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AnÂ operational amplifier, which is often calledanÂ op-ampisaÂ DC-coupledÂ high-gainÂ electronic voltageÂ amplifierÂ with a differential input and, usually, a single-ended output.Â An op-amp produces an output voltage that is typically millions of times larger than the voltage Â differenceÂ between its input terminals.
Typically the op-amp's very large gain is controlled byÂ negative feedback, which largely determines the magnitude of its output ("closed-loop") voltage gain in amplifier applications, or theÂ transfer functionÂ required (inÂ analog computers). Without negative feedback, and perhaps withÂ positive feedbackÂ forÂ regeneration, an op-amp essentially acts as aÂ comparator. High inputÂ impedanceÂ 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 asÂ integrated circuitÂ cells; patterns that can be reprinted several times on one chip as part of a more complex device.
The op-amp is one type ofÂ differential amplifier. Other types of differential amplifier include theÂ fully differential amplifierÂ (similar to the op-amp, but with two outputs), theÂ instrumentation amplifierÂ (usually built from three op-amps), theÂ isolation amplifierÂ (similar to the instrumentation amplifier, but with tolerance to common-mode voltages that would destroy an ordinary op-amp), andÂ negative feedback amplifierÂ (usually built from one or more op-amps and a resistive feedback network).
Various op-ampÂ ICsÂ in eight-pinÂ dual in-line packagesÂ ("DIPs")
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
: positive power supply
: negative power supply
The power supply pins (Â andÂ ) can be labeled in different ways. 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.
The amplifier's differential inputs consist of aÂ Â input and aÂ Â input, and ideally the op-amp amplifies only the difference in voltage between the two, which is called theÂ differential input voltage. The output voltage of the op-amp is given by the equation,
whereÂ Â is the voltage at the non-inverting terminal,Â Â is the voltage at the inverting terminal andÂ Gopen-loopÂ is theÂ open-loopÂ gain of the amplifier. (The term open-loop refers to the absence of a feedback loop from the output to the input.)
The magnitude ofÂ Gopen-loopÂ is typically very large-seldom less than a million-and therefore even a quite small difference betweenÂ Â andÂ (a few micro volts 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.Â Finley's lawÂ 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 ofÂ Gopen-loopÂ is not well controlled by the manufacturing process, and so it is impractical to use an operational amplifier as a stand-aloneÂ differential amplifier. If linear operation is desired,Â negative feedbackÂ must be used, usually achieved by applying a portion of the output voltage 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 introduceÂ hysteresisÂ 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.
Golden rules of op-amp negative feedback
IfÂ there is negative feedback andÂ ifÂ the output is not saturated,
both inputs are at the same voltage;
no current flows in or out of either input.
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 otherÂ op-amp imperfections, discussed in a later section.)
As a consequence of the first rule, theÂ input impedanceÂ 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 op-amp and is about oneÂ megaohmÂ in practice.
With no negative feedback, the op-amp acts as a switch. The inverting input is held at ground (0 V) by the resistor, so if the VinÂ applied to the non-inverting input is positive, the output will be maximum positive, and if VinÂ is negative, the output will be maximum negative. Since there is no feedback from the output to either input, this is anÂ open loopÂ circuit. The circuit's gain is just the Gopen-loopÂ of the op-amp.
Adding negative feedback via RfÂ reduces the gain. Equilibrium will be established when VoutÂ is just sufficient to reach around and pull the inverting input to the same voltage as Vin. As a simple example, if VinÂ = 1â€‰V and RfÂ = Rg, Voutwill be 2â€‰V, the amount required to keep V-Â at 1â€‰V. Because of the feedback provided by Rf, this is aÂ closed loopÂ circuit. Its over-all gain VoutÂ /Â VinÂ is called theÂ closed-loop gainÂ Gclosed-loop. Because the feedback is negative, in this case Gclosed-loopÂ is less than the Gopen-loopÂ of the op-amp.
Ideal and real op-amps
An equivalent circuit of an operational amplifier that models some resistive non-ideal parameters.
An ideal op-amp is usually considered to have the following properties, and they are considered to hold for all input voltages:
InfiniteÂ open-loop gainÂ (when doing theoretical analysis, aÂ limitÂ may be taken as open loop gainÂ GÂ goes to infinity)
Infinite voltage range available at the output (vout) (in practice the voltages available from the output are limited by the supply voltagesÂ Â andÂ )
InfiniteÂ bandwidthÂ (i.e., the frequency magnitude response is considered to be flat everywhere with zeroÂ phase shift).
