feedback via Rf puts us in a different universe. Equilibrium will be established when Vout is just
sufficient to reach around and pull the inverting input to the same voltage as V in. As a simple example, if Vin = 1 V and
Rf = Rg, Vout will be 2 V, the amount required to keep V– at 1 V. Because of the feedback provided by Rf, this is
a closed loop circuit. Its over-all gain Vout / Vin is called the closed-loop gain Gclosed-loop. Because the feedback is
negative, in this case Gclosed-loop is less than the Gopen-loop of the op-amp.
The magnitude of Gopen-loop is typically very large—seldom less than a million—and therefore even a
quite small difference between
and
(a few microvolts or less) will result in amplifier
saturation, where the output voltage goes to either the extreme maximum or minimum end of its
range, which is set approximately by the power supply voltages. Finley's law states that "When the
inverting and non-inverting inputs of an op-amp are not equal, its output is in saturation."
Additionally, the precise magnitude of
Gopen-loop is not well controlled by the manufacturing process,
and so it is impractical to use an operational amplifier as a stand-alone differential amplifier. If linear
operation is desired, negative-feedback must be used, usually achieved by applying a portion of the
output to the inverting input. The feedback enables the output of the amplifier to keep the inputs at
or near the same voltage so that saturation does not occur. Another benefit is that if much negative
feedback is used, the circuit's overall gain and other parameters become determined more by the
feedback network than by the op-amp itself. If the feedback network is made of components with
relatively constant, predictable, values such as resistors, capacitors and inductors, the
unpredictability and inconstancy of the op-amp's parameters (typical of semiconductor devices) do
not seriously affect the circuit's performance.
If no negative feedback is used, the op-amp functions as a switch or comparator.
Positive feedback may be used to introduce hysteresis or oscillation.
Returning to a consideration of linear (negative feedback) operation, the high open-loop gain and
low input leakage current of the op-amp imply two "golden rules" that are highly useful in analysing
linear op-amp circuits.
[edit]Golden
rules of op-amp negative feedback
If there is negative feedback and if the output is not saturated,
1. both inputs are at the same voltage;
2. no current flows in or out of either input.[2]
These rules are true of the ideal op-amp and for practical purposes are true of real op-amps unless
very high-speed or high-precision performance is being contemplated (in which case account must
be taken of things such as input capacitance, input bias currents and voltages, finite speed, and
other op-amp imperfections, discussed in a later section.)
As a consequence of the first rule, the input impedance of the two inputs will be nearly infinite. That
is, even if the open-loop impedance between the two inputs is low, the closed-loop input impedance
will be high because the inputs will be held at nearly the same voltage. This impedance is
considered as infinite for an ideal opamp and is about one megohm in practice.
[edit]Real
and Ideal op-amps
Equivalent circuit of an operational amplifier.
Shown on the right is an equivalent circuit model of an operational amplifier. The main part in the
amplifier is the dependent voltage source that increases in relation to the voltage across
amplifying the voltage difference between
Supply voltages
positive supply
and
and
Rin, thus
.
are used internally to power the dependent voltage source. The
sets an upper bound on the output, and the negative source
sets a lower
bound on the output.
More sophisticated equivalent circuit models can also be constructed which include things such as
input capacitance and input bias. On the other hand, one can imagine an even simpler "ideal" opamp by assuming Rin to be infinite and Rout
= 0.
An ideal op-amp is usually considered the following properties, and they are considered to hold for
any input voltages:
 Infinite open-loop gain (i.e., when doing theoretical analysis, limit should be taken as open loop
gain Gopen-loop goes to infinity)
 Infinite bandwidth (i.e., the frequency magnitude response is flat everywhere with zero phase
shift).
 Infinite input impedance (so, in the diagram,
