# Operational amplifier applications

Last updated

This article illustrates some typical operational amplifier applications. A non-ideal operational amplifier's equivalent circuit has a finite input impedance, a non-zero output impedance, and a finite gain. A real op-amp has a number of non-ideal features as shown in the diagram, but here a simplified schematic notation is used, many details such as device selection and power supply connections are not shown. Operational amplifiers are optimised for use with negative feedback, and this article discusses only negative-feedback applications. When positive feedback is required, a comparator is usually more appropriate. See Comparator applications for further information.

In electronics, a comparator is a device that compares two voltages or currents and outputs a digital signal indicating which is larger. It has two analog input terminals and and one binary digital output . The output is ideally

A comparator is an electronic component that compares two input voltages. Comparators are closely related to operational amplifiers, but a comparator is designed to operate with positive feedback and with its output saturated at one power rail or the other. An op-amp can be pressed into service as a poorly performing comparator if necessary, but its slew rate will be impaired.

## Practical considerations

### Operational amplifiers parameter requirements

In order for a particular device to be used in an application, it must satisfy certain requirements. The operational amplifier must

• have large open-loop signal gain (voltage gain of 200,000 is obtained in early integrated circuit exemplars), and
• have input impedance large with respect to values present in the feedback network.

With these requirements satisfied, the op-amp is considered ideal, and one can use the method of virtual ground to quickly and intuitively grasp the 'behavior' of any of the op-amp circuits below.

In electronics, a virtual ground is a node of a circuit that is maintained at a steady reference potential, without being connected directly to the reference potential. In some cases the reference potential is considered to be that of the surface of the earth, and the reference node is called "ground" or "earth" as a consequence.

### Component specification

Resistors used in practical solid-state op-amp circuits are typically in the kΩ range. Resistors much greater than 1 MΩ cause excessive thermal noise and make the circuit operation susceptible to significant errors due to bias or leakage currents.

### Input bias currents and input offset

Practical operational amplifiers draw a small current from each of their inputs due to bias requirements (in the case of bipolar junction transistor-based inputs) or leakage (in the case of MOSFET-based inputs).

These currents flow through the resistances connected to the inputs and produce small voltage drops across those resistances. Appropriate design of the feedback network can alleviate problems associated with input bias currents and common-mode gain, as explained below. The heuristic rule is to ensure that the impedance "looking out" of each input terminal is identical.

To the extent that the input bias currents do not match, there will be an effective input offset voltage present, which can lead to problems in circuit performance. Many commercial op-amp offerings provide a method for tuning the operational amplifier to balance the inputs (e.g., "offset null" or "balance" pins that can interact with an external voltage source attached to a potentiometer). Alternatively, 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.

The input offset voltage is a parameter defining the differential DC voltage required between the inputs of an amplifier, especially an operational amplifier (op-amp), to make the output zero.

Operational amplifiers using MOSFET-based input stages have input leakage currents that will be, in many designs, negligible.

The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET), also known as the metal–oxide–silicon transistor (MOS transistor, or MOS), is a type of field-effect transistor that is fabricated by the controlled oxidation of a semiconductor, typically silicon. It has a covered gate, whose voltage determines the conductivity of the device. This ability to change conductivity with the amount of applied voltage can be used for amplifying or switching electronic signals. The MOSFET was invented by Egyptian engineer Mohamed M. Atalla and Korean engineer Dawon Kahng at Bell Labs in November 1959. It is the basic building block of modern electronics, and the most widely manufactured device in history, with an estimated total of 13 sextillion (1.3 × 1022) MOSFETs manufactured between 1960 and 2018.

### Power supply effects

Although power supplies are not indicated in the (simplified) operational amplifier designs below, they are nonetheless present and can be critical in operational amplifier circuit design.

#### Supply noise

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 specified power supply rejection ratio 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 appropriate use of bypass capacitors connected across each power supply pin and ground. When bursts of current are required by a component, the component can bypass the power supply by receiving the current directly from the nearby capacitor (which is then slowly recharged by the power supply).