InfiniteÂ input impedanceÂ (so, in the diagram,Â , and zero current flows fromÂ Â toÂ Â )
Zero input current (i.e., there is assumed to be noÂ leakageÂ orÂ biasÂ current into the device)
ZeroÂ input offset voltageÂ (i.e., when the input terminals are shorted so thatÂ , the output is aÂ virtual groundÂ orvoutÂ = 0).
InfiniteÂ slew rateÂ (i.e., the rate of change of the output voltage is unbounded) and power bandwidth (full output voltage and current available at all frequencies).
ZeroÂ output impedanceÂ (i.e.,Â RoutÂ = 0, so that output voltage does not vary with output current)
InfiniteÂ Common-mode rejection ratioÂ (CMRR)
InfiniteÂ Power supply rejection ratioÂ for both power supply rails.
In practice, none of these ideals can be realized, and various shortcomings and compromises have to be accepted. Depending on the parameters of interest, a real op-amp may be modeled to take account of some of the non-infinite or non-zero parameters using equivalent resistors and capacitors in the op-amp model. The designer can then include the effects of these undesirable, but real, effects into the overall performance of the final circuit. Some parameters may turn out to haveÂ negligibleÂ effect on the final design while others represent actual limitations of the final performance, which must be evaluated.
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 basicÂ circuitÂ 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.
Input offset problems
It is important to note that the equations shown below, pertaining to each type of circuit, assume that anÂ ideal op ampÂ is used. Those interested in construction of any of these circuits for practical use should consult a more detailed reference. See theÂ External linksÂ andÂ Further readingÂ sections.
Resistors used in practical solid-state op-amp circuits are typically in the kÎ© range. Resistors much greater than 1 MÎ© cause excessiveÂ thermal noiseÂ and make the circuit operation susceptible to significant errors due to bias or leakage currents.
Practical operational amplifiers draw a small current from each of their inputs due to bias requirements and leakage. These currents flow through the resistances connected to the inputs and produce small voltage drops across those resistances. In AC signal applications this seldom matters. If high-precision DC operation is required, however, these voltage drops need to be considered. The design technique is to try to ensure that these voltage drops are equal for both inputs, and therefore cancel. If these voltage drops are equal and theÂ common-mode rejection ratioÂ of the operational amplifier is good, there will be considerable cancellation and improvement in DC accuracy.
If the input currents into the operational amplifier are equal, to reduce offset voltage the designer must ensure that the DC resistance looking out of each input is also matched. In general input currents differ, the difference being called theÂ input offset current, Ios. Matched external input resistances RinÂ will still produce an input voltage error ofÂ RinÂ·Ios Most manufacturers provide a method for tuning the operational amplifier to balance the input currents (e.g., "offset null" or "balance" pins that can interact with an external voltage source attached to a potentiometer). Otherwise, a tunable external voltage can be added to one of the inputs in order to balance out the offset effect. In cases where a design calls for one input to be short-circuited to ground, that short circuit can be replaced with a variable resistance that can be tuned to mitigate the offset problem.â€‚Â .
Note that many operational amplifiers that haveÂ MOSFET-based input stages have input leakage currents that will truly be negligible to most designs.
Power supply effects
Although the power supplies are not shown in the operational amplifier designs below, they can be critical in operational amplifier design.
Power supply imperfections (e.g., power signal ripple, non-zero source impedance) may lead to noticeable deviations from ideal operational amplifier behavior. For example, operational amplifiers have a specifiedÂ power supply rejection ratioÂ that indicates how well the output can reject signals that appear on the power supply inputs. Power supply inputs are often noisy in large designs because the power supply is used by nearly every component in the design, and inductance effects prevent current from being instantaneously delivered to every component at once. As a consequence, when a component requires large injections of current (e.g., a digital component that is frequently switching from one state to another), nearby components can experience sagging at their connection to the power supply. This problem can be mitigated with copious use ofÂ bypass capacitorsÂ placed connected across each power supply pin and ground. When bursts of current are required by a component, the component canÂ bypassÂ the power supply by receiving the current directly from the nearby capacitor (which is then slowly charged by the power supply).
