Operational amplifier

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Commonly known as op-amp.

A discrete amplifier useful for a variety of circuit operations. It takes two inputs, subtracts one from the other and amplifies the result.

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Ideal Operational Amplifier

The ideal op-amp has infinite gain, infinite input impedance (does not draw any current from its inputs) and has zero output impedance. It's also perfectly linear and has infinite bandwidth.

An op-amp can be directly coupled to other op-amps.

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Real-world Operational Amplifiers

Real op-amps, of course, aren't perfect like the ideal but they are pretty good and the ideal can be used as a theoretical model for understanding.

A good comparison would be the LM741, the general purpose op-amp. Other specialised op*amps are usually available that are better in one aspect or other.

Typical LM741 Characteristics:

Open Loop Gain = 200 000
Input Impedance = 2 MΩ
Bandwidth = 1.5 MHz

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Basic Op-Amp Operation

An op-amp takes two inputs and multiplies the difference between them by the open loop gain. However, an op-amp is not usually used directly like this because this open loop gain is way too big for normal use and isn't very stable (it varies a lot depending on temperature and general mood of the op-amp).

The high gain does come as an advantage. If a sample of the output fed back to cancel out some of the input, the effective gain is reduced, but the gain becomes much more stable. This is called negative feedback and the more gain you have to start off with, the more stable you can make the effective gain.

The two main op-amp circuits are the inverting and non-inverting amplifiers, given further down the page.

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Applications of Op-Amps

With cheap ICs, op-amps have become building blocks almost as fundamental as resistors and capacitors themselves.

Aside from obvious audio applications, they are essential in all areas of modern analogue signal processing. They are used for filtering, waveshaping, etc.

Since op-amps draw very little load from their inputs, a buffer op-amp is used on practically any input from an outside measurement device, like a microphone or peltier element.

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Op-Amp Circuit Analysis

It isn't always necessary to be able to fully analyse an op-amp circuit since most of the time a common circuit like one on this page will be used. Knowing some techniques for analysing op amps can still be useful, if only for memorising and understanding the formulae.

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Finite Gain, Ideal Impedance Model

In this model the output voltage is given by:

V o u t = G (V + - V -)

Where $G$ is the open loop gain (very large) and $V +, V -$ are the non-inverting and inverting inputs respectively.

Take the non-inverting amplifier. In this circuit $V +$ is directly connected to the input voltage, $V i n$ and part of the output is returned to $V -$ through a voltage divider.

$V_{out} = G(V_+ - V_-) = G (V_{in} - \frac{V_{out}R_i}{R_i+R_f})$

Therefore,

$\frac{V_{out}}{G} = V_{in} - \frac{V_{out}R_i}{R_i+R_f}$

Therefore,

$V_{out}(\frac{1}{G} + \frac{R_i}{R_i+R_f}) = V_{in}$

Therefore,

$\frac{V_{out}}{V_{in}} = \frac{1}{\frac{1}{G} + \frac{R_i}{R_i+R_f}}=\frac{G(R_i+R_f)}{R_i+R_f+GR_i}$

However, in practice, since G is very large, the $1 / G$ term in the second last expression is approximated to zero and the whole cancels to

$\frac{V_{out}}{V_{in}} = 1 + \frac{R_f}{R_i}$

Which could be more easily gained by assuming $1 / G$ to be insignificant from the outset, as in the next model.

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Infinite Gain, Ideal Impedance Model

Taking the equation for the finite gain model and dividing through by the open loop gain, $G$ we have:

V o u t / G = V + - V -

So that if $G$ is considered ridiculously large,

V + - V - = 0

Ie.

V + = V -

This makes a lot of op-amp circuits much simpler to understand and is a very powerful approximation. It doesn't always work for some overdriven circuits, however.

For example, take the non-inverting amplifier. (See #Non-inverting amplifier) Using this model the voltage at the - input will be equal to the voltage at the + input, which is $V i n$ . Therefore a current $V i n / R i$ will flow through $R i$ . Since this model also says the op amp has infinite input impedance, this same current must also flow through $R f$ . $V o u t$ then follows from a voltage divider rule.

$V_{out} = I (R_i + R_f) = \frac{V_{in}}{R_i}(R_i + R_f) = V_{in} + V_{in} \frac{R_f}{R_i}$

Therefore

$\frac{V_{out}}{V_{in}} = 1 + \frac{R_f}{R_i}$

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Finite Gain, Finite Impedance Model

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OP-amp circuits

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Amplifiers

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Inverting amplifier

The gain of an inverting amplifier is given by ${V_{out} \over V_{in}} = {R_f \over R_1}$

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Non-inverting amplifier

The gain of an non-inverting amplifier is given by ${V_{out} \over V_{in}} = 1 + {R_f \over R_1}$

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Summing amplifier

A summing amplifier is an inverting amplifier with multiple inputs. More than two inputs can be used.

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Difference amplifier

A diffrential amplifier amplifies the difference between the inputs. The V_out is given by ${V_{out} =} V_{in2} {{(R_3 + R_1) R_4} \over {(R_4 + R_2) R_1}} - {V_1 {R_3 \over R_1}}$

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Voltage follower

The voltage follower acts as a buffer for weak signals, the op amp input impedance is high, but the output impedance is low, so you can draw more power but avoiding loading of the signal source. The voltage follower with an ideal op amp gives simply V_in = V_out

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Current to voltage amplifier

The current to voltage amplifier can convert small (>0,01µA) to a more easily measured proportional voltage. $V_{out} = {-I_{in} \times R_f}$

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Voltage to current amplifier

The voltage to current amplifier converts voltage to proportional current. The desired current is going through the load resistor R1. The I_out is given by $I_{out} = {V_{in} \over R2}$ .

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Precision diode

The precision diode is a circuit which can rectify signals smaller than a voltage drop of a silicon diode. It is usually followed by a voltage follower. The frequency response of this circuit is limited by the slew rate of the op amp. Distortion can occur if the op-amp swings to its negative supply voltage and can't get back fast enough. In this case the output will stay at zero too long, distorting the wave.