In electronic systems, power supply rejection ratio (PSRR), also supply-voltage rejection ratio, is a term widely used to describe the capability of an electronic circuit to suppress any power supply variations to its output signal.

A decoupling capacitor is a capacitor used to decouple one part of an electrical network (circuit) from another. Noise caused by other circuit elements is shunted through the capacitor, reducing the effect it has on the rest of the circuit. An alternative name is bypass capacitor as it is used to bypass the power supply or other high impedance component of a circuit.

#### Using power supply currents in the signal path

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 external pushpull amplifier 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. [1]

## Amplifiers

The first example is the differential amplifier, from which many of the other applications can be derived, including the inverting, non-inverting, and summing amplifier, the voltage follower, integrator, differentiator, and gyrator.

### Differential amplifier (difference amplifier)

Amplifies the difference in voltage between its inputs.

The name "differential amplifier" must not be confused with the "differentiator", which is also shown on this page.
The "instrumentation amplifier", which is also shown on this page, is a modification of the differential amplifier that also provides high input impedance.

The circuit shown computes the difference of two voltages, multiplied by some gain factor. The output voltage

${\displaystyle V_{\text{out}}={\frac {\left(R_{\text{f}}+R_{1}\right)R_{\text{g}}}{\left(R_{\text{g}}+R_{2}\right)R_{1}}}V_{2}-{\frac {R_{\text{f}}}{R_{1}}}V_{1}=\left({\frac {R_{1}+R_{\text{f}}}{R_{1}}}\right)\cdot \left({\frac {R_{\text{g}}}{R_{\text{g}}+R_{2}}}\right)V_{2}-{\frac {R_{\text{f}}}{R_{1}}}V_{1}.}$

Or, expressed as a function of the common-mode input Vcom and difference input Vdif:

${\displaystyle V_{\text{com}}=(V_{1}+V_{2})/2;V_{\text{dif}}=V_{2}-V_{1},}$

the output voltage is

${\displaystyle V_{\text{out}}{\frac {R_{1}}{R_{\text{f}}}}=V_{\text{com}}{\frac {R_{1}/R_{\text{f}}-R_{2}/R_{\text{g}}}{1+R_{2}/R_{\text{g}}}}+V_{\text{dif}}{\frac {1+(R_{2}/R_{\text{g}}+R_{1}/R_{\text{f}})/2}{1+R_{2}/R_{\text{g}}}}.}$

In order for this circuit to produce a signal proportional to the voltage difference of the input terminals, the coefficient of the Vcom term (the common-mode gain) must be zero, or

${\displaystyle R_{1}/R_{\text{f}}=R_{2}/R_{\text{g}}.}$

With this constraint [nb 1] in place, the common-mode rejection ratio of this circuit is infinitely large, and the output

${\displaystyle V_{\text{out}}={\frac {R_{\text{f}}}{R_{1}}}V_{\text{dif}}={\frac {R_{\text{f}}}{R_{1}}}\left(V_{2}-V_{1}\right),}$

where the simple expression Rf / R1 represents the closed-loop gain of the differential amplifier.

The special case when the closed-loop gain is unity is a differential follower, with

${\displaystyle V_{\text{out}}=V_{2}-V_{1}.}$

### Inverting amplifier

An inverting amplifier is a special case of the differential amplifier in which that circuit's non-inverting input V2 is grounded, and inverting input V1 is identified with Vin above. The closed-loop gain is Rf / Rin, hence

${\displaystyle V_{\text{out}}=-{\frac {R_{\text{f}}}{R_{\text{in}}}}V_{\text{in}}\!\,}$.

The simplified circuit above is like the differential amplifier in the limit of R2 and Rg very small. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between Rf and Rin.