Additionally, current drawn into the operational amplifier from the power supply can be used as inputs to external circuitry that augment the capabilities of the operational amplifier. For example, an operational amplifier may not be fit for a particular high-gain application because its output would be required to generate signals outside of the safe range generated by the amplifier. In this case, an externalÂ push-pull amplifierÂ can be controlled by the current into and out of the operational amplifier. Thus, the operational amplifier may itself operate within its factory specified bounds while still allowing the negative feedback path to include a large output signal well outside of those bounds.
Compares two voltages and switches its output to indicate which voltage is larger.
(whereÂ VsÂ is the supply voltage and the opamp is powered byÂ +Â VsÂ andÂ âˆ’Â Vs.)
An inverting amplifier uses negative feedback to invert andÂ amplifyÂ a voltage. The RfÂ resistor allows some of the output signal to be returned to the input. Since the output is 180Â° out of phase, this amount is effectively subtracted from the input, thereby reducing the input into the operational amplifier. This reduces the overall gain of the amplifier and is dubbed negative feedback.
ZinÂ =Â RinÂ (becauseÂ VÂ âˆ’Â is aÂ virtual ground)
A third resistor, of valueÂ , added between the non-inverting input and ground, while not necessary, minimizes errors due to input bias currents.
The gain of the amplifier is determined by the ratio of RfÂ to Rin. That is:
The presence of the negative sign is a convention indicating that the output is inverted. For example, if RfÂ is 10,000 Î© and RinÂ is 1,000 Î©, then the gain would be -10000Î©/1000Î©, which is -10.
Amplifies a voltage (multiplies by a constant greater than 1)
The input impedance isÂ at leastÂ the impedance between non-inverting (Â +Â ) and inverting (Â âˆ’Â ) inputs, which is typically 1 MÎ© to 10 TÎ©, plus the impedance of the path from the inverting (Â âˆ’Â ) input to ground (i.e.,Â R1Â in parallel withÂ R2).
Because negative feedback ensures that the non-inverting and inverting inputs match, the input impedance is actuallyÂ much higher.
Although this circuit has a large input impedance, it suffers from error of input bias current.
The non-inverting (Â +Â ) and inverting (Â âˆ’Â ) inputs draw small leakage currents into the operational amplifier.
These input currents generate voltages that act like unmodeled input offsets. These unmodeled effects can lead to noise on the output (e.g., offsets or drift).
Assuming that the two leaking currents areÂ matched,Â their effect can be mitigated by ensuring the DC impedance lookingÂ outÂ of each input is the same.
The voltage produced by each bias current is equal to the product of the bias current with the equivalent DC impedance looking out of each input. Making those impedances equal makes the offset voltage at each input equal, and so the non-zero bias currents will have no impact on theÂ differenceÂ between the two inputs.
A resistor of value
which is the equivalent resistance ofÂ R1Â in parallel withÂ R2, between theÂ VinÂ source and the non-inverting (Â +Â ) input will ensure the impedances lookingÂ outÂ of each input will be matched.
The matched bias currents will then generate matched offset voltages, and their effect will be hidden to the operational amplifier (which acts on the difference between its inputs) so long as theÂ CMRRÂ is good.
Very often, the input currents areÂ notÂ matched.
Most operational amplifiers provide some method of balancing the two input currents (e.g., by way of an externalÂ potentiometer).
Alternatively, an external offset can be added to the operational amplifier input to nullify the effect.
Another solution is to insert a variable resistor between theÂ VinÂ source and the non-inverting (Â +Â ) input. The resistance can be tuned until the offset voltages at each input are matched.
Operational amplifiers withÂ MOSFET-based input stages have input currents that are so small that they often can be neglected.
The circuit shown is used for finding theÂ differenceÂ of two voltages each multiplied by some constant (determined by the resistors).
DifferentialÂ ZinÂ (between the two input pins) =Â R1Â +Â R2Â (Note: this is approximate)
For common-mode rejection, anything done to one input must be done to the other. The addition of a compensation capacitor in parallel with Rf, for instance, must be balanced by an equivalent capacitor in parallel with Rg.
The "instrumentation amplifier", which is also shown on this page, is another form of differential amplifier that also provides high input impedance.
WheneverÂ Â andÂ , the differential gain is
Â Â and Â Â
WhenÂ Â andÂ Â the differential gain is A = 1 and the circuit acts as a differential follower:
Used as aÂ buffer amplifierÂ to eliminate loading effects (e.g., connecting a device with a highÂ source impedanceÂ to a device with a lowÂ input impedance).
