Having learned the basic laws and theorems for circuit analysis, we are now ready to study an active circuit element of paramount importance: the operational amplifier or op amp for short. The op amp is a versatile circuit building block.

It can also be used in making a voltage or current controlled current source. An op amp can sum signal, amplify signal, integrate it, or differentiate it. The ability of the op amp to perform these mathematical operations is the reason it is called an operational amplifier. It is also the reason for the widespread use of op amps in analog design. Op amps are popular in practical circuit designs because they are versatile, inexpensive, ease to use, and fun to work with.

We begin by discussing the ideal op amp and later consider the nonideal op amp. Using nodal analysis as a tool, we consider ideal op amp circuits such as the inverter, voltage follower, summer, and difference amplifier. Finally, we learn an op amp is used in digital-to-analog converters and instrumentation amplifiers.

An operational amplifier is designed so that it performs some mathematical operations when external components, such as resistors and capacitors, are connected to its terminals. Thus,

The op amp is an electronic device consisting of a complex arrangement of resistors, transistors, capacitors, and diodes. A full discussion of what is inside the op amp is beyond the scope of this block. It will suffice to treat the op amp as a circuit building block and simply study what takes place at its terminals.

Op amps are commercially available in integrated circuit packages in several forms. Figure 1 shows a typical op amp package. A typical one is the eight-pin dual in-line package (or DIP), shown in Figure 2(a). Pin or terminal 8 is unused, and terminals 1 and 5 are of little concern to us. The five important terminals are:

1. The inverting input, pin 2,

2. The noninverting input, pin 3,

3. The output, pin 6,

4. The positive power supply V+V+ size 12{V rSup { size 8{+{}} } } {}, pin 7,

5. The negative power supply V−V− size 12{V rSup { size 8{ - {}} } } {}, pin 4.

The circuit symbol for the op amp is the triangle in Figure 2(b); as shown, the op amp has two inputs and one output. The inputs are marked with minus (-) and plus (+) to specify inverting and noninverting inputs, respectively. An input applied to the to the noninverting terminal will appear with the same polarity at the output, while an input applied to the inverting terminal will appear inverted at the output.

As an active element, the op amp must be powered by a voltage supply as typically show in Figure 3. Although the power supplies are often ignored in op amp circuit diagrams for the sake of simplicity, the power supply currents must not be overlooked. By KCL,

The equivalent circuit model of an op amp is shown in Figure 4. The output section consists of a voltage-controlled source in series with the output resistance R0. It is evident from Figure 4 that the input resistance g, is the Thevenin equivalent resistance seen at the input terminals, while the output resistance R0R0 size 12{R rSub { size 8{0} } } {} is the Thevenin equivalent resistance seen at the output. The differential input voltage vdvd size 12{v rSub { size 8{d} } } {} is given by

Where v1v1 size 12{v rSub { size 8{1} } } {} is the voltage between the inverting terminal and ground and v2v2 size 12{v rSub { size 8{2} } } {} is the voltage between the noninverting terminal and ground. The op amp senses the difference between the two inputs, multiplies it by the gain A, and cause the resulting voltage to appear at the output. Thus, the output v0v0 size 12{v rSub { size 8{0} } } {} is given by

A is called the open-loop voltage gain because it is the gain of the op amp without any external feedback from output to input. Table 1 shows typical values of voltage gain A, input resistance RiRi size 12{R rSub { size 8{i} } } {}, output resistance R0R0 size 12{R rSub { size 8{0} } } {}, and supply voltage VCCVCC size 12{V rSub { size 8{ ital "CC"} } } {}.

Typical ranges for op amp parameters.

The concept of feedback is crucial to our understanding of op amp circuits. A negative feedback is achieved when the output is fed back to the inverting terminal of the op amp. When there is a feedback path from output to input of the op amp, the ratio of the output voltage to the input voltage is called the closed loop gain. As a result of the negative feedback, it can be shown that the closed-loop gain is almost insensitive to the open-loop gain A of the op amp. For this reason, op amps are used in circuit with feedback paths.

A practical limitation of the op amp is that the magnitude of its output voltage cannot exceed ∣VCC∣∣VCC∣ size 12{ lline V rSub { size 8{ ital "CC"} } rline } {}. In other words, the output voltage is dependent on and is limited by the power supply voltage. Figure 5 illustrates that the op amp can operate in three modes, depending on the differential input voltage vdvd size 12{v rSub { size 8{d} } } {}:

1. Positive saturation, v0=VCCv0=VCC size 12{v rSub { size 8{0} } =V rSub { size 8{ ital "CC"} } } {},

2. Linear region, −VCC≤v0=Avd≤VCC−VCC≤v0=Avd≤VCC size 12{ - V rSub { size 8{ ital "CC"} } <= v rSub { size 8{0} } = ital "Av" rSub { size 8{d} } <= V rSub { size 8{ ital "CC"} } } {},

3. Negative saturation, v0=−VCCv0=−VCC size 12{v rSub { size 8{0} } = - V rSub { size 8{ ital "CC"} } } {}.

Although we shall always operate the op amp in the linear region, the possibility of saturation must be borne in mind when one design with op amps, to avoid designing op amp circuits that will not work in the laboratory.

To facilitate the understanding of op amp circuits, we will assume ideal op amp. An op amp is ideal if it has the following characteristics:

Although assuming an ideal op amp provides only approximate analysis, most modern amplifiers have such large gain and input impedance that the approximate analysis is a good one. Unless stated otherwise, we will assume from now on that every op amp is ideal.

For circuit analysis, the ideal op amp is illustrated in Figure 6, which is derived from the nonideal model in Figure 4. Two important characteristics of the op amp are:

1. The current into both input terminals are zero:

and

This is due to infinite input resistance. An infinite resistance between the input terminals implies that an open circuit exists there and current cannot enter the op amp. But the output circuit is not necessary zero according to Equation 1.