, and zero current flows from
to
)
 Zero input current (i.e., there is no leakage or bias current into the device)
 Zero input offset voltage (i.e., when the input terminals are shorted so that
is a virtual ground).
, the output
 Infinite slew rate (i.e., the rate of change of the output voltage is unbounded) and power
bandwidth (full output voltage and current available at all frequencies).
 Zero output impedance (i.e., Rout
= 0, and so output voltage does not vary with output current)
 Zero noise
 Infinite Common-mode rejection ratio (CMRR)
 Infinite Power supply rejection ratio for both power supply rails.
Because of these properties, an op-amp can be modeled as a nullor.
[edit]History
GAP/R's K2-W: a vacuum-tube op-amp (1953).
ADI's HOS-050: a high speed hybrid IC op-amp (1979).
An op-amp in a modern DIP package.
[edit]1941:
First (vacuum tube) op-amp
An op-amp, defined as a general-purpose, DC-coupled, high gain, inverting feedbackamplifier, is
first found in US Patent 2,401,779 "Summing Amplifier" filed by Karl D. Swartzel Jr. of Bell labs in
1941. This design used three vacuum tubes to achieve a gain of 90dB and operated on voltage rails
of ±350V. It had a single inverting input rather than differential inverting and non-inverting inputs, as
are common in today's op-amps. Throughout World War II, Swartzel's design proved its value by
being liberally used in the M9 artillery directordesigned at Bell Labs. This artillery director worked
with the SCR584 radar system to achieve extraordinary hit rates (near 90%) that would not have
been possible otherwise.[3]
[edit]1947:
First op-amp with an explicit non-inverting input
In 1947, the operational amplifier was first formally defined and named in a paper by Professor John
R. Ragazzini of Columbia University. In this same paper a footnote mentioned an op-amp design by
a student that would turn out to be quite significant. This op-amp, designed by Loebe Julie, was
superior in a variety of ways. It had two major innovations. Its input stage used a longtailed triode pair with loads matched to reduce drift in the output and, far more importantly, it was the
first op-amp design to have two inputs (one inverting, the other non-inverting). The differential input
made a whole range of new functionality possible, but it would not be used for a long time due to the
rise of the chopper-stabilized amplifier.[4]
[edit]1948:
First chopper-stabilized op-amp
In 1949, Edwin A. Goldberg designed a chopper-stabilized op-amp. This set-up uses a normal opamp with an additional AC amplifier that goes alongside the op-amp. The chopper gets an AC signal
from DC by switching between the DC voltage and ground at a fast rate (60 Hz or 400 Hz). This
signal is then amplified, rectified, filtered and fed into the op-amp's non-inverting input. This vastly
improved the gain of the op-amp while significantly reducing the output drift and DC offset.
Unfortunately, any design that used a chopper couldn't use their non-inverting input for any other
purpose. Nevertheless, the much improved characteristics of the chopper-stabilized op-amp made it
the dominant way to use op-amps. Techniques that used the non-inverting input regularly would not
be very popular until the 1960s when op-amp ICs started to show up in the field.
In 1953, vacuum tube op-amps became commercially available with the release of the model K2-W
from George A. Philbrick Researches, Incorporated. The designation on the devices shown, GAP/R,
is a contraction for the complete company name. Two nine-pin 12AX7 vacuum tubes were mounted
in an octal package and had a model K2-P chopper add-on available that would effectively "use up"
the non-inverting input. This op-amp was based on a descendant of Loebe Julie's 1947 design and,
along with its successors, would start the widespread use of op-amps in industry.
[edit]1961:
First discrete IC op-amps
GAP/R's model P45: a solid-state, discrete op-amp (1961).
With the birth of the transistor in 1947, and the silicon transistor in 1954, the concept of ICs became
a reality. The introduction of the planar process in 1959 made transistors and ICs stable enough to
be commercially useful. By 1961, solid-state, discrete op-amps were being produced. These op-
amps were effectively small circuit boards with packages such as edge-connectors. They usually
