#### The Operational Amplifier, Physics tutorial

Operational Amplifier:

Operational amplifier, or op-amp is very high gain, high input impedance directly coupled positive feedback amplifier with low output impedance, that can amplify signals having frequency ranging from 0 Hz to little beyond 1 MHz. They are prepared with different internal configurations in linear ICs. OP-AMP is so named as it was initially designed to execute mathematical operations such as multiplication, summation, differentiation, subtraction, and integration etc. in analog computers. Present day practice is much wider in scope but popular name OP-AMP continues. Though OP-AMP is complete amplifier, it is so designed that external components (resistors, capacitors etc.) can be joined to its terminals to change external characteristics. Therefore, it is comparatively easy to modify this amplifier to fit particular application and it is, in fact, because of this versatility which OP-AMPs have become so popular in industry.

The Basic OP-AMP:

Each input results is either same or opposite polarity (or phase) output, depending on whether signal is applied to plus (+) or minus (-).

Single-Ended Input:

Single-ended input operation results when input signal is connected to one input with other input joined to ground. The input is applied to plus input (with minus input at ground), that results in output having same polarity as applied input signal. Figure(b) shows input signal applied to minus input, output then being opposite in phase to applied signal.

Double-Ended (Differential) Input:

Additionally to using only one input, it is probable to apply signals at each input - this being double-ended operation. Figure(a) given below shows an input, applied between two input terminal (recall that neither input is at ground), with resulting amplified output in phase with that applied between plus and minus inputs. Figure(b) shows same action resulting when two separate signals are applied to inputs, difference signal being Vi1 - Vi2.

Double-Ended Output:

The op-amp can also be operated with opposite outputs. The input applied to either inputs will result in outputs from both output terminals, such output always being opposite in polarity.

The same operation with the single output estimated between output terminal (not with respect to ground). This difference output signal is V01 - Vo2. Difference output is also referred to as floating signal as neither output terminal is ground (reference) terminal. The difference output is twice as large as either V01 or V02, as they are of opposite polarity and subtracting them is twice amplitude [i.e., 10v -(-10V)=20].

Common-Mode Operation:

When same input signals are applied to both inputs, common-mode operation results. Ideally, two inputs are equally amplified, and as they result in opposite polarity signals at output, these signals cancel, resulting in 0-V output. Practically, the small signal will result.

Common-Mode Rejection:

The significant feature of the differential connection is that signal that are opposite at input are highly amplified, while those which are common to two inputs are only slightly amplified -overall operation being to amplify difference signal while rejecting common signal at two inputs. As noise (any unwanted input signal) is usually common to both inputs, differential connection tends to give attenuation of this unwanted input while giving amplified output of difference signal applied to inputs. This operating characteristic, referred to as common mode rejection.

Differential and Common-Mode Operation:

One of the more significant characteristics of differential circuit connection, as given in an op-amp, is circuit's ability to really amplify signals which are opposite at two inputs, while only slightly amplifying signals which are common to both inputs. The op-amp gives output component which is because of amplification of te difference of signals applied to plus and minus inputs and component because of signal common to both inputs. As amplification of opposite signals is much greater than that of common input signals, circuit gives a common-mode rejection as described by numerical value known as common-mode rejection ratio (CMRR).

Differential Inputs:

When separate inputs are applied to op-amp, resulting difference signal is difference between two inputs.

Vd = Vi1 - Vi2

Common Inputs:

When both input signals are same, the common signal element because of two inputs can be stated as average of sum of two signals.

V0 = 1/2(Vi1 + Vi2)

Output Voltage:

As any signals applied to op-amp in general have both in-phase and out-of-phase components, resulting output can be stated as:

Where Vd = difference voltage

Vc = common voltage

Ad = difference gain of amplifier

Ac = common-mode gain of amplifier

Opposite Polarity Inputs:

If opposite polarity inputs applied to the op-amp are preferably opposite signals, Vi1 = -Vi2 = Vs, resulting difference voltage is:

Vd = Vi1 - Vi2 = Vs -(-Vs) = 2Vs

While resulting common voltage is:

V0 = 1/2(Vi1 + Vi2) = 1/2[Vs + (-Vs)] = 0

So that resulting output voltage is:

This shows that when inputs are ideal opposite signal (no common element), output is differential gain times twice input signal applied to one of the inputs.

Same Polarity Inputs:

If same polarity inputs are applied to the op-amp, Vi1 = Vin = Vs, resulting difference voltage is:

Vd = Vi1 - Vi2 = Vs - Vs = 0

As resulting common voltage is:

V0 = 1/2(Vi1 + Vi2) = 1/2[Vs + Vs] = Vs

So that resulting output voltage is:

This shows that when inputs are ideal in-phase signals (no difference signal), output is common-mode gain times the input signal, Vs, that shows that only common-mode operation takes place.

Practical OP-AMP Circuits:

The op-amp can be joined in the large number of circuits to give different operating characteristics. Few most common of these circuit connections are explained below:

Inverting Amplifier:

The most extensively utilized constant-gain amplifier circuit is inverting amplifier. Output is attained by multiplying input by fixed or constant gain, set by input resistor (R1) and feedback resistor (RF) - this output also being inverted from input. We can write that:

V0 = -(Rf/R1)V1

Non-Inverting Amplifier:

It has better frequency stability. To find out voltage gain of circuit, we can use equivalent representation. Note that voltage across R1 is V1 since Vi = 0V. This should be equal to output voltage, through voltage divider of R1 and Rf, so that

V1 = (R1/(R1 + Rf))V0 that results in

V0/V1 = (R1 + Rf)/R1 = 1 + Rf/R1

Unity Follower:

The unity follower circuit gives gain of unity with no polarity or phase reversal. From equivalent circuit, it is clear that

V0 = Vi

And that output is same polarity and magnitude as input. Circuit works like emitter- or source-follower circuit except that gain is exactly unity.

Summing Amplifier:

In this there are a three-input amplifier circuit that gives a means of algebraically summing (adding) three voltages, each multiplied by the constant-gain factor. By using equivalent representation output voltage can be stated in terms of inputs as:

V0 = -((Rf/R1)V1 + (Rf/R2)V2 + (Rf/R1)V3)

In other words, every input adds voltage to output multiplied by separate constant-gain multiplier. If more inputs are utilized, they each add additional component to output.

Integrator:

If feedback component utilized is capacitor, resulting connection is known as integrator. Expression for voltage between input and output can be derived in terms of current I, from input to output. Capacitive impedance can be stated as:

Xc = 1/jωC = 1/sC

Where s = jω is in Laplace notation. Solving for v0/v1 yields

I = V1/R = -V0/Xc = -V0/1/sC = -sCV0

V0/V1 = -1/sCR

Expression above can be rewritten in time domain as

v0(t) = -1/RC∫v1(t)dt

It shows that output is integral of input, with the inversion and scale multiplier of 1/RC. Ability to integrate given signal gives analog computer with ability to solve differential equations and hence gives ability to electrically solve analogs of physical system operation.

Differentiator:

The differentiator does give a useful operation, resulting relation for circuit being

V0(t) = -RC(dv1(t)/dt)

Where scale factor is -RC

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