To intuitively see the gain equation above, calculate the current in Rin:

${\displaystyle i_{\text{in}}={\frac {V_{\text{in}}}{R_{\text{in}}}}}$

then recall that this same current must be passing through Rf, therefore (because V = V+ = 0):

${\displaystyle V_{\text{out}}=-i_{\text{in}}R_{\text{f}}=-V_{\text{in}}{\frac {R_{\text{f}}}{R_{\text{in}}}}}$

A mechanical analogy is a seesaw, with the V node (between Rin and Rf) as the fulcrum, at ground potential. Vin is at a length Rin from the fulcrum; Vout is at a length Rf. When Vin descends "below ground", the output Vout rises proportionately to balance the seesaw, and vice versa. [2]

As the negative input of the op-amp acts as a virtual ground, the input impedance of this circuit is equal to Rin.

### Non-inverting amplifier

A non-inverting amplifier is a special case of the differential amplifier in which that circuit's inverting input V1 is grounded, and non-inverting input V2 is identified with Vin above, with R1R2. Referring to the circuit immediately above,

${\displaystyle V_{\text{out}}=\left(1+{\frac {R_{\text{2}}}{R_{\text{1}}}}\right)V_{\text{in}}\!\,}$.

To intuitively see this gain equation, use the virtual ground technique to calculate the current in resistor R1:

${\displaystyle i_{1}={\frac {V_{\text{in}}}{R_{1}}}\,,}$

then recall that this same current must be passing through R2, therefore:

${\displaystyle V_{\text{out}}=V_{\text{in}}+i_{1}R_{2}=V_{\text{in}}\left(1+{\frac {R_{2}}{R_{1}}}\right)}$

Unlike the inverting amplifier, a non-inverting amplifier cannot have a gain of less than 1.

A mechanical analogy is a class-2 lever, with one terminal of R1 as the fulcrum, at ground potential. Vin is at a length R1 from the fulcrum; Vout is at a length R2 further along. When Vin ascends "above ground", the output Vout rises proportionately with the lever.

The input impedance of the simplified non-inverting amplifier is high:

${\displaystyle Z_{\text{in}}=(1+A_{\text{OL}}B)Z_{\text{dif}}}$

where Zdif is the op-amp's input impedance to differential signals, and AOL is the open-loop voltage gain of the op-amp (which varies with frequency), and B is the feedback factor (the fraction of the output signal that returns to the input). [3] [4] In the case of the ideal op-amp, with AOL infinite and Zdif infinite, the input impedance is also infinite. In this case, though, the circuit will be susceptible to input bias current drift because of the mismatch between the impedances driving the V+ and V op-amp inputs.

The feedback loop similarly decreases the output impedance:

${\displaystyle Z_{\text{out}}={\frac {Z_{\text{OL}}}{1+A_{\text{OL}}B}}}$

where Zout is the output impedance with feedback, and ZOL is the open-loop output impedance. [4]

### Voltage follower (unity buffer amplifier)

Used as a buffer amplifier to eliminate loading effects (e.g., connecting a device with a high source impedance to a device with a low input impedance).

${\displaystyle V_{\text{out}}=V_{\text{in}}\!}$
${\displaystyle Z_{\text{in}}=\infty }$ (realistically, the differential input impedance of the op-amp itself (1 MΩ to 1 TΩ), multiplied by the open-loop gain of the op-amp)

Due to the strong (i.e., unity gain) feedback and certain non-ideal characteristics of real operational amplifiers, this feedback system is prone to have poor stability margins. Consequently, the system may be unstable when connected to sufficiently capacitive loads. In these cases, a lag compensation network (e.g., connecting the load to the voltage follower through a resistor) can be used to restore stability. The manufacturer data sheet 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.

The input and output impedance are affected by the feedback loop in the same way as the non-inverting amplifier, with B=1.

