About electronics: Operational amplifier applications

Monday, September 17, 2007

Operational amplifier applications

This article illustrates some typical applications of solid-state integrated circuit operational amplifiers. A simplified schematic notation is used, and the reader is reminded that many details such as device selection and power supply connections are not shown.

The resistors used in these configurations are typically in the kΩ range. <1>1 MΩ range resistors cause excessive thermal noise and make the circuit operation susceptible to significant errors due to bias currents.

Note: It is important to realize that the equations shown below, pertaining to each type of circuit, assume that it is an ideal op amp. Those interested in construction of any of these circuits for practical use should consult a more detailed reference. See the External links and References sections.

[]

1 Linear circuit applications
- 1.1 Differential amplifier
  - 1.1.1 Amplified difference
  - 1.1.2 Difference amplifier
- 1.2 Inverting amplifier
- 1.3 Non-inverting amplifier
- 1.4 Voltage follower
- 1.5 Summing amplifier
- 1.6 Integrator
- 1.7 Differentiator
- 1.8 Comparator
- 1.9 Instrumentation amplifier
- 1.10 Schmitt trigger
- 1.11 Inductance gyrator
- 1.12 Zero level detector
- 1.13 Negative impedance converter (NIC)
2 Non-linear configurations
- 2.1 Precision rectifier
- 2.2 Peak detector
- 2.3 Logarithmic output
- 2.4 Exponential output
3 Other applications
4 See also
5 References
6 External links

[] Linear circuit applications

[] Differential amplifier

Differential amplifier

Main article: Differential amplifier

The circuit shown is used for finding the difference of two voltages each multiplied by some constant (determined by the resistors).

The name "differential amplifier" should not be confused with the "differentiator", also shown on this page.

$V_\mathrm{out} = V_2 \left( { \left( R_\mathrm{f} + R_1 \right) R_\mathrm{g} \over \left( R_\mathrm{g} + R_2 \right) R_1} \right) - V_1 \left( {R_\mathrm{f} \over R_1} \right)$

Differential $Z in$ (between the two input pins) = $R 1 + R 2$

[] Amplified difference

Whenever $R 1 = R 2$ and $R f = R g$ ,

$V_\mathrm{out} = {R_\mathrm{f} \over R_1} \left( V_2 - V_1 \right)$

[] Difference amplifier

When $R 1 = R f = R 2 = R g$ :

$V_\mathrm{out} = V_2 - V_1 \,\!$

[] Inverting amplifier

Inverting amplifier

Inverts and amplifies a voltage (multiplies by a negative constant)

$V_\mathrm{out} = -V_\mathrm{in} ( R_\mathrm{f} / R_\mathrm{in} ) \!\$

$Z in = R in$ (because $V -$ is a virtual ground)
A third resistor, of value $R_\mathrm{f} \| R_\mathrm{in} = R_\mathrm{f} R_\mathrm{in} / (R_\mathrm{f} + R_\mathrm{in})$ , added between the non-inverting input and ground, while not necessary, minimizes errors due to input bias currents.

[] Non-inverting amplifier

Non-inverting amplifier

Amplifies a voltage (multiplies by a constant greater than 1)

$V_\mathrm{out} = V_\mathrm{in} \left( 1 + {R_2 \over R_1} \right)$

$Z_\mathrm{in} = \infin$ (realistically, the input impedance of the opamp itself, 1 MΩ to 10 TΩ)
A third resistor, of value $R_\mathrm{f} \| R_\mathrm{in}$ , added between the $V in$ source and the non-inverting input, while not necessary, minimizes errors due to input bias currents.

[] Voltage follower

Voltage follower

Used as a buffer amplifier, to eliminate loading effects or to interface impedances (connecting a device with a high source impedance to a device with a low input impedance)

$V_\mathrm{out} = V_\mathrm{in} \!\$

$Z_\mathrm{in} = \infin$ (realistically, the differential input impedance of the op-amp itself, 1 MΩ to 1 TΩ)

[] Summing amplifier

Summing amplifier

Sums several (weighted) voltages

$V_\mathrm{out} = - R_\mathrm{f} \left( { V_1 \over R_1 } + { V_2 \over R_2 } + \cdots + {V_n \over R_n} \right)$

When $R_1 = R_2 = \cdots = R_n$ , and $R f$ independent

$V_\mathrm{out} = - \left( {R_\mathrm{f} \over R_1} \right) (V_1 + V_2 + \cdots + V_n ) \!\$

When $R_1 = R_2 = \cdots = R_n = R_\mathrm{f}$

$V_\mathrm{out} = - ( V_1 + V_2 + \cdots + V_n ) \!\$

Output is inverted
Input impedance $Z n = R n$ , for each input ( $V -$ is a virtual ground)

[] Integrator

Integrating amplifier

Integrates the (inverted) signal over time

$V_\mathrm{out} = \int_0^t - {V_\mathrm{in} \over RC} \, dt + V_\mathrm{initial}$

(where $V in$ and $V out$ are functions of time, $V initial$ is the output voltage of the integrator at time t = 0.)