had hand-selected resistors in order to improve things such as voltage offset and drift. The P45
(1961) had a gain of 94 dB and ran on ±15 V rails. It was intended to deal with signals in the range
of ±10 V.
[edit]1962:
First op-amps in potted modules
GAP/R's model PP65: a solid-state op-amp in a potted module (1962).
By 1962, several companies were producing modular potted packages that could be plugged
into printed circuit boards.[citation needed] These packages were crucially important as they made the
operational amplifier into a single black box which could be easily treated as a component in a larger
circuit.
[edit]1963:
First monolithic IC op-amp
In 1963, the first monolithic IC op-amp, the μA702 designed by Bob Widlar at Fairchild
Semiconductor, was released. Monolithic ICs consist of a single chip as opposed to a chip and
discrete parts (a discrete IC) or multiple chips bonded and connected on a circuit board (a hybrid
IC). Almost all modern op-amps are monolithic ICs; however, this first IC did not meet with much
success. Issues such as an uneven supply voltage, low gain and a small dynamic range held off the
dominance of monolithic op-amps until 1965 when the μA709[5] (also designed by Bob Widlar) was
released.
[edit]1968:
Release of the μA741 – would be seen as a nearly ubiquitous
chip
The popularity of monolithic op-amps was further improved upon the release of the LM101 in 1967,
which solved a variety of issues, and the subsequent release of the μA741 in 1968. The μA741 was
extremely similar to the LM101 except that Fairchild's facilities allowed them to include a 30 pF
compensation capacitor inside the chip instead of requiring external compensation. This simple
difference has made the 741the canonical op-amp and many modern amps base their pinout on the
741s.The μA741 is still in production, and has become ubiquitous in electronics—many
manufacturers produce a version of this classic chip, recognizable by part numbers containing 741.
[edit]1966:
First varactor bridge op-amps
Since the 741, there have been many different directions taken in op-amp design. Varactor bridge
op-amps started to be produced in the late 1960s; they were designed to have extremely small input
current and are still amongst the best op-amps available in terms of common-mode rejection with
the ability to correctly deal with hundreds of volts at their inputs.
[edit]1970:
First high-speed, low-input current FET design
In the 1970s high speed, low-input current designs started to be made by using FETs. These would
be largely replaced by op-amps made withMOSFETs in the 1980s. During the 1970s single sided
supply op-amps also became available.
[edit]1972:
Single sided supply op-amps being produced
A single sided supply op-amp is one where the input and output voltages can be as low as the
negative power supply voltage instead of needing to be at least two volts above it. The result is that
it can operate in many applications with the negative supply pin on the op-amp being connected to
the signal ground, thus eliminating the need for a separate negative power supply.
The LM324 (released in 1972) was one such op-amp that came in a quad package and became an
industry standard. In addition to packaging multiple op-amps in a single package, the 1970s also
saw the birth of op-amps in hybrid packages. These op-amps were generally improved versions of
existing monolithic op-amps. As the properties of monolithic op-amps improved, the more complex
hybrid ICs were quickly relegated to systems that are required to have extremely long service lives
or other specialty systems.
[edit]Recent
trends
Recently supply voltages in analog circuits have decreased (as they have in digital logic) and lowvoltage opamps have been introduced reflecting this. Supplies of ±5V and increasingly 5V are
common. To maximize the signal range modern op-amps commonly have rail-to-rail inputs (the
input signals can range from the lowest supply voltage to the highest) and sometimes rail-to-rail
outputs.
[edit]Classification
of Operational Amplifiers
Op-amps may be classified by their construction:
 discrete (built from individual transistors or tubes/valves)
 IC (fabricated in an Integrated circuit) - most common
 hybrid
IC op-amps may be classified in many ways, including:
 Military, Industrial, or Commercial grade (for example: the LM301 is the commercial grade