Â (realistically, the differential input impedance of the op-amp itself, 1 MÎ© to 1 TÎ©)
Due to the strong (i.e.,Â unityÂ gain) feedback and certain non-ideal characteristics of real operational amplifiers, this feedback system is prone to have poorÂ stability margins. Consequently, the system may beÂ unstableÂ when connected to sufficiently capacitive loads. In these cases, aÂ lag compensationÂ network (e.g., connecting the load to the voltage follower through a resistor) can be used to restore stability. The manufacturer data sheetÂ for the operational amplifier may provide guidance for the selection of components in external compensation networks. Alternatively, another operational amplifier can be chosen that has more appropriate internal compensation.
A summing amplifer sums several (weighted) voltages:
WhenÂ , andÂ RfÂ independent
Output is inverted
Input impedance of theÂ nthÂ input isÂ ZnÂ =Â RnÂ (VÂ âˆ’Â is aÂ virtual ground)
IntegratesÂ the (inverted) signal over time
(whereÂ VinÂ andÂ VoutÂ are functions of time,Â VinitialÂ is the output voltage of the integrator at timeÂ tÂ = 0.)
Note that this can also be viewed as aÂ low-passÂ electronic filter. It is a filter with a singleÂ poleÂ at DC (i.e., whereÂ Ï‰ = 0) and gain.
There are several potential problems with this circuit.
It is usually assumed that the inputÂ VinÂ has zero DC component (i.e., has a zero average value). Otherwise, unless the capacitor is periodically discharged, the output will drift outside of the operational amplifier's operating range.
Even whenÂ VinÂ has no offset, the leakage or bias currents into the operational amplifier inputs can add an unexpected offset voltage toÂ VinÂ that causes the output to drift. Balancing input currentsÂ andÂ replacing the non-inverting (Â +Â ) short-circuit to ground with a resistor with resistanceÂ RÂ can reduce the severity of this problem.
Because this circuit provides no DC feedback (i.e., the capacitor appears like an open circuit to signals withÂ Ï‰ = 0), the offset of the output may not agree with expectations (i.e.,VinitialÂ may be out of the designer's control with the present circuit).
Many of these problems can be made less severe by adding aÂ largeÂ resistorÂ RFÂ in parallel with the feedback capacitor. At significantly high frequencies, this resistor will have negligible effect. However, at low frequencies where there are drift and offset problems, the resistor provides the necessary feedback to hold the output steady at the correct value. In effect, this resistor reduces the DC gain of the "integrator"Â - it goes from infinite to some finite valueÂ RFÂ /Â R.
DifferentiatesÂ the (inverted) signal over time.
Note that this can also be viewed as aÂ high-passÂ electronic filter. It is a filter with a singleÂ zeroÂ at DC (i.e., whereÂ Ï‰ = 0) and gain. The high pass characteristics of a differentiating amplifier can lead to unstable behavior when the circuit is used in an analog servo loop. For this reason the system function would be re-formulated to use integrators.
Combines very highÂ input impedance, highÂ common-mode rejection, lowÂ DC offset, and other properties used in making very accurate, low-noise measurements
Is made by adding aÂ non-invertingÂ bufferÂ to each input of theÂ differential amplifierÂ to increase the input impedance.
AÂ bistable multivibratorÂ implemented as aÂ comparatorÂ withÂ hysteresis.
Non-inverting Schmitt trigger
In this configuration, the input voltage is applied through the resistorÂ R1Â (which may be the source internal resistance) to the non-inverting input and the inverting input is grounded or referenced. The hysteresis curve is non-inverting and the switching thresholds areÂ Â whereÂ VsatÂ is the greatest output magnitude of the operational amplifier.
Inverting Schmitt trigger
Alternatively, the input source and theÂ groundÂ may be swapped. Now the input voltage is applied directly to the inverting input and the non-inverting input is grounded or referenced. The hysteresis curve is inverting and the switching thresholds areÂ . Such a configuration is used in theÂ relaxation oscillatorÂ shown below.
Relaxation oscillator implemented with inverting Schmitt trigger and RC network
By using anÂ RC networkÂ to add slowÂ negative feedbackÂ to the invertingÂ Schmitt trigger, aÂ relaxation oscillatorÂ is formed. The feedback through the RC network causes the Schmitt trigger output toÂ oscillateÂ in an endless symmetricÂ square waveÂ (i.e., the Schmitt trigger in this configuration is anÂ astable multivibrator).