### Summing amplifier

A summing amplifier sums several (weighted) voltages:

${\displaystyle V_{\text{out}}=-R_{\text{f}}\left({\frac {V_{1}}{R_{1}}}+{\frac {V_{2}}{R_{2}}}+\cdots +{\frac {V_{n}}{R_{n}}}\right)}$
• When ${\displaystyle R_{1}=R_{2}=\cdots =R_{n}}$, and ${\displaystyle R_{\text{f}}}$ independent
${\displaystyle V_{\text{out}}=-{\frac {R_{\text{f}}}{R_{1}}}(V_{1}+V_{2}+\cdots +V_{n})\!}$
• When ${\displaystyle R_{1}=R_{2}=\cdots =R_{n}=R_{\text{f}}}$
${\displaystyle V_{\text{out}}=-(V_{1}+V_{2}+\cdots +V_{n})\!}$
• Output is inverted
• Input impedance of the nth input is ${\displaystyle Z_{n}=R_{n}}$ (${\displaystyle V_{-}}$ is a virtual ground)

### Instrumentation amplifier

Combines very high input impedance, high common-mode rejection, low DC offset, and other properties used in making very accurate, low-noise measurements

## Oscillators

### Wien bridge oscillator

Produces a very low distortion sine wave. Uses negative temperature compensation in the form of a light bulb or diode.

## Filters

Operational amplifiers can be used in construction of active filters, providing high-pass, low-pass, band-pass, reject and delay functions. The high input impedance and gain of an op-amp allow straightforward calculation of element values, allowing accurate implementation of any desired filter topology with little concern for the loading effects of stages in the filter or of subsequent stages. However, the frequencies at which active filters can be implemented is limited; when the behavior of the amplifiers departs significantly from the ideal behavior assumed in elementary design of the filters, filter performance is degraded.

## Comparator

An operational amplifier can, if necessary, be forced to act as a comparator. The smallest difference between the input voltages will be amplified enormously, causing the output to swing to nearly the supply voltage. However, it is usually better to use a dedicated comparator for this purpose, as its output has a higher slew rate and can reach either power supply rail. Some op-amps have clamping diodes on the input that prevent use as a comparator. [5]

## Integration and differentiation

### Inverting integrator

The integrator is mostly used in analog computers, analog-to-digital converters and wave-shaping circuits.

Integrates (and inverts) the input signal Vin(t) over a time interval t, t0 < t < t1, yielding an output voltage at time t = t1 of

${\displaystyle V_{\text{out}}(t_{1})=V_{\text{out}}(t_{0})-{\frac {1}{RC}}\int _{t_{0}}^{t_{1}}V_{\text{in}}(t)\,dt,}$

where Vout(t0) represents the output voltage of the circuit at time t = t0. This is the same as saying that the output voltage changes over time t0 < t < t1 by an amount proportional to the time integral of the input voltage:

${\displaystyle -{\frac {1}{RC}}\int _{t_{0}}^{t_{1}}V_{\text{in}}(t)\,dt.}$

This circuit can be viewed as a low-pass electronic filter, one with a single pole at DC (i.e., where ${\displaystyle \omega =0}$) and with gain.

In a practical application one encounters a significant difficulty: unless the capacitor C is periodically discharged, the output voltage will eventually drift outside of the operational amplifier's operating range. This can be due to any combination of:

• The input Vin has a non-zero DC component,
• Input bias current is non-zero,
• Input offset voltage is non-zero. [6]

A slightly more complex circuit can ameliorate the second two problems, and in some cases, the first as well.

Here, the feedback resistor Rf provides a discharge path for capacitor Cf, while the series resistor at the non-inverting input Rn, when of the correct value, alleviates input bias current and common-mode problems. That value is the parallel resistance of Ri and Rf, or using the shorthand notation ||:

${\displaystyle R_{\text{n}}={\frac {1}{{\frac {1}{R_{\text{i}}}}+{\frac {1}{R_{\text{f}}}}}}=R_{\text{i}}||R_{\text{f}}.}$

The relationship between input signal and output signal is now

${\displaystyle V_{\text{out}}(t_{1})=V_{\text{out}}(t_{0})-{\frac {1}{R_{\text{i}}C_{\text{f}}}}\int _{t_{0}}^{t_{1}}V_{\text{in}}(t)\,dt.}$

### Inverting differentiator

Differentiates the (inverted) signal over time:

${\displaystyle V_{\text{out}}=-RC{\frac {dV_{\text{in}}}{dt}},}$

where ${\displaystyle V_{\text{in}}}$ and ${\displaystyle V_{\text{out}}}$ are functions of time.