version of the LM101, the LM201 is the industrial version). This may define operating
temperature ranges and other environmental or quality factors.
 Classification by package type may also affect environmental hardiness, as well as
manufacturing options; DIP, and other through-hole packages are tending to be replaced
by Surface-mount devices.
 Classification by internal compensation: op-amps may suffer from high frequency instability in
some negative feedback circuits unless a small compensation capacitor modifies the phase-
and frequency- responses; op-amps with capacitor built in are termed compensated, or
perhaps compensated for closed-loop gains down to (say) 5, others: uncompensated.
 Single, dual and quad versions of many commercial op-amp IC are available, meaning 1, 2 or 4
operational amplifiers are included in the same package.
 Rail-to-rail input (and/or output) op-amps can work with input (and/or output) signals very close to
the power supply rails.
 CMOS op-amps (such as the CA3140E) provide extremely high input resistances, higher
than JFET-input op-amps, which are normally higher than bipolar-input op-amps.
 other varieties of op-amp include programmable op-amps (simply meaning the quiescent current,
gain, bandwidth and so on can be adjusted slightly by an external resistor).
 manufacturers often tabulate their op-amps according to purpose, such as low-noise preamplifiers, wide bandwidth amplifiers, and so on.
[edit]Applications
DIP pinout for 741-type operational amplifier
Main article: Operational amplifier applications
[edit]Use in electronics system design
The use of op-amps as circuit blocks is much easier and clearer than specifying all their individual
circuit elements (transistors, resistors, etc.), whether the amplifiers used are integrated or discrete.
In the first approximation op-amps can be used as if they were ideal differential gain blocks; at a
later stage limits can be placed on the acceptable range of parameters for each op-amp.
Circuit design follows the same lines for all electronic circuits. A specification is drawn up governing
what the circuit is required to do, with allowable limits. For example, the gain may be required to be
100 times, with a tolerance of 5% but drift of less than 1% in a specified temperature range; the
input impedance not less than one megohm; etc.
A basic circuit is designed, often with the help of circuit modeling (on a computer). Specific
commercially available op-amps and other components are then chosen that meet the design
criteria within the specified tolerances at acceptable cost. If not all criteria can be met, the
specification may need to be modified.
A prototype is then built and tested; changes to meet or improve the specification, alter functionality,
or reduce the cost, may be made.
[edit]Positive
feedback configurations
Another typical configuration of op-amps is the positive feedback, which takes a fraction of the
output signal back to the non-inverting input. An important application of it is the comparator with
hysteresis (i.e., the Schmitt trigger).
[edit]Basic
single stage amplifiers
[edit]Non-inverting amplifier
An op-amp connected in the non-inverting amplifier configuration
The gain equation for the op-amp is:
However, in this circuit V– is a function of Vout because of the negative feedback through
the R1R2 network.
R1 and R2 form a voltage divider with reduction factor
Since the V– input is a high-impedance input, it does not load the voltage divider
appreciably, so:
Substituting this into the gain equation, we obtain:
Solving for Vout:
If Gopen-loop is very large, this simplifies to
.
[edit]Inverting amplifier
Because it does not require a differential input, this negative
feedback connection was the most typical use of an op-amp in the
days of analog computers.[citation needed] It remains very popular.[citation
needed]
An op-amp connected in the inverting amplifier configuration
This circuit is easily analysed with the help of the two "golden
rules".
Since the non-inverting input is grounded, rule 1 tells us that the
inverting input will also be at ground potential (0 Volts):
The current through Rin is then:
Rule 2 tells us that no current enters the inverting input.
Then, by Kirchoff's current lawthe current
through Rf must be the same as the current through Rin.
The voltage drop across Rf is then given by Ohm's law:
Since V- is zero volts, Vout is just − VRf:
[6]