Note that this can also be viewed as a type of electronic filter.

[] Differentiator

Differentiating amplifier

Differentiates the (inverted) signal over time.

The name "differentiator" should not be confused with the "differential amplifier", also shown on this page.

$V_\mathrm{out} = - RC \left( {dV_\mathrm{in} \over dt} \right)$

(where $V in$ and $V out$ are functions of time)

Note that this can also be viewed as a type of electronic filter.

[] Comparator

Comparator

Main article: Comparator

Compares two voltages and outputs one of two states depending on which is greater

$\text{[math]}$ V_2 \\ V_\mathrm{S-} & V_1 < src="http://upload.wikimedia.org/math/4/8/d/48dc7cae7c62efe5fe79c917fc700380.png">

[] Instrumentation amplifier

Instrumentation amplifier

Main article: 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

Is made by adding a inverting buffer to each input of the differential amplifier to increase the input impedance.

[] Schmitt trigger

Schmitt trigger

Main article: Schmitt trigger

A comparator with hysteresis

Hysteresis from $\frac{-R_1}{R_2}V_{sat}$ to $\frac{R_1}{R_2}V_{sat}$ .

[] Inductance gyrator

Inductance gyrator

Main article: Gyrator

Simulates an inductor.

[] Zero level detector

Voltage divider reference

Zener sets reference voltage

[] Negative impedance converter (NIC)

Negative impedance converter

Main article: Negative impedance converter

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:

$R_\mathrm{in} = - R_3 \frac{R_1}{R_2}$

for more information see the main article Negative impedance converter.

[] Non-linear configurations

[ Precision rectifier

Super diode

Main article: Precision rectifier

Behaves like an ideal diode for the load, which is here represented by a generic resistor $R L$ .

This basic configuration has some limitations. For more information and to know the configuration that is actually used, see the main article.

[] Peak detector

Peak detector

When the switch is closed, the output goes to zero volts. When the switch is opened for a certain time interval, the capacitor will charge to the maximum input voltage attained during that time interval.

The charging time of the capacitor must be much shorter than the period of the highest appreciable frequency component of the input voltage.

[] Logarithmic output

Logarithmic configuration

The relationship between the input voltage $v in$ and the output voltage $v out$ is given by:

$v_\mathrm{out} = -V_{\gamma} \ln \left( \frac{v_\mathrm{in}}{I_\mathrm{S} \cdot R} \right)$

where $I S$ is the saturation current.

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:

$\frac{v_\mathrm{in}}{R} = I_\mathrm{R} = I_\mathrm{D}$

where $I D$ is the current through the diode. As known, the relationship between the current and the voltage for a diode is:

$I_\mathrm{D} = I_\mathrm{S} \left( e^{\frac{V_\mathrm{D}}{V_{\gamma}}} - 1 \right)$

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

$I_\mathrm{D} \simeq I_\mathrm{S} e^{V_\mathrm{D} \over V_{\gamma}}$

Putting these two formulae together and considering that the output voltage $V out$ is the inverse of the voltage across the diode $V D$ , the relationship is proven.

Note that this implementation does not consider temperature stability and other non-ideal effects.

[] Exponential output

Exponential configuration

The relationship between the input voltage $v in$ and the output voltage $v out$ is given by:

$v_\mathrm{out} = - R I_\mathrm{S} e^{v_\mathrm{in} \over V_{\gamma}}$

where $I S$ is the saturation current.

Considering the operational amplifier ideal, then the negative pin is virtually grounded, so the current through the diode is given by:

$I_\mathrm{D} = I_\mathrm{S} \left( e^{\frac{V_\mathrm{D}}{V_{\gamma}}} - 1 \right)$

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

$I_\mathrm{D} \simeq I_\mathrm{S} e^{V_\mathrm{D} \over V_{\gamma}}$

The output voltage is given by:

$v_\mathrm{out} = -R I_\mathrm{D}\,$

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