The transfer function of the inverting differentiator has a single zero in the origin (i.e., where angular frequency ${\displaystyle \omega =0}$). The high-pass characteristics of a differentiating amplifier can lead to stability challenges when the circuit is used in an analog servo loop (e.g., in a PID controller with a significant derivative gain). In particular, as a root locus analysis would show, increasing feedback gain will drive a closed-loop pole toward marginal stability at the DC zero introduced by the differentiator.

## Synthetic elements

### Inductance gyrator

Simulates an inductor (i.e., provides inductance 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. Applications where this circuit may be superior to a physical inductor are simulating a variable inductance or simulating a very large inductance.

This circuit is of limited use in applications relying on the back EMF property of an inductor as this effect will be limited in a gyrator circuit to the voltage supplies of the op-amp.

### Negative impedance converter (NIC)

Creates a resistor 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:

${\displaystyle R_{\text{in}}=-R_{3}{\frac {R_{1}}{R_{2}}}}$

In general, the components ${\displaystyle R_{1}}$, ${\displaystyle R_{2}}$, and ${\displaystyle R_{3}}$ need not be resistors; they can be any component that can be described with an impedance.

## Non-linear

### Precision rectifier

The voltage drop VF 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 VF is compensated and the circuit behaves very nearly as an ideal (super) diode with VF = 0 V.

The circuit has speed limitations at high frequency because of the slow negative feedback and due to the low slew rate of many non-ideal op-amps.

### Logarithmic output

• The relationship between the input voltage Vin and the output voltage Vout is given by:
${\displaystyle V_{\text{out}}=-V_{\text{T}}\ln \left({\frac {V_{\text{in}}}{I_{\text{S}}\,R}}\right)}$
where IS is the saturation current and VT is the thermal voltage.
• If the operational amplifier is considered ideal, the inverting input 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:
${\displaystyle {\frac {V_{\text{in}}}{R}}=I_{\text{R}}=I_{\text{D}}}$
where ID is the current through the diode. As known, the relationship between the current and the voltage for a diode is:
${\displaystyle I_{\text{D}}=I_{\text{S}}\left(e^{\frac {V_{\text{D}}}{V_{\text{T}}}}-1\right).}$
This, when the voltage is greater than zero, can be approximated by:
${\displaystyle I_{\text{D}}\simeq I_{\text{S}}e^{\frac {V_{\text{D}}}{V_{\text{T}}}}.}$
Putting these two formulae together and considering that the output voltage is the negative of the voltage across the diode (Vout = VD), the relationship is proven.

This implementation does not consider temperature stability and other non-ideal effects.

### Exponential output

• The relationship between the input voltage ${\displaystyle V_{\text{in}}}$ and the output voltage ${\displaystyle V_{\text{out}}}$ is given by:
${\displaystyle V_{\text{out}}=-RI_{\text{S}}e^{\frac {V{\text{in}}}{V_{\text{T}}}}}$

where ${\displaystyle I_{\text{S}}}$ is the saturation current and ${\displaystyle V_{\text{T}}}$ is the thermal voltage.

• Considering the operational amplifier ideal, then the negative pin is virtually grounded, so the current through the diode is given by:
${\displaystyle I_{\text{D}}=I_{\text{S}}\left(e^{\frac {V_{\text{D}}}{V_{\text{T}}}}-1\right)}$

when the voltage is greater than zero, it can be approximated by:

${\displaystyle I_{\text{D}}\simeq I_{\text{S}}e^{\frac {V_{\text{D}}}{V_{\text{T}}}}.}$

The output voltage is given by:

${\displaystyle V_{\text{out}}=-RI_{\text{D}}.\,}$

## Notes

1. If you think of the left-hand side of the relation as the closed-loop gain of the inverting input, and the right-hand side as the gain of the non-inverting input, then matching these two quantities provides an output insensitive to the common-mode voltage of ${\displaystyle V_{1}}$ and ${\displaystyle V_{2}}$.