2. The voltage across the input terminals is negligibly small; i.e.,

Or

Thus, an ideal op amp has zero current into its two input terminals and negligibly small voltage between the two input terminals. Equation 4 and Equation 6 are extremely important and should be regarded as the key handles to analyzing op amp circuits.

In this and following sections, we consider some useful op amp circuits that often serve as modules for designing more complex circuits. The first of such op amp circuits is the inverting amplifier shown in Equation 9. In this circuit, the noninverting circuit is grounded. vi is connected to the inverting input through RiRi size 12{R rSub { size 8{i} } } {}, and the feedback resistor RfRf size 12{R rSub { size 8{f} } } {} is connected between the inverting input and output. Our goal is to obtain the relationship the input voltage vi and the output voltage v0. Applying KCL at node 1,

But v1=v2=0v1=v2=0 size 12{v rSub { size 8{1} } =v rSub { size 8{2} } =0} {} for an ideal op amp, since the noninverting terminal is grounded. Hence,

v i R i = − v 0 R f v i R i = − v 0 R f size 12{ { {v rSub { size 8{i} } } over {R rSub { size 8{i} } } } = { { - v rSub { size 8{0} } } over {R rSub { size 8{f} } } } } {}

Or

The voltage gain is Av=v0/vi=−Rf/RiAv=v0/vi=−Rf/Ri size 12{A rSub { size 8{v} } = {v rSub { size 8{0} } } slash {v rSub { size 8{i} } = - {R rSub { size 8{f} } } slash {R rSub { size 8{i} } } } } {}. The designation of the circuit in Figure 7 as an inverter arises from the negative sign. Thus,

Note that the gain is the feedback resistance divided by the input resistance which means that the gain depends only on the external elements connected to the op amp. In view of Equation 8, an equivalent circuit for the inverting amplifier is shown in Figure 8. The inverting amplifier is used, for example, in current to voltage converter.

Another important application of the op amp is the noninverting amplifier shown in Figure 9. In this case, the input voltage vivi size 12{v rSub { size 8{i} } } {} is applied directly at the noninverting input terminal and resistor RiRi size 12{R rSub { size 8{i} } } {} is connected between ground and the inverting terminal. We are interested in the output voltage and the voltage gain. Application of KCL at the inverting terminal gives

But v1=v2=viv1=v2=vi size 12{v rSub { size 8{1} } =v rSub { size 8{2} } =v rSub { size 8{i} } } {}. Equation 9 becomes

− v i R 1 = v i − v 0 R f − v i R 1 = v i − v 0 R f size 12{ { { - v rSub { size 8{i} } } over {R rSub { size 8{1} } } } = { {v rSub { size 8{i} } - v rSub { size 8{0} } } over {R rSub { size 8{f} } } } } {}

Or

The voltage gain is Av=v0/vi=1+Rf/R1Av=v0/vi=1+Rf/R1 size 12{A rSub { size 8{v} } = {v rSub { size 8{0} } } slash {v rSub { size 8{i} } =1+ {R rSub { size 8{f} } } slash {R rSub { size 8{1} } } } } {}, which does not have a negative sign. Thus, the output has the same polarity as the input.

Again we notice that the gain depends only on the external resistors.

Notice that if feedback resistor Rf=0Rf=0 size 12{R rSub { size 8{f} } =0} {} (short circuit) or R1=∞R1=∞ size 12{R rSub { size 8{1} } = infinity } {} (open circuit) or both, the gain becomes 1. Under these conditions ( Rf=0Rf=0 size 12{R rSub { size 8{f} } =0} {} and R1=∞R1=∞ size 12{R rSub { size 8{1} } = infinity } {} ), the circuit in Figure 9 becomes that shown in Figure 10, which is called a voltage follower (or unity gain amplifier) because the output follows the input. Thus, for a voltage follower

Such circuit has very high input impedance and is therefore useful as an intermediate-stage (or buffer) amplifier to isolate one circuit from another as portrayed in Figure 11. The voltage follower minimizes interaction between the two stages and eliminates interstage loading.

Besides amplification, the op amp can perform addition and subtraction. The addition is performed by the summing amplifier covered in this section; the subtraction is performed by the difference amplifier covered in the next section.

The summing amplifier, shown in Figure 12, is variation of the inverting amplifier. It takes advantage of the fact that the inverting configuration can handle many inputs at the same time. We keep in mind that the current entering each op amp input is zero. Applying KCL at node a gives

But

i 1 = v 1 − v a R 1 , i 2 = v 2 − v a R 2 i 1 = v 1 − v a R 1 , i 2 = v 2 − v a R 2 size 12{i rSub { size 8{1} } = { {v rSub { size 8{1} } - v rSub { size 8{a} } } over {R rSub { size 8{1} } } } ,i rSub { size 8{2} } = { {v rSub { size 8{2} } - v rSub { size 8{a} } } over {R rSub { size 8{2} } } } } {}

We note that va=0va=0 size 12{v rSub { size 8{a} } =0} {} and substitute Equation 13 into Equation 12. We get

Indicating that the output voltage is a weighted sum of the inputs. For this reason, the circuit in Figure 18 is called a summer. Needless to say, the summer can have more than three inputs.

Difference (or differential) amplifiers are used in various applications where there is need to amplify the difference between two input signals. They are first cousins of the instrumentation amplifier, the most useful and popular amplifier, which we will discuss in section 9.

Consider the op amp circuit shown in Figure 13. Keep in mind that zero currents enter the op amp terminals. Applying KCL to node a,

v 1 − v a R 1 = v a