Some Variations:

A resistor is often inserted between
the non-inverting input and ground
(so both inputs "see" similar
resistances), reducing the input offset
voltage due to different voltage drops
due to bias current, and may reduce
distortion in some op-amps.

A DC-blocking capacitor may be
inserted in series with the input
resistor when a frequency
response down to DC is not needed
and any DC voltage on the input is
unwanted. That is, the capacitive
component of the input impedance
inserts a DC zero and a lowfrequencypole that gives the circuit
a bandpass or highpass characteristic.
[edit]Other

applications
audio- and video-frequency preamplifiers and buffers

voltage comparators

differential amplifiers

differentiators and integrators

filters

precision rectifiers

precision peak detectors

voltage and current regulators

analog calculators

analog-to-digital converters

digital-to-analog converter

voltage clamps

oscillators and waveform generators
Most single, dual and quad op-amps available
have a standardized pin-out which permits
one type to be substituted for another without
wiring changes. A specific op-amp may be
chosen for its open loop gain, bandwidth,
noise performance, input impedance, power
consumption, or a compromise between any
of these factors.
[edit]Limitations
of real op-amps
Real op-amps differ from the ideal model in
various respects.
IC op-amps as implemented in practice are
moderately complex integrated circuits; see
the internal circuitry for the relatively simple
741 op-amp below, for example.
[edit]DC
imperfections
Real operational amplifiers suffer from
several non-ideal effects:
Finite gain
Open-loop gain is infinite in the ideal operational amplifier but finite in real operational amplifiers.
Typical devices exhibit open-loop DC gain ranging from 100,000 to over 1 million. So long as
the loop gain (i.e., the product of open-loop and feedback gains) is very large, the circuit gain will
be determined entirely by the amount of negative feedback (i.e., it will be independent of openloop gain). In cases where closed-loop gain must be very high, the feedback gain will be very low,
and the low feedback gain causes low loop gain; in these cases, the operational amplifier will
cease to behave ideally.
Finite input impedance
The input impedance of the operational amplifier is defined as the impedance between its two
inputs. It is not the impedance from each input to ground. In the typical high-gain negativefeedback applications, the feedback ensures that the two inputs sit at the same voltage, and so
the impedance between them is made artificially very high. Hence, this parameter is rarely an
important design parameter. BecauseMOSFET-input operational amplifiers often have protection
circuits that effectively short circuit any input differences greater than a small threshold, the input
impedance can appear to be very low in some tests. However, as long as these operational
amplifiers are used in a typical high-gain negative feedback application, these protection circuits
will be inactive and the negative feedback will render the input impedance to be practically
infinite. The input bias and leakage currents described below are a more important design
parameter for typical operational amplifier applications.
Non-zero output impedance
Low output impedance is important for low resistance loads; for these loads, the voltage drop
across the output impedance of the amplifier will be significant. Hence, the output impedance of
the amplifier reflects the maximum power that can be provided. If the output voltage is fed back
negatively, the output impedance of the amplifier is effectively lowered; thus, in linear
applications, op-amps usually exhibit a very low output impedance indeed. Negative feedback
can not, however, reduce the limitations that Rload in conjunction with Rout place on the maximum
and minimum possible output voltages; it can only reduce output errors within that range.
Low-impedance outputs
typically require
high quiescent (i.e., idle)
current in the output stage
and will dissipate more
power. So low-power designs
may purposely sacrifice lowimpedance outputs.
Input current
Due to biasing requirements or leakage, a small amount of current (typically ~10 nanoamperes
for bipolar op-amps, tens of picoamperes forJFET input stages, and only a few pA
for MOSFET input stages) flows into the inputs. When large resistors or sources with high output
impedances are used in the circuit, these small currents can produce large unmodeled voltage
drops. If the input currents are matched andthe impedance looking out of both inputs
are matched, then the voltages produced at each input will be equal. Because the operational
amplifier operates on the difference between its inputs, these matched voltages will have no
effect (unless the operational amplifier has poorCMRR, which is described below). It is more
common for the input currents (or the impedances looking out of each input) to be slightly
mismatched, and so a small offset voltage can be produced. This offset voltage can create
offsets or drifting in the operational amplifier. It can often be nulled externally; however, many
operational amplifiers include offset null or balance pins and some procedure for using them to
remove this offset. Some operational amplifiers attempt to nullify this offset automatically.
Input offset voltage
This voltage, which is what is required across the op-amp's input terminals to drive the output
voltage to zero,[7][nb 1] is related to the mismatches in input bias current. In the perfect amplifier,
there would be no input offset voltage. However, it exists in actual op-amps because of
imperfections in the differential amplifier that constitutes the input stage of the vast majority of
these devices. Input offset voltage creates two problems: First, due to the amplifier's high voltage
gain, it virtually assures that the amplifier output will go into saturation if it is operated without
negative feedback, even when the input terminals are wired together. Second, in a closed loop,
negative feedback configuration, the input offset voltage is amplified along with the signal and this
may pose a problem if high precision DC amplification is required or if the input signal is very
small.[nb 2]
Common mode
gain
A perfect operational amplifier amplifies only the voltage difference between its two inputs,
completely rejecting all voltages that are common to both. However, the differential input stage of
an operational amplifier is never perfect, leading to the amplification of these identical voltages to
some degree. The standard measure of this defect is called the common-mode rejection
ratio (denoted CMRR). Minimization of common mode gain is usually important in non-inverting
amplifiers (described below) that operate at high amplification.
Temperatur
e effects
All parameters change with temperature. Temperature drift of the input offset voltage is especially
important.
Powersupply
rejectio
n
The output of a perfect operational amplifier will be completely independent from ripples that
arrive on its power supply inputs. Every real operational amplifier has a specified power supply
rejection ratio (PSRR) that reflects how well the op-amp can reject changes in its supply voltage.
Copious use of bypass capacitors can improve the PSRR of many devices, including the
operational amplifier.
Dr
ift
Real op-amp parameters are subject to slow change over time and with changes in temperature,
input conditions, etc.
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