## Related Research Articles

An operational amplifier is a DC-coupled high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. In this configuration, an op-amp produces an output potential that is typically hundreds of thousands of times larger than the potential difference between its input terminals. Operational amplifiers had their origins in analog computers, where they were used to perform mathematical operations in many linear, non-linear, and frequency-dependent circuits.

In electronics, gain is a measure of the ability of a two-port circuit to increase the power or amplitude of a signal from the input to the output port by adding energy converted from some power supply to the signal. It is usually defined as the mean ratio of the signal amplitude or power at the output port to the amplitude or power at the input port. It is often expressed using the logarithmic decibel (dB) units. A gain greater than one, that is amplification, is the defining property of an active component or circuit, while a passive circuit will have a gain of less than one.

A Negative-feedback amplifier is an electronic amplifier that subtracts a fraction of its output from its input, so that negative feedback opposes the original signal. The applied negative feedback can improve its performance and reduces sensitivity to parameter variations due to manufacturing or environment. Because of these advantages, many amplifiers and control systems use negative feedback.

An instrumentationamplifier is a type of differential amplifier that has been outfitted with input buffer amplifiers, which eliminate the need for input impedance matching and thus make the amplifier particularly suitable for use in measurement and test equipment. Additional characteristics include very low DC offset, low drift, low noise, very high open-loop gain, very high common-mode rejection ratio, and very high input impedances. Instrumentation amplifiers are used where great accuracy and stability of the circuit both short and long-term are required.

A differential amplifier is a type of electronic amplifier that amplifies the difference between two input voltages but suppresses any voltage common to the two inputs. It is an analog circuit with two inputs and and one output in which the output is ideally proportional to the difference between the two voltages

In electronics, a Schmitt trigger is a comparator circuit with hysteresis implemented by applying positive feedback to the noninverting input of a comparator or differential amplifier. It is an active circuit which converts an analog input signal to a digital output signal. The circuit is named a "trigger" because the output retains its value until the input changes sufficiently to trigger a change. In the non-inverting configuration, when the input is higher than a chosen threshold, the output is high. When the input is below a different (lower) chosen threshold the output is low, and when the input is between the two levels the output retains its value. This dual threshold action is called hysteresis and implies that the Schmitt trigger possesses memory and can act as a bistable multivibrator. There is a close relation between the two kinds of circuits: a Schmitt trigger can be converted into a latch and a latch can be converted into a Schmitt trigger.

A buffer amplifier is one that provides electrical impedance transformation from one circuit to another, with the aim of preventing the signal source from being affected by whatever currents that the load may be produced with. The signal is 'buffered from' load currents. Two main types of buffer exist: the voltage buffer and the current buffer.

In electronics, a common-emitter amplifier is one of three basic single-stage bipolar-junction-transistor (BJT) amplifier topologies, typically used as the voltage amplifier.

The open-loop gain of an amplifier is the gain obtained when no overall feedback is used in the circuit. Open loop gain, in some amplifiers, can be exceedingly high. An ideal operational amplifier (op-amp) has infinite open-loop gain. Typically an op-amp may have a maximal open-loop gain of around . The very high open-loop gain of the op-amp allows a wide range of feedback levels to be applied to achieve the desired performance.

The negative impedance converter (NIC) is a one-port op-amp circuit acting as a negative load which injects energy into circuits in contrast to an ordinary load that consumes energy from them. This is achieved by adding or subtracting excessive varying voltage in series to the voltage drop across an equivalent positive impedance. This reverses the voltage polarity or the current direction of the port and introduces a phase shift of 180° (inversion) between the voltage and the current for any signal generator. The two versions obtained are accordingly a negative impedance converter with voltage inversion (VNIC) and a negative impedance converter with current inversion (INIC). The basic circuit of an INIC and its analysis is shown below.

