12/25/03

Operational Amplifiers

An operational amplifier with voltage gain , A , can be represented by the following symbol:

The voltages to be amplified are v − ( t ) and v + ( t ) , which are the inverting (minus) and non-inverting (plus) inputs, respectively. The amplified voltage output is v o u t ( t ) .

An ideal operational amplifier is one for which

• the output voltage is v o u t ( t )     =   A ( v + ( t )   −   v − ( t ) )
• the currents at the "plus" and "minus" input terminals are absolutely zero
• the voltage gain A   →   ∞ .

The operational amplifier is an example of a differential amplifier in that its output voltage, v o u t ( t ) , is proportional to the difference between its two input voltages, v + ( t ) and v − ( t ) . Practical operational amplifiers approximate the behavior of ideal amplifiers fairly well over the audio frequency range.

As our first example application of operational amplifiers, consider the following circuit:

In this circuit, R and v i n ( t ) represent the source of the input voltage to the amplifier, which might be, for example, a microphone or some other sensor, or even another amplifier. Note that this circuit feeds the entire output voltage back into the negative input terminal of the operational amplifier. This circuit therefore employs maximum negative voltage feedback.

To analyze this circuit, recall that, for ideal operational amplifiers,

v o u t ( t )     =   A ( v + ( t )   −   v − ( t ) )

For our particular circuit, v − is the same as v o u t :

v −   ≡   v o u t

Because the ideal operational amplifier draws no current through its input terminals, (and hence develops no voltage drop across the input resistor, R ), then

v +   ≡   v i n

so that

v o u t ( t )     =   A ( v i n ( t )   −   v o u t ( t ) )

or

( 1   +   A )   v o u t ( t )     =   A   v i n ( t )

so that

  v o u t ( t )     =   A ( A   +   1 )   v i n ( t )

As A   →   ∞ ,

  v o u t ( t )     =     v i n ( t )

The fact that the output voltage follows the input voltage shows that our circuit is an example of a voltage follower. The main desirable feature of such circuits is that they draw little current at their inputs, but can supply hefty currents at their output terminals. If it weren't for this advantage, we could accomplish just as much with a piece of wire.

As a second example, consider the following circuit:

In comparison with the first circuit, not that only a fraction of the output voltage is fed back into the negative input terminal of the operational amplifier in this circuit. Again, recall

v o u t ( t )     =   A ( v + ( t )   −   v − ( t ) )

Now, as above,

v +   ≡   v i n

Because the ideal operational amplifier draws no current at its input terminals, v - ( t ) is simply a voltage-divided copy of v o u t ( t ) :

v − ( t )   =   R 1 R 1   +   R 2   v o u t ( t )

Thus,

v o u t ( t )   =   A   ( v i n ( t )   −   R 1 R 1   +   R 2   v o u t ( t ) )

v o u t ( t )   ( 1   +   A   R 1 R 1   +   R 2   v o u t ( t ) )   =   A   v i n ( t )

v o u t ( t )   =   A 1   +   A   R 1 R 1   +   R 2   v i n ( t )

v o u t ( t )     =   1     R 1 R 1   +   R 2   v i n ( t )

v o u t ( t )     =   ( 1   +   R 2 R 1 )   v i n ( t )

Because the input and output voltages have the same sign (move up and down together), this amplifier is called a non-inverting amplifier . The voltage gain, G , of this amplifier is:

G   =   1   +   R 2 R 1

Note that we can use an operational amplifier to realize an amplifier of any gain we wish simply by adjusting the ratio of a couple of resistors external to the IC operational amplifier chip. The significance of this result is as an example of how an operational amplifier chip, which includes the overwhelming majority of the circuit components (approaching, perhaps, 100 ) in the total circuit, can be mass produced at a very low cost and then customized for a specific purpose (realizing an amplifier with a particular gain, in this case) by the addition of a few external components. When the application of operational amplifiers was limited (in so-called analog computers) mainly to synthesizing integrators, their cost was several hundred dollars, each. Now, because we realize their versatility and employ them in many diverse applications, demand is large enough to drive the price for some common types below one dollar.

Operational amplifiers are, indeed, wonderful devices. We should note, however, that available operational amplifiers handle mainly low power, audio frequency, signals. At high frequencies and high powers, amplifiers that use several individual transistors reign.

As our third example, consider the following circuit:

As A   →   ∞ , analysis shows that

v o u t   =   −   R F R   +   R 1   v i n F

Because the output and input voltages for this amplifier have opposite signs, this amplifier is called an inverting amplifier. The gain, G , of the circuit is

G   =   −   R F R   +   R 1

Once again, we can realize any reasonable gain by choosing suitable values for the resistors in the circuit.

As our fourth example, consider:

This circuit is an inverting amplifier with two input voltages. As A   →   ∞ , analysis shows

v o u t   =   −   ( R F R 1   v 1   +   R F R 2   v 2 )

This configuration, therefore, is an inverting summing amplifier: its output is a linear combination of the two inputs, v 1 and v 2 . (Summing amplifiers are not easily made with the non-inverting amplifier configuration. The "virtual ground" at the "minus" terminal of the inverting configuration turns out to be essential for the summing process.) These results are easily generalized to include inverting amplifiers with three or more inputs. An interesting application of such a multiple-input summing amplifier is in mixing audio signals from microphones used by, for example, a mixed group of vocalists and instrumentalists. The individual voices or instruments can be emphasized or de-emphasized by adjusting the individual resistors, R 1 and R 2 , In practical audio mixers, these resistors take the form of "pots" (potentiometers), resistors whose value can be varied by turning their knobs. Once the individual resistors have been adjusted to give the appropriate audio mix, the resistor, R F , serves as a master control to adjust the overall sound level.

As our final example, consider the following operational amplifier circuit:

If the capacitor is initially discharged so that v o u t ( 0 )   =   0 , then analysis shows that

v o u t ( t )   =   −   1 ( R   +   R 1 )   C   ∫   0   t v i n ( τ )   d τ

For this circuit, the output voltage clearly is proportional to the integral of the input voltage. As a consequence, this circuit is called an integrator . Such circuits formed the basis of analog computers. During the 1940's and 1950's, such computers were engineers' main tools for solving the coupled sets of differential equations that described complicated engineering systems. Ultimately, digital computers replaced analog computers because digital computers offer much more accuracy and flexibility. During World War II, however, all of the newly sophisticated control systems for guns, radar and aircraft were designed with analog computers.