Simulates anÂ inductorÂ (i.e., providesÂ inductanceÂ without the use of a possibly costly inductor). The circuit exploits the fact that the current flowing through a capacitor behaves through time as the voltage across an inductor. The capacitor used in this circuit is smaller than the inductor it simulates and its capacitance is less subject to changes in value due to environmental changes.
Zero level detector
Voltage divider reference
Zener sets reference voltage
Negative impedance converter (NIC)
Negative impedance converter
Creates aÂ resistorÂ having a negative value for any signal generator
In this case, the ratio between the input voltage and the input current (thus the input resistance) is given by:
In general, the componentsÂ R1,Â R2, andÂ R3Â need not be resistors; they can be any component that can be described with animpedance.
Wien bridge oscillator
Wien bridge oscillator
Produces a pureÂ sine wave.
The voltage drop VFÂ across the forward biased diode in the circuit of a passive rectifier is undesired. In this active version, the problem is solved by connecting the diode in the negative feedback loop. The op-amp compares the output voltage across the load with the input voltage and increases its own output voltage with the value of VF. As a result, the voltage drop VFÂ is compensated and the circuit behaves as an ideal (super)Â diodeÂ with VFÂ = 0 V.
The circuit has speed limitations at high frequency because of the slow negative feedback.
The relationship between the input voltageÂ vinÂ and the output voltageÂ voutÂ is given by:
whereÂ ISÂ is theÂ saturation currentÂ andÂ VTÂ is theÂ thermal voltage.
If the operational amplifier is considered ideal, the negative pin is virtually grounded, so the current flowing into the resistor from the source (and thus through the diode to the output, since the op-amp inputs draw no current) is:
whereÂ IDÂ is the current through the diode. As known,Â the relationship between the current and the voltageÂ for aÂ diodeÂ is:
This, when the voltage is greater than zero, can be approximated by:
Putting these two formulae together and considering that the output voltage is the negative of the voltage across the diode (VoutÂ = âˆ’Â VD), the relationship is proven.
Note that this implementation does not consider temperature stability and other non-ideal effects.
The relationship between the input voltageÂ vinÂ and the output voltageÂ voutÂ is given by:
whereÂ ISÂ is theÂ saturation currentÂ andÂ VTÂ is theÂ thermal voltage.
Considering the operational amplifier ideal, then the negative pin is virtually grounded, so the current through the diode is given by:
when the voltage is greater than zero, it can be approximated by:
The output voltage is given by:
1. An operational amplifier, comprising: a dynamic biasing stage operable to convert a differential input voltage to a dynamic drive current; an input stage responsively coupled to the differential input voltage and the dynamic drive current, said input stage configured as a folded cascode circuit including a differential pair of transistors; an output gain stage responsively coupled to the differential pair of transistors and having a pair of bipolar output transistors, the output transistors having emitters each having a load resistor and being coupled to an output; and a compensation capacitor coupled to the output of the output gain stage and directly coupled to a supply voltage.Â
2. The amplifier as specified in claim 1 wherein the output gain stage transistors each have a base emitter voltage and a collector current, wherein the collector currents increase exponentially as the respective base emitter voltage increases.Â
3. The amplifier as specified in claim 1 wherein the output transistors are about half the size as the differential pair of transistors.Â
4. The amplifier as specified in claim 1 further comprising an output buffer coupled to an amplifier output.Â
5. An operational amplifier, comprising: a dynamic biasing stage operable to convert a differential input voltage to a dynamic drive current; an input stage responsively coupled to the differential input voltage and the dynamic drive current, said input stage configured as a folded cascode circuit including a differential pair of transistors; an output gain stage responsively coupled to the input stage and having a pair of transistors each having a base emitter voltage and a collector current, wherein the collector currents increase exponentially as the respective base emitter voltage increases; and a compensation capacitor coupled to an output of the output gain stage and directly coupled to the supply voltage.Â
6. The amplifier as specified in claim 5 wherein the output gain stage transistors each have a load resistor.Â
7. The amplifier as specified in claim 5 wherein the output gain stage transistors are about half the size as the differential pair of transistors.Â
8. The amplifier as specified in claim 5 further comprising an output buffer coupled to an amplifier output.Â