Parasitic capacitance, or stray capacitance is an unavoidable and usually unwanted capacitance that exists between the parts of an electronic component or circuit simply because of their proximity to each other. When two electrical conductors at different voltages are close together, the electric field between them causes electric charge to be stored on them; this effect is parasitic capacitance. All actual circuit elements such as inductors, diodes, and transistors have internal capacitance, which can cause their behavior to depart from that of 'ideal' circuit elements. Additionally, there is always non-zero capacitance between any two conductors; this can be significant at higher frequencies with closely spaced conductors, such as wires or printed circuit board traces. Parasitic capacitance is a significant problem in high frequency circuits and is often the factor limiting the operating frequency and bandwidth of electronic components and circuits.

The operational transconductance amplifier (OTA) is an amplifier whose differential input voltage produces an output current. Thus, it is a voltage controlled current source (VCCS). There is usually an additional input for a current to control the amplifier's transconductance. The OTA is similar to a standard operational amplifier in that it has a high impedance differential input stage and that it may be used with negative feedback.

In electronics, a differentiator is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change of the input. A true differentiator cannot be physically realized, because it has infinite gain at infinite frequency. A similar effect can be achieved, however, by limiting the gain above some frequency.
An active differentiator includes some form of amplifier, while a passive differentiator is made only of resistors, capacitors and inductors.

A fully differential amplifier (FDA) is a DC-coupled high-gain electronic voltage amplifier with differential inputs and differential outputs. In its ordinary usage, the output of the FDA is controlled by two feedback paths which, because of the amplifier's high gain, almost completely determine the output voltage for any given input.

A log amplifier is an amplifier for which the output voltage Vout is K times the natural log of the input voltage Vin. This can be expressed as,

The Miller theorem refers to the process of creating equivalent circuits. It asserts that a floating impedance element, supplied by two voltage sources connected in series, may be split into two grounded elements with corresponding impedances. There is also a dual Miller theorem with regards to impedance supplied by two current sources connected in parallel. The two versions are based on the two Kirchhoff's circuit laws.

In electronics, a transimpedance amplifier, (TIA) is a current to voltage converter, almost exclusively implemented with one or more operational amplifiers. The TIA can be used to amplify the current output of Geiger–Müller tubes, photo multiplier tubes, accelerometers, photo detectors and other types of sensors to a usable voltage. Current to voltage converters are used with sensors that have a current response that is more linear than the voltage response. This is the case with photodiodes where it is not uncommon for the current response to have better than 1% nonlinearity over a wide range of light input. The transimpedance amplifier presents a low impedance to the photodiode and isolates it from the output voltage of the operational amplifier. In its simplest form a transimpedance amplifier has just a large valued feedback resistor, Rf. The gain of the amplifier is set by this resistor and because the amplifier is in an inverting configuration, has a value of -Rf. There are several different configurations of transimpedance amplifiers, each suited to a particular application. The one factor they all have in common is the requirement to convert the low-level current of a sensor to a voltage. The gain, bandwidth, as well as current and voltage offsets change with different types of sensors, requiring different configurations of transimpedance amplifiers.

The operational amplifier integrator is an electronic integration circuit. Based on the operational amplifier (op-amp), it performs the mathematical operation of integration with respect to time; that is, its output voltage is proportional to the input voltage integrated over time.

## References

1. Paul Horowitz and Winfield Hill, The Art of Electronics . 2nd ed. Cambridge University Press, Cambridge, 1989 ISBN   0-521-37095-7
2. Basic Electronics Theory, Delton T. Horn, 4th ed. McGraw-Hill Professional, 1994, p. 342343.
3. "Benefits of Negative Feedback". HyperPhysics. Retrieved 2018-05-07.
4. Simpson, Robert E. (1987). "7.2 Negative Voltage Feedback". Introductory electronics for scientists and engineers (2nd ed.). Boston: Allyn and Bacon. p. 291. ISBN   0205083773. OCLC   13821010.
5. "AN1177 Op-Amp Precision Design: DC Errors" (PDF). Microchip. 2 January 2008. Archived (PDF) from the original on 2013-01-11. Retrieved 26 December 2012.